Hindawi Publishing CorporationAdvances in MeteorologyVolume 2013 Article ID 981271 11 pageshttpdxdoiorg1011552013981271

Research ArticleThe Sensitivity of Characteristics of Large ScaleBaroclinic Unstable Waves in Southern Hemisphere tothe Underlying Climate

Sergei Soldatenko and Chris Tingwell

Centre for Australian Weather and Climate Research 700 Collins Street Docklands VIC 3008 Australia

Correspondence should be addressed to Sergei Soldatenko ssoldatenkobomgovau

Received 26 September 2013 Accepted 18 November 2013

Academic Editor Luis Gimeno

Copyright copy 2013 S Soldatenko and C Tingwell This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The sensitivity of the main characteristics of baroclinically unstable waves with respect to fundamental parameters of theatmosphere (the static stability parameter 120590

0and vertical shear of a zonal wind Λ) is theoretically explored Two types of waves

are considered synoptic scale waves and planetary scale (ultralong) waves based on an Eady-type model and model with verticallyaveraged primitive equations Sensitivity functions are obtained that estimate the impact of variations in 120590

0and Λ on the growth

rate and other characteristics of unstable waves and demonstrate that waves belonging to the short-wave part of the spectrum ofunstable waves are more sensitive to changes in the static stability parameter than waves belonging to the long-wave part of thespectrumThe obtained theoretical results show that the increase of the static stability and decrease of the meridional temperaturegradient in midlatitude baroclinic zones in some areas of the southern hemisphere lead to a slowing of the growth rate of baroclinicunstable waves and an increasing wavelength of baroclinic unstable wavemaximum growth rate that is a spectrum shift of unstablewaves towards longer wavelengths These might affect the favorable conditions for the development of baroclinic instability andtherefore the intensity of cyclone generation activity

1 Introduction

Static stability and the meridional temperature gradient(MTG) are among the most important fundamental param-eters characterizing the state of the atmosphere and inparticular midlatitude large-scale eddy dynamics [1 2] Staticstability and MTG play a significant role in the developmentof baroclinic instability which is the dominant mechanismfor generating large-scale atmospheric eddies (cyclones) thatform the storm tracks in midlatitudes The physical natureof baroclinic instability is well understood and explainedin the scientific literature including text books on dynamicmeteorology (eg [1ndash4]) Baroclinic instability can be viewedas sloping convection where growing perturbations drawupon the available potential energy which is proportional toa meridional temperature gradient Since the publication ofthe pioneering theoretical works of Charney [5] and Eady[6] in which the fundamental baroclinic mechanism of

the atmospheric large-scale instability was first describedmany scientific papers have been published that examine thegrowth of initially infinitesimal perturbations in the atmo-sphere and ocean caused by baroclinic effects Both lineartheory for the onset of baroclinic instability and its nonlinearsaturation have been explored in many research articlesA thorough review of baroclinic instability is presented in[7] and to some extent in [8 9] Theoretical models ofbaroclinic instability typically represent linearized dynamicsequations and the instability problem is examined as aneigenvalue problem The nonmodal instability analysis inboth linear and nonlinear formulations is more generalthan eigenanalysis This technique suggests the solution ofa Cauchy (initial value) problem (eg [10ndash12]) Analyticalweakly nonlinear theories of baroclinic instability represent afurther extension of research (eg [13ndash16]) which focuses onthe finite-amplitude behaviour of unstable baroclinic wavesOther areas of studies have explored the life-cycle behaviour

2 Advances in Meteorology

of baroclinic unstable waves to understand how initiallyinfinitesimal perturbations grow to large but finite amplitudemodifying the mean flow This kind of research requiresnumerical integration of the atmospheric nonlinear modelequations (eg [17ndash20])

It is important to underline the significant role of thetwo-layer model of Philips [21 22] and its modifications(eg [13 23ndash28]) in developing baroclinic instability theoriesHowever for this purpose a two-layer model was usuallyused in the quasigeostrophic approximation A generalizedbaroclinic instability analysis based on two-level primitive-equation model has been described in [29] and applied todetailed consideration of the baroclinic instability mecha-nism

There is evidence of an increase over the past decades ofthe static stability in the extratropics [30] and a pole-wardmovement of the midlatitude precipitation zones and stormtracks which are uniquely linked to zones with strong MTG-baroclinic zones (eg [31ndash35]) Recent studies have also indi-cated an increase in intensity of extratropical cyclones anda decrease in frequency [36ndash44] Changes in geographicallocations and intensity of storm tracks aremore distinct in thesouthern hemisphere (SH) (eg [45]) affecting the essentialfeatures of weather patterns over large territories such asAustralia [46ndash54] and indicating a change of favourableconditions for baroclinic instability because the key role ofbaroclinic instability in the development of midlatitudecyclones is well established

To study how baroclinic instability has changed in theSH in recent decades a comparative analysis of climates forperiods 1949ndash1968 and 1975ndash1994 has been carried out in[50 54] usingNational Centres for Environmental Prediction(NCEP) reanalyses and the European Centre for Medium-Range Weather Forecasting (ECMWF) Reanalysis (ERA40)data As a measure of the baroclinic instability the Phillipscriterion [55] adapted to spherical geometry was chosen

119906(300 hPa)

minus 119906(700 hPa)

ge

119887119896119888119901120590

1198860Ω

cos120593sin2120593

(1)

where 119906(300 hPa) and 119906(700 hPa) are the zonal wind velocities

at 300 hPa and 700 hPa isobaric levels respectively 120590 is thestatic stabilitymeasure for a given reference state 119888

119901is the spe-

cific heat of air at constant pressureΩ is the angular rotationspeed of the earth 119886

0is the earthrsquos radius 119887

119896is a dimensionless

constant and 120593 is a latitude J S Frederiksen and C SFrederiksen [50] obtained a significant decreasing trend inbaroclinic instability over the middle latitudes of the SHThemost significant negative trend is localized between 30S and40S For instance the difference between the two periods1949ndash1968 and 1975ndash1994 reached a maximum of 17 in theSH subtropical jet stream (for the July climate) In additionit was found that further poleward the baroclinic instabilityintensified creating favourable conditions for cyclogenesisUsing a two-level linearized primitive equation model Fred-eriksen et al [51] studied the growth rate of unstable modesfor the July reference state averaged over the abovementionedperiodsThey identified a 30 reduction in the growth rate ofcyclone-scale modes between the two periods

It should be pointed out that since the beginning of the1980s exploration of the essential features of the atmosphericgeneral circulation in the SH has been facilitated by theavailability of plentiful satellite upper-air data In particularthe importance of baroclinic instability and large-scale eddiesin the formation of zonal wind characteristics such as atropospheric double-jet phenomenonwas studied in [56 57]

Taking into account the significant role of baroclinicinstability in the development ofmidlatitude cyclones and theintensity changes of baroclinic instability in some areas of theSH in recent decades this paper examines the sensitivity ofthemain characteristics of baroclinically unstable waves (egthe growth rates of unstable waves as function of wavelength)to fundamental atmospheric parameters the static stabilityparameter120590

0which is characterized by temperature lapse rate

Γ = minus120597119879120597119911 and zonal wind vertical shear Λ which by ther-mal wind balance characterizes the meridional temperaturegradient Two classes of waves [55] are considered The firstone is characterized by the Rossby number Ro equiv 119880(119891119871) sim

01 and the representative horizontal length scale 119871 which issmaller than the Earthrsquos radius 119886

0 that is119871119886

0sim 01 Here119880 is

a horizontal velocity scale and119891 is theCoriolis parameter Forthis class of waves 120577 ≫ 119863 where 120577 is the vertical componentof relative vorticity and119863 is the horizontal divergence Synop-tic scale waves belong to this type of motion [58]The secondclass which includes planetary or ultralong waves [59] ischaracterized by Ro sim 001 and 119871119886

0sim 1 For these waves the

magnitude of horizontal divergence is comparable with thevertical component of vorticity that is 120577 asymp 119863 To study syn-optic scale waves the Eady-type model is used with uniformzonal wind shear between upper and lower boundaries on an119891-plane In this context parameters 120590

0and Λ are considered

to be variables that control the development of baroclinicinstability in the atmosphere Ultralong waves are investi-gated based on a model with vertically averaged equations[60 61] with a 120573-plane approximation Simplified modelssuch as the Eady model of baroclinic instability and modelswith vertically averaged equations despite their simplicityallow solutions to be obtained that clearly illustrate realphysical processes in the atmosphere

2 Synoptic Scale BaroclinicallyUnstable Waves

21 The Model Equations We consider the inviscid primitiveequation atmospheric model in normalized isobaric coordi-nates (119909 119910 120585) on an 119891-plane in the following form

(

120597

120597119905

+ u sdot nabla + 120596

1198750

120597

120597120585

) u + 119891k times u = minusnablaΦ

(

120597

120597119905

+ u sdot nabla)119879 minus 119878120585120596 =

119876

119888119901

nabla sdot u + 1

1198750

120597120596

120597120585

= 0

120597Φ

120597120585

= minus

119877119879

120585

(2)

Advances in Meteorology 3

The state variables of the model are the horizontal velocityvector u = (119906 V)119879 the vertical pressure velocity 120596 equiv 119889119901119889119905where 119901 is pressure the geopotentialΦ and the temperature119879 The operator nabla equiv (120597120597119909 120597120597119910) is applied to the horizontalcoordinates 119909 and 119910 directed eastward and northwardrespectively The normalized pressure 120585 = 119901119875

0 where 119875

0=

1000 hPa is a ldquostandardrdquo pressure approximately equal to thesurface pressure is taken as the vertical coordinate while thetime is denoted by 119905 Other notations are the diabatic heatingrate per unit time per unitmass119876 the gas constant119877 the unitvector in the vertical direction k and the reference state staticstability measure 119878

120585in the normalized isobaric coordinate

system

119878120585=

119877119879

1198921198750120585

(Γ119889minus Γ) (3)

where 119879 is a reference temperature 119892 is the gravity accelera-tion Γ

119889is the dry adiabatic lapse rate and Γ is the reference

state lapse rateWe employ the119891-plane approximation so thatthe Coriolis parameter 119891 is a constant 119891 = 119891

0= 2Ω sin120593

0

with 1205930being the latitude of interest Hereafter we consider

only adiabatic process and thus assume zero heating rate 119876The following boundary conditions are used for the pressurevelocity

120596 = 0 at 120585 = 0 120585 = 1 (4)

The atmospheric reference state defined by 119906 V120596119879 andΦ is steady and satisfies the following relations

119906 = minus

1

1198910

120597Φ

120597119910

V = 0

120596 = 0

120597Φ

120597120585

= minus

119877119879

120585

(5)

where 119879 = 119879(119910 120585) By substituting (5) into the set of (2) onecan see that (5) is a solution of (2) that describes the zonalflow

120597119906

120597120585

=

119877

1198910120585

120597119879

120597119910

(6)

which matches the specified distribution of the zonally aver-aged temperature 119879(119910 120585) and represents thermal wind bal-ance To consider only the baroclinicmechanism of the atmo-spheric instability meridional variability of the basic zonalflow is excluded In other words the barotropic impact on theinstability of the basic zonal flow is not taken into accountTherefore we assume that the velocity of the basic zonalflow does not depend on the horizontal 119910-coordinate that is119906 = 119906(120585)Thus the problem now is the study of the instabilityof the basic zonal flow (6) with respect to infinitesimalperturbations For this purpose the system (2) is linearizedaround the basic state (5) Representing the state variables as120595(119909 119910 120585 119905) = 120595(120585) + 120595

1015840(119909 120585 119905) where 120595 is a basic state and

1205951015840 is an infinitesimal perturbation and taking into account

the hydrostatic equations and the thermal wind relationship(6) the linearized system can be written as

1205971199061015840

120597119905

+ 119906

120597119906

120597119909

+

1205961015840

1198750

120597119906

120597120585

= minus

120597Φ1015840

120597119909

+ 1198910V1015840

120597V1015840

120597119905

+ 119906

120597V1015840

120597119909

+ 11989101199061015840= 0

1205971199061015840

120597119909

+

1

1198750

1205971205961015840

120597120585

= 0

120597

120597119905

(

120597Φ1015840

120597120585

) + 119906

120597

120597119909

(

120597Φ1015840

120597120585

) minus V10158401198910

120597119906

120597120585

+ 120590011987501205961015840= 0

(7)

The static stability parameter 1205900is expressed as

1205900= minus

120572

1198750

120597 lnΘ120597120585

=

1198772119879

1198921198752

01205852(Γ119889minus Γ) (8)

where 120572 is a specific volume and Θ = 119879120585minus119877119888119901 is a reference

potential temperature Suppose that 120597119906120597120585 = minusΛ120585= const

Applying the method of separation of variables we assumethe solutions of the form

1205951015840(119909 120585 119905) = (120585) 119890

119894119896(119909minus119888119905) (9)

where (120585) is a function of 120585 only 119896 is a wave number and 119888is a phase velocity of perturbations which in general is a com-plex value 119888 = 119888

119903+ 119894119888119894 After substituting (9) into (7) we can

finally obtain the following single equation for

(119906 minus 119888) [1 minus

1198962

1198912

0

(119906 minus 119888) (119906 minus 119888)]

1205972

1205971205852

+ 2Λ120585

120597

120597120585

minus 12059001198752

0

1198962

1198912

0

(119906 minus 119888) = 0

(10)

Similar equations were considered in a number of publica-tions (eg [4 62 63])The analytical solution of this equationcan be obtained by eliminating gravity waves from the con-sideration and assuming that 120590

0= const For this particular

case (10) can be transformed into

(119906 minus 119888)

1205972

1205971205852+ 2Λ120577

120597

120597120585

minus (119906 minus 119888) 12059001198752

0

1198962

1198912

0

= 0 (11)

The boundary conditions for are specified as

= 0 at 120585 = 0 120585 = 1 (12)

Equation (11) together with boundary conditions (12) rep-resents the eigenvalue problem for the complex phase speed119888 The fundamental solutions of (11) are expressed throughthe Bessel functions of the first and second kinds With thehomogeneous boundary conditions (12) the following twodiscrete eigenvalues can be obtained [4]

11988812=

Λ120585

2

[1 plusmn radic1 minus

4

1205782(120578 coth (120578) minus 1)] (13)

4 Advances in Meteorology

where

120578 = 1198750radic1205900 (

119896

1198910

) (14)

The phase velocity 119888 will amplify exponentially if 119888 has animaginary part 119888

119894 From (13) we can see that this will occur if

the discriminant in (13) is less than zero

4

1205782(120578 coth (120578) minus 1) gt 1 (15)

which gives by theNewtonrsquos iteration algorithm the necessarycondition for instability 120578 lt 120578

119888asymp 23994 Besides two dis-

crete eigenvalues (13) the eigenvalue problem (11)-(12) has acontinuous spectrum of eigenvalues 119888 isin (0 119906(120585)) that are realand therefore can be neglected in the problem of baroclinicinstability [4] The growth rate of unstable waves 120594

119896equiv 119896119888119894is

calculated by the following expression

120594119896=

Λ1205851198910

21198750radic1205900

radic10038161003816100381610038161205782minus 4 (120578 coth (120578) minus 1)100381610038161003816

1003816 (16)

As we can see from (16) at a given latitude the growth rate120594119896is a function of 120590

0(static stability) Λ

120585(wind shear) and

quantity 120578 that depends on the horizontal wavenumber 119896 and1205900

22 Impact of Static Stability andVerticalWind Shear on Baro-clinic Instability Within the Eady problem framework thestatic stability parameter 120590

0and the vertical wind shear Λ

120585

represent the main control variables By varying 1205900and Λ

120585

one can obtain estimates of the impact of these parameters onthe development of baroclinic instability in the atmosphereIn this research parameters corresponding to the basicstate are given the following values Λ

120585= 40msminus1 [50]

and 1205900= 2times10

minus6m2 Paminus2 sminus2 [1]These parameter values canbe used as an approximation to describe the zonal-averagedatmospheric conditions for JJA (June July andAugust) in theSH [50] The latitude of interest is assumed to be 120593

0= 45 S

which gives 1198910= minus1028 times 10

minus4 sminus1Figure 1 shows plot of Eady growth rate versus zonal

wavenumber obtained with (16)The growth rate has a short-wave instability cutoff beyond which waves are stable Let119871min be the wavelength that corresponds to a short-wavecutoff Value of 119871min can be obtained from (14) when 120578 = 120578

119888

which gives 119871min = 3592 km To calculate the wavelength ofmaximum growth rate 119871

120594max one can take 120597120594119896120597119896 and set

the result equal to zero which gives 120578 = 120578119898asymp 16061 Then by

using (14) we can obtain 119871120594max = 5366 km

The influence of the static stability parameter on thewavelength of maximum growth rate 119871

120594max and the shortwave cut-off 119871min are shown in Figure 2 In general anincrease in the parameter 120590

0leads to an increase in both

119871119896119888max and 119871min The functional dependences between 119871min

and 1205900 and between 119871

119896119888max and 1205900 are almost linear plusmn10departure of static stability parameter Δ120590

0from its nominal

value 1205900= 20times10

minus6m2 Paminus2 sminus2 results in about plusmn5 changefor both 119871

119896119888max and 119871min with respect to the nominal value1205900 For instance if Δ120590 = 01 times 120590

0 then 119871

119896119888max = 5628 and

09

06

03

000 2 4 6 8

kz

Xk

(dayminus1)

Mode with maximum Short-wavegrowth rate cutoff

Figure 1 Growth rate 120594119896versus zonal wavenumber 119896

119911for 1205900= 2 times

10minus6m2 Paminus2 s minus 2 and Δ

120585= 40msminus1

6

4

2

(km

)

1 2 3(m2 Paminus2 sminus2 )

LminLkcmax

1205900 times 106

Ltimes10minus3

Figure 2 Length of waves with maximum growth rates 119871119896119888max and

short-wave cutoff 119871min as functions of the static stability parameter1205900

119871min = 3768 km and if Δ120590 = minus011205900 then 119871

119896119888max = 5091

and 119871min = 3408 kmFigure 3 illustrates the growth rate of unstable waves

versus the static stability parameter at different values of Λ120585

Parameters 1205900and Λ

120585influence the growth rate 120594

119896in the

opposite direction growth rate decreases if 1205900increases and

if Λ120585decreases Note that the decrease of the parameter Λ

120585

indicates the weakening of the intensity of the barocliniczone that is reduction of the MTG In nature both of theseprocesses take place which leads to a synergistic effect For

Advances in Meteorology 5

20

15

10

05

0 1 2 3

Xk

(dayminus1)

Λ120576 = 30Λ120576 = 40

Λ120576 = 50Λ120576 = 60

(m2 Paminus2 sminus2 )1205900 times 106

Figure 3 Growth rate 120594119896versus static stability parameter 120590

0for

different values of parameter Λ120585(units m sminus1)

instance if Λ120585decreases by 10 and the static stability

parameter increases by 10 the growth rate 120594119896decreases by

14Since 120594

119896is a nonlinear function of 120590

0(16) to estimate the

influence of infinitesimal perturbations in 1205900on variations in

120594119896 the sensitivity function

119878120590=

120597120594119896

1205971205900

(17)

and the relative sensitivity function

119878119877

120590=

120597120594119896120594119896

12059712059001205900

=

1205900

120594119896

120597120594119896

1205971205900

(18)

can be used The function 119878120590shows changes in 120594

119896due to

variations in 1205900 The relative sensitivity function 119878119877

120590is used

to compare model parameters to find out what parameter isthemost important for a certain percent change in the param-eters Sensitivity functions (17) and (18) are evaluated in thevicinity of some nominal value of the parameter 120590

0 We can

select several nominal values to cover some range of changesin 1205900 Differentiating (16) with respect to control parameter

1205900 we can obtain the expression for 119878

120590

119878120590=

120594119896

21205900

(1198750radic1205900

119896

1198910

120578 minus 2 coth (120578) + 2120578csch2 (120578)1205782minus 4 (120578 coth (120578) minus 1)

minus 1)

(19)

Sensitivity 119878120590versus zonal wavenumbers for different values

of 1205900with Λ

120585= 40msminus1 are shown in Figure 4 The absolute

value of the sensitivity of 120594119896with respect to 120590

0exponentially

minus5

minus10

minus15

minus20

0 2 4 6 8 10

S120590

kz

1205900 = 101205900 = 201205900 = 30

Figure 4 Sensitivity function 119878120590versus zonal wavenumber 119896

119911for

different nominal values of the static stability parameter 1205900(units

106m2 Paminus2 sminus2)

increases with decreasing wavelength For planetary scalewaves (zonal wave numbers 1ndash4) the absolute value of thesensitivity of 120594

119896with respect to 120590

0is palpably less than

sensitivity for synoptic scale waves (zonal wave numbers ge5)Absolute and relative sensitivity functions 119878

120590and 119878119877

120590calcu-

lated for different values of1205900for various zonal wave numbers

are shown in Tables 1 and 2 respectivelyThe expression for sensitivity function 119878

Λcan be eas-

ily obtained by differentiating (16) with respect to controlparameter Λ

120585

119878Λ=

1198910

21198750radic1205900

radic10038161003816100381610038161205782minus 4 (120578 coth (120578) minus 1)100381610038161003816

1003816 (20)

The function 119878Λversus zonalwave number 119896

119911for different

1205900is plotted in Figure 5 It is clear to see that for a given value

of the parameter 1205900the graph of function 119878

Λ(119896119911) is verymuch

like the classic picture of the growth rates 120594119896versus zonal

wavenumber 119896119911[5] It is interesting that the relative sensitivity

function 119878119877Λdoes not depend on the wavelength (wavenum-

ber) and for all of the unstable waves is equal to unity

119878119877

Λ=

120597120594119896120594119896

120597Λ120585Λ120585

=

Λ120585

120594119896

120597120594119896

120597Λ120585

= 1 (21)

Since relative sensitivity functions allow direct comparison ofthe importance of model parameters on the growth rate 120594

119896

we can see that because 119878119877Λ= 1 the parameter Λ

120585(ie the

meridional temperature gradient) is more important than

6 Advances in Meteorology

Table 1 Absolute sensitivity 119878120590as a function of zonal wavenumber 119896

119911for different values of static stability parameter 120590

0

1205900m2 Paminus2 sminus2 Zonal wave number 119896

119911

1 2 3 4 5 6 710 times 10

minus6minus00157 minus01251 minus04175 minus09772 minus18879 minus32470 minus52005

15 times 10minus6

minus00157 minus01245 minus04139 minus09675 minus18794 minus33014 minus5592720 times 10

minus6minus00157 minus01240 minus04110 minus09626 minus18979 minus34964 minus69371

25 times 10minus6

minus00157 minus01234 minus04088 minus09629 minus19509 minus39665 minus23003130 times 10

minus6minus00157 minus01230 minus04071 minus09691 minus20540 minus53057

Table 2 Relative sensitivity 119878119877120590as a function of zonal wavenumber 119896

119911for different values of static stability parameter 120590

0

1205900m2 Paminus2 sminus2 Zonal wave number 119896

119911

1 2 3 4 5 6 710 times 10

minus6minus00062 minus00250 minus00575 minus01057 minus01735 minus02692 minus04095

15 times 10minus6

minus00093 minus00379 minus00881 minus01656 minus02837 minus04749 minus0837120 times 10

minus6minus00124 minus00509 minus01201 minus02375 minus04220 minus08004 minus19939

25 times 10minus6

minus00156 minus00642 minus01539 minus03087 minus06071 minus14399 minus31424330 times 10

minus6minus00187 minus00777 minus01898 minus03974 minus08759 minus34322

3

2

1

00 2 4 6 8 10 12

kz

1205900 = 101205900 = 201205900 = 30

107timesSΛ

Figure 5 Sensitivity function 119878Λversus zonal wavenumber 119896

119911for

different nominal values of the static stability parameter 1205900(units

106m2 Paminus2 sminus2)

the static stability parameter 1205900except for 120590

0gt 2 times

10minus6m2 Paminus2 sminus2 for waves with 119896

119911ge 6 (see Table 2) Wher-

ever the midlatitude values in Table 2 are less than one thegrowth rate is more sensitive to the meridional temperaturegradient (ie Λ

120585) than the static stability (120590

0)

The obtained results are consistent with observations[50 51 54] an increase in static stability and a decrease of

the MTG have occurred over the past few decades in someareas of the SH which has led to a decrease in the growthrate of baroclinic unstable waves a shift of the spectrum ofunstable waves in the long wavelength part of spectrum andaweakened intensity of cyclogenesis Naturally these changesaffect favourable conditions for the development of baroclinicinstability and the essential features of weather patterns overlarge territories particularly over Australia

3 Planetary Scale Waves

To study the influence of the static stability parameter onthe dynamics of planetary scale (ultralong) waves a thinfilm approximation is applied This approximation employsa specific averaging technique over the vertical coordinate tothe system of primitive equations [60] As a result a two-dimensional set of equations can be obtained that describesthe dynamics of a two-dimensional baroclinic film Theseequations reproduce all the wavelike solutions that corre-spond to the main weather-forming modes of three-dimen-sional models and therefore can be used in theoretical studiesof large-scale dynamic processes in the atmosphere Thesystem of vertically averaged equations can be written as [60]

120597119906

120597119905

+ V sdot nabla119906 = 119891V minus119877

120587

120597

120597119909

(120587119879)

120597V120597119905

+ V sdot nablaV = minus119891119906 minus119877

120587

120597

120597119910

(120587119879)

120597120587

120597119905

+ nabla sdot (120587V) = 0

120597119879

120597119905

+ V sdot nabla119879 +

119877

119888119901

119879nabla sdot V = 0

(22)

where 120587 = 1199011199041198750 For instance if the original primitive equa-

tions are written in the Phillipsrsquo vertical coordinate system

Advances in Meteorology 7

120590 = 119901119901119904[64] the operator for vertical averaging is intro-

duced as 120595 = int

1

0120595119889120590 and state variables are represented as

120595 = 120595 + 1205951015840 Equations (22) are obtained by neglecting the

orography and terms 1199061015840V1015840 V1015840V1015840 and 1198791015840V1015840 [65] A detailedlinear analysis of the vertically averaged equation (22) isrepresented in [60] In particular two types of wave solutionswere found fast waves that propagate westward and slowwaves that move eastward Within the framework of thismodel ultralong waves are always neutral for any verticallyaveraged zonal wind velocity Indeed linearizing (22) aroundthe following basic state

1198790= 1198790(119910) 119906

0= minus(

119877

1198910

)

1205971198790

120597119910

V0= 0 120587

0= 1

(23)

and assuming the beta plane approximation 119891 = 1198910+ 120573119910

where 120573 = (2Ω1198860) cos120593

0 and representing the solution in

the form (9) one can finally obtain under different asymp-totics the following expressions for four wave solutions [60]

(a) acoustic waves

11988812= 1199060plusmnradic1198882

0+

1198912

0

1198962

(24)

(b) Rossby wave

1198883= 1199060minus

120573

1198962+ 1198912

01198882

0

(25)

(c) baroclinic wave

1198884= 1199060minus

1198912

01199062

0

1198882

0120573

(26)

Here 11988820= (1+120581)119877119879

0and 120581 = 119877119888

119901 These results however are

valid only for the specific case of a neutral atmosphere withΓ = Γ

119889[60 61] To take into account the atmospheric static

stability on the behaviour of ultralong waves the polytropicmodel of the atmosphere can be used for which

119879 (119909 119910 119911 119905) = 1198790(119909 119910 119905) minus Γ (119909 119910 119905) 119911 (27)

where 1198790is the temperature at the surface and Γ is a vertical

temperature gradient Integrating (27) with respect to verticalcoordinate we can obtain 119879

0= 119879(1 + 119877Γ119892) [60] Assuming

the geostrophic approximation on a 120573-plane the set ofvertically averaged equations can be written as [61]

120597119879

120597119905

+

119877119879

1198912

0

1205721

120587

(120587 119879) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

1

1205722

(119879 Γ)

+ 120581

1205731198772119879

2

1198921198912

0

1205724

1205722

2

120597Γ

120597119909

minus

120573119877119879

1198912

0

(1205723+ 120581

12057211205724

1205722

)

1

120587

120597 (120587119879)

120597119909

minus 120581

120573119877119879

1198912

0

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

12057211205724

1205722

120597119879

120597119909

= 0

120597Γ

120597119905

minus

119877119879

1198910

1205722

1

1205722

1

120587

(Γ 120587) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

119877

1198910

1205721

1205722

(119879 Γ)

minus 120581

120573119877119879

1198912

0

12057211205724

1205722

2

120597Γ

120597119909

minus 120581

120573119892

1198912

0

120572112057231205724

1205722

1

120587

120597 (120587119879)

120597119909

+ 120581

120573119892

1198912

0

1205722

11205724

1205722

120597119879

120597119909

= 0

120597120587

120597119905

minus

120573119877

1198912

0

120597 (120587119879)

120597119909

= 0

(28)

Here 1205721= 1 + 119877(Γ119892) 120572

2= 1 + 2119877(Γ119892) 120572

3= 119877(Γ119892) 120572

4=

1 minus (ΓΓ119889) and the operator (119860 119861) = (120597119860120597119909)(120597119861120597119910) minus (120597119860

120597119910)(120597119861120597119909) The basic state is defined as a stationary solutionof system (28) for which

120597119879

120597119909

= 0

120597Γ

120597119909

= 0

120597120587

120597119909

= 0 (29)

or in other words

119879 = 1198790(119910) Γ = Γ

0(119910) 120587 = 120587

0(119910) (30)

Linearizing (28) around the basic state (30) the followingcubic characteristic equation can be obtained in which thesecond order terms are neglected [61]

1198883+ 1198882

1205721

1205722

[120582 (1205722+ 3120581120572

4) minus 1199060]

+ 119888 120582

1205721

1205722

[1205821205811205724(2 + 120572

3+ 120581

1205724

1205722

(21205721minus

1

1205722

))

minus1199060(2 + 120581120572

4+ 31205723) ]

+ (120582

1205721

1205722

1199062

0minus 1205822120581

1205722

11205724

1205722

1199060+ 2120582312058121205722

112057231205722

4

1205723

2

)

= 0

(31)

where 120582 = 12057311987711987901198912

0and 119906

0= minus(119877119891

01205870)((120597(120587

01198790))120597119910) If

the discriminant of this equation is positive then the wavesolution is unstable The domain of zonal flow instabilitycan be found numerically (see diagram in [61]) In Figure 6we reproduce only for the 1st quadrant of a Cartesianplane the domain of instability calculated as a function ofvertically averaged zonal wind velocity 119906

0and dimensionless

temperature lapse rate ΓΓ119889

The imaginary part of phase velocity 119888119894which charac-

terises the growth rate of unstable waves 120594119896equiv 119896119888119894is displayed

in Figure 7 as a function of dimensionless temperature lapserate ΓΓ

119889for different values of vertically averaged zonal

wind velocity 1199060 A maximum phase velocity 119888

119894exists for

given values of 1199060 that is dependent on the ratio of ΓΓ

119889

For instance if 1199060= 20msminus1 then the maximum value

8 Advances in Meteorology

10

08

06

04

02

000 50 100 150 200 250 300

ΓΓd

u0 (m sminus1)

Figure 6 Domain of instability (filled) as a function of dimension-less temperature lapse rate ΓΓ

119889and vertically averaged zonal wind

velocity 1199060

(119888119894)max asymp 834msminus1 is reached at ΓΓ

119889asymp 055 Figure 6 shows

that increasing vertically averaged zonal wind 1199060is associated

with increasing 119888119894 This is further evident in Figure 8 which

shows 119888119894as a function of 119906

0for a range of ΓΓ

119889values The

lower ΓΓ119889

and the larger 119888119894 that is 119888

119894 increases with

decreasing static stability

4 Concluding Remarks

We have studied theoretically the impact of variations in thestatic stability parameter 120590

0and zonal wind shear Λ

120585on the

characteristics of baroclinically unstable waves of synopticscales using Eady-type model with the uniform Λ

120585between

upper and lower boundaries on an 119891-plane Quantitativeestimates of variations in 120590

0and Λ

120585on the growth rate 120594

119896

wavelength of maximum growth rate 119871120594max and short-wave

cutoff 119871min were obtainedAnalytical expressions are derived for sensitivity func-

tions for the growth rate 120594119896with respect to variations in static

stability parameter andwind shear velocityThese expressionsallow estimating to a first-order approximation the influenceof changes in 120590

0and Λ

120585on 120594119896 Analytical expressions for

relative sensitivity functions allow estimating the significanceof variations in 120590

0andΛ

120585on the growth rate of baroclinically

unstable waves with a given zonal wave numberTo study the impact of variations in atmospheric static

stability and zonal wind velocity on the instability of plan-etary scale waves the model with vertically averaged prim-itive equations with 120573-plane approximation was applied Ascontrol parameters we have used dimensionless temperature

15

10

5

0minus02 00 02 04 06 08 10

ΓΓd

u0 = 20u0 = 30u0 = 40

ci

(m sminus

1)

Figure 7 Imaginary part of phase speed 119888119894versus dimensionless

temperature lapse rate ΓΓ119889for different values of vertically averaged

zonal wind velocity 1199060

lapse rate ΓΓ119889and vertically averaged zonal wind velocity

1199060 We have estimated the influence of ΓΓ

119889and 119906

0on the

imaginary part of phase speed 119888119894 whichwas used as ameasure

of instabilityThe obtained results are qualitatively consistent with

changes in the essential weather patterns that occurred overthe last several decades in some areas of the SH and inparticular over Australia (eg [49 50 52ndash54]) Climatechange results suggest SH midlatitude static stability 120590

0may

increase and the MTG (the vertical wind shear Λ120585) may

decrease which according to our linear theoretical modelsleads to a slowing of the growth rate of baroclinic unstablewaves 120594

119896and an increasing wavelength of baroclinic unstable

wave with maximum growth rate 119871120594max that is a spectrum

shift of unstable waves towards longer wavelengths Thesemight affect the favourable conditions for the developmentof baroclinic instability and therefore the rate of cyclogenesisand a reduction in cyclone intensity The obtained sensitivityfunctions demonstrate that waves belonging to the short-wave part of the spectrum of unstable waves are moresensitive to changes in the static stability parameter thanwaves belonging to the long-wave part of the spectrum

To obtain more realistic estimates of the sensitivity of thegrowth rate of unstable waves with respect to static stabilityparameter and MTG numerical modeling based on a fullGCM is required It is hoped to carry out such work in thefuture

Advances in Meteorology 9

ΓΓd = 01ΓΓd = 04

ΓΓd = 23ΓΓd = 09

20

10

00 20 40 60

u0 (mmiddotsminus1)

ci(m

sminus1)

Figure 8 Imaginary part of phase speed 119888119894versus vertically

averaged zonal wind velocity 1199060for different values of dimensionless

temperature lapse rate ΓΓ119889

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors thank Dr I Pisnichenko for clarification ofcertain questions relevant to this paper

References

[1] J R Holton An Introduction to Dynamic Meteorology Aca-demic Press 3rd edition 1992

[2] G K Vallis Atmospheric and Oceanic Fluid Dynamics Cam-bridge University Press 2006

[3] M Mak Atmospheric Dynamics Cambridge University Press2011

[4] V P Dymnikov Stability and Predictability of the Large ScaleAtmospheric Processes Institute of Numerical Mathematics ofthe Russian Academy of Sciences Moscow Russia 2007

[5] J G Charney ldquoThe dynamics of long waves in baroclinicwesterly currentrdquo Journal of Meteorology vol 4 pp 135ndash1621947

[6] E T Eady ldquoLong waves and cyclone wavesrdquo Tellus vol 1 pp33ndash52 1949

[7] R T Pierrehumbert and K L Swanson ldquoBaroclinic instabilityrdquoAnnual Review of Fluid Mechanics vol 27 no 1 pp 419ndash4671995

[8] R Gall ldquoA comparison of linear instability theory with theeddy statistics of a general circulation modelrdquo Journal of theAtmospheric Sciences vol 33 no 3 pp 349ndash373 1976

[9] I M Held ldquoProgress and problems in large-scale atmosphericdynamicsrdquo in The Global Circulation of the Atmosphere TSchneider and A Sobel Eds pp 1ndash21 Princeton UniversityPress Princeton NJ USA 2007

[10] B Farrell ldquoModal and non-modal baroclinic wavesrdquo Journal ofthe Atmospheric Sciences vol 41 no 4 pp 668ndash673 1984

[11] M K Tippett ldquoTransient moist baroclinic instabilityrdquo Tellus Avol 51 no 2 pp 273ndash288 1999

[12] D Hodyss and R Grotjahn ldquoNonmodal and unstable normalmode baroclinic growth as a function of horizontal scalerdquoDynamics of Atmospheres and Oceans vol 37 no 1 pp 1ndash242003

[13] J Pedlosky ldquoFinite amplitude baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 27 no 1 pp 15ndash30 1970

[14] R D Romea ldquoThe effects of friction and beta on finite ampli-tude baroclinic wavesrdquo Journal of the Atmospheric Sciences vol34 pp 1689ndash1695 1977

[15] J Pedlosky ldquoFinite-amplitude baroclinic waves at minimumcritical shearrdquo Journal of the Atmospheric Sciences vol 39 no3 pp 555ndash562 1982

[16] T Warn and P Gauthier ldquoPotential vorticity mixing bymarginally unstable baroclinic disturbancesrdquo Tellus A vol 41no 2 pp 115ndash131 1989

[17] A J Simmons and B J Hoskins ldquoThe lifecycles of some non-linear wavesrdquo Journal of the Atmospheric Sciences vol 35 pp414ndash432 1978

[18] A J Simmons and B J Hoskins ldquoBarotropic influences of thegrowth and decay of nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 37 pp 1679ndash1684 1980

[19] S B Feldstein and I M Held ldquoBarotropic decay of baroclinicwavesrdquo Journal of the Atmospheric Sciences vol 46 pp 1679ndash1684 1989

[20] C D Thorncroft B J Hoskins and M E McIntyre ldquoTwoparadigms of baroclinic-wave life-cycle behaviourrdquo QuaterlyJournal vol 119 no 509 pp 17ndash55 1993

[21] N A Phillips ldquoA simple three-dimensional model for the studyof largescale extratropical flow patternsrdquo Journal ofMeteorologyvol 8 pp 381ndash394 1951

[22] N A Phillips ldquoEnergy transformations and meridional circu-lations associated with simple baroclinic waves in a two-levelquasi-geostrophic modelrdquo Tellus vol 6 pp 273ndash286 1954

[23] L M Polvani and J Pedlosky ldquoThe effect of dissipation onspatially growing nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 45 no 14 pp 1977ndash1989 1988

[24] T G Shepherd ldquoNonlinear saturation of baroclinic instabilityPart I the two-layermodelrdquo Journal of the Atmospheric Sciencesvol 45 no 14 pp 2014ndash2025 1988

[25] G Balasubramanian and M K Yau ldquoBaroclinic instability ina two-layer model with parameterized slantwise convectionrdquoJournal of the Atmospheric Sciences vol 51 no 7 pp 971ndash9901994

[26] A Wiin-Nielsen ldquoOn the structure of atmospheric waves inmiddle latitudesrdquo Atmosfera vol 16 no 2 pp 83ndash102 2003

[27] D D Holm and B A Wingate ldquoBaroclinic instabilities of thetwo-layer quasigeostrophic alpha modelrdquo Journal of PhysicalOceanography vol 35 no 7 pp 1287ndash1296 2005

10 Advances in Meteorology

[28] J Egger ldquoBaroclinic instability in the two-layermodel interpre-tationsrdquo Meteorologische Zeitschrift vol 18 no 5 pp 559ndash5652009

[29] J Lamberts G Lapeyre and V Zeitlin ldquoMoist versus dry baro-clinic instability in a simplified two-layer atmospheric modelwith condensation and latent heat releaserdquo Journal of theAtmospheric Sciences vol 69 pp 1405ndash1426 2012

[30] D M W Frierson ldquoRobust increases in midlatitude static sta-bility in simulations of global warmingrdquo Geophysical ResearchLetters vol 33 no 24 Article ID L24816 2006

[31] N M J Hall B J Hoskins P J Valdes and C A Senior ldquoStormtracks in a high-resolution GCMwith doubled carbon dioxiderdquoQuarterly Journal vol 120 no 519 pp 1209ndash1230 1994

[32] S B Feldstein ldquoThe recent trend and variance increase of theannularmoderdquo Journal of Climate vol 15 no 3 pp 88ndash94 2002

[33] G J Marshall ldquoTrends in the southern annular mode fromobservations and reanalysesrdquo Journal of Climate vol 16 pp4134ndash4143 2003

[34] G M Ostermeier and J M Wallace ldquoTrends in the NorthAtlantic Oscillation-Northern Hemisphere annular mode dur-ing the twentieth centuryrdquo Journal of Climate vol 16 no 2 pp336ndash341 2003

[35] D W J Thompson and S Solomon ldquoInterpretation of recentSouthern Hemisphere climate changerdquo Science vol 296 no5569 pp 895ndash899 2002

[36] S J Lambert ldquoThe effect of enhanced greenhouse warming onwinter cyclone frequencies and strengthsrdquo Journal of Climatevol 8 no 5 pp 1447ndash1462 1995

[37] F Lunkeit K Fraedrich and S E Bauer ldquoStorm tracks ina warmer climate sensitivity studies with a simplified globalcirculation modelrdquo Climate Dynamics vol 14 no 11 pp 813ndash826 1998

[38] D J Lorenz and D L Hartmann ldquoEddy-zonal flow feedback inthe Southern Hemisphererdquo Journal of the Atmospheric Sciencesvol 58 no 21 pp 3312ndash3327 2001

[39] Q Geng and M Sugi ldquoPossible change of extratropical cycloneactivity due to enhanced greenhouse gases and sulphateaerosols study with a high-resolution AGCMrdquo Journal ofClimate vol 16 pp 2262ndash2274 2003

[40] J C Fyfe ldquoExtratropical Southern Hemisphere cyclonesharbingers of climate changerdquo Journal of Climate vol 16 pp2802ndash2805 2003

[41] J H Yin ldquoA consistent poleward shift of the storm tracksin simulations of 21st century climaterdquo Geophysical ResearchLetters vol 32 no 18 Article ID L18701 pp 1ndash4 2005

[42] S J Lambert and J C Fyfe ldquoChanges in winter cyclonefrequencies and strengths simulated in enhanced greenhousewarming experiments results from the models participating inthe IPCC diagnostic exerciserdquo Climate Dynamics vol 26 no7-8 pp 713ndash728 2006

[43] G Gastineau and B J Soden ldquoModel projected changes ofextreme wind events in response to global warmingrdquo Geophys-ical Research Letters vol 36 no 10 Article ID L10810 2009

[44] Y Wu M Ting R Seager H-P Huang and M A CaneldquoChanges in storm tracks and energy transports in a warmer cli-mate simulated by the GFDL CM21 modelrdquo Climate Dynamicsvol 37 no 1 pp 53ndash72 2011

[45] L Bengtsson K I Hodges and E Roeckner ldquoStorm tracks andclimate changerdquo Journal of Climate vol 19 no 15 pp 3518ndash35432006

[46] I Smith ldquoAn assessment of recent trends in Australian rainfallrdquoAustralian Meteorological Magazine vol 53 no 3 pp 163ndash1732004

[47] N Nicholls Detecting Understanding and Attributing ClimateChange Australian Greenhouse Office Publication 2007

[48] B C Bates P Hope B Ryan I Smith and S Charles ldquoKey find-ings from the Indian Ocean Climate Initiative and their impacton policy development in Australiardquo Climatic Change vol89 no 3-4 pp 339ndash354 2008

[49] J S Frederiksen and C S Frederiksen ldquoDecadal changes inSouthern Hemisphere winter cyclogenesisrdquo CSIROMarine andAtmospheric Research Paper 002 2005

[50] J S Frederiksen and C S Frederiksen ldquoInterdecadal changes insouthern hemisphere winter storm track modesrdquo Tellus A vol59 no 5 pp 599ndash617 2007

[51] J S Frederiksen C S Frederiksen and S L Osbrough ldquoMod-elling of changes in Southern Hemisphere weather systemsduring the 20th centuryrdquo in Proceedings of the 18th WorldIMACS Congress and International Congress on Modelling andSimulation pp 2562ndash2568 Cairns Australia July 2009

[52] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoCauses of changing Southern Hemispheric weathersystemsrdquo in Managing Climate Change I Jupp P Holper andW Cai Eds pp 85ndash98 CSIRO Publishing 2010

[53] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoChanges in Southern Hemisphere rainfall circulationand weather systemsrdquo in Proceedings of the 19th InternationalCongress on Modelling and Simulation (MODSIM rsquo11) pp 2712ndash2718 Perth Australia December 2011

[54] C S Frederiksen J S Frederiksen J M Sisson and S LOsbrough ldquoObserved and projected changes in the annual cycleof Southern Hemisphere baroclinicity for storm formationrdquoin Proceedings of the 19th International Congress on Modellingand Simulation (MODSIMrsquo 11) pp 2719ndash2725 Perth AustraliaDecember 2011

[55] N A Phillips ldquoGeostrophicmotionrdquoReviews of Geophysics vol1 no 2 pp 123ndash176 1963

[56] I Bordi K Fraedrich F Lunkeit and A Sutera ldquoTroposphericdouble jets meridional cells and eddies a case study andidealized simulationsrdquoMonthly Weather Review vol 135 no 9pp 3118ndash3133 2007

[57] I Bordi K Fraedrich M Ghil and A Sutera ldquoZonal flowregime changes in a GCM and in a simple quasigeostrophicmodel the role of stratospheric dynamicsrdquo Journal of theAtmospheric Sciences vol 66 no 5 pp 1366ndash1383 2009

[58] J G Charney ldquoOn the scale of atmosphericmotionsrdquoGeofysiskePublikasjoner vol 17 no 2 pp 3ndash17 1948

[59] A Burger ldquoScale consideration of planetary motions of theatmosphererdquo Tellus vol 10 pp 195ndash205 1958

[60] I A Pisnichenko ldquoUltralong-wave dynamics in a two-dimen-sional baroclinic atmosphere modelrdquo Atmospheric and OceanicPhysics vol 16 no 9 pp 883ndash892 1980

[61] I A Pisnichenko ldquoInfluence variable static stability on thedynamics of ultralong waves in two-dimensional baroclinicmodel of the atmosphererdquoAtmospheric andOceanic Physics vol19 no 11 pp 1223ndash1226 1983

[62] A Wiin-Nielsen ldquoOn barotropic and baroclinic models withspecial emphasis onultra-longwavesrdquoMonthlyWeather Reviewvol 87 pp 171ndash183 1959

[63] B W BinWang and A Barcilon ldquoMoist stability of a barocliniczonal flow with conditionally unstable stratificationrdquo Journal ofthe Atmospheric Sciences vol 43 no 7 pp 705ndash719 1986

Advances in Meteorology 11

[64] N A Phillips ldquoA coordinate system having some specialadvantages for numerical forecastingrdquo Journal of Meteorologyvol 14 pp 184ndash185 1957

[65] D M Alishaev ldquoDynamics of a two-dimensional baroclinicatmosphererdquoAtmospheric and Oceanic Physics vol 16 no 2 pp99ndash107 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

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Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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International Journal of

Geophysics

OceanographyInternational Journal of

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Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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MineralogyInternational Journal of

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Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

2 Advances in Meteorology

of baroclinic unstable waves to understand how initiallyinfinitesimal perturbations grow to large but finite amplitudemodifying the mean flow This kind of research requiresnumerical integration of the atmospheric nonlinear modelequations (eg [17ndash20])

It is important to underline the significant role of thetwo-layer model of Philips [21 22] and its modifications(eg [13 23ndash28]) in developing baroclinic instability theoriesHowever for this purpose a two-layer model was usuallyused in the quasigeostrophic approximation A generalizedbaroclinic instability analysis based on two-level primitive-equation model has been described in [29] and applied todetailed consideration of the baroclinic instability mecha-nism

There is evidence of an increase over the past decades ofthe static stability in the extratropics [30] and a pole-wardmovement of the midlatitude precipitation zones and stormtracks which are uniquely linked to zones with strong MTG-baroclinic zones (eg [31ndash35]) Recent studies have also indi-cated an increase in intensity of extratropical cyclones anda decrease in frequency [36ndash44] Changes in geographicallocations and intensity of storm tracks aremore distinct in thesouthern hemisphere (SH) (eg [45]) affecting the essentialfeatures of weather patterns over large territories such asAustralia [46ndash54] and indicating a change of favourableconditions for baroclinic instability because the key role ofbaroclinic instability in the development of midlatitudecyclones is well established

To study how baroclinic instability has changed in theSH in recent decades a comparative analysis of climates forperiods 1949ndash1968 and 1975ndash1994 has been carried out in[50 54] usingNational Centres for Environmental Prediction(NCEP) reanalyses and the European Centre for Medium-Range Weather Forecasting (ECMWF) Reanalysis (ERA40)data As a measure of the baroclinic instability the Phillipscriterion [55] adapted to spherical geometry was chosen

119906(300 hPa)

minus 119906(700 hPa)

ge

119887119896119888119901120590

1198860Ω

cos120593sin2120593

(1)

where 119906(300 hPa) and 119906(700 hPa) are the zonal wind velocities

at 300 hPa and 700 hPa isobaric levels respectively 120590 is thestatic stabilitymeasure for a given reference state 119888

119901is the spe-

cific heat of air at constant pressureΩ is the angular rotationspeed of the earth 119886

0is the earthrsquos radius 119887

119896is a dimensionless

constant and 120593 is a latitude J S Frederiksen and C SFrederiksen [50] obtained a significant decreasing trend inbaroclinic instability over the middle latitudes of the SHThemost significant negative trend is localized between 30S and40S For instance the difference between the two periods1949ndash1968 and 1975ndash1994 reached a maximum of 17 in theSH subtropical jet stream (for the July climate) In additionit was found that further poleward the baroclinic instabilityintensified creating favourable conditions for cyclogenesisUsing a two-level linearized primitive equation model Fred-eriksen et al [51] studied the growth rate of unstable modesfor the July reference state averaged over the abovementionedperiodsThey identified a 30 reduction in the growth rate ofcyclone-scale modes between the two periods

It should be pointed out that since the beginning of the1980s exploration of the essential features of the atmosphericgeneral circulation in the SH has been facilitated by theavailability of plentiful satellite upper-air data In particularthe importance of baroclinic instability and large-scale eddiesin the formation of zonal wind characteristics such as atropospheric double-jet phenomenonwas studied in [56 57]

Taking into account the significant role of baroclinicinstability in the development ofmidlatitude cyclones and theintensity changes of baroclinic instability in some areas of theSH in recent decades this paper examines the sensitivity ofthemain characteristics of baroclinically unstable waves (egthe growth rates of unstable waves as function of wavelength)to fundamental atmospheric parameters the static stabilityparameter120590

0which is characterized by temperature lapse rate

Γ = minus120597119879120597119911 and zonal wind vertical shear Λ which by ther-mal wind balance characterizes the meridional temperaturegradient Two classes of waves [55] are considered The firstone is characterized by the Rossby number Ro equiv 119880(119891119871) sim

01 and the representative horizontal length scale 119871 which issmaller than the Earthrsquos radius 119886

0 that is119871119886

0sim 01 Here119880 is

a horizontal velocity scale and119891 is theCoriolis parameter Forthis class of waves 120577 ≫ 119863 where 120577 is the vertical componentof relative vorticity and119863 is the horizontal divergence Synop-tic scale waves belong to this type of motion [58]The secondclass which includes planetary or ultralong waves [59] ischaracterized by Ro sim 001 and 119871119886

0sim 1 For these waves the

magnitude of horizontal divergence is comparable with thevertical component of vorticity that is 120577 asymp 119863 To study syn-optic scale waves the Eady-type model is used with uniformzonal wind shear between upper and lower boundaries on an119891-plane In this context parameters 120590

0and Λ are considered

to be variables that control the development of baroclinicinstability in the atmosphere Ultralong waves are investi-gated based on a model with vertically averaged equations[60 61] with a 120573-plane approximation Simplified modelssuch as the Eady model of baroclinic instability and modelswith vertically averaged equations despite their simplicityallow solutions to be obtained that clearly illustrate realphysical processes in the atmosphere

2 Synoptic Scale BaroclinicallyUnstable Waves

21 The Model Equations We consider the inviscid primitiveequation atmospheric model in normalized isobaric coordi-nates (119909 119910 120585) on an 119891-plane in the following form

(

120597

120597119905

+ u sdot nabla + 120596

1198750

120597

120597120585

) u + 119891k times u = minusnablaΦ

(

120597

120597119905

+ u sdot nabla)119879 minus 119878120585120596 =

119876

119888119901

nabla sdot u + 1

1198750

120597120596

120597120585

= 0

120597Φ

120597120585

= minus

119877119879

120585

(2)

Advances in Meteorology 3

The state variables of the model are the horizontal velocityvector u = (119906 V)119879 the vertical pressure velocity 120596 equiv 119889119901119889119905where 119901 is pressure the geopotentialΦ and the temperature119879 The operator nabla equiv (120597120597119909 120597120597119910) is applied to the horizontalcoordinates 119909 and 119910 directed eastward and northwardrespectively The normalized pressure 120585 = 119901119875

0 where 119875

0=

1000 hPa is a ldquostandardrdquo pressure approximately equal to thesurface pressure is taken as the vertical coordinate while thetime is denoted by 119905 Other notations are the diabatic heatingrate per unit time per unitmass119876 the gas constant119877 the unitvector in the vertical direction k and the reference state staticstability measure 119878

120585in the normalized isobaric coordinate

system

119878120585=

119877119879

1198921198750120585

(Γ119889minus Γ) (3)

where 119879 is a reference temperature 119892 is the gravity accelera-tion Γ

119889is the dry adiabatic lapse rate and Γ is the reference

state lapse rateWe employ the119891-plane approximation so thatthe Coriolis parameter 119891 is a constant 119891 = 119891

0= 2Ω sin120593

0

with 1205930being the latitude of interest Hereafter we consider

only adiabatic process and thus assume zero heating rate 119876The following boundary conditions are used for the pressurevelocity

120596 = 0 at 120585 = 0 120585 = 1 (4)

The atmospheric reference state defined by 119906 V120596119879 andΦ is steady and satisfies the following relations

119906 = minus

1

1198910

120597Φ

120597119910

V = 0

120596 = 0

120597Φ

120597120585

= minus

119877119879

120585

(5)

where 119879 = 119879(119910 120585) By substituting (5) into the set of (2) onecan see that (5) is a solution of (2) that describes the zonalflow

120597119906

120597120585

=

119877

1198910120585

120597119879

120597119910

(6)

which matches the specified distribution of the zonally aver-aged temperature 119879(119910 120585) and represents thermal wind bal-ance To consider only the baroclinicmechanism of the atmo-spheric instability meridional variability of the basic zonalflow is excluded In other words the barotropic impact on theinstability of the basic zonal flow is not taken into accountTherefore we assume that the velocity of the basic zonalflow does not depend on the horizontal 119910-coordinate that is119906 = 119906(120585)Thus the problem now is the study of the instabilityof the basic zonal flow (6) with respect to infinitesimalperturbations For this purpose the system (2) is linearizedaround the basic state (5) Representing the state variables as120595(119909 119910 120585 119905) = 120595(120585) + 120595

1015840(119909 120585 119905) where 120595 is a basic state and

1205951015840 is an infinitesimal perturbation and taking into account

the hydrostatic equations and the thermal wind relationship(6) the linearized system can be written as

1205971199061015840

120597119905

+ 119906

120597119906

120597119909

+

1205961015840

1198750

120597119906

120597120585

= minus

120597Φ1015840

120597119909

+ 1198910V1015840

120597V1015840

120597119905

+ 119906

120597V1015840

120597119909

+ 11989101199061015840= 0

1205971199061015840

120597119909

+

1

1198750

1205971205961015840

120597120585

= 0

120597

120597119905

(

120597Φ1015840

120597120585

) + 119906

120597

120597119909

(

120597Φ1015840

120597120585

) minus V10158401198910

120597119906

120597120585

+ 120590011987501205961015840= 0

(7)

The static stability parameter 1205900is expressed as

1205900= minus

120572

1198750

120597 lnΘ120597120585

=

1198772119879

1198921198752

01205852(Γ119889minus Γ) (8)

where 120572 is a specific volume and Θ = 119879120585minus119877119888119901 is a reference

potential temperature Suppose that 120597119906120597120585 = minusΛ120585= const

Applying the method of separation of variables we assumethe solutions of the form

1205951015840(119909 120585 119905) = (120585) 119890

119894119896(119909minus119888119905) (9)

where (120585) is a function of 120585 only 119896 is a wave number and 119888is a phase velocity of perturbations which in general is a com-plex value 119888 = 119888

119903+ 119894119888119894 After substituting (9) into (7) we can

finally obtain the following single equation for

(119906 minus 119888) [1 minus

1198962

1198912

0

(119906 minus 119888) (119906 minus 119888)]

1205972

1205971205852

+ 2Λ120585

120597

120597120585

minus 12059001198752

0

1198962

1198912

0

(119906 minus 119888) = 0

(10)

Similar equations were considered in a number of publica-tions (eg [4 62 63])The analytical solution of this equationcan be obtained by eliminating gravity waves from the con-sideration and assuming that 120590

0= const For this particular

case (10) can be transformed into

(119906 minus 119888)

1205972

1205971205852+ 2Λ120577

120597

120597120585

minus (119906 minus 119888) 12059001198752

0

1198962

1198912

0

= 0 (11)

The boundary conditions for are specified as

= 0 at 120585 = 0 120585 = 1 (12)

Equation (11) together with boundary conditions (12) rep-resents the eigenvalue problem for the complex phase speed119888 The fundamental solutions of (11) are expressed throughthe Bessel functions of the first and second kinds With thehomogeneous boundary conditions (12) the following twodiscrete eigenvalues can be obtained [4]

11988812=

Λ120585

2

[1 plusmn radic1 minus

4

1205782(120578 coth (120578) minus 1)] (13)

4 Advances in Meteorology

where

120578 = 1198750radic1205900 (

119896

1198910

) (14)

The phase velocity 119888 will amplify exponentially if 119888 has animaginary part 119888

119894 From (13) we can see that this will occur if

the discriminant in (13) is less than zero

4

1205782(120578 coth (120578) minus 1) gt 1 (15)

which gives by theNewtonrsquos iteration algorithm the necessarycondition for instability 120578 lt 120578

119888asymp 23994 Besides two dis-

crete eigenvalues (13) the eigenvalue problem (11)-(12) has acontinuous spectrum of eigenvalues 119888 isin (0 119906(120585)) that are realand therefore can be neglected in the problem of baroclinicinstability [4] The growth rate of unstable waves 120594

119896equiv 119896119888119894is

calculated by the following expression

120594119896=

Λ1205851198910

21198750radic1205900

radic10038161003816100381610038161205782minus 4 (120578 coth (120578) minus 1)100381610038161003816

1003816 (16)

As we can see from (16) at a given latitude the growth rate120594119896is a function of 120590

0(static stability) Λ

120585(wind shear) and

quantity 120578 that depends on the horizontal wavenumber 119896 and1205900

22 Impact of Static Stability andVerticalWind Shear on Baro-clinic Instability Within the Eady problem framework thestatic stability parameter 120590

0and the vertical wind shear Λ

120585

represent the main control variables By varying 1205900and Λ

120585

one can obtain estimates of the impact of these parameters onthe development of baroclinic instability in the atmosphereIn this research parameters corresponding to the basicstate are given the following values Λ

120585= 40msminus1 [50]

and 1205900= 2times10

minus6m2 Paminus2 sminus2 [1]These parameter values canbe used as an approximation to describe the zonal-averagedatmospheric conditions for JJA (June July andAugust) in theSH [50] The latitude of interest is assumed to be 120593

0= 45 S

which gives 1198910= minus1028 times 10

minus4 sminus1Figure 1 shows plot of Eady growth rate versus zonal

wavenumber obtained with (16)The growth rate has a short-wave instability cutoff beyond which waves are stable Let119871min be the wavelength that corresponds to a short-wavecutoff Value of 119871min can be obtained from (14) when 120578 = 120578

119888

which gives 119871min = 3592 km To calculate the wavelength ofmaximum growth rate 119871

120594max one can take 120597120594119896120597119896 and set

the result equal to zero which gives 120578 = 120578119898asymp 16061 Then by

using (14) we can obtain 119871120594max = 5366 km

The influence of the static stability parameter on thewavelength of maximum growth rate 119871

120594max and the shortwave cut-off 119871min are shown in Figure 2 In general anincrease in the parameter 120590

0leads to an increase in both

119871119896119888max and 119871min The functional dependences between 119871min

and 1205900 and between 119871

119896119888max and 1205900 are almost linear plusmn10departure of static stability parameter Δ120590

0from its nominal

value 1205900= 20times10

minus6m2 Paminus2 sminus2 results in about plusmn5 changefor both 119871

119896119888max and 119871min with respect to the nominal value1205900 For instance if Δ120590 = 01 times 120590

0 then 119871

119896119888max = 5628 and

09

06

03

000 2 4 6 8

kz

Xk

(dayminus1)

Mode with maximum Short-wavegrowth rate cutoff

Figure 1 Growth rate 120594119896versus zonal wavenumber 119896

119911for 1205900= 2 times

10minus6m2 Paminus2 s minus 2 and Δ

120585= 40msminus1

6

4

2

(km

)

1 2 3(m2 Paminus2 sminus2 )

LminLkcmax

1205900 times 106

Ltimes10minus3

Figure 2 Length of waves with maximum growth rates 119871119896119888max and

short-wave cutoff 119871min as functions of the static stability parameter1205900

119871min = 3768 km and if Δ120590 = minus011205900 then 119871

119896119888max = 5091

and 119871min = 3408 kmFigure 3 illustrates the growth rate of unstable waves

versus the static stability parameter at different values of Λ120585

Parameters 1205900and Λ

120585influence the growth rate 120594

119896in the

opposite direction growth rate decreases if 1205900increases and

if Λ120585decreases Note that the decrease of the parameter Λ

120585

indicates the weakening of the intensity of the barocliniczone that is reduction of the MTG In nature both of theseprocesses take place which leads to a synergistic effect For

Advances in Meteorology 5

20

15

10

05

0 1 2 3

Xk

(dayminus1)

Λ120576 = 30Λ120576 = 40

Λ120576 = 50Λ120576 = 60

(m2 Paminus2 sminus2 )1205900 times 106

Figure 3 Growth rate 120594119896versus static stability parameter 120590

0for

different values of parameter Λ120585(units m sminus1)

instance if Λ120585decreases by 10 and the static stability

parameter increases by 10 the growth rate 120594119896decreases by

14Since 120594

119896is a nonlinear function of 120590

0(16) to estimate the

influence of infinitesimal perturbations in 1205900on variations in

120594119896 the sensitivity function

119878120590=

120597120594119896

1205971205900

(17)

and the relative sensitivity function

119878119877

120590=

120597120594119896120594119896

12059712059001205900

=

1205900

120594119896

120597120594119896

1205971205900

(18)

can be used The function 119878120590shows changes in 120594

119896due to

variations in 1205900 The relative sensitivity function 119878119877

120590is used

to compare model parameters to find out what parameter isthemost important for a certain percent change in the param-eters Sensitivity functions (17) and (18) are evaluated in thevicinity of some nominal value of the parameter 120590

0 We can

select several nominal values to cover some range of changesin 1205900 Differentiating (16) with respect to control parameter

1205900 we can obtain the expression for 119878

120590

119878120590=

120594119896

21205900

(1198750radic1205900

119896

1198910

120578 minus 2 coth (120578) + 2120578csch2 (120578)1205782minus 4 (120578 coth (120578) minus 1)

minus 1)

(19)

Sensitivity 119878120590versus zonal wavenumbers for different values

of 1205900with Λ

120585= 40msminus1 are shown in Figure 4 The absolute

value of the sensitivity of 120594119896with respect to 120590

0exponentially

minus5

minus10

minus15

minus20

0 2 4 6 8 10

S120590

kz

1205900 = 101205900 = 201205900 = 30

Figure 4 Sensitivity function 119878120590versus zonal wavenumber 119896

119911for

different nominal values of the static stability parameter 1205900(units

106m2 Paminus2 sminus2)

increases with decreasing wavelength For planetary scalewaves (zonal wave numbers 1ndash4) the absolute value of thesensitivity of 120594

119896with respect to 120590

0is palpably less than

sensitivity for synoptic scale waves (zonal wave numbers ge5)Absolute and relative sensitivity functions 119878

120590and 119878119877

120590calcu-

lated for different values of1205900for various zonal wave numbers

are shown in Tables 1 and 2 respectivelyThe expression for sensitivity function 119878

Λcan be eas-

ily obtained by differentiating (16) with respect to controlparameter Λ

120585

119878Λ=

1198910

21198750radic1205900

radic10038161003816100381610038161205782minus 4 (120578 coth (120578) minus 1)100381610038161003816

1003816 (20)

The function 119878Λversus zonalwave number 119896

119911for different

1205900is plotted in Figure 5 It is clear to see that for a given value

of the parameter 1205900the graph of function 119878

Λ(119896119911) is verymuch

like the classic picture of the growth rates 120594119896versus zonal

wavenumber 119896119911[5] It is interesting that the relative sensitivity

function 119878119877Λdoes not depend on the wavelength (wavenum-

ber) and for all of the unstable waves is equal to unity

119878119877

Λ=

120597120594119896120594119896

120597Λ120585Λ120585

=

Λ120585

120594119896

120597120594119896

120597Λ120585

= 1 (21)

Since relative sensitivity functions allow direct comparison ofthe importance of model parameters on the growth rate 120594

119896

we can see that because 119878119877Λ= 1 the parameter Λ

120585(ie the

meridional temperature gradient) is more important than

6 Advances in Meteorology

Table 1 Absolute sensitivity 119878120590as a function of zonal wavenumber 119896

119911for different values of static stability parameter 120590

0

1205900m2 Paminus2 sminus2 Zonal wave number 119896

119911

1 2 3 4 5 6 710 times 10

minus6minus00157 minus01251 minus04175 minus09772 minus18879 minus32470 minus52005

15 times 10minus6

minus00157 minus01245 minus04139 minus09675 minus18794 minus33014 minus5592720 times 10

minus6minus00157 minus01240 minus04110 minus09626 minus18979 minus34964 minus69371

25 times 10minus6

minus00157 minus01234 minus04088 minus09629 minus19509 minus39665 minus23003130 times 10

minus6minus00157 minus01230 minus04071 minus09691 minus20540 minus53057

Table 2 Relative sensitivity 119878119877120590as a function of zonal wavenumber 119896

119911for different values of static stability parameter 120590

0

1205900m2 Paminus2 sminus2 Zonal wave number 119896

119911

1 2 3 4 5 6 710 times 10

minus6minus00062 minus00250 minus00575 minus01057 minus01735 minus02692 minus04095

15 times 10minus6

minus00093 minus00379 minus00881 minus01656 minus02837 minus04749 minus0837120 times 10

minus6minus00124 minus00509 minus01201 minus02375 minus04220 minus08004 minus19939

25 times 10minus6

minus00156 minus00642 minus01539 minus03087 minus06071 minus14399 minus31424330 times 10

minus6minus00187 minus00777 minus01898 minus03974 minus08759 minus34322

3

2

1

00 2 4 6 8 10 12

kz

1205900 = 101205900 = 201205900 = 30

107timesSΛ

Figure 5 Sensitivity function 119878Λversus zonal wavenumber 119896

119911for

different nominal values of the static stability parameter 1205900(units

106m2 Paminus2 sminus2)

the static stability parameter 1205900except for 120590

0gt 2 times

10minus6m2 Paminus2 sminus2 for waves with 119896

119911ge 6 (see Table 2) Wher-

ever the midlatitude values in Table 2 are less than one thegrowth rate is more sensitive to the meridional temperaturegradient (ie Λ

120585) than the static stability (120590

0)

The obtained results are consistent with observations[50 51 54] an increase in static stability and a decrease of

the MTG have occurred over the past few decades in someareas of the SH which has led to a decrease in the growthrate of baroclinic unstable waves a shift of the spectrum ofunstable waves in the long wavelength part of spectrum andaweakened intensity of cyclogenesis Naturally these changesaffect favourable conditions for the development of baroclinicinstability and the essential features of weather patterns overlarge territories particularly over Australia

3 Planetary Scale Waves

To study the influence of the static stability parameter onthe dynamics of planetary scale (ultralong) waves a thinfilm approximation is applied This approximation employsa specific averaging technique over the vertical coordinate tothe system of primitive equations [60] As a result a two-dimensional set of equations can be obtained that describesthe dynamics of a two-dimensional baroclinic film Theseequations reproduce all the wavelike solutions that corre-spond to the main weather-forming modes of three-dimen-sional models and therefore can be used in theoretical studiesof large-scale dynamic processes in the atmosphere Thesystem of vertically averaged equations can be written as [60]

120597119906

120597119905

+ V sdot nabla119906 = 119891V minus119877

120587

120597

120597119909

(120587119879)

120597V120597119905

+ V sdot nablaV = minus119891119906 minus119877

120587

120597

120597119910

(120587119879)

120597120587

120597119905

+ nabla sdot (120587V) = 0

120597119879

120597119905

+ V sdot nabla119879 +

119877

119888119901

119879nabla sdot V = 0

(22)

where 120587 = 1199011199041198750 For instance if the original primitive equa-

tions are written in the Phillipsrsquo vertical coordinate system

Advances in Meteorology 7

120590 = 119901119901119904[64] the operator for vertical averaging is intro-

duced as 120595 = int

1

0120595119889120590 and state variables are represented as

120595 = 120595 + 1205951015840 Equations (22) are obtained by neglecting the

orography and terms 1199061015840V1015840 V1015840V1015840 and 1198791015840V1015840 [65] A detailedlinear analysis of the vertically averaged equation (22) isrepresented in [60] In particular two types of wave solutionswere found fast waves that propagate westward and slowwaves that move eastward Within the framework of thismodel ultralong waves are always neutral for any verticallyaveraged zonal wind velocity Indeed linearizing (22) aroundthe following basic state

1198790= 1198790(119910) 119906

0= minus(

119877

1198910

)

1205971198790

120597119910

V0= 0 120587

0= 1

(23)

and assuming the beta plane approximation 119891 = 1198910+ 120573119910

where 120573 = (2Ω1198860) cos120593

0 and representing the solution in

the form (9) one can finally obtain under different asymp-totics the following expressions for four wave solutions [60]

(a) acoustic waves

11988812= 1199060plusmnradic1198882

0+

1198912

0

1198962

(24)

(b) Rossby wave

1198883= 1199060minus

120573

1198962+ 1198912

01198882

0

(25)

(c) baroclinic wave

1198884= 1199060minus

1198912

01199062

0

1198882

0120573

(26)

Here 11988820= (1+120581)119877119879

0and 120581 = 119877119888

119901 These results however are

valid only for the specific case of a neutral atmosphere withΓ = Γ

119889[60 61] To take into account the atmospheric static

stability on the behaviour of ultralong waves the polytropicmodel of the atmosphere can be used for which

119879 (119909 119910 119911 119905) = 1198790(119909 119910 119905) minus Γ (119909 119910 119905) 119911 (27)

where 1198790is the temperature at the surface and Γ is a vertical

temperature gradient Integrating (27) with respect to verticalcoordinate we can obtain 119879

0= 119879(1 + 119877Γ119892) [60] Assuming

the geostrophic approximation on a 120573-plane the set ofvertically averaged equations can be written as [61]

120597119879

120597119905

+

119877119879

1198912

0

1205721

120587

(120587 119879) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

1

1205722

(119879 Γ)

+ 120581

1205731198772119879

2

1198921198912

0

1205724

1205722

2

120597Γ

120597119909

minus

120573119877119879

1198912

0

(1205723+ 120581

12057211205724

1205722

)

1

120587

120597 (120587119879)

120597119909

minus 120581

120573119877119879

1198912

0

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

12057211205724

1205722

120597119879

120597119909

= 0

120597Γ

120597119905

minus

119877119879

1198910

1205722

1

1205722

1

120587

(Γ 120587) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

119877

1198910

1205721

1205722

(119879 Γ)

minus 120581

120573119877119879

1198912

0

12057211205724

1205722

2

120597Γ

120597119909

minus 120581

120573119892

1198912

0

120572112057231205724

1205722

1

120587

120597 (120587119879)

120597119909

+ 120581

120573119892

1198912

0

1205722

11205724

1205722

120597119879

120597119909

= 0

120597120587

120597119905

minus

120573119877

1198912

0

120597 (120587119879)

120597119909

= 0

(28)

Here 1205721= 1 + 119877(Γ119892) 120572

2= 1 + 2119877(Γ119892) 120572

3= 119877(Γ119892) 120572

4=

1 minus (ΓΓ119889) and the operator (119860 119861) = (120597119860120597119909)(120597119861120597119910) minus (120597119860

120597119910)(120597119861120597119909) The basic state is defined as a stationary solutionof system (28) for which

120597119879

120597119909

= 0

120597Γ

120597119909

= 0

120597120587

120597119909

= 0 (29)

or in other words

119879 = 1198790(119910) Γ = Γ

0(119910) 120587 = 120587

0(119910) (30)

Linearizing (28) around the basic state (30) the followingcubic characteristic equation can be obtained in which thesecond order terms are neglected [61]

1198883+ 1198882

1205721

1205722

[120582 (1205722+ 3120581120572

4) minus 1199060]

+ 119888 120582

1205721

1205722

[1205821205811205724(2 + 120572

3+ 120581

1205724

1205722

(21205721minus

1

1205722

))

minus1199060(2 + 120581120572

4+ 31205723) ]

+ (120582

1205721

1205722

1199062

0minus 1205822120581

1205722

11205724

1205722

1199060+ 2120582312058121205722

112057231205722

4

1205723

2

)

= 0

(31)

where 120582 = 12057311987711987901198912

0and 119906

0= minus(119877119891

01205870)((120597(120587

01198790))120597119910) If

the discriminant of this equation is positive then the wavesolution is unstable The domain of zonal flow instabilitycan be found numerically (see diagram in [61]) In Figure 6we reproduce only for the 1st quadrant of a Cartesianplane the domain of instability calculated as a function ofvertically averaged zonal wind velocity 119906

0and dimensionless

temperature lapse rate ΓΓ119889

The imaginary part of phase velocity 119888119894which charac-

terises the growth rate of unstable waves 120594119896equiv 119896119888119894is displayed

in Figure 7 as a function of dimensionless temperature lapserate ΓΓ

119889for different values of vertically averaged zonal

wind velocity 1199060 A maximum phase velocity 119888

119894exists for

given values of 1199060 that is dependent on the ratio of ΓΓ

119889

For instance if 1199060= 20msminus1 then the maximum value

8 Advances in Meteorology

10

08

06

04

02

000 50 100 150 200 250 300

ΓΓd

u0 (m sminus1)

Figure 6 Domain of instability (filled) as a function of dimension-less temperature lapse rate ΓΓ

119889and vertically averaged zonal wind

velocity 1199060

(119888119894)max asymp 834msminus1 is reached at ΓΓ

119889asymp 055 Figure 6 shows

that increasing vertically averaged zonal wind 1199060is associated

with increasing 119888119894 This is further evident in Figure 8 which

shows 119888119894as a function of 119906

0for a range of ΓΓ

119889values The

lower ΓΓ119889

and the larger 119888119894 that is 119888

119894 increases with

decreasing static stability

4 Concluding Remarks

We have studied theoretically the impact of variations in thestatic stability parameter 120590

0and zonal wind shear Λ

120585on the

characteristics of baroclinically unstable waves of synopticscales using Eady-type model with the uniform Λ

120585between

upper and lower boundaries on an 119891-plane Quantitativeestimates of variations in 120590

0and Λ

120585on the growth rate 120594

119896

wavelength of maximum growth rate 119871120594max and short-wave

cutoff 119871min were obtainedAnalytical expressions are derived for sensitivity func-

tions for the growth rate 120594119896with respect to variations in static

stability parameter andwind shear velocityThese expressionsallow estimating to a first-order approximation the influenceof changes in 120590

0and Λ

120585on 120594119896 Analytical expressions for

relative sensitivity functions allow estimating the significanceof variations in 120590

0andΛ

120585on the growth rate of baroclinically

unstable waves with a given zonal wave numberTo study the impact of variations in atmospheric static

stability and zonal wind velocity on the instability of plan-etary scale waves the model with vertically averaged prim-itive equations with 120573-plane approximation was applied Ascontrol parameters we have used dimensionless temperature

15

10

5

0minus02 00 02 04 06 08 10

ΓΓd

u0 = 20u0 = 30u0 = 40

ci

(m sminus

1)

Figure 7 Imaginary part of phase speed 119888119894versus dimensionless

temperature lapse rate ΓΓ119889for different values of vertically averaged

zonal wind velocity 1199060

lapse rate ΓΓ119889and vertically averaged zonal wind velocity

1199060 We have estimated the influence of ΓΓ

119889and 119906

0on the

imaginary part of phase speed 119888119894 whichwas used as ameasure

of instabilityThe obtained results are qualitatively consistent with

changes in the essential weather patterns that occurred overthe last several decades in some areas of the SH and inparticular over Australia (eg [49 50 52ndash54]) Climatechange results suggest SH midlatitude static stability 120590

0may

increase and the MTG (the vertical wind shear Λ120585) may

decrease which according to our linear theoretical modelsleads to a slowing of the growth rate of baroclinic unstablewaves 120594

119896and an increasing wavelength of baroclinic unstable

wave with maximum growth rate 119871120594max that is a spectrum

shift of unstable waves towards longer wavelengths Thesemight affect the favourable conditions for the developmentof baroclinic instability and therefore the rate of cyclogenesisand a reduction in cyclone intensity The obtained sensitivityfunctions demonstrate that waves belonging to the short-wave part of the spectrum of unstable waves are moresensitive to changes in the static stability parameter thanwaves belonging to the long-wave part of the spectrum

To obtain more realistic estimates of the sensitivity of thegrowth rate of unstable waves with respect to static stabilityparameter and MTG numerical modeling based on a fullGCM is required It is hoped to carry out such work in thefuture

Advances in Meteorology 9

ΓΓd = 01ΓΓd = 04

ΓΓd = 23ΓΓd = 09

20

10

00 20 40 60

u0 (mmiddotsminus1)

ci(m

sminus1)

Figure 8 Imaginary part of phase speed 119888119894versus vertically

averaged zonal wind velocity 1199060for different values of dimensionless

temperature lapse rate ΓΓ119889

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors thank Dr I Pisnichenko for clarification ofcertain questions relevant to this paper

References

[1] J R Holton An Introduction to Dynamic Meteorology Aca-demic Press 3rd edition 1992

[2] G K Vallis Atmospheric and Oceanic Fluid Dynamics Cam-bridge University Press 2006

[3] M Mak Atmospheric Dynamics Cambridge University Press2011

[4] V P Dymnikov Stability and Predictability of the Large ScaleAtmospheric Processes Institute of Numerical Mathematics ofthe Russian Academy of Sciences Moscow Russia 2007

[5] J G Charney ldquoThe dynamics of long waves in baroclinicwesterly currentrdquo Journal of Meteorology vol 4 pp 135ndash1621947

[6] E T Eady ldquoLong waves and cyclone wavesrdquo Tellus vol 1 pp33ndash52 1949

[7] R T Pierrehumbert and K L Swanson ldquoBaroclinic instabilityrdquoAnnual Review of Fluid Mechanics vol 27 no 1 pp 419ndash4671995

[8] R Gall ldquoA comparison of linear instability theory with theeddy statistics of a general circulation modelrdquo Journal of theAtmospheric Sciences vol 33 no 3 pp 349ndash373 1976

[9] I M Held ldquoProgress and problems in large-scale atmosphericdynamicsrdquo in The Global Circulation of the Atmosphere TSchneider and A Sobel Eds pp 1ndash21 Princeton UniversityPress Princeton NJ USA 2007

[10] B Farrell ldquoModal and non-modal baroclinic wavesrdquo Journal ofthe Atmospheric Sciences vol 41 no 4 pp 668ndash673 1984

[11] M K Tippett ldquoTransient moist baroclinic instabilityrdquo Tellus Avol 51 no 2 pp 273ndash288 1999

[12] D Hodyss and R Grotjahn ldquoNonmodal and unstable normalmode baroclinic growth as a function of horizontal scalerdquoDynamics of Atmospheres and Oceans vol 37 no 1 pp 1ndash242003

[13] J Pedlosky ldquoFinite amplitude baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 27 no 1 pp 15ndash30 1970

[14] R D Romea ldquoThe effects of friction and beta on finite ampli-tude baroclinic wavesrdquo Journal of the Atmospheric Sciences vol34 pp 1689ndash1695 1977

[15] J Pedlosky ldquoFinite-amplitude baroclinic waves at minimumcritical shearrdquo Journal of the Atmospheric Sciences vol 39 no3 pp 555ndash562 1982

[16] T Warn and P Gauthier ldquoPotential vorticity mixing bymarginally unstable baroclinic disturbancesrdquo Tellus A vol 41no 2 pp 115ndash131 1989

[17] A J Simmons and B J Hoskins ldquoThe lifecycles of some non-linear wavesrdquo Journal of the Atmospheric Sciences vol 35 pp414ndash432 1978

[18] A J Simmons and B J Hoskins ldquoBarotropic influences of thegrowth and decay of nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 37 pp 1679ndash1684 1980

[19] S B Feldstein and I M Held ldquoBarotropic decay of baroclinicwavesrdquo Journal of the Atmospheric Sciences vol 46 pp 1679ndash1684 1989

[20] C D Thorncroft B J Hoskins and M E McIntyre ldquoTwoparadigms of baroclinic-wave life-cycle behaviourrdquo QuaterlyJournal vol 119 no 509 pp 17ndash55 1993

[21] N A Phillips ldquoA simple three-dimensional model for the studyof largescale extratropical flow patternsrdquo Journal ofMeteorologyvol 8 pp 381ndash394 1951

[22] N A Phillips ldquoEnergy transformations and meridional circu-lations associated with simple baroclinic waves in a two-levelquasi-geostrophic modelrdquo Tellus vol 6 pp 273ndash286 1954

[23] L M Polvani and J Pedlosky ldquoThe effect of dissipation onspatially growing nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 45 no 14 pp 1977ndash1989 1988

[24] T G Shepherd ldquoNonlinear saturation of baroclinic instabilityPart I the two-layermodelrdquo Journal of the Atmospheric Sciencesvol 45 no 14 pp 2014ndash2025 1988

[25] G Balasubramanian and M K Yau ldquoBaroclinic instability ina two-layer model with parameterized slantwise convectionrdquoJournal of the Atmospheric Sciences vol 51 no 7 pp 971ndash9901994

[26] A Wiin-Nielsen ldquoOn the structure of atmospheric waves inmiddle latitudesrdquo Atmosfera vol 16 no 2 pp 83ndash102 2003

[27] D D Holm and B A Wingate ldquoBaroclinic instabilities of thetwo-layer quasigeostrophic alpha modelrdquo Journal of PhysicalOceanography vol 35 no 7 pp 1287ndash1296 2005

10 Advances in Meteorology

[28] J Egger ldquoBaroclinic instability in the two-layermodel interpre-tationsrdquo Meteorologische Zeitschrift vol 18 no 5 pp 559ndash5652009

[29] J Lamberts G Lapeyre and V Zeitlin ldquoMoist versus dry baro-clinic instability in a simplified two-layer atmospheric modelwith condensation and latent heat releaserdquo Journal of theAtmospheric Sciences vol 69 pp 1405ndash1426 2012

[30] D M W Frierson ldquoRobust increases in midlatitude static sta-bility in simulations of global warmingrdquo Geophysical ResearchLetters vol 33 no 24 Article ID L24816 2006

[31] N M J Hall B J Hoskins P J Valdes and C A Senior ldquoStormtracks in a high-resolution GCMwith doubled carbon dioxiderdquoQuarterly Journal vol 120 no 519 pp 1209ndash1230 1994

[32] S B Feldstein ldquoThe recent trend and variance increase of theannularmoderdquo Journal of Climate vol 15 no 3 pp 88ndash94 2002

[33] G J Marshall ldquoTrends in the southern annular mode fromobservations and reanalysesrdquo Journal of Climate vol 16 pp4134ndash4143 2003

[34] G M Ostermeier and J M Wallace ldquoTrends in the NorthAtlantic Oscillation-Northern Hemisphere annular mode dur-ing the twentieth centuryrdquo Journal of Climate vol 16 no 2 pp336ndash341 2003

[35] D W J Thompson and S Solomon ldquoInterpretation of recentSouthern Hemisphere climate changerdquo Science vol 296 no5569 pp 895ndash899 2002

[36] S J Lambert ldquoThe effect of enhanced greenhouse warming onwinter cyclone frequencies and strengthsrdquo Journal of Climatevol 8 no 5 pp 1447ndash1462 1995

[37] F Lunkeit K Fraedrich and S E Bauer ldquoStorm tracks ina warmer climate sensitivity studies with a simplified globalcirculation modelrdquo Climate Dynamics vol 14 no 11 pp 813ndash826 1998

[38] D J Lorenz and D L Hartmann ldquoEddy-zonal flow feedback inthe Southern Hemisphererdquo Journal of the Atmospheric Sciencesvol 58 no 21 pp 3312ndash3327 2001

[39] Q Geng and M Sugi ldquoPossible change of extratropical cycloneactivity due to enhanced greenhouse gases and sulphateaerosols study with a high-resolution AGCMrdquo Journal ofClimate vol 16 pp 2262ndash2274 2003

[40] J C Fyfe ldquoExtratropical Southern Hemisphere cyclonesharbingers of climate changerdquo Journal of Climate vol 16 pp2802ndash2805 2003

[41] J H Yin ldquoA consistent poleward shift of the storm tracksin simulations of 21st century climaterdquo Geophysical ResearchLetters vol 32 no 18 Article ID L18701 pp 1ndash4 2005

[42] S J Lambert and J C Fyfe ldquoChanges in winter cyclonefrequencies and strengths simulated in enhanced greenhousewarming experiments results from the models participating inthe IPCC diagnostic exerciserdquo Climate Dynamics vol 26 no7-8 pp 713ndash728 2006

[43] G Gastineau and B J Soden ldquoModel projected changes ofextreme wind events in response to global warmingrdquo Geophys-ical Research Letters vol 36 no 10 Article ID L10810 2009

[44] Y Wu M Ting R Seager H-P Huang and M A CaneldquoChanges in storm tracks and energy transports in a warmer cli-mate simulated by the GFDL CM21 modelrdquo Climate Dynamicsvol 37 no 1 pp 53ndash72 2011

[45] L Bengtsson K I Hodges and E Roeckner ldquoStorm tracks andclimate changerdquo Journal of Climate vol 19 no 15 pp 3518ndash35432006

[46] I Smith ldquoAn assessment of recent trends in Australian rainfallrdquoAustralian Meteorological Magazine vol 53 no 3 pp 163ndash1732004

[47] N Nicholls Detecting Understanding and Attributing ClimateChange Australian Greenhouse Office Publication 2007

[48] B C Bates P Hope B Ryan I Smith and S Charles ldquoKey find-ings from the Indian Ocean Climate Initiative and their impacton policy development in Australiardquo Climatic Change vol89 no 3-4 pp 339ndash354 2008

[49] J S Frederiksen and C S Frederiksen ldquoDecadal changes inSouthern Hemisphere winter cyclogenesisrdquo CSIROMarine andAtmospheric Research Paper 002 2005

[50] J S Frederiksen and C S Frederiksen ldquoInterdecadal changes insouthern hemisphere winter storm track modesrdquo Tellus A vol59 no 5 pp 599ndash617 2007

[51] J S Frederiksen C S Frederiksen and S L Osbrough ldquoMod-elling of changes in Southern Hemisphere weather systemsduring the 20th centuryrdquo in Proceedings of the 18th WorldIMACS Congress and International Congress on Modelling andSimulation pp 2562ndash2568 Cairns Australia July 2009

[52] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoCauses of changing Southern Hemispheric weathersystemsrdquo in Managing Climate Change I Jupp P Holper andW Cai Eds pp 85ndash98 CSIRO Publishing 2010

[53] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoChanges in Southern Hemisphere rainfall circulationand weather systemsrdquo in Proceedings of the 19th InternationalCongress on Modelling and Simulation (MODSIM rsquo11) pp 2712ndash2718 Perth Australia December 2011

[54] C S Frederiksen J S Frederiksen J M Sisson and S LOsbrough ldquoObserved and projected changes in the annual cycleof Southern Hemisphere baroclinicity for storm formationrdquoin Proceedings of the 19th International Congress on Modellingand Simulation (MODSIMrsquo 11) pp 2719ndash2725 Perth AustraliaDecember 2011

[55] N A Phillips ldquoGeostrophicmotionrdquoReviews of Geophysics vol1 no 2 pp 123ndash176 1963

[56] I Bordi K Fraedrich F Lunkeit and A Sutera ldquoTroposphericdouble jets meridional cells and eddies a case study andidealized simulationsrdquoMonthly Weather Review vol 135 no 9pp 3118ndash3133 2007

[57] I Bordi K Fraedrich M Ghil and A Sutera ldquoZonal flowregime changes in a GCM and in a simple quasigeostrophicmodel the role of stratospheric dynamicsrdquo Journal of theAtmospheric Sciences vol 66 no 5 pp 1366ndash1383 2009

[58] J G Charney ldquoOn the scale of atmosphericmotionsrdquoGeofysiskePublikasjoner vol 17 no 2 pp 3ndash17 1948

[59] A Burger ldquoScale consideration of planetary motions of theatmosphererdquo Tellus vol 10 pp 195ndash205 1958

[60] I A Pisnichenko ldquoUltralong-wave dynamics in a two-dimen-sional baroclinic atmosphere modelrdquo Atmospheric and OceanicPhysics vol 16 no 9 pp 883ndash892 1980

[61] I A Pisnichenko ldquoInfluence variable static stability on thedynamics of ultralong waves in two-dimensional baroclinicmodel of the atmosphererdquoAtmospheric andOceanic Physics vol19 no 11 pp 1223ndash1226 1983

[62] A Wiin-Nielsen ldquoOn barotropic and baroclinic models withspecial emphasis onultra-longwavesrdquoMonthlyWeather Reviewvol 87 pp 171ndash183 1959

[63] B W BinWang and A Barcilon ldquoMoist stability of a barocliniczonal flow with conditionally unstable stratificationrdquo Journal ofthe Atmospheric Sciences vol 43 no 7 pp 705ndash719 1986

Advances in Meteorology 11

[64] N A Phillips ldquoA coordinate system having some specialadvantages for numerical forecastingrdquo Journal of Meteorologyvol 14 pp 184ndash185 1957

[65] D M Alishaev ldquoDynamics of a two-dimensional baroclinicatmosphererdquoAtmospheric and Oceanic Physics vol 16 no 2 pp99ndash107 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 3

The state variables of the model are the horizontal velocityvector u = (119906 V)119879 the vertical pressure velocity 120596 equiv 119889119901119889119905where 119901 is pressure the geopotentialΦ and the temperature119879 The operator nabla equiv (120597120597119909 120597120597119910) is applied to the horizontalcoordinates 119909 and 119910 directed eastward and northwardrespectively The normalized pressure 120585 = 119901119875

0 where 119875

0=

1000 hPa is a ldquostandardrdquo pressure approximately equal to thesurface pressure is taken as the vertical coordinate while thetime is denoted by 119905 Other notations are the diabatic heatingrate per unit time per unitmass119876 the gas constant119877 the unitvector in the vertical direction k and the reference state staticstability measure 119878

120585in the normalized isobaric coordinate

system

119878120585=

119877119879

1198921198750120585

(Γ119889minus Γ) (3)

where 119879 is a reference temperature 119892 is the gravity accelera-tion Γ

119889is the dry adiabatic lapse rate and Γ is the reference

state lapse rateWe employ the119891-plane approximation so thatthe Coriolis parameter 119891 is a constant 119891 = 119891

0= 2Ω sin120593

0

with 1205930being the latitude of interest Hereafter we consider

only adiabatic process and thus assume zero heating rate 119876The following boundary conditions are used for the pressurevelocity

120596 = 0 at 120585 = 0 120585 = 1 (4)

The atmospheric reference state defined by 119906 V120596119879 andΦ is steady and satisfies the following relations

119906 = minus

1

1198910

120597Φ

120597119910

V = 0

120596 = 0

120597Φ

120597120585

= minus

119877119879

120585

(5)

where 119879 = 119879(119910 120585) By substituting (5) into the set of (2) onecan see that (5) is a solution of (2) that describes the zonalflow

120597119906

120597120585

=

119877

1198910120585

120597119879

120597119910

(6)

which matches the specified distribution of the zonally aver-aged temperature 119879(119910 120585) and represents thermal wind bal-ance To consider only the baroclinicmechanism of the atmo-spheric instability meridional variability of the basic zonalflow is excluded In other words the barotropic impact on theinstability of the basic zonal flow is not taken into accountTherefore we assume that the velocity of the basic zonalflow does not depend on the horizontal 119910-coordinate that is119906 = 119906(120585)Thus the problem now is the study of the instabilityof the basic zonal flow (6) with respect to infinitesimalperturbations For this purpose the system (2) is linearizedaround the basic state (5) Representing the state variables as120595(119909 119910 120585 119905) = 120595(120585) + 120595

1015840(119909 120585 119905) where 120595 is a basic state and

1205951015840 is an infinitesimal perturbation and taking into account

the hydrostatic equations and the thermal wind relationship(6) the linearized system can be written as

1205971199061015840

120597119905

+ 119906

120597119906

120597119909

+

1205961015840

1198750

120597119906

120597120585

= minus

120597Φ1015840

120597119909

+ 1198910V1015840

120597V1015840

120597119905

+ 119906

120597V1015840

120597119909

+ 11989101199061015840= 0

1205971199061015840

120597119909

+

1

1198750

1205971205961015840

120597120585

= 0

120597

120597119905

(

120597Φ1015840

120597120585

) + 119906

120597

120597119909

(

120597Φ1015840

120597120585

) minus V10158401198910

120597119906

120597120585

+ 120590011987501205961015840= 0

(7)

The static stability parameter 1205900is expressed as

1205900= minus

120572

1198750

120597 lnΘ120597120585

=

1198772119879

1198921198752

01205852(Γ119889minus Γ) (8)

where 120572 is a specific volume and Θ = 119879120585minus119877119888119901 is a reference

potential temperature Suppose that 120597119906120597120585 = minusΛ120585= const

Applying the method of separation of variables we assumethe solutions of the form

1205951015840(119909 120585 119905) = (120585) 119890

119894119896(119909minus119888119905) (9)

where (120585) is a function of 120585 only 119896 is a wave number and 119888is a phase velocity of perturbations which in general is a com-plex value 119888 = 119888

119903+ 119894119888119894 After substituting (9) into (7) we can

finally obtain the following single equation for

(119906 minus 119888) [1 minus

1198962

1198912

0

(119906 minus 119888) (119906 minus 119888)]

1205972

1205971205852

+ 2Λ120585

120597

120597120585

minus 12059001198752

0

1198962

1198912

0

(119906 minus 119888) = 0

(10)

Similar equations were considered in a number of publica-tions (eg [4 62 63])The analytical solution of this equationcan be obtained by eliminating gravity waves from the con-sideration and assuming that 120590

0= const For this particular

case (10) can be transformed into

(119906 minus 119888)

1205972

1205971205852+ 2Λ120577

120597

120597120585

minus (119906 minus 119888) 12059001198752

0

1198962

1198912

0

= 0 (11)

The boundary conditions for are specified as

= 0 at 120585 = 0 120585 = 1 (12)

Equation (11) together with boundary conditions (12) rep-resents the eigenvalue problem for the complex phase speed119888 The fundamental solutions of (11) are expressed throughthe Bessel functions of the first and second kinds With thehomogeneous boundary conditions (12) the following twodiscrete eigenvalues can be obtained [4]

11988812=

Λ120585

2

[1 plusmn radic1 minus

4

1205782(120578 coth (120578) minus 1)] (13)

4 Advances in Meteorology

where

120578 = 1198750radic1205900 (

119896

1198910

) (14)

The phase velocity 119888 will amplify exponentially if 119888 has animaginary part 119888

119894 From (13) we can see that this will occur if

the discriminant in (13) is less than zero

4

1205782(120578 coth (120578) minus 1) gt 1 (15)

which gives by theNewtonrsquos iteration algorithm the necessarycondition for instability 120578 lt 120578

119888asymp 23994 Besides two dis-

crete eigenvalues (13) the eigenvalue problem (11)-(12) has acontinuous spectrum of eigenvalues 119888 isin (0 119906(120585)) that are realand therefore can be neglected in the problem of baroclinicinstability [4] The growth rate of unstable waves 120594

119896equiv 119896119888119894is

calculated by the following expression

120594119896=

Λ1205851198910

21198750radic1205900

radic10038161003816100381610038161205782minus 4 (120578 coth (120578) minus 1)100381610038161003816

1003816 (16)

As we can see from (16) at a given latitude the growth rate120594119896is a function of 120590

0(static stability) Λ

120585(wind shear) and

quantity 120578 that depends on the horizontal wavenumber 119896 and1205900

22 Impact of Static Stability andVerticalWind Shear on Baro-clinic Instability Within the Eady problem framework thestatic stability parameter 120590

0and the vertical wind shear Λ

120585

represent the main control variables By varying 1205900and Λ

120585

one can obtain estimates of the impact of these parameters onthe development of baroclinic instability in the atmosphereIn this research parameters corresponding to the basicstate are given the following values Λ

120585= 40msminus1 [50]

and 1205900= 2times10

minus6m2 Paminus2 sminus2 [1]These parameter values canbe used as an approximation to describe the zonal-averagedatmospheric conditions for JJA (June July andAugust) in theSH [50] The latitude of interest is assumed to be 120593

0= 45 S

which gives 1198910= minus1028 times 10

minus4 sminus1Figure 1 shows plot of Eady growth rate versus zonal

wavenumber obtained with (16)The growth rate has a short-wave instability cutoff beyond which waves are stable Let119871min be the wavelength that corresponds to a short-wavecutoff Value of 119871min can be obtained from (14) when 120578 = 120578

119888

which gives 119871min = 3592 km To calculate the wavelength ofmaximum growth rate 119871

120594max one can take 120597120594119896120597119896 and set

the result equal to zero which gives 120578 = 120578119898asymp 16061 Then by

using (14) we can obtain 119871120594max = 5366 km

The influence of the static stability parameter on thewavelength of maximum growth rate 119871

120594max and the shortwave cut-off 119871min are shown in Figure 2 In general anincrease in the parameter 120590

0leads to an increase in both

119871119896119888max and 119871min The functional dependences between 119871min

and 1205900 and between 119871

119896119888max and 1205900 are almost linear plusmn10departure of static stability parameter Δ120590

0from its nominal

value 1205900= 20times10

minus6m2 Paminus2 sminus2 results in about plusmn5 changefor both 119871

119896119888max and 119871min with respect to the nominal value1205900 For instance if Δ120590 = 01 times 120590

0 then 119871

119896119888max = 5628 and

09

06

03

000 2 4 6 8

kz

Xk

(dayminus1)

Mode with maximum Short-wavegrowth rate cutoff

Figure 1 Growth rate 120594119896versus zonal wavenumber 119896

119911for 1205900= 2 times

10minus6m2 Paminus2 s minus 2 and Δ

120585= 40msminus1

6

4

2

(km

)

1 2 3(m2 Paminus2 sminus2 )

LminLkcmax

1205900 times 106

Ltimes10minus3

Figure 2 Length of waves with maximum growth rates 119871119896119888max and

short-wave cutoff 119871min as functions of the static stability parameter1205900

119871min = 3768 km and if Δ120590 = minus011205900 then 119871

119896119888max = 5091

and 119871min = 3408 kmFigure 3 illustrates the growth rate of unstable waves

versus the static stability parameter at different values of Λ120585

Parameters 1205900and Λ

120585influence the growth rate 120594

119896in the

opposite direction growth rate decreases if 1205900increases and

if Λ120585decreases Note that the decrease of the parameter Λ

120585

indicates the weakening of the intensity of the barocliniczone that is reduction of the MTG In nature both of theseprocesses take place which leads to a synergistic effect For

Advances in Meteorology 5

20

15

10

05

0 1 2 3

Xk

(dayminus1)

Λ120576 = 30Λ120576 = 40

Λ120576 = 50Λ120576 = 60

(m2 Paminus2 sminus2 )1205900 times 106

Figure 3 Growth rate 120594119896versus static stability parameter 120590

0for

different values of parameter Λ120585(units m sminus1)

instance if Λ120585decreases by 10 and the static stability

parameter increases by 10 the growth rate 120594119896decreases by

14Since 120594

119896is a nonlinear function of 120590

0(16) to estimate the

influence of infinitesimal perturbations in 1205900on variations in

120594119896 the sensitivity function

119878120590=

120597120594119896

1205971205900

(17)

and the relative sensitivity function

119878119877

120590=

120597120594119896120594119896

12059712059001205900

=

1205900

120594119896

120597120594119896

1205971205900

(18)

can be used The function 119878120590shows changes in 120594

119896due to

variations in 1205900 The relative sensitivity function 119878119877

120590is used

to compare model parameters to find out what parameter isthemost important for a certain percent change in the param-eters Sensitivity functions (17) and (18) are evaluated in thevicinity of some nominal value of the parameter 120590

0 We can

select several nominal values to cover some range of changesin 1205900 Differentiating (16) with respect to control parameter

1205900 we can obtain the expression for 119878

120590

119878120590=

120594119896

21205900

(1198750radic1205900

119896

1198910

120578 minus 2 coth (120578) + 2120578csch2 (120578)1205782minus 4 (120578 coth (120578) minus 1)

minus 1)

(19)

Sensitivity 119878120590versus zonal wavenumbers for different values

of 1205900with Λ

120585= 40msminus1 are shown in Figure 4 The absolute

value of the sensitivity of 120594119896with respect to 120590

0exponentially

minus5

minus10

minus15

minus20

0 2 4 6 8 10

S120590

kz

1205900 = 101205900 = 201205900 = 30

Figure 4 Sensitivity function 119878120590versus zonal wavenumber 119896

119911for

different nominal values of the static stability parameter 1205900(units

106m2 Paminus2 sminus2)

increases with decreasing wavelength For planetary scalewaves (zonal wave numbers 1ndash4) the absolute value of thesensitivity of 120594

119896with respect to 120590

0is palpably less than

sensitivity for synoptic scale waves (zonal wave numbers ge5)Absolute and relative sensitivity functions 119878

120590and 119878119877

120590calcu-

lated for different values of1205900for various zonal wave numbers

are shown in Tables 1 and 2 respectivelyThe expression for sensitivity function 119878

Λcan be eas-

ily obtained by differentiating (16) with respect to controlparameter Λ

120585

119878Λ=

1198910

21198750radic1205900

radic10038161003816100381610038161205782minus 4 (120578 coth (120578) minus 1)100381610038161003816

1003816 (20)

The function 119878Λversus zonalwave number 119896

119911for different

1205900is plotted in Figure 5 It is clear to see that for a given value

of the parameter 1205900the graph of function 119878

Λ(119896119911) is verymuch

like the classic picture of the growth rates 120594119896versus zonal

wavenumber 119896119911[5] It is interesting that the relative sensitivity

function 119878119877Λdoes not depend on the wavelength (wavenum-

ber) and for all of the unstable waves is equal to unity

119878119877

Λ=

120597120594119896120594119896

120597Λ120585Λ120585

=

Λ120585

120594119896

120597120594119896

120597Λ120585

= 1 (21)

Since relative sensitivity functions allow direct comparison ofthe importance of model parameters on the growth rate 120594

119896

we can see that because 119878119877Λ= 1 the parameter Λ

120585(ie the

meridional temperature gradient) is more important than

6 Advances in Meteorology

Table 1 Absolute sensitivity 119878120590as a function of zonal wavenumber 119896

119911for different values of static stability parameter 120590

0

1205900m2 Paminus2 sminus2 Zonal wave number 119896

119911

1 2 3 4 5 6 710 times 10

minus6minus00157 minus01251 minus04175 minus09772 minus18879 minus32470 minus52005

15 times 10minus6

minus00157 minus01245 minus04139 minus09675 minus18794 minus33014 minus5592720 times 10

minus6minus00157 minus01240 minus04110 minus09626 minus18979 minus34964 minus69371

25 times 10minus6

minus00157 minus01234 minus04088 minus09629 minus19509 minus39665 minus23003130 times 10

minus6minus00157 minus01230 minus04071 minus09691 minus20540 minus53057

Table 2 Relative sensitivity 119878119877120590as a function of zonal wavenumber 119896

119911for different values of static stability parameter 120590

0

1205900m2 Paminus2 sminus2 Zonal wave number 119896

119911

1 2 3 4 5 6 710 times 10

minus6minus00062 minus00250 minus00575 minus01057 minus01735 minus02692 minus04095

15 times 10minus6

minus00093 minus00379 minus00881 minus01656 minus02837 minus04749 minus0837120 times 10

minus6minus00124 minus00509 minus01201 minus02375 minus04220 minus08004 minus19939

25 times 10minus6

minus00156 minus00642 minus01539 minus03087 minus06071 minus14399 minus31424330 times 10

minus6minus00187 minus00777 minus01898 minus03974 minus08759 minus34322

3

2

1

00 2 4 6 8 10 12

kz

1205900 = 101205900 = 201205900 = 30

107timesSΛ

Figure 5 Sensitivity function 119878Λversus zonal wavenumber 119896

119911for

different nominal values of the static stability parameter 1205900(units

106m2 Paminus2 sminus2)

the static stability parameter 1205900except for 120590

0gt 2 times

10minus6m2 Paminus2 sminus2 for waves with 119896

119911ge 6 (see Table 2) Wher-

ever the midlatitude values in Table 2 are less than one thegrowth rate is more sensitive to the meridional temperaturegradient (ie Λ

120585) than the static stability (120590

0)

The obtained results are consistent with observations[50 51 54] an increase in static stability and a decrease of

the MTG have occurred over the past few decades in someareas of the SH which has led to a decrease in the growthrate of baroclinic unstable waves a shift of the spectrum ofunstable waves in the long wavelength part of spectrum andaweakened intensity of cyclogenesis Naturally these changesaffect favourable conditions for the development of baroclinicinstability and the essential features of weather patterns overlarge territories particularly over Australia

3 Planetary Scale Waves

To study the influence of the static stability parameter onthe dynamics of planetary scale (ultralong) waves a thinfilm approximation is applied This approximation employsa specific averaging technique over the vertical coordinate tothe system of primitive equations [60] As a result a two-dimensional set of equations can be obtained that describesthe dynamics of a two-dimensional baroclinic film Theseequations reproduce all the wavelike solutions that corre-spond to the main weather-forming modes of three-dimen-sional models and therefore can be used in theoretical studiesof large-scale dynamic processes in the atmosphere Thesystem of vertically averaged equations can be written as [60]

120597119906

120597119905

+ V sdot nabla119906 = 119891V minus119877

120587

120597

120597119909

(120587119879)

120597V120597119905

+ V sdot nablaV = minus119891119906 minus119877

120587

120597

120597119910

(120587119879)

120597120587

120597119905

+ nabla sdot (120587V) = 0

120597119879

120597119905

+ V sdot nabla119879 +

119877

119888119901

119879nabla sdot V = 0

(22)

where 120587 = 1199011199041198750 For instance if the original primitive equa-

tions are written in the Phillipsrsquo vertical coordinate system

Advances in Meteorology 7

120590 = 119901119901119904[64] the operator for vertical averaging is intro-

duced as 120595 = int

1

0120595119889120590 and state variables are represented as

120595 = 120595 + 1205951015840 Equations (22) are obtained by neglecting the

orography and terms 1199061015840V1015840 V1015840V1015840 and 1198791015840V1015840 [65] A detailedlinear analysis of the vertically averaged equation (22) isrepresented in [60] In particular two types of wave solutionswere found fast waves that propagate westward and slowwaves that move eastward Within the framework of thismodel ultralong waves are always neutral for any verticallyaveraged zonal wind velocity Indeed linearizing (22) aroundthe following basic state

1198790= 1198790(119910) 119906

0= minus(

119877

1198910

)

1205971198790

120597119910

V0= 0 120587

0= 1

(23)

and assuming the beta plane approximation 119891 = 1198910+ 120573119910

where 120573 = (2Ω1198860) cos120593

0 and representing the solution in

the form (9) one can finally obtain under different asymp-totics the following expressions for four wave solutions [60]

(a) acoustic waves

11988812= 1199060plusmnradic1198882

0+

1198912

0

1198962

(24)

(b) Rossby wave

1198883= 1199060minus

120573

1198962+ 1198912

01198882

0

(25)

(c) baroclinic wave

1198884= 1199060minus

1198912

01199062

0

1198882

0120573

(26)

Here 11988820= (1+120581)119877119879

0and 120581 = 119877119888

119901 These results however are

valid only for the specific case of a neutral atmosphere withΓ = Γ

119889[60 61] To take into account the atmospheric static

stability on the behaviour of ultralong waves the polytropicmodel of the atmosphere can be used for which

119879 (119909 119910 119911 119905) = 1198790(119909 119910 119905) minus Γ (119909 119910 119905) 119911 (27)

where 1198790is the temperature at the surface and Γ is a vertical

temperature gradient Integrating (27) with respect to verticalcoordinate we can obtain 119879

0= 119879(1 + 119877Γ119892) [60] Assuming

the geostrophic approximation on a 120573-plane the set ofvertically averaged equations can be written as [61]

120597119879

120597119905

+

119877119879

1198912

0

1205721

120587

(120587 119879) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

1

1205722

(119879 Γ)

+ 120581

1205731198772119879

2

1198921198912

0

1205724

1205722

2

120597Γ

120597119909

minus

120573119877119879

1198912

0

(1205723+ 120581

12057211205724

1205722

)

1

120587

120597 (120587119879)

120597119909

minus 120581

120573119877119879

1198912

0

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

12057211205724

1205722

120597119879

120597119909

= 0

120597Γ

120597119905

minus

119877119879

1198910

1205722

1

1205722

1

120587

(Γ 120587) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

119877

1198910

1205721

1205722

(119879 Γ)

minus 120581

120573119877119879

1198912

0

12057211205724

1205722

2

120597Γ

120597119909

minus 120581

120573119892

1198912

0

120572112057231205724

1205722

1

120587

120597 (120587119879)

120597119909

+ 120581

120573119892

1198912

0

1205722

11205724

1205722

120597119879

120597119909

= 0

120597120587

120597119905

minus

120573119877

1198912

0

120597 (120587119879)

120597119909

= 0

(28)

Here 1205721= 1 + 119877(Γ119892) 120572

2= 1 + 2119877(Γ119892) 120572

3= 119877(Γ119892) 120572

4=

1 minus (ΓΓ119889) and the operator (119860 119861) = (120597119860120597119909)(120597119861120597119910) minus (120597119860

120597119910)(120597119861120597119909) The basic state is defined as a stationary solutionof system (28) for which

120597119879

120597119909

= 0

120597Γ

120597119909

= 0

120597120587

120597119909

= 0 (29)

or in other words

119879 = 1198790(119910) Γ = Γ

0(119910) 120587 = 120587

0(119910) (30)

Linearizing (28) around the basic state (30) the followingcubic characteristic equation can be obtained in which thesecond order terms are neglected [61]

1198883+ 1198882

1205721

1205722

[120582 (1205722+ 3120581120572

4) minus 1199060]

+ 119888 120582

1205721

1205722

[1205821205811205724(2 + 120572

3+ 120581

1205724

1205722

(21205721minus

1

1205722

))

minus1199060(2 + 120581120572

4+ 31205723) ]

+ (120582

1205721

1205722

1199062

0minus 1205822120581

1205722

11205724

1205722

1199060+ 2120582312058121205722

112057231205722

4

1205723

2

)

= 0

(31)

where 120582 = 12057311987711987901198912

0and 119906

0= minus(119877119891

01205870)((120597(120587

01198790))120597119910) If

the discriminant of this equation is positive then the wavesolution is unstable The domain of zonal flow instabilitycan be found numerically (see diagram in [61]) In Figure 6we reproduce only for the 1st quadrant of a Cartesianplane the domain of instability calculated as a function ofvertically averaged zonal wind velocity 119906

0and dimensionless

temperature lapse rate ΓΓ119889

The imaginary part of phase velocity 119888119894which charac-

terises the growth rate of unstable waves 120594119896equiv 119896119888119894is displayed

in Figure 7 as a function of dimensionless temperature lapserate ΓΓ

119889for different values of vertically averaged zonal

wind velocity 1199060 A maximum phase velocity 119888

119894exists for

given values of 1199060 that is dependent on the ratio of ΓΓ

119889

For instance if 1199060= 20msminus1 then the maximum value

8 Advances in Meteorology

10

08

06

04

02

000 50 100 150 200 250 300

ΓΓd

u0 (m sminus1)

Figure 6 Domain of instability (filled) as a function of dimension-less temperature lapse rate ΓΓ

119889and vertically averaged zonal wind

velocity 1199060

(119888119894)max asymp 834msminus1 is reached at ΓΓ

119889asymp 055 Figure 6 shows

that increasing vertically averaged zonal wind 1199060is associated

with increasing 119888119894 This is further evident in Figure 8 which

shows 119888119894as a function of 119906

0for a range of ΓΓ

119889values The

lower ΓΓ119889

and the larger 119888119894 that is 119888

119894 increases with

decreasing static stability

4 Concluding Remarks

We have studied theoretically the impact of variations in thestatic stability parameter 120590

0and zonal wind shear Λ

120585on the

characteristics of baroclinically unstable waves of synopticscales using Eady-type model with the uniform Λ

120585between

upper and lower boundaries on an 119891-plane Quantitativeestimates of variations in 120590

0and Λ

120585on the growth rate 120594

119896

wavelength of maximum growth rate 119871120594max and short-wave

cutoff 119871min were obtainedAnalytical expressions are derived for sensitivity func-

tions for the growth rate 120594119896with respect to variations in static

stability parameter andwind shear velocityThese expressionsallow estimating to a first-order approximation the influenceof changes in 120590

0and Λ

120585on 120594119896 Analytical expressions for

relative sensitivity functions allow estimating the significanceof variations in 120590

0andΛ

120585on the growth rate of baroclinically

unstable waves with a given zonal wave numberTo study the impact of variations in atmospheric static

stability and zonal wind velocity on the instability of plan-etary scale waves the model with vertically averaged prim-itive equations with 120573-plane approximation was applied Ascontrol parameters we have used dimensionless temperature

15

10

5

0minus02 00 02 04 06 08 10

ΓΓd

u0 = 20u0 = 30u0 = 40

ci

(m sminus

1)

Figure 7 Imaginary part of phase speed 119888119894versus dimensionless

temperature lapse rate ΓΓ119889for different values of vertically averaged

zonal wind velocity 1199060

lapse rate ΓΓ119889and vertically averaged zonal wind velocity

1199060 We have estimated the influence of ΓΓ

119889and 119906

0on the

imaginary part of phase speed 119888119894 whichwas used as ameasure

of instabilityThe obtained results are qualitatively consistent with

changes in the essential weather patterns that occurred overthe last several decades in some areas of the SH and inparticular over Australia (eg [49 50 52ndash54]) Climatechange results suggest SH midlatitude static stability 120590

0may

increase and the MTG (the vertical wind shear Λ120585) may

decrease which according to our linear theoretical modelsleads to a slowing of the growth rate of baroclinic unstablewaves 120594

119896and an increasing wavelength of baroclinic unstable

wave with maximum growth rate 119871120594max that is a spectrum

shift of unstable waves towards longer wavelengths Thesemight affect the favourable conditions for the developmentof baroclinic instability and therefore the rate of cyclogenesisand a reduction in cyclone intensity The obtained sensitivityfunctions demonstrate that waves belonging to the short-wave part of the spectrum of unstable waves are moresensitive to changes in the static stability parameter thanwaves belonging to the long-wave part of the spectrum

To obtain more realistic estimates of the sensitivity of thegrowth rate of unstable waves with respect to static stabilityparameter and MTG numerical modeling based on a fullGCM is required It is hoped to carry out such work in thefuture

Advances in Meteorology 9

ΓΓd = 01ΓΓd = 04

ΓΓd = 23ΓΓd = 09

20

10

00 20 40 60

u0 (mmiddotsminus1)

ci(m

sminus1)

Figure 8 Imaginary part of phase speed 119888119894versus vertically

averaged zonal wind velocity 1199060for different values of dimensionless

temperature lapse rate ΓΓ119889

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors thank Dr I Pisnichenko for clarification ofcertain questions relevant to this paper

References

[1] J R Holton An Introduction to Dynamic Meteorology Aca-demic Press 3rd edition 1992

[2] G K Vallis Atmospheric and Oceanic Fluid Dynamics Cam-bridge University Press 2006

[3] M Mak Atmospheric Dynamics Cambridge University Press2011

[4] V P Dymnikov Stability and Predictability of the Large ScaleAtmospheric Processes Institute of Numerical Mathematics ofthe Russian Academy of Sciences Moscow Russia 2007

[5] J G Charney ldquoThe dynamics of long waves in baroclinicwesterly currentrdquo Journal of Meteorology vol 4 pp 135ndash1621947

[6] E T Eady ldquoLong waves and cyclone wavesrdquo Tellus vol 1 pp33ndash52 1949

[7] R T Pierrehumbert and K L Swanson ldquoBaroclinic instabilityrdquoAnnual Review of Fluid Mechanics vol 27 no 1 pp 419ndash4671995

[8] R Gall ldquoA comparison of linear instability theory with theeddy statistics of a general circulation modelrdquo Journal of theAtmospheric Sciences vol 33 no 3 pp 349ndash373 1976

[9] I M Held ldquoProgress and problems in large-scale atmosphericdynamicsrdquo in The Global Circulation of the Atmosphere TSchneider and A Sobel Eds pp 1ndash21 Princeton UniversityPress Princeton NJ USA 2007

[10] B Farrell ldquoModal and non-modal baroclinic wavesrdquo Journal ofthe Atmospheric Sciences vol 41 no 4 pp 668ndash673 1984

[11] M K Tippett ldquoTransient moist baroclinic instabilityrdquo Tellus Avol 51 no 2 pp 273ndash288 1999

[12] D Hodyss and R Grotjahn ldquoNonmodal and unstable normalmode baroclinic growth as a function of horizontal scalerdquoDynamics of Atmospheres and Oceans vol 37 no 1 pp 1ndash242003

[13] J Pedlosky ldquoFinite amplitude baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 27 no 1 pp 15ndash30 1970

[14] R D Romea ldquoThe effects of friction and beta on finite ampli-tude baroclinic wavesrdquo Journal of the Atmospheric Sciences vol34 pp 1689ndash1695 1977

[15] J Pedlosky ldquoFinite-amplitude baroclinic waves at minimumcritical shearrdquo Journal of the Atmospheric Sciences vol 39 no3 pp 555ndash562 1982

[16] T Warn and P Gauthier ldquoPotential vorticity mixing bymarginally unstable baroclinic disturbancesrdquo Tellus A vol 41no 2 pp 115ndash131 1989

[17] A J Simmons and B J Hoskins ldquoThe lifecycles of some non-linear wavesrdquo Journal of the Atmospheric Sciences vol 35 pp414ndash432 1978

[18] A J Simmons and B J Hoskins ldquoBarotropic influences of thegrowth and decay of nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 37 pp 1679ndash1684 1980

[19] S B Feldstein and I M Held ldquoBarotropic decay of baroclinicwavesrdquo Journal of the Atmospheric Sciences vol 46 pp 1679ndash1684 1989

[20] C D Thorncroft B J Hoskins and M E McIntyre ldquoTwoparadigms of baroclinic-wave life-cycle behaviourrdquo QuaterlyJournal vol 119 no 509 pp 17ndash55 1993

[21] N A Phillips ldquoA simple three-dimensional model for the studyof largescale extratropical flow patternsrdquo Journal ofMeteorologyvol 8 pp 381ndash394 1951

[22] N A Phillips ldquoEnergy transformations and meridional circu-lations associated with simple baroclinic waves in a two-levelquasi-geostrophic modelrdquo Tellus vol 6 pp 273ndash286 1954

[23] L M Polvani and J Pedlosky ldquoThe effect of dissipation onspatially growing nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 45 no 14 pp 1977ndash1989 1988

[24] T G Shepherd ldquoNonlinear saturation of baroclinic instabilityPart I the two-layermodelrdquo Journal of the Atmospheric Sciencesvol 45 no 14 pp 2014ndash2025 1988

[25] G Balasubramanian and M K Yau ldquoBaroclinic instability ina two-layer model with parameterized slantwise convectionrdquoJournal of the Atmospheric Sciences vol 51 no 7 pp 971ndash9901994

[26] A Wiin-Nielsen ldquoOn the structure of atmospheric waves inmiddle latitudesrdquo Atmosfera vol 16 no 2 pp 83ndash102 2003

[27] D D Holm and B A Wingate ldquoBaroclinic instabilities of thetwo-layer quasigeostrophic alpha modelrdquo Journal of PhysicalOceanography vol 35 no 7 pp 1287ndash1296 2005

10 Advances in Meteorology

[28] J Egger ldquoBaroclinic instability in the two-layermodel interpre-tationsrdquo Meteorologische Zeitschrift vol 18 no 5 pp 559ndash5652009

[29] J Lamberts G Lapeyre and V Zeitlin ldquoMoist versus dry baro-clinic instability in a simplified two-layer atmospheric modelwith condensation and latent heat releaserdquo Journal of theAtmospheric Sciences vol 69 pp 1405ndash1426 2012

[30] D M W Frierson ldquoRobust increases in midlatitude static sta-bility in simulations of global warmingrdquo Geophysical ResearchLetters vol 33 no 24 Article ID L24816 2006

[31] N M J Hall B J Hoskins P J Valdes and C A Senior ldquoStormtracks in a high-resolution GCMwith doubled carbon dioxiderdquoQuarterly Journal vol 120 no 519 pp 1209ndash1230 1994

[32] S B Feldstein ldquoThe recent trend and variance increase of theannularmoderdquo Journal of Climate vol 15 no 3 pp 88ndash94 2002

[33] G J Marshall ldquoTrends in the southern annular mode fromobservations and reanalysesrdquo Journal of Climate vol 16 pp4134ndash4143 2003

[34] G M Ostermeier and J M Wallace ldquoTrends in the NorthAtlantic Oscillation-Northern Hemisphere annular mode dur-ing the twentieth centuryrdquo Journal of Climate vol 16 no 2 pp336ndash341 2003

[35] D W J Thompson and S Solomon ldquoInterpretation of recentSouthern Hemisphere climate changerdquo Science vol 296 no5569 pp 895ndash899 2002

[36] S J Lambert ldquoThe effect of enhanced greenhouse warming onwinter cyclone frequencies and strengthsrdquo Journal of Climatevol 8 no 5 pp 1447ndash1462 1995

[37] F Lunkeit K Fraedrich and S E Bauer ldquoStorm tracks ina warmer climate sensitivity studies with a simplified globalcirculation modelrdquo Climate Dynamics vol 14 no 11 pp 813ndash826 1998

[38] D J Lorenz and D L Hartmann ldquoEddy-zonal flow feedback inthe Southern Hemisphererdquo Journal of the Atmospheric Sciencesvol 58 no 21 pp 3312ndash3327 2001

[39] Q Geng and M Sugi ldquoPossible change of extratropical cycloneactivity due to enhanced greenhouse gases and sulphateaerosols study with a high-resolution AGCMrdquo Journal ofClimate vol 16 pp 2262ndash2274 2003

[40] J C Fyfe ldquoExtratropical Southern Hemisphere cyclonesharbingers of climate changerdquo Journal of Climate vol 16 pp2802ndash2805 2003

[41] J H Yin ldquoA consistent poleward shift of the storm tracksin simulations of 21st century climaterdquo Geophysical ResearchLetters vol 32 no 18 Article ID L18701 pp 1ndash4 2005

[42] S J Lambert and J C Fyfe ldquoChanges in winter cyclonefrequencies and strengths simulated in enhanced greenhousewarming experiments results from the models participating inthe IPCC diagnostic exerciserdquo Climate Dynamics vol 26 no7-8 pp 713ndash728 2006

[43] G Gastineau and B J Soden ldquoModel projected changes ofextreme wind events in response to global warmingrdquo Geophys-ical Research Letters vol 36 no 10 Article ID L10810 2009

[44] Y Wu M Ting R Seager H-P Huang and M A CaneldquoChanges in storm tracks and energy transports in a warmer cli-mate simulated by the GFDL CM21 modelrdquo Climate Dynamicsvol 37 no 1 pp 53ndash72 2011

[45] L Bengtsson K I Hodges and E Roeckner ldquoStorm tracks andclimate changerdquo Journal of Climate vol 19 no 15 pp 3518ndash35432006

[46] I Smith ldquoAn assessment of recent trends in Australian rainfallrdquoAustralian Meteorological Magazine vol 53 no 3 pp 163ndash1732004

[47] N Nicholls Detecting Understanding and Attributing ClimateChange Australian Greenhouse Office Publication 2007

[48] B C Bates P Hope B Ryan I Smith and S Charles ldquoKey find-ings from the Indian Ocean Climate Initiative and their impacton policy development in Australiardquo Climatic Change vol89 no 3-4 pp 339ndash354 2008

[49] J S Frederiksen and C S Frederiksen ldquoDecadal changes inSouthern Hemisphere winter cyclogenesisrdquo CSIROMarine andAtmospheric Research Paper 002 2005

[50] J S Frederiksen and C S Frederiksen ldquoInterdecadal changes insouthern hemisphere winter storm track modesrdquo Tellus A vol59 no 5 pp 599ndash617 2007

[51] J S Frederiksen C S Frederiksen and S L Osbrough ldquoMod-elling of changes in Southern Hemisphere weather systemsduring the 20th centuryrdquo in Proceedings of the 18th WorldIMACS Congress and International Congress on Modelling andSimulation pp 2562ndash2568 Cairns Australia July 2009

[52] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoCauses of changing Southern Hemispheric weathersystemsrdquo in Managing Climate Change I Jupp P Holper andW Cai Eds pp 85ndash98 CSIRO Publishing 2010

[53] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoChanges in Southern Hemisphere rainfall circulationand weather systemsrdquo in Proceedings of the 19th InternationalCongress on Modelling and Simulation (MODSIM rsquo11) pp 2712ndash2718 Perth Australia December 2011

[54] C S Frederiksen J S Frederiksen J M Sisson and S LOsbrough ldquoObserved and projected changes in the annual cycleof Southern Hemisphere baroclinicity for storm formationrdquoin Proceedings of the 19th International Congress on Modellingand Simulation (MODSIMrsquo 11) pp 2719ndash2725 Perth AustraliaDecember 2011

[55] N A Phillips ldquoGeostrophicmotionrdquoReviews of Geophysics vol1 no 2 pp 123ndash176 1963

[56] I Bordi K Fraedrich F Lunkeit and A Sutera ldquoTroposphericdouble jets meridional cells and eddies a case study andidealized simulationsrdquoMonthly Weather Review vol 135 no 9pp 3118ndash3133 2007

[57] I Bordi K Fraedrich M Ghil and A Sutera ldquoZonal flowregime changes in a GCM and in a simple quasigeostrophicmodel the role of stratospheric dynamicsrdquo Journal of theAtmospheric Sciences vol 66 no 5 pp 1366ndash1383 2009

[58] J G Charney ldquoOn the scale of atmosphericmotionsrdquoGeofysiskePublikasjoner vol 17 no 2 pp 3ndash17 1948

[59] A Burger ldquoScale consideration of planetary motions of theatmosphererdquo Tellus vol 10 pp 195ndash205 1958

[60] I A Pisnichenko ldquoUltralong-wave dynamics in a two-dimen-sional baroclinic atmosphere modelrdquo Atmospheric and OceanicPhysics vol 16 no 9 pp 883ndash892 1980

[61] I A Pisnichenko ldquoInfluence variable static stability on thedynamics of ultralong waves in two-dimensional baroclinicmodel of the atmosphererdquoAtmospheric andOceanic Physics vol19 no 11 pp 1223ndash1226 1983

[62] A Wiin-Nielsen ldquoOn barotropic and baroclinic models withspecial emphasis onultra-longwavesrdquoMonthlyWeather Reviewvol 87 pp 171ndash183 1959

[63] B W BinWang and A Barcilon ldquoMoist stability of a barocliniczonal flow with conditionally unstable stratificationrdquo Journal ofthe Atmospheric Sciences vol 43 no 7 pp 705ndash719 1986

Advances in Meteorology 11

[64] N A Phillips ldquoA coordinate system having some specialadvantages for numerical forecastingrdquo Journal of Meteorologyvol 14 pp 184ndash185 1957

[65] D M Alishaev ldquoDynamics of a two-dimensional baroclinicatmosphererdquoAtmospheric and Oceanic Physics vol 16 no 2 pp99ndash107 1980

Submit your manuscripts athttpwwwhindawicom

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4 Advances in Meteorology

where

120578 = 1198750radic1205900 (

119896

1198910

) (14)

The phase velocity 119888 will amplify exponentially if 119888 has animaginary part 119888

119894 From (13) we can see that this will occur if

the discriminant in (13) is less than zero

4

1205782(120578 coth (120578) minus 1) gt 1 (15)

which gives by theNewtonrsquos iteration algorithm the necessarycondition for instability 120578 lt 120578

119888asymp 23994 Besides two dis-

crete eigenvalues (13) the eigenvalue problem (11)-(12) has acontinuous spectrum of eigenvalues 119888 isin (0 119906(120585)) that are realand therefore can be neglected in the problem of baroclinicinstability [4] The growth rate of unstable waves 120594

119896equiv 119896119888119894is

calculated by the following expression

120594119896=

Λ1205851198910

21198750radic1205900

radic10038161003816100381610038161205782minus 4 (120578 coth (120578) minus 1)100381610038161003816

1003816 (16)

As we can see from (16) at a given latitude the growth rate120594119896is a function of 120590

0(static stability) Λ

120585(wind shear) and

quantity 120578 that depends on the horizontal wavenumber 119896 and1205900

22 Impact of Static Stability andVerticalWind Shear on Baro-clinic Instability Within the Eady problem framework thestatic stability parameter 120590

0and the vertical wind shear Λ

120585

represent the main control variables By varying 1205900and Λ

120585

one can obtain estimates of the impact of these parameters onthe development of baroclinic instability in the atmosphereIn this research parameters corresponding to the basicstate are given the following values Λ

120585= 40msminus1 [50]

and 1205900= 2times10

minus6m2 Paminus2 sminus2 [1]These parameter values canbe used as an approximation to describe the zonal-averagedatmospheric conditions for JJA (June July andAugust) in theSH [50] The latitude of interest is assumed to be 120593

0= 45 S

which gives 1198910= minus1028 times 10

minus4 sminus1Figure 1 shows plot of Eady growth rate versus zonal

wavenumber obtained with (16)The growth rate has a short-wave instability cutoff beyond which waves are stable Let119871min be the wavelength that corresponds to a short-wavecutoff Value of 119871min can be obtained from (14) when 120578 = 120578

119888

which gives 119871min = 3592 km To calculate the wavelength ofmaximum growth rate 119871

120594max one can take 120597120594119896120597119896 and set

the result equal to zero which gives 120578 = 120578119898asymp 16061 Then by

using (14) we can obtain 119871120594max = 5366 km

The influence of the static stability parameter on thewavelength of maximum growth rate 119871

120594max and the shortwave cut-off 119871min are shown in Figure 2 In general anincrease in the parameter 120590

0leads to an increase in both

119871119896119888max and 119871min The functional dependences between 119871min

and 1205900 and between 119871

119896119888max and 1205900 are almost linear plusmn10departure of static stability parameter Δ120590

0from its nominal

value 1205900= 20times10

minus6m2 Paminus2 sminus2 results in about plusmn5 changefor both 119871

119896119888max and 119871min with respect to the nominal value1205900 For instance if Δ120590 = 01 times 120590

0 then 119871

119896119888max = 5628 and

09

06

03

000 2 4 6 8

kz

Xk

(dayminus1)

Mode with maximum Short-wavegrowth rate cutoff

Figure 1 Growth rate 120594119896versus zonal wavenumber 119896

119911for 1205900= 2 times

10minus6m2 Paminus2 s minus 2 and Δ

120585= 40msminus1

6

4

2

(km

)

1 2 3(m2 Paminus2 sminus2 )

LminLkcmax

1205900 times 106

Ltimes10minus3

Figure 2 Length of waves with maximum growth rates 119871119896119888max and

short-wave cutoff 119871min as functions of the static stability parameter1205900

119871min = 3768 km and if Δ120590 = minus011205900 then 119871

119896119888max = 5091

and 119871min = 3408 kmFigure 3 illustrates the growth rate of unstable waves

versus the static stability parameter at different values of Λ120585

Parameters 1205900and Λ

120585influence the growth rate 120594

119896in the

opposite direction growth rate decreases if 1205900increases and

if Λ120585decreases Note that the decrease of the parameter Λ

120585

indicates the weakening of the intensity of the barocliniczone that is reduction of the MTG In nature both of theseprocesses take place which leads to a synergistic effect For

Advances in Meteorology 5

20

15

10

05

0 1 2 3

Xk

(dayminus1)

Λ120576 = 30Λ120576 = 40

Λ120576 = 50Λ120576 = 60

(m2 Paminus2 sminus2 )1205900 times 106

Figure 3 Growth rate 120594119896versus static stability parameter 120590

0for

different values of parameter Λ120585(units m sminus1)

instance if Λ120585decreases by 10 and the static stability

parameter increases by 10 the growth rate 120594119896decreases by

14Since 120594

119896is a nonlinear function of 120590

0(16) to estimate the

influence of infinitesimal perturbations in 1205900on variations in

120594119896 the sensitivity function

119878120590=

120597120594119896

1205971205900

(17)

and the relative sensitivity function

119878119877

120590=

120597120594119896120594119896

12059712059001205900

=

1205900

120594119896

120597120594119896

1205971205900

(18)

can be used The function 119878120590shows changes in 120594

119896due to

variations in 1205900 The relative sensitivity function 119878119877

120590is used

to compare model parameters to find out what parameter isthemost important for a certain percent change in the param-eters Sensitivity functions (17) and (18) are evaluated in thevicinity of some nominal value of the parameter 120590

0 We can

select several nominal values to cover some range of changesin 1205900 Differentiating (16) with respect to control parameter

1205900 we can obtain the expression for 119878

120590

119878120590=

120594119896

21205900

(1198750radic1205900

119896

1198910

120578 minus 2 coth (120578) + 2120578csch2 (120578)1205782minus 4 (120578 coth (120578) minus 1)

minus 1)

(19)

Sensitivity 119878120590versus zonal wavenumbers for different values

of 1205900with Λ

120585= 40msminus1 are shown in Figure 4 The absolute

value of the sensitivity of 120594119896with respect to 120590

0exponentially

minus5

minus10

minus15

minus20

0 2 4 6 8 10

S120590

kz

1205900 = 101205900 = 201205900 = 30

Figure 4 Sensitivity function 119878120590versus zonal wavenumber 119896

119911for

different nominal values of the static stability parameter 1205900(units

106m2 Paminus2 sminus2)

increases with decreasing wavelength For planetary scalewaves (zonal wave numbers 1ndash4) the absolute value of thesensitivity of 120594

119896with respect to 120590

0is palpably less than

sensitivity for synoptic scale waves (zonal wave numbers ge5)Absolute and relative sensitivity functions 119878

120590and 119878119877

120590calcu-

lated for different values of1205900for various zonal wave numbers

are shown in Tables 1 and 2 respectivelyThe expression for sensitivity function 119878

Λcan be eas-

ily obtained by differentiating (16) with respect to controlparameter Λ

120585

119878Λ=

1198910

21198750radic1205900

radic10038161003816100381610038161205782minus 4 (120578 coth (120578) minus 1)100381610038161003816

1003816 (20)

The function 119878Λversus zonalwave number 119896

119911for different

1205900is plotted in Figure 5 It is clear to see that for a given value

of the parameter 1205900the graph of function 119878

Λ(119896119911) is verymuch

like the classic picture of the growth rates 120594119896versus zonal

wavenumber 119896119911[5] It is interesting that the relative sensitivity

function 119878119877Λdoes not depend on the wavelength (wavenum-

ber) and for all of the unstable waves is equal to unity

119878119877

Λ=

120597120594119896120594119896

120597Λ120585Λ120585

=

Λ120585

120594119896

120597120594119896

120597Λ120585

= 1 (21)

Since relative sensitivity functions allow direct comparison ofthe importance of model parameters on the growth rate 120594

119896

we can see that because 119878119877Λ= 1 the parameter Λ

120585(ie the

meridional temperature gradient) is more important than

6 Advances in Meteorology

Table 1 Absolute sensitivity 119878120590as a function of zonal wavenumber 119896

119911for different values of static stability parameter 120590

0

1205900m2 Paminus2 sminus2 Zonal wave number 119896

119911

1 2 3 4 5 6 710 times 10

minus6minus00157 minus01251 minus04175 minus09772 minus18879 minus32470 minus52005

15 times 10minus6

minus00157 minus01245 minus04139 minus09675 minus18794 minus33014 minus5592720 times 10

minus6minus00157 minus01240 minus04110 minus09626 minus18979 minus34964 minus69371

25 times 10minus6

minus00157 minus01234 minus04088 minus09629 minus19509 minus39665 minus23003130 times 10

minus6minus00157 minus01230 minus04071 minus09691 minus20540 minus53057

Table 2 Relative sensitivity 119878119877120590as a function of zonal wavenumber 119896

119911for different values of static stability parameter 120590

0

1205900m2 Paminus2 sminus2 Zonal wave number 119896

119911

1 2 3 4 5 6 710 times 10

minus6minus00062 minus00250 minus00575 minus01057 minus01735 minus02692 minus04095

15 times 10minus6

minus00093 minus00379 minus00881 minus01656 minus02837 minus04749 minus0837120 times 10

minus6minus00124 minus00509 minus01201 minus02375 minus04220 minus08004 minus19939

25 times 10minus6

minus00156 minus00642 minus01539 minus03087 minus06071 minus14399 minus31424330 times 10

minus6minus00187 minus00777 minus01898 minus03974 minus08759 minus34322

3

2

1

00 2 4 6 8 10 12

kz

1205900 = 101205900 = 201205900 = 30

107timesSΛ

Figure 5 Sensitivity function 119878Λversus zonal wavenumber 119896

119911for

different nominal values of the static stability parameter 1205900(units

106m2 Paminus2 sminus2)

the static stability parameter 1205900except for 120590

0gt 2 times

10minus6m2 Paminus2 sminus2 for waves with 119896

119911ge 6 (see Table 2) Wher-

ever the midlatitude values in Table 2 are less than one thegrowth rate is more sensitive to the meridional temperaturegradient (ie Λ

120585) than the static stability (120590

0)

The obtained results are consistent with observations[50 51 54] an increase in static stability and a decrease of

the MTG have occurred over the past few decades in someareas of the SH which has led to a decrease in the growthrate of baroclinic unstable waves a shift of the spectrum ofunstable waves in the long wavelength part of spectrum andaweakened intensity of cyclogenesis Naturally these changesaffect favourable conditions for the development of baroclinicinstability and the essential features of weather patterns overlarge territories particularly over Australia

3 Planetary Scale Waves

To study the influence of the static stability parameter onthe dynamics of planetary scale (ultralong) waves a thinfilm approximation is applied This approximation employsa specific averaging technique over the vertical coordinate tothe system of primitive equations [60] As a result a two-dimensional set of equations can be obtained that describesthe dynamics of a two-dimensional baroclinic film Theseequations reproduce all the wavelike solutions that corre-spond to the main weather-forming modes of three-dimen-sional models and therefore can be used in theoretical studiesof large-scale dynamic processes in the atmosphere Thesystem of vertically averaged equations can be written as [60]

120597119906

120597119905

+ V sdot nabla119906 = 119891V minus119877

120587

120597

120597119909

(120587119879)

120597V120597119905

+ V sdot nablaV = minus119891119906 minus119877

120587

120597

120597119910

(120587119879)

120597120587

120597119905

+ nabla sdot (120587V) = 0

120597119879

120597119905

+ V sdot nabla119879 +

119877

119888119901

119879nabla sdot V = 0

(22)

where 120587 = 1199011199041198750 For instance if the original primitive equa-

tions are written in the Phillipsrsquo vertical coordinate system

Advances in Meteorology 7

120590 = 119901119901119904[64] the operator for vertical averaging is intro-

duced as 120595 = int

1

0120595119889120590 and state variables are represented as

120595 = 120595 + 1205951015840 Equations (22) are obtained by neglecting the

orography and terms 1199061015840V1015840 V1015840V1015840 and 1198791015840V1015840 [65] A detailedlinear analysis of the vertically averaged equation (22) isrepresented in [60] In particular two types of wave solutionswere found fast waves that propagate westward and slowwaves that move eastward Within the framework of thismodel ultralong waves are always neutral for any verticallyaveraged zonal wind velocity Indeed linearizing (22) aroundthe following basic state

1198790= 1198790(119910) 119906

0= minus(

119877

1198910

)

1205971198790

120597119910

V0= 0 120587

0= 1

(23)

and assuming the beta plane approximation 119891 = 1198910+ 120573119910

where 120573 = (2Ω1198860) cos120593

0 and representing the solution in

the form (9) one can finally obtain under different asymp-totics the following expressions for four wave solutions [60]

(a) acoustic waves

11988812= 1199060plusmnradic1198882

0+

1198912

0

1198962

(24)

(b) Rossby wave

1198883= 1199060minus

120573

1198962+ 1198912

01198882

0

(25)

(c) baroclinic wave

1198884= 1199060minus

1198912

01199062

0

1198882

0120573

(26)

Here 11988820= (1+120581)119877119879

0and 120581 = 119877119888

119901 These results however are

valid only for the specific case of a neutral atmosphere withΓ = Γ

119889[60 61] To take into account the atmospheric static

stability on the behaviour of ultralong waves the polytropicmodel of the atmosphere can be used for which

119879 (119909 119910 119911 119905) = 1198790(119909 119910 119905) minus Γ (119909 119910 119905) 119911 (27)

where 1198790is the temperature at the surface and Γ is a vertical

temperature gradient Integrating (27) with respect to verticalcoordinate we can obtain 119879

0= 119879(1 + 119877Γ119892) [60] Assuming

the geostrophic approximation on a 120573-plane the set ofvertically averaged equations can be written as [61]

120597119879

120597119905

+

119877119879

1198912

0

1205721

120587

(120587 119879) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

1

1205722

(119879 Γ)

+ 120581

1205731198772119879

2

1198921198912

0

1205724

1205722

2

120597Γ

120597119909

minus

120573119877119879

1198912

0

(1205723+ 120581

12057211205724

1205722

)

1

120587

120597 (120587119879)

120597119909

minus 120581

120573119877119879

1198912

0

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

12057211205724

1205722

120597119879

120597119909

= 0

120597Γ

120597119905

minus

119877119879

1198910

1205722

1

1205722

1

120587

(Γ 120587) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

119877

1198910

1205721

1205722

(119879 Γ)

minus 120581

120573119877119879

1198912

0

12057211205724

1205722

2

120597Γ

120597119909

minus 120581

120573119892

1198912

0

120572112057231205724

1205722

1

120587

120597 (120587119879)

120597119909

+ 120581

120573119892

1198912

0

1205722

11205724

1205722

120597119879

120597119909

= 0

120597120587

120597119905

minus

120573119877

1198912

0

120597 (120587119879)

120597119909

= 0

(28)

Here 1205721= 1 + 119877(Γ119892) 120572

2= 1 + 2119877(Γ119892) 120572

3= 119877(Γ119892) 120572

4=

1 minus (ΓΓ119889) and the operator (119860 119861) = (120597119860120597119909)(120597119861120597119910) minus (120597119860

120597119910)(120597119861120597119909) The basic state is defined as a stationary solutionof system (28) for which

120597119879

120597119909

= 0

120597Γ

120597119909

= 0

120597120587

120597119909

= 0 (29)

or in other words

119879 = 1198790(119910) Γ = Γ

0(119910) 120587 = 120587

0(119910) (30)

Linearizing (28) around the basic state (30) the followingcubic characteristic equation can be obtained in which thesecond order terms are neglected [61]

1198883+ 1198882

1205721

1205722

[120582 (1205722+ 3120581120572

4) minus 1199060]

+ 119888 120582

1205721

1205722

[1205821205811205724(2 + 120572

3+ 120581

1205724

1205722

(21205721minus

1

1205722

))

minus1199060(2 + 120581120572

4+ 31205723) ]

+ (120582

1205721

1205722

1199062

0minus 1205822120581

1205722

11205724

1205722

1199060+ 2120582312058121205722

112057231205722

4

1205723

2

)

= 0

(31)

where 120582 = 12057311987711987901198912

0and 119906

0= minus(119877119891

01205870)((120597(120587

01198790))120597119910) If

the discriminant of this equation is positive then the wavesolution is unstable The domain of zonal flow instabilitycan be found numerically (see diagram in [61]) In Figure 6we reproduce only for the 1st quadrant of a Cartesianplane the domain of instability calculated as a function ofvertically averaged zonal wind velocity 119906

0and dimensionless

temperature lapse rate ΓΓ119889

The imaginary part of phase velocity 119888119894which charac-

terises the growth rate of unstable waves 120594119896equiv 119896119888119894is displayed

in Figure 7 as a function of dimensionless temperature lapserate ΓΓ

119889for different values of vertically averaged zonal

wind velocity 1199060 A maximum phase velocity 119888

119894exists for

given values of 1199060 that is dependent on the ratio of ΓΓ

119889

For instance if 1199060= 20msminus1 then the maximum value

8 Advances in Meteorology

10

08

06

04

02

000 50 100 150 200 250 300

ΓΓd

u0 (m sminus1)

Figure 6 Domain of instability (filled) as a function of dimension-less temperature lapse rate ΓΓ

119889and vertically averaged zonal wind

velocity 1199060

(119888119894)max asymp 834msminus1 is reached at ΓΓ

119889asymp 055 Figure 6 shows

that increasing vertically averaged zonal wind 1199060is associated

with increasing 119888119894 This is further evident in Figure 8 which

shows 119888119894as a function of 119906

0for a range of ΓΓ

119889values The

lower ΓΓ119889

and the larger 119888119894 that is 119888

119894 increases with

decreasing static stability

4 Concluding Remarks

We have studied theoretically the impact of variations in thestatic stability parameter 120590

0and zonal wind shear Λ

120585on the

characteristics of baroclinically unstable waves of synopticscales using Eady-type model with the uniform Λ

120585between

upper and lower boundaries on an 119891-plane Quantitativeestimates of variations in 120590

0and Λ

120585on the growth rate 120594

119896

wavelength of maximum growth rate 119871120594max and short-wave

cutoff 119871min were obtainedAnalytical expressions are derived for sensitivity func-

tions for the growth rate 120594119896with respect to variations in static

stability parameter andwind shear velocityThese expressionsallow estimating to a first-order approximation the influenceof changes in 120590

0and Λ

120585on 120594119896 Analytical expressions for

relative sensitivity functions allow estimating the significanceof variations in 120590

0andΛ

120585on the growth rate of baroclinically

unstable waves with a given zonal wave numberTo study the impact of variations in atmospheric static

stability and zonal wind velocity on the instability of plan-etary scale waves the model with vertically averaged prim-itive equations with 120573-plane approximation was applied Ascontrol parameters we have used dimensionless temperature

15

10

5

0minus02 00 02 04 06 08 10

ΓΓd

u0 = 20u0 = 30u0 = 40

ci

(m sminus

1)

Figure 7 Imaginary part of phase speed 119888119894versus dimensionless

temperature lapse rate ΓΓ119889for different values of vertically averaged

zonal wind velocity 1199060

lapse rate ΓΓ119889and vertically averaged zonal wind velocity

1199060 We have estimated the influence of ΓΓ

119889and 119906

0on the

imaginary part of phase speed 119888119894 whichwas used as ameasure

of instabilityThe obtained results are qualitatively consistent with

changes in the essential weather patterns that occurred overthe last several decades in some areas of the SH and inparticular over Australia (eg [49 50 52ndash54]) Climatechange results suggest SH midlatitude static stability 120590

0may

increase and the MTG (the vertical wind shear Λ120585) may

decrease which according to our linear theoretical modelsleads to a slowing of the growth rate of baroclinic unstablewaves 120594

119896and an increasing wavelength of baroclinic unstable

wave with maximum growth rate 119871120594max that is a spectrum

shift of unstable waves towards longer wavelengths Thesemight affect the favourable conditions for the developmentof baroclinic instability and therefore the rate of cyclogenesisand a reduction in cyclone intensity The obtained sensitivityfunctions demonstrate that waves belonging to the short-wave part of the spectrum of unstable waves are moresensitive to changes in the static stability parameter thanwaves belonging to the long-wave part of the spectrum

To obtain more realistic estimates of the sensitivity of thegrowth rate of unstable waves with respect to static stabilityparameter and MTG numerical modeling based on a fullGCM is required It is hoped to carry out such work in thefuture

Advances in Meteorology 9

ΓΓd = 01ΓΓd = 04

ΓΓd = 23ΓΓd = 09

20

10

00 20 40 60

u0 (mmiddotsminus1)

ci(m

sminus1)

Figure 8 Imaginary part of phase speed 119888119894versus vertically

averaged zonal wind velocity 1199060for different values of dimensionless

temperature lapse rate ΓΓ119889

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors thank Dr I Pisnichenko for clarification ofcertain questions relevant to this paper

References

[1] J R Holton An Introduction to Dynamic Meteorology Aca-demic Press 3rd edition 1992

[2] G K Vallis Atmospheric and Oceanic Fluid Dynamics Cam-bridge University Press 2006

[3] M Mak Atmospheric Dynamics Cambridge University Press2011

[4] V P Dymnikov Stability and Predictability of the Large ScaleAtmospheric Processes Institute of Numerical Mathematics ofthe Russian Academy of Sciences Moscow Russia 2007

[5] J G Charney ldquoThe dynamics of long waves in baroclinicwesterly currentrdquo Journal of Meteorology vol 4 pp 135ndash1621947

[6] E T Eady ldquoLong waves and cyclone wavesrdquo Tellus vol 1 pp33ndash52 1949

[7] R T Pierrehumbert and K L Swanson ldquoBaroclinic instabilityrdquoAnnual Review of Fluid Mechanics vol 27 no 1 pp 419ndash4671995

[8] R Gall ldquoA comparison of linear instability theory with theeddy statistics of a general circulation modelrdquo Journal of theAtmospheric Sciences vol 33 no 3 pp 349ndash373 1976

[9] I M Held ldquoProgress and problems in large-scale atmosphericdynamicsrdquo in The Global Circulation of the Atmosphere TSchneider and A Sobel Eds pp 1ndash21 Princeton UniversityPress Princeton NJ USA 2007

[10] B Farrell ldquoModal and non-modal baroclinic wavesrdquo Journal ofthe Atmospheric Sciences vol 41 no 4 pp 668ndash673 1984

[11] M K Tippett ldquoTransient moist baroclinic instabilityrdquo Tellus Avol 51 no 2 pp 273ndash288 1999

[12] D Hodyss and R Grotjahn ldquoNonmodal and unstable normalmode baroclinic growth as a function of horizontal scalerdquoDynamics of Atmospheres and Oceans vol 37 no 1 pp 1ndash242003

[13] J Pedlosky ldquoFinite amplitude baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 27 no 1 pp 15ndash30 1970

[14] R D Romea ldquoThe effects of friction and beta on finite ampli-tude baroclinic wavesrdquo Journal of the Atmospheric Sciences vol34 pp 1689ndash1695 1977

[15] J Pedlosky ldquoFinite-amplitude baroclinic waves at minimumcritical shearrdquo Journal of the Atmospheric Sciences vol 39 no3 pp 555ndash562 1982

[16] T Warn and P Gauthier ldquoPotential vorticity mixing bymarginally unstable baroclinic disturbancesrdquo Tellus A vol 41no 2 pp 115ndash131 1989

[17] A J Simmons and B J Hoskins ldquoThe lifecycles of some non-linear wavesrdquo Journal of the Atmospheric Sciences vol 35 pp414ndash432 1978

[18] A J Simmons and B J Hoskins ldquoBarotropic influences of thegrowth and decay of nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 37 pp 1679ndash1684 1980

[19] S B Feldstein and I M Held ldquoBarotropic decay of baroclinicwavesrdquo Journal of the Atmospheric Sciences vol 46 pp 1679ndash1684 1989

[20] C D Thorncroft B J Hoskins and M E McIntyre ldquoTwoparadigms of baroclinic-wave life-cycle behaviourrdquo QuaterlyJournal vol 119 no 509 pp 17ndash55 1993

[21] N A Phillips ldquoA simple three-dimensional model for the studyof largescale extratropical flow patternsrdquo Journal ofMeteorologyvol 8 pp 381ndash394 1951

[22] N A Phillips ldquoEnergy transformations and meridional circu-lations associated with simple baroclinic waves in a two-levelquasi-geostrophic modelrdquo Tellus vol 6 pp 273ndash286 1954

[23] L M Polvani and J Pedlosky ldquoThe effect of dissipation onspatially growing nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 45 no 14 pp 1977ndash1989 1988

[24] T G Shepherd ldquoNonlinear saturation of baroclinic instabilityPart I the two-layermodelrdquo Journal of the Atmospheric Sciencesvol 45 no 14 pp 2014ndash2025 1988

[25] G Balasubramanian and M K Yau ldquoBaroclinic instability ina two-layer model with parameterized slantwise convectionrdquoJournal of the Atmospheric Sciences vol 51 no 7 pp 971ndash9901994

[26] A Wiin-Nielsen ldquoOn the structure of atmospheric waves inmiddle latitudesrdquo Atmosfera vol 16 no 2 pp 83ndash102 2003

[27] D D Holm and B A Wingate ldquoBaroclinic instabilities of thetwo-layer quasigeostrophic alpha modelrdquo Journal of PhysicalOceanography vol 35 no 7 pp 1287ndash1296 2005

10 Advances in Meteorology

[28] J Egger ldquoBaroclinic instability in the two-layermodel interpre-tationsrdquo Meteorologische Zeitschrift vol 18 no 5 pp 559ndash5652009

[29] J Lamberts G Lapeyre and V Zeitlin ldquoMoist versus dry baro-clinic instability in a simplified two-layer atmospheric modelwith condensation and latent heat releaserdquo Journal of theAtmospheric Sciences vol 69 pp 1405ndash1426 2012

[30] D M W Frierson ldquoRobust increases in midlatitude static sta-bility in simulations of global warmingrdquo Geophysical ResearchLetters vol 33 no 24 Article ID L24816 2006

[31] N M J Hall B J Hoskins P J Valdes and C A Senior ldquoStormtracks in a high-resolution GCMwith doubled carbon dioxiderdquoQuarterly Journal vol 120 no 519 pp 1209ndash1230 1994

[32] S B Feldstein ldquoThe recent trend and variance increase of theannularmoderdquo Journal of Climate vol 15 no 3 pp 88ndash94 2002

[33] G J Marshall ldquoTrends in the southern annular mode fromobservations and reanalysesrdquo Journal of Climate vol 16 pp4134ndash4143 2003

[34] G M Ostermeier and J M Wallace ldquoTrends in the NorthAtlantic Oscillation-Northern Hemisphere annular mode dur-ing the twentieth centuryrdquo Journal of Climate vol 16 no 2 pp336ndash341 2003

[35] D W J Thompson and S Solomon ldquoInterpretation of recentSouthern Hemisphere climate changerdquo Science vol 296 no5569 pp 895ndash899 2002

[36] S J Lambert ldquoThe effect of enhanced greenhouse warming onwinter cyclone frequencies and strengthsrdquo Journal of Climatevol 8 no 5 pp 1447ndash1462 1995

[37] F Lunkeit K Fraedrich and S E Bauer ldquoStorm tracks ina warmer climate sensitivity studies with a simplified globalcirculation modelrdquo Climate Dynamics vol 14 no 11 pp 813ndash826 1998

[38] D J Lorenz and D L Hartmann ldquoEddy-zonal flow feedback inthe Southern Hemisphererdquo Journal of the Atmospheric Sciencesvol 58 no 21 pp 3312ndash3327 2001

[39] Q Geng and M Sugi ldquoPossible change of extratropical cycloneactivity due to enhanced greenhouse gases and sulphateaerosols study with a high-resolution AGCMrdquo Journal ofClimate vol 16 pp 2262ndash2274 2003

[40] J C Fyfe ldquoExtratropical Southern Hemisphere cyclonesharbingers of climate changerdquo Journal of Climate vol 16 pp2802ndash2805 2003

[41] J H Yin ldquoA consistent poleward shift of the storm tracksin simulations of 21st century climaterdquo Geophysical ResearchLetters vol 32 no 18 Article ID L18701 pp 1ndash4 2005

[42] S J Lambert and J C Fyfe ldquoChanges in winter cyclonefrequencies and strengths simulated in enhanced greenhousewarming experiments results from the models participating inthe IPCC diagnostic exerciserdquo Climate Dynamics vol 26 no7-8 pp 713ndash728 2006

[43] G Gastineau and B J Soden ldquoModel projected changes ofextreme wind events in response to global warmingrdquo Geophys-ical Research Letters vol 36 no 10 Article ID L10810 2009

[44] Y Wu M Ting R Seager H-P Huang and M A CaneldquoChanges in storm tracks and energy transports in a warmer cli-mate simulated by the GFDL CM21 modelrdquo Climate Dynamicsvol 37 no 1 pp 53ndash72 2011

[45] L Bengtsson K I Hodges and E Roeckner ldquoStorm tracks andclimate changerdquo Journal of Climate vol 19 no 15 pp 3518ndash35432006

[46] I Smith ldquoAn assessment of recent trends in Australian rainfallrdquoAustralian Meteorological Magazine vol 53 no 3 pp 163ndash1732004

[47] N Nicholls Detecting Understanding and Attributing ClimateChange Australian Greenhouse Office Publication 2007

[48] B C Bates P Hope B Ryan I Smith and S Charles ldquoKey find-ings from the Indian Ocean Climate Initiative and their impacton policy development in Australiardquo Climatic Change vol89 no 3-4 pp 339ndash354 2008

[49] J S Frederiksen and C S Frederiksen ldquoDecadal changes inSouthern Hemisphere winter cyclogenesisrdquo CSIROMarine andAtmospheric Research Paper 002 2005

[50] J S Frederiksen and C S Frederiksen ldquoInterdecadal changes insouthern hemisphere winter storm track modesrdquo Tellus A vol59 no 5 pp 599ndash617 2007

[51] J S Frederiksen C S Frederiksen and S L Osbrough ldquoMod-elling of changes in Southern Hemisphere weather systemsduring the 20th centuryrdquo in Proceedings of the 18th WorldIMACS Congress and International Congress on Modelling andSimulation pp 2562ndash2568 Cairns Australia July 2009

[52] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoCauses of changing Southern Hemispheric weathersystemsrdquo in Managing Climate Change I Jupp P Holper andW Cai Eds pp 85ndash98 CSIRO Publishing 2010

[53] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoChanges in Southern Hemisphere rainfall circulationand weather systemsrdquo in Proceedings of the 19th InternationalCongress on Modelling and Simulation (MODSIM rsquo11) pp 2712ndash2718 Perth Australia December 2011

[54] C S Frederiksen J S Frederiksen J M Sisson and S LOsbrough ldquoObserved and projected changes in the annual cycleof Southern Hemisphere baroclinicity for storm formationrdquoin Proceedings of the 19th International Congress on Modellingand Simulation (MODSIMrsquo 11) pp 2719ndash2725 Perth AustraliaDecember 2011

[55] N A Phillips ldquoGeostrophicmotionrdquoReviews of Geophysics vol1 no 2 pp 123ndash176 1963

[56] I Bordi K Fraedrich F Lunkeit and A Sutera ldquoTroposphericdouble jets meridional cells and eddies a case study andidealized simulationsrdquoMonthly Weather Review vol 135 no 9pp 3118ndash3133 2007

[57] I Bordi K Fraedrich M Ghil and A Sutera ldquoZonal flowregime changes in a GCM and in a simple quasigeostrophicmodel the role of stratospheric dynamicsrdquo Journal of theAtmospheric Sciences vol 66 no 5 pp 1366ndash1383 2009

[58] J G Charney ldquoOn the scale of atmosphericmotionsrdquoGeofysiskePublikasjoner vol 17 no 2 pp 3ndash17 1948

[59] A Burger ldquoScale consideration of planetary motions of theatmosphererdquo Tellus vol 10 pp 195ndash205 1958

[60] I A Pisnichenko ldquoUltralong-wave dynamics in a two-dimen-sional baroclinic atmosphere modelrdquo Atmospheric and OceanicPhysics vol 16 no 9 pp 883ndash892 1980

[61] I A Pisnichenko ldquoInfluence variable static stability on thedynamics of ultralong waves in two-dimensional baroclinicmodel of the atmosphererdquoAtmospheric andOceanic Physics vol19 no 11 pp 1223ndash1226 1983

[62] A Wiin-Nielsen ldquoOn barotropic and baroclinic models withspecial emphasis onultra-longwavesrdquoMonthlyWeather Reviewvol 87 pp 171ndash183 1959

[63] B W BinWang and A Barcilon ldquoMoist stability of a barocliniczonal flow with conditionally unstable stratificationrdquo Journal ofthe Atmospheric Sciences vol 43 no 7 pp 705ndash719 1986

Advances in Meteorology 11

[64] N A Phillips ldquoA coordinate system having some specialadvantages for numerical forecastingrdquo Journal of Meteorologyvol 14 pp 184ndash185 1957

[65] D M Alishaev ldquoDynamics of a two-dimensional baroclinicatmosphererdquoAtmospheric and Oceanic Physics vol 16 no 2 pp99ndash107 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 5

20

15

10

05

0 1 2 3

Xk

(dayminus1)

Λ120576 = 30Λ120576 = 40

Λ120576 = 50Λ120576 = 60

(m2 Paminus2 sminus2 )1205900 times 106

Figure 3 Growth rate 120594119896versus static stability parameter 120590

0for

different values of parameter Λ120585(units m sminus1)

instance if Λ120585decreases by 10 and the static stability

parameter increases by 10 the growth rate 120594119896decreases by

14Since 120594

119896is a nonlinear function of 120590

0(16) to estimate the

influence of infinitesimal perturbations in 1205900on variations in

120594119896 the sensitivity function

119878120590=

120597120594119896

1205971205900

(17)

and the relative sensitivity function

119878119877

120590=

120597120594119896120594119896

12059712059001205900

=

1205900

120594119896

120597120594119896

1205971205900

(18)

can be used The function 119878120590shows changes in 120594

119896due to

variations in 1205900 The relative sensitivity function 119878119877

120590is used

to compare model parameters to find out what parameter isthemost important for a certain percent change in the param-eters Sensitivity functions (17) and (18) are evaluated in thevicinity of some nominal value of the parameter 120590

0 We can

select several nominal values to cover some range of changesin 1205900 Differentiating (16) with respect to control parameter

1205900 we can obtain the expression for 119878

120590

119878120590=

120594119896

21205900

(1198750radic1205900

119896

1198910

120578 minus 2 coth (120578) + 2120578csch2 (120578)1205782minus 4 (120578 coth (120578) minus 1)

minus 1)

(19)

Sensitivity 119878120590versus zonal wavenumbers for different values

of 1205900with Λ

120585= 40msminus1 are shown in Figure 4 The absolute

value of the sensitivity of 120594119896with respect to 120590

0exponentially

minus5

minus10

minus15

minus20

0 2 4 6 8 10

S120590

kz

1205900 = 101205900 = 201205900 = 30

Figure 4 Sensitivity function 119878120590versus zonal wavenumber 119896

119911for

different nominal values of the static stability parameter 1205900(units

106m2 Paminus2 sminus2)

increases with decreasing wavelength For planetary scalewaves (zonal wave numbers 1ndash4) the absolute value of thesensitivity of 120594

119896with respect to 120590

0is palpably less than

sensitivity for synoptic scale waves (zonal wave numbers ge5)Absolute and relative sensitivity functions 119878

120590and 119878119877

120590calcu-

lated for different values of1205900for various zonal wave numbers

are shown in Tables 1 and 2 respectivelyThe expression for sensitivity function 119878

Λcan be eas-

ily obtained by differentiating (16) with respect to controlparameter Λ

120585

119878Λ=

1198910

21198750radic1205900

radic10038161003816100381610038161205782minus 4 (120578 coth (120578) minus 1)100381610038161003816

1003816 (20)

The function 119878Λversus zonalwave number 119896

119911for different

1205900is plotted in Figure 5 It is clear to see that for a given value

of the parameter 1205900the graph of function 119878

Λ(119896119911) is verymuch

like the classic picture of the growth rates 120594119896versus zonal

wavenumber 119896119911[5] It is interesting that the relative sensitivity

function 119878119877Λdoes not depend on the wavelength (wavenum-

ber) and for all of the unstable waves is equal to unity

119878119877

Λ=

120597120594119896120594119896

120597Λ120585Λ120585

=

Λ120585

120594119896

120597120594119896

120597Λ120585

= 1 (21)

Since relative sensitivity functions allow direct comparison ofthe importance of model parameters on the growth rate 120594

119896

we can see that because 119878119877Λ= 1 the parameter Λ

120585(ie the

meridional temperature gradient) is more important than

6 Advances in Meteorology

Table 1 Absolute sensitivity 119878120590as a function of zonal wavenumber 119896

119911for different values of static stability parameter 120590

0

1205900m2 Paminus2 sminus2 Zonal wave number 119896

119911

1 2 3 4 5 6 710 times 10

minus6minus00157 minus01251 minus04175 minus09772 minus18879 minus32470 minus52005

15 times 10minus6

minus00157 minus01245 minus04139 minus09675 minus18794 minus33014 minus5592720 times 10

minus6minus00157 minus01240 minus04110 minus09626 minus18979 minus34964 minus69371

25 times 10minus6

minus00157 minus01234 minus04088 minus09629 minus19509 minus39665 minus23003130 times 10

minus6minus00157 minus01230 minus04071 minus09691 minus20540 minus53057

Table 2 Relative sensitivity 119878119877120590as a function of zonal wavenumber 119896

119911for different values of static stability parameter 120590

0

1205900m2 Paminus2 sminus2 Zonal wave number 119896

119911

1 2 3 4 5 6 710 times 10

minus6minus00062 minus00250 minus00575 minus01057 minus01735 minus02692 minus04095

15 times 10minus6

minus00093 minus00379 minus00881 minus01656 minus02837 minus04749 minus0837120 times 10

minus6minus00124 minus00509 minus01201 minus02375 minus04220 minus08004 minus19939

25 times 10minus6

minus00156 minus00642 minus01539 minus03087 minus06071 minus14399 minus31424330 times 10

minus6minus00187 minus00777 minus01898 minus03974 minus08759 minus34322

3

2

1

00 2 4 6 8 10 12

kz

1205900 = 101205900 = 201205900 = 30

107timesSΛ

Figure 5 Sensitivity function 119878Λversus zonal wavenumber 119896

119911for

different nominal values of the static stability parameter 1205900(units

106m2 Paminus2 sminus2)

the static stability parameter 1205900except for 120590

0gt 2 times

10minus6m2 Paminus2 sminus2 for waves with 119896

119911ge 6 (see Table 2) Wher-

ever the midlatitude values in Table 2 are less than one thegrowth rate is more sensitive to the meridional temperaturegradient (ie Λ

120585) than the static stability (120590

0)

The obtained results are consistent with observations[50 51 54] an increase in static stability and a decrease of

the MTG have occurred over the past few decades in someareas of the SH which has led to a decrease in the growthrate of baroclinic unstable waves a shift of the spectrum ofunstable waves in the long wavelength part of spectrum andaweakened intensity of cyclogenesis Naturally these changesaffect favourable conditions for the development of baroclinicinstability and the essential features of weather patterns overlarge territories particularly over Australia

3 Planetary Scale Waves

To study the influence of the static stability parameter onthe dynamics of planetary scale (ultralong) waves a thinfilm approximation is applied This approximation employsa specific averaging technique over the vertical coordinate tothe system of primitive equations [60] As a result a two-dimensional set of equations can be obtained that describesthe dynamics of a two-dimensional baroclinic film Theseequations reproduce all the wavelike solutions that corre-spond to the main weather-forming modes of three-dimen-sional models and therefore can be used in theoretical studiesof large-scale dynamic processes in the atmosphere Thesystem of vertically averaged equations can be written as [60]

120597119906

120597119905

+ V sdot nabla119906 = 119891V minus119877

120587

120597

120597119909

(120587119879)

120597V120597119905

+ V sdot nablaV = minus119891119906 minus119877

120587

120597

120597119910

(120587119879)

120597120587

120597119905

+ nabla sdot (120587V) = 0

120597119879

120597119905

+ V sdot nabla119879 +

119877

119888119901

119879nabla sdot V = 0

(22)

where 120587 = 1199011199041198750 For instance if the original primitive equa-

tions are written in the Phillipsrsquo vertical coordinate system

Advances in Meteorology 7

120590 = 119901119901119904[64] the operator for vertical averaging is intro-

duced as 120595 = int

1

0120595119889120590 and state variables are represented as

120595 = 120595 + 1205951015840 Equations (22) are obtained by neglecting the

orography and terms 1199061015840V1015840 V1015840V1015840 and 1198791015840V1015840 [65] A detailedlinear analysis of the vertically averaged equation (22) isrepresented in [60] In particular two types of wave solutionswere found fast waves that propagate westward and slowwaves that move eastward Within the framework of thismodel ultralong waves are always neutral for any verticallyaveraged zonal wind velocity Indeed linearizing (22) aroundthe following basic state

1198790= 1198790(119910) 119906

0= minus(

119877

1198910

)

1205971198790

120597119910

V0= 0 120587

0= 1

(23)

and assuming the beta plane approximation 119891 = 1198910+ 120573119910

where 120573 = (2Ω1198860) cos120593

0 and representing the solution in

the form (9) one can finally obtain under different asymp-totics the following expressions for four wave solutions [60]

(a) acoustic waves

11988812= 1199060plusmnradic1198882

0+

1198912

0

1198962

(24)

(b) Rossby wave

1198883= 1199060minus

120573

1198962+ 1198912

01198882

0

(25)

(c) baroclinic wave

1198884= 1199060minus

1198912

01199062

0

1198882

0120573

(26)

Here 11988820= (1+120581)119877119879

0and 120581 = 119877119888

119901 These results however are

valid only for the specific case of a neutral atmosphere withΓ = Γ

119889[60 61] To take into account the atmospheric static

stability on the behaviour of ultralong waves the polytropicmodel of the atmosphere can be used for which

119879 (119909 119910 119911 119905) = 1198790(119909 119910 119905) minus Γ (119909 119910 119905) 119911 (27)

where 1198790is the temperature at the surface and Γ is a vertical

temperature gradient Integrating (27) with respect to verticalcoordinate we can obtain 119879

0= 119879(1 + 119877Γ119892) [60] Assuming

the geostrophic approximation on a 120573-plane the set ofvertically averaged equations can be written as [61]

120597119879

120597119905

+

119877119879

1198912

0

1205721

120587

(120587 119879) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

1

1205722

(119879 Γ)

+ 120581

1205731198772119879

2

1198921198912

0

1205724

1205722

2

120597Γ

120597119909

minus

120573119877119879

1198912

0

(1205723+ 120581

12057211205724

1205722

)

1

120587

120597 (120587119879)

120597119909

minus 120581

120573119877119879

1198912

0

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

12057211205724

1205722

120597119879

120597119909

= 0

120597Γ

120597119905

minus

119877119879

1198910

1205722

1

1205722

1

120587

(Γ 120587) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

119877

1198910

1205721

1205722

(119879 Γ)

minus 120581

120573119877119879

1198912

0

12057211205724

1205722

2

120597Γ

120597119909

minus 120581

120573119892

1198912

0

120572112057231205724

1205722

1

120587

120597 (120587119879)

120597119909

+ 120581

120573119892

1198912

0

1205722

11205724

1205722

120597119879

120597119909

= 0

120597120587

120597119905

minus

120573119877

1198912

0

120597 (120587119879)

120597119909

= 0

(28)

Here 1205721= 1 + 119877(Γ119892) 120572

2= 1 + 2119877(Γ119892) 120572

3= 119877(Γ119892) 120572

4=

1 minus (ΓΓ119889) and the operator (119860 119861) = (120597119860120597119909)(120597119861120597119910) minus (120597119860

120597119910)(120597119861120597119909) The basic state is defined as a stationary solutionof system (28) for which

120597119879

120597119909

= 0

120597Γ

120597119909

= 0

120597120587

120597119909

= 0 (29)

or in other words

119879 = 1198790(119910) Γ = Γ

0(119910) 120587 = 120587

0(119910) (30)

Linearizing (28) around the basic state (30) the followingcubic characteristic equation can be obtained in which thesecond order terms are neglected [61]

1198883+ 1198882

1205721

1205722

[120582 (1205722+ 3120581120572

4) minus 1199060]

+ 119888 120582

1205721

1205722

[1205821205811205724(2 + 120572

3+ 120581

1205724

1205722

(21205721minus

1

1205722

))

minus1199060(2 + 120581120572

4+ 31205723) ]

+ (120582

1205721

1205722

1199062

0minus 1205822120581

1205722

11205724

1205722

1199060+ 2120582312058121205722

112057231205722

4

1205723

2

)

= 0

(31)

where 120582 = 12057311987711987901198912

0and 119906

0= minus(119877119891

01205870)((120597(120587

01198790))120597119910) If

the discriminant of this equation is positive then the wavesolution is unstable The domain of zonal flow instabilitycan be found numerically (see diagram in [61]) In Figure 6we reproduce only for the 1st quadrant of a Cartesianplane the domain of instability calculated as a function ofvertically averaged zonal wind velocity 119906

0and dimensionless

temperature lapse rate ΓΓ119889

The imaginary part of phase velocity 119888119894which charac-

terises the growth rate of unstable waves 120594119896equiv 119896119888119894is displayed

in Figure 7 as a function of dimensionless temperature lapserate ΓΓ

119889for different values of vertically averaged zonal

wind velocity 1199060 A maximum phase velocity 119888

119894exists for

given values of 1199060 that is dependent on the ratio of ΓΓ

119889

For instance if 1199060= 20msminus1 then the maximum value

8 Advances in Meteorology

10

08

06

04

02

000 50 100 150 200 250 300

ΓΓd

u0 (m sminus1)

Figure 6 Domain of instability (filled) as a function of dimension-less temperature lapse rate ΓΓ

119889and vertically averaged zonal wind

velocity 1199060

(119888119894)max asymp 834msminus1 is reached at ΓΓ

119889asymp 055 Figure 6 shows

that increasing vertically averaged zonal wind 1199060is associated

with increasing 119888119894 This is further evident in Figure 8 which

shows 119888119894as a function of 119906

0for a range of ΓΓ

119889values The

lower ΓΓ119889

and the larger 119888119894 that is 119888

119894 increases with

decreasing static stability

4 Concluding Remarks

We have studied theoretically the impact of variations in thestatic stability parameter 120590

0and zonal wind shear Λ

120585on the

characteristics of baroclinically unstable waves of synopticscales using Eady-type model with the uniform Λ

120585between

upper and lower boundaries on an 119891-plane Quantitativeestimates of variations in 120590

0and Λ

120585on the growth rate 120594

119896

wavelength of maximum growth rate 119871120594max and short-wave

cutoff 119871min were obtainedAnalytical expressions are derived for sensitivity func-

tions for the growth rate 120594119896with respect to variations in static

stability parameter andwind shear velocityThese expressionsallow estimating to a first-order approximation the influenceof changes in 120590

0and Λ

120585on 120594119896 Analytical expressions for

relative sensitivity functions allow estimating the significanceof variations in 120590

0andΛ

120585on the growth rate of baroclinically

unstable waves with a given zonal wave numberTo study the impact of variations in atmospheric static

stability and zonal wind velocity on the instability of plan-etary scale waves the model with vertically averaged prim-itive equations with 120573-plane approximation was applied Ascontrol parameters we have used dimensionless temperature

15

10

5

0minus02 00 02 04 06 08 10

ΓΓd

u0 = 20u0 = 30u0 = 40

ci

(m sminus

1)

Figure 7 Imaginary part of phase speed 119888119894versus dimensionless

temperature lapse rate ΓΓ119889for different values of vertically averaged

zonal wind velocity 1199060

lapse rate ΓΓ119889and vertically averaged zonal wind velocity

1199060 We have estimated the influence of ΓΓ

119889and 119906

0on the

imaginary part of phase speed 119888119894 whichwas used as ameasure

of instabilityThe obtained results are qualitatively consistent with

changes in the essential weather patterns that occurred overthe last several decades in some areas of the SH and inparticular over Australia (eg [49 50 52ndash54]) Climatechange results suggest SH midlatitude static stability 120590

0may

increase and the MTG (the vertical wind shear Λ120585) may

decrease which according to our linear theoretical modelsleads to a slowing of the growth rate of baroclinic unstablewaves 120594

119896and an increasing wavelength of baroclinic unstable

wave with maximum growth rate 119871120594max that is a spectrum

shift of unstable waves towards longer wavelengths Thesemight affect the favourable conditions for the developmentof baroclinic instability and therefore the rate of cyclogenesisand a reduction in cyclone intensity The obtained sensitivityfunctions demonstrate that waves belonging to the short-wave part of the spectrum of unstable waves are moresensitive to changes in the static stability parameter thanwaves belonging to the long-wave part of the spectrum

To obtain more realistic estimates of the sensitivity of thegrowth rate of unstable waves with respect to static stabilityparameter and MTG numerical modeling based on a fullGCM is required It is hoped to carry out such work in thefuture

Advances in Meteorology 9

ΓΓd = 01ΓΓd = 04

ΓΓd = 23ΓΓd = 09

20

10

00 20 40 60

u0 (mmiddotsminus1)

ci(m

sminus1)

Figure 8 Imaginary part of phase speed 119888119894versus vertically

averaged zonal wind velocity 1199060for different values of dimensionless

temperature lapse rate ΓΓ119889

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors thank Dr I Pisnichenko for clarification ofcertain questions relevant to this paper

References

[1] J R Holton An Introduction to Dynamic Meteorology Aca-demic Press 3rd edition 1992

[2] G K Vallis Atmospheric and Oceanic Fluid Dynamics Cam-bridge University Press 2006

[3] M Mak Atmospheric Dynamics Cambridge University Press2011

[4] V P Dymnikov Stability and Predictability of the Large ScaleAtmospheric Processes Institute of Numerical Mathematics ofthe Russian Academy of Sciences Moscow Russia 2007

[5] J G Charney ldquoThe dynamics of long waves in baroclinicwesterly currentrdquo Journal of Meteorology vol 4 pp 135ndash1621947

[6] E T Eady ldquoLong waves and cyclone wavesrdquo Tellus vol 1 pp33ndash52 1949

[7] R T Pierrehumbert and K L Swanson ldquoBaroclinic instabilityrdquoAnnual Review of Fluid Mechanics vol 27 no 1 pp 419ndash4671995

[8] R Gall ldquoA comparison of linear instability theory with theeddy statistics of a general circulation modelrdquo Journal of theAtmospheric Sciences vol 33 no 3 pp 349ndash373 1976

[9] I M Held ldquoProgress and problems in large-scale atmosphericdynamicsrdquo in The Global Circulation of the Atmosphere TSchneider and A Sobel Eds pp 1ndash21 Princeton UniversityPress Princeton NJ USA 2007

[10] B Farrell ldquoModal and non-modal baroclinic wavesrdquo Journal ofthe Atmospheric Sciences vol 41 no 4 pp 668ndash673 1984

[11] M K Tippett ldquoTransient moist baroclinic instabilityrdquo Tellus Avol 51 no 2 pp 273ndash288 1999

[12] D Hodyss and R Grotjahn ldquoNonmodal and unstable normalmode baroclinic growth as a function of horizontal scalerdquoDynamics of Atmospheres and Oceans vol 37 no 1 pp 1ndash242003

[13] J Pedlosky ldquoFinite amplitude baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 27 no 1 pp 15ndash30 1970

[14] R D Romea ldquoThe effects of friction and beta on finite ampli-tude baroclinic wavesrdquo Journal of the Atmospheric Sciences vol34 pp 1689ndash1695 1977

[15] J Pedlosky ldquoFinite-amplitude baroclinic waves at minimumcritical shearrdquo Journal of the Atmospheric Sciences vol 39 no3 pp 555ndash562 1982

[16] T Warn and P Gauthier ldquoPotential vorticity mixing bymarginally unstable baroclinic disturbancesrdquo Tellus A vol 41no 2 pp 115ndash131 1989

[17] A J Simmons and B J Hoskins ldquoThe lifecycles of some non-linear wavesrdquo Journal of the Atmospheric Sciences vol 35 pp414ndash432 1978

[18] A J Simmons and B J Hoskins ldquoBarotropic influences of thegrowth and decay of nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 37 pp 1679ndash1684 1980

[19] S B Feldstein and I M Held ldquoBarotropic decay of baroclinicwavesrdquo Journal of the Atmospheric Sciences vol 46 pp 1679ndash1684 1989

[20] C D Thorncroft B J Hoskins and M E McIntyre ldquoTwoparadigms of baroclinic-wave life-cycle behaviourrdquo QuaterlyJournal vol 119 no 509 pp 17ndash55 1993

[21] N A Phillips ldquoA simple three-dimensional model for the studyof largescale extratropical flow patternsrdquo Journal ofMeteorologyvol 8 pp 381ndash394 1951

[22] N A Phillips ldquoEnergy transformations and meridional circu-lations associated with simple baroclinic waves in a two-levelquasi-geostrophic modelrdquo Tellus vol 6 pp 273ndash286 1954

[23] L M Polvani and J Pedlosky ldquoThe effect of dissipation onspatially growing nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 45 no 14 pp 1977ndash1989 1988

[24] T G Shepherd ldquoNonlinear saturation of baroclinic instabilityPart I the two-layermodelrdquo Journal of the Atmospheric Sciencesvol 45 no 14 pp 2014ndash2025 1988

[25] G Balasubramanian and M K Yau ldquoBaroclinic instability ina two-layer model with parameterized slantwise convectionrdquoJournal of the Atmospheric Sciences vol 51 no 7 pp 971ndash9901994

[26] A Wiin-Nielsen ldquoOn the structure of atmospheric waves inmiddle latitudesrdquo Atmosfera vol 16 no 2 pp 83ndash102 2003

[27] D D Holm and B A Wingate ldquoBaroclinic instabilities of thetwo-layer quasigeostrophic alpha modelrdquo Journal of PhysicalOceanography vol 35 no 7 pp 1287ndash1296 2005

10 Advances in Meteorology

[28] J Egger ldquoBaroclinic instability in the two-layermodel interpre-tationsrdquo Meteorologische Zeitschrift vol 18 no 5 pp 559ndash5652009

[29] J Lamberts G Lapeyre and V Zeitlin ldquoMoist versus dry baro-clinic instability in a simplified two-layer atmospheric modelwith condensation and latent heat releaserdquo Journal of theAtmospheric Sciences vol 69 pp 1405ndash1426 2012

[30] D M W Frierson ldquoRobust increases in midlatitude static sta-bility in simulations of global warmingrdquo Geophysical ResearchLetters vol 33 no 24 Article ID L24816 2006

[31] N M J Hall B J Hoskins P J Valdes and C A Senior ldquoStormtracks in a high-resolution GCMwith doubled carbon dioxiderdquoQuarterly Journal vol 120 no 519 pp 1209ndash1230 1994

[32] S B Feldstein ldquoThe recent trend and variance increase of theannularmoderdquo Journal of Climate vol 15 no 3 pp 88ndash94 2002

[33] G J Marshall ldquoTrends in the southern annular mode fromobservations and reanalysesrdquo Journal of Climate vol 16 pp4134ndash4143 2003

[34] G M Ostermeier and J M Wallace ldquoTrends in the NorthAtlantic Oscillation-Northern Hemisphere annular mode dur-ing the twentieth centuryrdquo Journal of Climate vol 16 no 2 pp336ndash341 2003

[35] D W J Thompson and S Solomon ldquoInterpretation of recentSouthern Hemisphere climate changerdquo Science vol 296 no5569 pp 895ndash899 2002

[36] S J Lambert ldquoThe effect of enhanced greenhouse warming onwinter cyclone frequencies and strengthsrdquo Journal of Climatevol 8 no 5 pp 1447ndash1462 1995

[37] F Lunkeit K Fraedrich and S E Bauer ldquoStorm tracks ina warmer climate sensitivity studies with a simplified globalcirculation modelrdquo Climate Dynamics vol 14 no 11 pp 813ndash826 1998

[38] D J Lorenz and D L Hartmann ldquoEddy-zonal flow feedback inthe Southern Hemisphererdquo Journal of the Atmospheric Sciencesvol 58 no 21 pp 3312ndash3327 2001

[39] Q Geng and M Sugi ldquoPossible change of extratropical cycloneactivity due to enhanced greenhouse gases and sulphateaerosols study with a high-resolution AGCMrdquo Journal ofClimate vol 16 pp 2262ndash2274 2003

[40] J C Fyfe ldquoExtratropical Southern Hemisphere cyclonesharbingers of climate changerdquo Journal of Climate vol 16 pp2802ndash2805 2003

[41] J H Yin ldquoA consistent poleward shift of the storm tracksin simulations of 21st century climaterdquo Geophysical ResearchLetters vol 32 no 18 Article ID L18701 pp 1ndash4 2005

[42] S J Lambert and J C Fyfe ldquoChanges in winter cyclonefrequencies and strengths simulated in enhanced greenhousewarming experiments results from the models participating inthe IPCC diagnostic exerciserdquo Climate Dynamics vol 26 no7-8 pp 713ndash728 2006

[43] G Gastineau and B J Soden ldquoModel projected changes ofextreme wind events in response to global warmingrdquo Geophys-ical Research Letters vol 36 no 10 Article ID L10810 2009

[44] Y Wu M Ting R Seager H-P Huang and M A CaneldquoChanges in storm tracks and energy transports in a warmer cli-mate simulated by the GFDL CM21 modelrdquo Climate Dynamicsvol 37 no 1 pp 53ndash72 2011

[45] L Bengtsson K I Hodges and E Roeckner ldquoStorm tracks andclimate changerdquo Journal of Climate vol 19 no 15 pp 3518ndash35432006

[46] I Smith ldquoAn assessment of recent trends in Australian rainfallrdquoAustralian Meteorological Magazine vol 53 no 3 pp 163ndash1732004

[47] N Nicholls Detecting Understanding and Attributing ClimateChange Australian Greenhouse Office Publication 2007

[48] B C Bates P Hope B Ryan I Smith and S Charles ldquoKey find-ings from the Indian Ocean Climate Initiative and their impacton policy development in Australiardquo Climatic Change vol89 no 3-4 pp 339ndash354 2008

[49] J S Frederiksen and C S Frederiksen ldquoDecadal changes inSouthern Hemisphere winter cyclogenesisrdquo CSIROMarine andAtmospheric Research Paper 002 2005

[50] J S Frederiksen and C S Frederiksen ldquoInterdecadal changes insouthern hemisphere winter storm track modesrdquo Tellus A vol59 no 5 pp 599ndash617 2007

[51] J S Frederiksen C S Frederiksen and S L Osbrough ldquoMod-elling of changes in Southern Hemisphere weather systemsduring the 20th centuryrdquo in Proceedings of the 18th WorldIMACS Congress and International Congress on Modelling andSimulation pp 2562ndash2568 Cairns Australia July 2009

[52] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoCauses of changing Southern Hemispheric weathersystemsrdquo in Managing Climate Change I Jupp P Holper andW Cai Eds pp 85ndash98 CSIRO Publishing 2010

[53] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoChanges in Southern Hemisphere rainfall circulationand weather systemsrdquo in Proceedings of the 19th InternationalCongress on Modelling and Simulation (MODSIM rsquo11) pp 2712ndash2718 Perth Australia December 2011

[54] C S Frederiksen J S Frederiksen J M Sisson and S LOsbrough ldquoObserved and projected changes in the annual cycleof Southern Hemisphere baroclinicity for storm formationrdquoin Proceedings of the 19th International Congress on Modellingand Simulation (MODSIMrsquo 11) pp 2719ndash2725 Perth AustraliaDecember 2011

[55] N A Phillips ldquoGeostrophicmotionrdquoReviews of Geophysics vol1 no 2 pp 123ndash176 1963

[56] I Bordi K Fraedrich F Lunkeit and A Sutera ldquoTroposphericdouble jets meridional cells and eddies a case study andidealized simulationsrdquoMonthly Weather Review vol 135 no 9pp 3118ndash3133 2007

[57] I Bordi K Fraedrich M Ghil and A Sutera ldquoZonal flowregime changes in a GCM and in a simple quasigeostrophicmodel the role of stratospheric dynamicsrdquo Journal of theAtmospheric Sciences vol 66 no 5 pp 1366ndash1383 2009

[58] J G Charney ldquoOn the scale of atmosphericmotionsrdquoGeofysiskePublikasjoner vol 17 no 2 pp 3ndash17 1948

[59] A Burger ldquoScale consideration of planetary motions of theatmosphererdquo Tellus vol 10 pp 195ndash205 1958

[60] I A Pisnichenko ldquoUltralong-wave dynamics in a two-dimen-sional baroclinic atmosphere modelrdquo Atmospheric and OceanicPhysics vol 16 no 9 pp 883ndash892 1980

[61] I A Pisnichenko ldquoInfluence variable static stability on thedynamics of ultralong waves in two-dimensional baroclinicmodel of the atmosphererdquoAtmospheric andOceanic Physics vol19 no 11 pp 1223ndash1226 1983

[62] A Wiin-Nielsen ldquoOn barotropic and baroclinic models withspecial emphasis onultra-longwavesrdquoMonthlyWeather Reviewvol 87 pp 171ndash183 1959

[63] B W BinWang and A Barcilon ldquoMoist stability of a barocliniczonal flow with conditionally unstable stratificationrdquo Journal ofthe Atmospheric Sciences vol 43 no 7 pp 705ndash719 1986

Advances in Meteorology 11

[64] N A Phillips ldquoA coordinate system having some specialadvantages for numerical forecastingrdquo Journal of Meteorologyvol 14 pp 184ndash185 1957

[65] D M Alishaev ldquoDynamics of a two-dimensional baroclinicatmosphererdquoAtmospheric and Oceanic Physics vol 16 no 2 pp99ndash107 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

6 Advances in Meteorology

Table 1 Absolute sensitivity 119878120590as a function of zonal wavenumber 119896

119911for different values of static stability parameter 120590

0

1205900m2 Paminus2 sminus2 Zonal wave number 119896

119911

1 2 3 4 5 6 710 times 10

minus6minus00157 minus01251 minus04175 minus09772 minus18879 minus32470 minus52005

15 times 10minus6

minus00157 minus01245 minus04139 minus09675 minus18794 minus33014 minus5592720 times 10

minus6minus00157 minus01240 minus04110 minus09626 minus18979 minus34964 minus69371

25 times 10minus6

minus00157 minus01234 minus04088 minus09629 minus19509 minus39665 minus23003130 times 10

minus6minus00157 minus01230 minus04071 minus09691 minus20540 minus53057

Table 2 Relative sensitivity 119878119877120590as a function of zonal wavenumber 119896

119911for different values of static stability parameter 120590

0

1205900m2 Paminus2 sminus2 Zonal wave number 119896

119911

1 2 3 4 5 6 710 times 10

minus6minus00062 minus00250 minus00575 minus01057 minus01735 minus02692 minus04095

15 times 10minus6

minus00093 minus00379 minus00881 minus01656 minus02837 minus04749 minus0837120 times 10

minus6minus00124 minus00509 minus01201 minus02375 minus04220 minus08004 minus19939

25 times 10minus6

minus00156 minus00642 minus01539 minus03087 minus06071 minus14399 minus31424330 times 10

minus6minus00187 minus00777 minus01898 minus03974 minus08759 minus34322

3

2

1

00 2 4 6 8 10 12

kz

1205900 = 101205900 = 201205900 = 30

107timesSΛ

Figure 5 Sensitivity function 119878Λversus zonal wavenumber 119896

119911for

different nominal values of the static stability parameter 1205900(units

106m2 Paminus2 sminus2)

the static stability parameter 1205900except for 120590

0gt 2 times

10minus6m2 Paminus2 sminus2 for waves with 119896

119911ge 6 (see Table 2) Wher-

ever the midlatitude values in Table 2 are less than one thegrowth rate is more sensitive to the meridional temperaturegradient (ie Λ

120585) than the static stability (120590

0)

The obtained results are consistent with observations[50 51 54] an increase in static stability and a decrease of

the MTG have occurred over the past few decades in someareas of the SH which has led to a decrease in the growthrate of baroclinic unstable waves a shift of the spectrum ofunstable waves in the long wavelength part of spectrum andaweakened intensity of cyclogenesis Naturally these changesaffect favourable conditions for the development of baroclinicinstability and the essential features of weather patterns overlarge territories particularly over Australia

3 Planetary Scale Waves

To study the influence of the static stability parameter onthe dynamics of planetary scale (ultralong) waves a thinfilm approximation is applied This approximation employsa specific averaging technique over the vertical coordinate tothe system of primitive equations [60] As a result a two-dimensional set of equations can be obtained that describesthe dynamics of a two-dimensional baroclinic film Theseequations reproduce all the wavelike solutions that corre-spond to the main weather-forming modes of three-dimen-sional models and therefore can be used in theoretical studiesof large-scale dynamic processes in the atmosphere Thesystem of vertically averaged equations can be written as [60]

120597119906

120597119905

+ V sdot nabla119906 = 119891V minus119877

120587

120597

120597119909

(120587119879)

120597V120597119905

+ V sdot nablaV = minus119891119906 minus119877

120587

120597

120597119910

(120587119879)

120597120587

120597119905

+ nabla sdot (120587V) = 0

120597119879

120597119905

+ V sdot nabla119879 +

119877

119888119901

119879nabla sdot V = 0

(22)

where 120587 = 1199011199041198750 For instance if the original primitive equa-

tions are written in the Phillipsrsquo vertical coordinate system

Advances in Meteorology 7

120590 = 119901119901119904[64] the operator for vertical averaging is intro-

duced as 120595 = int

1

0120595119889120590 and state variables are represented as

120595 = 120595 + 1205951015840 Equations (22) are obtained by neglecting the

orography and terms 1199061015840V1015840 V1015840V1015840 and 1198791015840V1015840 [65] A detailedlinear analysis of the vertically averaged equation (22) isrepresented in [60] In particular two types of wave solutionswere found fast waves that propagate westward and slowwaves that move eastward Within the framework of thismodel ultralong waves are always neutral for any verticallyaveraged zonal wind velocity Indeed linearizing (22) aroundthe following basic state

1198790= 1198790(119910) 119906

0= minus(

119877

1198910

)

1205971198790

120597119910

V0= 0 120587

0= 1

(23)

and assuming the beta plane approximation 119891 = 1198910+ 120573119910

where 120573 = (2Ω1198860) cos120593

0 and representing the solution in

the form (9) one can finally obtain under different asymp-totics the following expressions for four wave solutions [60]

(a) acoustic waves

11988812= 1199060plusmnradic1198882

0+

1198912

0

1198962

(24)

(b) Rossby wave

1198883= 1199060minus

120573

1198962+ 1198912

01198882

0

(25)

(c) baroclinic wave

1198884= 1199060minus

1198912

01199062

0

1198882

0120573

(26)

Here 11988820= (1+120581)119877119879

0and 120581 = 119877119888

119901 These results however are

valid only for the specific case of a neutral atmosphere withΓ = Γ

119889[60 61] To take into account the atmospheric static

stability on the behaviour of ultralong waves the polytropicmodel of the atmosphere can be used for which

119879 (119909 119910 119911 119905) = 1198790(119909 119910 119905) minus Γ (119909 119910 119905) 119911 (27)

where 1198790is the temperature at the surface and Γ is a vertical

temperature gradient Integrating (27) with respect to verticalcoordinate we can obtain 119879

0= 119879(1 + 119877Γ119892) [60] Assuming

the geostrophic approximation on a 120573-plane the set ofvertically averaged equations can be written as [61]

120597119879

120597119905

+

119877119879

1198912

0

1205721

120587

(120587 119879) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

1

1205722

(119879 Γ)

+ 120581

1205731198772119879

2

1198921198912

0

1205724

1205722

2

120597Γ

120597119909

minus

120573119877119879

1198912

0

(1205723+ 120581

12057211205724

1205722

)

1

120587

120597 (120587119879)

120597119909

minus 120581

120573119877119879

1198912

0

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

12057211205724

1205722

120597119879

120597119909

= 0

120597Γ

120597119905

minus

119877119879

1198910

1205722

1

1205722

1

120587

(Γ 120587) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

119877

1198910

1205721

1205722

(119879 Γ)

minus 120581

120573119877119879

1198912

0

12057211205724

1205722

2

120597Γ

120597119909

minus 120581

120573119892

1198912

0

120572112057231205724

1205722

1

120587

120597 (120587119879)

120597119909

+ 120581

120573119892

1198912

0

1205722

11205724

1205722

120597119879

120597119909

= 0

120597120587

120597119905

minus

120573119877

1198912

0

120597 (120587119879)

120597119909

= 0

(28)

Here 1205721= 1 + 119877(Γ119892) 120572

2= 1 + 2119877(Γ119892) 120572

3= 119877(Γ119892) 120572

4=

1 minus (ΓΓ119889) and the operator (119860 119861) = (120597119860120597119909)(120597119861120597119910) minus (120597119860

120597119910)(120597119861120597119909) The basic state is defined as a stationary solutionof system (28) for which

120597119879

120597119909

= 0

120597Γ

120597119909

= 0

120597120587

120597119909

= 0 (29)

or in other words

119879 = 1198790(119910) Γ = Γ

0(119910) 120587 = 120587

0(119910) (30)

Linearizing (28) around the basic state (30) the followingcubic characteristic equation can be obtained in which thesecond order terms are neglected [61]

1198883+ 1198882

1205721

1205722

[120582 (1205722+ 3120581120572

4) minus 1199060]

+ 119888 120582

1205721

1205722

[1205821205811205724(2 + 120572

3+ 120581

1205724

1205722

(21205721minus

1

1205722

))

minus1199060(2 + 120581120572

4+ 31205723) ]

+ (120582

1205721

1205722

1199062

0minus 1205822120581

1205722

11205724

1205722

1199060+ 2120582312058121205722

112057231205722

4

1205723

2

)

= 0

(31)

where 120582 = 12057311987711987901198912

0and 119906

0= minus(119877119891

01205870)((120597(120587

01198790))120597119910) If

the discriminant of this equation is positive then the wavesolution is unstable The domain of zonal flow instabilitycan be found numerically (see diagram in [61]) In Figure 6we reproduce only for the 1st quadrant of a Cartesianplane the domain of instability calculated as a function ofvertically averaged zonal wind velocity 119906

0and dimensionless

temperature lapse rate ΓΓ119889

The imaginary part of phase velocity 119888119894which charac-

terises the growth rate of unstable waves 120594119896equiv 119896119888119894is displayed

in Figure 7 as a function of dimensionless temperature lapserate ΓΓ

119889for different values of vertically averaged zonal

wind velocity 1199060 A maximum phase velocity 119888

119894exists for

given values of 1199060 that is dependent on the ratio of ΓΓ

119889

For instance if 1199060= 20msminus1 then the maximum value

8 Advances in Meteorology

10

08

06

04

02

000 50 100 150 200 250 300

ΓΓd

u0 (m sminus1)

Figure 6 Domain of instability (filled) as a function of dimension-less temperature lapse rate ΓΓ

119889and vertically averaged zonal wind

velocity 1199060

(119888119894)max asymp 834msminus1 is reached at ΓΓ

119889asymp 055 Figure 6 shows

that increasing vertically averaged zonal wind 1199060is associated

with increasing 119888119894 This is further evident in Figure 8 which

shows 119888119894as a function of 119906

0for a range of ΓΓ

119889values The

lower ΓΓ119889

and the larger 119888119894 that is 119888

119894 increases with

decreasing static stability

4 Concluding Remarks

We have studied theoretically the impact of variations in thestatic stability parameter 120590

0and zonal wind shear Λ

120585on the

characteristics of baroclinically unstable waves of synopticscales using Eady-type model with the uniform Λ

120585between

upper and lower boundaries on an 119891-plane Quantitativeestimates of variations in 120590

0and Λ

120585on the growth rate 120594

119896

wavelength of maximum growth rate 119871120594max and short-wave

cutoff 119871min were obtainedAnalytical expressions are derived for sensitivity func-

tions for the growth rate 120594119896with respect to variations in static

stability parameter andwind shear velocityThese expressionsallow estimating to a first-order approximation the influenceof changes in 120590

0and Λ

120585on 120594119896 Analytical expressions for

relative sensitivity functions allow estimating the significanceof variations in 120590

0andΛ

120585on the growth rate of baroclinically

unstable waves with a given zonal wave numberTo study the impact of variations in atmospheric static

stability and zonal wind velocity on the instability of plan-etary scale waves the model with vertically averaged prim-itive equations with 120573-plane approximation was applied Ascontrol parameters we have used dimensionless temperature

15

10

5

0minus02 00 02 04 06 08 10

ΓΓd

u0 = 20u0 = 30u0 = 40

ci

(m sminus

1)

Figure 7 Imaginary part of phase speed 119888119894versus dimensionless

temperature lapse rate ΓΓ119889for different values of vertically averaged

zonal wind velocity 1199060

lapse rate ΓΓ119889and vertically averaged zonal wind velocity

1199060 We have estimated the influence of ΓΓ

119889and 119906

0on the

imaginary part of phase speed 119888119894 whichwas used as ameasure

of instabilityThe obtained results are qualitatively consistent with

changes in the essential weather patterns that occurred overthe last several decades in some areas of the SH and inparticular over Australia (eg [49 50 52ndash54]) Climatechange results suggest SH midlatitude static stability 120590

0may

increase and the MTG (the vertical wind shear Λ120585) may

decrease which according to our linear theoretical modelsleads to a slowing of the growth rate of baroclinic unstablewaves 120594

119896and an increasing wavelength of baroclinic unstable

wave with maximum growth rate 119871120594max that is a spectrum

shift of unstable waves towards longer wavelengths Thesemight affect the favourable conditions for the developmentof baroclinic instability and therefore the rate of cyclogenesisand a reduction in cyclone intensity The obtained sensitivityfunctions demonstrate that waves belonging to the short-wave part of the spectrum of unstable waves are moresensitive to changes in the static stability parameter thanwaves belonging to the long-wave part of the spectrum

To obtain more realistic estimates of the sensitivity of thegrowth rate of unstable waves with respect to static stabilityparameter and MTG numerical modeling based on a fullGCM is required It is hoped to carry out such work in thefuture

Advances in Meteorology 9

ΓΓd = 01ΓΓd = 04

ΓΓd = 23ΓΓd = 09

20

10

00 20 40 60

u0 (mmiddotsminus1)

ci(m

sminus1)

Figure 8 Imaginary part of phase speed 119888119894versus vertically

averaged zonal wind velocity 1199060for different values of dimensionless

temperature lapse rate ΓΓ119889

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors thank Dr I Pisnichenko for clarification ofcertain questions relevant to this paper

References

[1] J R Holton An Introduction to Dynamic Meteorology Aca-demic Press 3rd edition 1992

[2] G K Vallis Atmospheric and Oceanic Fluid Dynamics Cam-bridge University Press 2006

[3] M Mak Atmospheric Dynamics Cambridge University Press2011

[4] V P Dymnikov Stability and Predictability of the Large ScaleAtmospheric Processes Institute of Numerical Mathematics ofthe Russian Academy of Sciences Moscow Russia 2007

[5] J G Charney ldquoThe dynamics of long waves in baroclinicwesterly currentrdquo Journal of Meteorology vol 4 pp 135ndash1621947

[6] E T Eady ldquoLong waves and cyclone wavesrdquo Tellus vol 1 pp33ndash52 1949

[7] R T Pierrehumbert and K L Swanson ldquoBaroclinic instabilityrdquoAnnual Review of Fluid Mechanics vol 27 no 1 pp 419ndash4671995

[8] R Gall ldquoA comparison of linear instability theory with theeddy statistics of a general circulation modelrdquo Journal of theAtmospheric Sciences vol 33 no 3 pp 349ndash373 1976

[9] I M Held ldquoProgress and problems in large-scale atmosphericdynamicsrdquo in The Global Circulation of the Atmosphere TSchneider and A Sobel Eds pp 1ndash21 Princeton UniversityPress Princeton NJ USA 2007

[10] B Farrell ldquoModal and non-modal baroclinic wavesrdquo Journal ofthe Atmospheric Sciences vol 41 no 4 pp 668ndash673 1984

[11] M K Tippett ldquoTransient moist baroclinic instabilityrdquo Tellus Avol 51 no 2 pp 273ndash288 1999

[12] D Hodyss and R Grotjahn ldquoNonmodal and unstable normalmode baroclinic growth as a function of horizontal scalerdquoDynamics of Atmospheres and Oceans vol 37 no 1 pp 1ndash242003

[13] J Pedlosky ldquoFinite amplitude baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 27 no 1 pp 15ndash30 1970

[14] R D Romea ldquoThe effects of friction and beta on finite ampli-tude baroclinic wavesrdquo Journal of the Atmospheric Sciences vol34 pp 1689ndash1695 1977

[15] J Pedlosky ldquoFinite-amplitude baroclinic waves at minimumcritical shearrdquo Journal of the Atmospheric Sciences vol 39 no3 pp 555ndash562 1982

[16] T Warn and P Gauthier ldquoPotential vorticity mixing bymarginally unstable baroclinic disturbancesrdquo Tellus A vol 41no 2 pp 115ndash131 1989

[17] A J Simmons and B J Hoskins ldquoThe lifecycles of some non-linear wavesrdquo Journal of the Atmospheric Sciences vol 35 pp414ndash432 1978

[18] A J Simmons and B J Hoskins ldquoBarotropic influences of thegrowth and decay of nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 37 pp 1679ndash1684 1980

[19] S B Feldstein and I M Held ldquoBarotropic decay of baroclinicwavesrdquo Journal of the Atmospheric Sciences vol 46 pp 1679ndash1684 1989

[20] C D Thorncroft B J Hoskins and M E McIntyre ldquoTwoparadigms of baroclinic-wave life-cycle behaviourrdquo QuaterlyJournal vol 119 no 509 pp 17ndash55 1993

[21] N A Phillips ldquoA simple three-dimensional model for the studyof largescale extratropical flow patternsrdquo Journal ofMeteorologyvol 8 pp 381ndash394 1951

[22] N A Phillips ldquoEnergy transformations and meridional circu-lations associated with simple baroclinic waves in a two-levelquasi-geostrophic modelrdquo Tellus vol 6 pp 273ndash286 1954

[23] L M Polvani and J Pedlosky ldquoThe effect of dissipation onspatially growing nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 45 no 14 pp 1977ndash1989 1988

[24] T G Shepherd ldquoNonlinear saturation of baroclinic instabilityPart I the two-layermodelrdquo Journal of the Atmospheric Sciencesvol 45 no 14 pp 2014ndash2025 1988

[25] G Balasubramanian and M K Yau ldquoBaroclinic instability ina two-layer model with parameterized slantwise convectionrdquoJournal of the Atmospheric Sciences vol 51 no 7 pp 971ndash9901994

[26] A Wiin-Nielsen ldquoOn the structure of atmospheric waves inmiddle latitudesrdquo Atmosfera vol 16 no 2 pp 83ndash102 2003

[27] D D Holm and B A Wingate ldquoBaroclinic instabilities of thetwo-layer quasigeostrophic alpha modelrdquo Journal of PhysicalOceanography vol 35 no 7 pp 1287ndash1296 2005

10 Advances in Meteorology

[28] J Egger ldquoBaroclinic instability in the two-layermodel interpre-tationsrdquo Meteorologische Zeitschrift vol 18 no 5 pp 559ndash5652009

[29] J Lamberts G Lapeyre and V Zeitlin ldquoMoist versus dry baro-clinic instability in a simplified two-layer atmospheric modelwith condensation and latent heat releaserdquo Journal of theAtmospheric Sciences vol 69 pp 1405ndash1426 2012

[30] D M W Frierson ldquoRobust increases in midlatitude static sta-bility in simulations of global warmingrdquo Geophysical ResearchLetters vol 33 no 24 Article ID L24816 2006

[31] N M J Hall B J Hoskins P J Valdes and C A Senior ldquoStormtracks in a high-resolution GCMwith doubled carbon dioxiderdquoQuarterly Journal vol 120 no 519 pp 1209ndash1230 1994

[32] S B Feldstein ldquoThe recent trend and variance increase of theannularmoderdquo Journal of Climate vol 15 no 3 pp 88ndash94 2002

[33] G J Marshall ldquoTrends in the southern annular mode fromobservations and reanalysesrdquo Journal of Climate vol 16 pp4134ndash4143 2003

[34] G M Ostermeier and J M Wallace ldquoTrends in the NorthAtlantic Oscillation-Northern Hemisphere annular mode dur-ing the twentieth centuryrdquo Journal of Climate vol 16 no 2 pp336ndash341 2003

[35] D W J Thompson and S Solomon ldquoInterpretation of recentSouthern Hemisphere climate changerdquo Science vol 296 no5569 pp 895ndash899 2002

[36] S J Lambert ldquoThe effect of enhanced greenhouse warming onwinter cyclone frequencies and strengthsrdquo Journal of Climatevol 8 no 5 pp 1447ndash1462 1995

[37] F Lunkeit K Fraedrich and S E Bauer ldquoStorm tracks ina warmer climate sensitivity studies with a simplified globalcirculation modelrdquo Climate Dynamics vol 14 no 11 pp 813ndash826 1998

[38] D J Lorenz and D L Hartmann ldquoEddy-zonal flow feedback inthe Southern Hemisphererdquo Journal of the Atmospheric Sciencesvol 58 no 21 pp 3312ndash3327 2001

[39] Q Geng and M Sugi ldquoPossible change of extratropical cycloneactivity due to enhanced greenhouse gases and sulphateaerosols study with a high-resolution AGCMrdquo Journal ofClimate vol 16 pp 2262ndash2274 2003

[40] J C Fyfe ldquoExtratropical Southern Hemisphere cyclonesharbingers of climate changerdquo Journal of Climate vol 16 pp2802ndash2805 2003

[41] J H Yin ldquoA consistent poleward shift of the storm tracksin simulations of 21st century climaterdquo Geophysical ResearchLetters vol 32 no 18 Article ID L18701 pp 1ndash4 2005

[42] S J Lambert and J C Fyfe ldquoChanges in winter cyclonefrequencies and strengths simulated in enhanced greenhousewarming experiments results from the models participating inthe IPCC diagnostic exerciserdquo Climate Dynamics vol 26 no7-8 pp 713ndash728 2006

[43] G Gastineau and B J Soden ldquoModel projected changes ofextreme wind events in response to global warmingrdquo Geophys-ical Research Letters vol 36 no 10 Article ID L10810 2009

[44] Y Wu M Ting R Seager H-P Huang and M A CaneldquoChanges in storm tracks and energy transports in a warmer cli-mate simulated by the GFDL CM21 modelrdquo Climate Dynamicsvol 37 no 1 pp 53ndash72 2011

[45] L Bengtsson K I Hodges and E Roeckner ldquoStorm tracks andclimate changerdquo Journal of Climate vol 19 no 15 pp 3518ndash35432006

[46] I Smith ldquoAn assessment of recent trends in Australian rainfallrdquoAustralian Meteorological Magazine vol 53 no 3 pp 163ndash1732004

[47] N Nicholls Detecting Understanding and Attributing ClimateChange Australian Greenhouse Office Publication 2007

[48] B C Bates P Hope B Ryan I Smith and S Charles ldquoKey find-ings from the Indian Ocean Climate Initiative and their impacton policy development in Australiardquo Climatic Change vol89 no 3-4 pp 339ndash354 2008

[49] J S Frederiksen and C S Frederiksen ldquoDecadal changes inSouthern Hemisphere winter cyclogenesisrdquo CSIROMarine andAtmospheric Research Paper 002 2005

[50] J S Frederiksen and C S Frederiksen ldquoInterdecadal changes insouthern hemisphere winter storm track modesrdquo Tellus A vol59 no 5 pp 599ndash617 2007

[51] J S Frederiksen C S Frederiksen and S L Osbrough ldquoMod-elling of changes in Southern Hemisphere weather systemsduring the 20th centuryrdquo in Proceedings of the 18th WorldIMACS Congress and International Congress on Modelling andSimulation pp 2562ndash2568 Cairns Australia July 2009

[52] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoCauses of changing Southern Hemispheric weathersystemsrdquo in Managing Climate Change I Jupp P Holper andW Cai Eds pp 85ndash98 CSIRO Publishing 2010

[53] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoChanges in Southern Hemisphere rainfall circulationand weather systemsrdquo in Proceedings of the 19th InternationalCongress on Modelling and Simulation (MODSIM rsquo11) pp 2712ndash2718 Perth Australia December 2011

[54] C S Frederiksen J S Frederiksen J M Sisson and S LOsbrough ldquoObserved and projected changes in the annual cycleof Southern Hemisphere baroclinicity for storm formationrdquoin Proceedings of the 19th International Congress on Modellingand Simulation (MODSIMrsquo 11) pp 2719ndash2725 Perth AustraliaDecember 2011

[55] N A Phillips ldquoGeostrophicmotionrdquoReviews of Geophysics vol1 no 2 pp 123ndash176 1963

[56] I Bordi K Fraedrich F Lunkeit and A Sutera ldquoTroposphericdouble jets meridional cells and eddies a case study andidealized simulationsrdquoMonthly Weather Review vol 135 no 9pp 3118ndash3133 2007

[57] I Bordi K Fraedrich M Ghil and A Sutera ldquoZonal flowregime changes in a GCM and in a simple quasigeostrophicmodel the role of stratospheric dynamicsrdquo Journal of theAtmospheric Sciences vol 66 no 5 pp 1366ndash1383 2009

[58] J G Charney ldquoOn the scale of atmosphericmotionsrdquoGeofysiskePublikasjoner vol 17 no 2 pp 3ndash17 1948

[59] A Burger ldquoScale consideration of planetary motions of theatmosphererdquo Tellus vol 10 pp 195ndash205 1958

[60] I A Pisnichenko ldquoUltralong-wave dynamics in a two-dimen-sional baroclinic atmosphere modelrdquo Atmospheric and OceanicPhysics vol 16 no 9 pp 883ndash892 1980

[61] I A Pisnichenko ldquoInfluence variable static stability on thedynamics of ultralong waves in two-dimensional baroclinicmodel of the atmosphererdquoAtmospheric andOceanic Physics vol19 no 11 pp 1223ndash1226 1983

[62] A Wiin-Nielsen ldquoOn barotropic and baroclinic models withspecial emphasis onultra-longwavesrdquoMonthlyWeather Reviewvol 87 pp 171ndash183 1959

[63] B W BinWang and A Barcilon ldquoMoist stability of a barocliniczonal flow with conditionally unstable stratificationrdquo Journal ofthe Atmospheric Sciences vol 43 no 7 pp 705ndash719 1986

Advances in Meteorology 11

[64] N A Phillips ldquoA coordinate system having some specialadvantages for numerical forecastingrdquo Journal of Meteorologyvol 14 pp 184ndash185 1957

[65] D M Alishaev ldquoDynamics of a two-dimensional baroclinicatmosphererdquoAtmospheric and Oceanic Physics vol 16 no 2 pp99ndash107 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 7

120590 = 119901119901119904[64] the operator for vertical averaging is intro-

duced as 120595 = int

1

0120595119889120590 and state variables are represented as

120595 = 120595 + 1205951015840 Equations (22) are obtained by neglecting the

orography and terms 1199061015840V1015840 V1015840V1015840 and 1198791015840V1015840 [65] A detailedlinear analysis of the vertically averaged equation (22) isrepresented in [60] In particular two types of wave solutionswere found fast waves that propagate westward and slowwaves that move eastward Within the framework of thismodel ultralong waves are always neutral for any verticallyaveraged zonal wind velocity Indeed linearizing (22) aroundthe following basic state

1198790= 1198790(119910) 119906

0= minus(

119877

1198910

)

1205971198790

120597119910

V0= 0 120587

0= 1

(23)

and assuming the beta plane approximation 119891 = 1198910+ 120573119910

where 120573 = (2Ω1198860) cos120593

0 and representing the solution in

the form (9) one can finally obtain under different asymp-totics the following expressions for four wave solutions [60]

(a) acoustic waves

11988812= 1199060plusmnradic1198882

0+

1198912

0

1198962

(24)

(b) Rossby wave

1198883= 1199060minus

120573

1198962+ 1198912

01198882

0

(25)

(c) baroclinic wave

1198884= 1199060minus

1198912

01199062

0

1198882

0120573

(26)

Here 11988820= (1+120581)119877119879

0and 120581 = 119877119888

119901 These results however are

valid only for the specific case of a neutral atmosphere withΓ = Γ

119889[60 61] To take into account the atmospheric static

stability on the behaviour of ultralong waves the polytropicmodel of the atmosphere can be used for which

119879 (119909 119910 119911 119905) = 1198790(119909 119910 119905) minus Γ (119909 119910 119905) 119911 (27)

where 1198790is the temperature at the surface and Γ is a vertical

temperature gradient Integrating (27) with respect to verticalcoordinate we can obtain 119879

0= 119879(1 + 119877Γ119892) [60] Assuming

the geostrophic approximation on a 120573-plane the set ofvertically averaged equations can be written as [61]

120597119879

120597119905

+

119877119879

1198912

0

1205721

120587

(120587 119879) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

1

1205722

(119879 Γ)

+ 120581

1205731198772119879

2

1198921198912

0

1205724

1205722

2

120597Γ

120597119909

minus

120573119877119879

1198912

0

(1205723+ 120581

12057211205724

1205722

)

1

120587

120597 (120587119879)

120597119909

minus 120581

120573119877119879

1198912

0

1205723

1205722120587

(120587 Γ) minus

1198772119879

1198921198910

12057211205724

1205722

120597119879

120597119909

= 0

120597Γ

120597119905

minus

119877119879

1198910

1205722

1

1205722

1

120587

(Γ 120587) +

1198772119879

2

1198921198910

1205723

1205722120587

(120587 Γ) minus

119877

1198910

1205721

1205722

(119879 Γ)

minus 120581

120573119877119879

1198912

0

12057211205724

1205722

2

120597Γ

120597119909

minus 120581

120573119892

1198912

0

120572112057231205724

1205722

1

120587

120597 (120587119879)

120597119909

+ 120581

120573119892

1198912

0

1205722

11205724

1205722

120597119879

120597119909

= 0

120597120587

120597119905

minus

120573119877

1198912

0

120597 (120587119879)

120597119909

= 0

(28)

Here 1205721= 1 + 119877(Γ119892) 120572

2= 1 + 2119877(Γ119892) 120572

3= 119877(Γ119892) 120572

4=

1 minus (ΓΓ119889) and the operator (119860 119861) = (120597119860120597119909)(120597119861120597119910) minus (120597119860

120597119910)(120597119861120597119909) The basic state is defined as a stationary solutionof system (28) for which

120597119879

120597119909

= 0

120597Γ

120597119909

= 0

120597120587

120597119909

= 0 (29)

or in other words

119879 = 1198790(119910) Γ = Γ

0(119910) 120587 = 120587

0(119910) (30)

Linearizing (28) around the basic state (30) the followingcubic characteristic equation can be obtained in which thesecond order terms are neglected [61]

1198883+ 1198882

1205721

1205722

[120582 (1205722+ 3120581120572

4) minus 1199060]

+ 119888 120582

1205721

1205722

[1205821205811205724(2 + 120572

3+ 120581

1205724

1205722

(21205721minus

1

1205722

))

minus1199060(2 + 120581120572

4+ 31205723) ]

+ (120582

1205721

1205722

1199062

0minus 1205822120581

1205722

11205724

1205722

1199060+ 2120582312058121205722

112057231205722

4

1205723

2

)

= 0

(31)

where 120582 = 12057311987711987901198912

0and 119906

0= minus(119877119891

01205870)((120597(120587

01198790))120597119910) If

the discriminant of this equation is positive then the wavesolution is unstable The domain of zonal flow instabilitycan be found numerically (see diagram in [61]) In Figure 6we reproduce only for the 1st quadrant of a Cartesianplane the domain of instability calculated as a function ofvertically averaged zonal wind velocity 119906

0and dimensionless

temperature lapse rate ΓΓ119889

The imaginary part of phase velocity 119888119894which charac-

terises the growth rate of unstable waves 120594119896equiv 119896119888119894is displayed

in Figure 7 as a function of dimensionless temperature lapserate ΓΓ

119889for different values of vertically averaged zonal

wind velocity 1199060 A maximum phase velocity 119888

119894exists for

given values of 1199060 that is dependent on the ratio of ΓΓ

119889

For instance if 1199060= 20msminus1 then the maximum value

8 Advances in Meteorology

10

08

06

04

02

000 50 100 150 200 250 300

ΓΓd

u0 (m sminus1)

Figure 6 Domain of instability (filled) as a function of dimension-less temperature lapse rate ΓΓ

119889and vertically averaged zonal wind

velocity 1199060

(119888119894)max asymp 834msminus1 is reached at ΓΓ

119889asymp 055 Figure 6 shows

that increasing vertically averaged zonal wind 1199060is associated

with increasing 119888119894 This is further evident in Figure 8 which

shows 119888119894as a function of 119906

0for a range of ΓΓ

119889values The

lower ΓΓ119889

and the larger 119888119894 that is 119888

119894 increases with

decreasing static stability

4 Concluding Remarks

We have studied theoretically the impact of variations in thestatic stability parameter 120590

0and zonal wind shear Λ

120585on the

characteristics of baroclinically unstable waves of synopticscales using Eady-type model with the uniform Λ

120585between

upper and lower boundaries on an 119891-plane Quantitativeestimates of variations in 120590

0and Λ

120585on the growth rate 120594

119896

wavelength of maximum growth rate 119871120594max and short-wave

cutoff 119871min were obtainedAnalytical expressions are derived for sensitivity func-

tions for the growth rate 120594119896with respect to variations in static

stability parameter andwind shear velocityThese expressionsallow estimating to a first-order approximation the influenceof changes in 120590

0and Λ

120585on 120594119896 Analytical expressions for

relative sensitivity functions allow estimating the significanceof variations in 120590

0andΛ

120585on the growth rate of baroclinically

unstable waves with a given zonal wave numberTo study the impact of variations in atmospheric static

stability and zonal wind velocity on the instability of plan-etary scale waves the model with vertically averaged prim-itive equations with 120573-plane approximation was applied Ascontrol parameters we have used dimensionless temperature

15

10

5

0minus02 00 02 04 06 08 10

ΓΓd

u0 = 20u0 = 30u0 = 40

ci

(m sminus

1)

Figure 7 Imaginary part of phase speed 119888119894versus dimensionless

temperature lapse rate ΓΓ119889for different values of vertically averaged

zonal wind velocity 1199060

lapse rate ΓΓ119889and vertically averaged zonal wind velocity

1199060 We have estimated the influence of ΓΓ

119889and 119906

0on the

imaginary part of phase speed 119888119894 whichwas used as ameasure

of instabilityThe obtained results are qualitatively consistent with

changes in the essential weather patterns that occurred overthe last several decades in some areas of the SH and inparticular over Australia (eg [49 50 52ndash54]) Climatechange results suggest SH midlatitude static stability 120590

0may

increase and the MTG (the vertical wind shear Λ120585) may

decrease which according to our linear theoretical modelsleads to a slowing of the growth rate of baroclinic unstablewaves 120594

119896and an increasing wavelength of baroclinic unstable

wave with maximum growth rate 119871120594max that is a spectrum

shift of unstable waves towards longer wavelengths Thesemight affect the favourable conditions for the developmentof baroclinic instability and therefore the rate of cyclogenesisand a reduction in cyclone intensity The obtained sensitivityfunctions demonstrate that waves belonging to the short-wave part of the spectrum of unstable waves are moresensitive to changes in the static stability parameter thanwaves belonging to the long-wave part of the spectrum

To obtain more realistic estimates of the sensitivity of thegrowth rate of unstable waves with respect to static stabilityparameter and MTG numerical modeling based on a fullGCM is required It is hoped to carry out such work in thefuture

Advances in Meteorology 9

ΓΓd = 01ΓΓd = 04

ΓΓd = 23ΓΓd = 09

20

10

00 20 40 60

u0 (mmiddotsminus1)

ci(m

sminus1)

Figure 8 Imaginary part of phase speed 119888119894versus vertically

averaged zonal wind velocity 1199060for different values of dimensionless

temperature lapse rate ΓΓ119889

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors thank Dr I Pisnichenko for clarification ofcertain questions relevant to this paper

References

[1] J R Holton An Introduction to Dynamic Meteorology Aca-demic Press 3rd edition 1992

[2] G K Vallis Atmospheric and Oceanic Fluid Dynamics Cam-bridge University Press 2006

[3] M Mak Atmospheric Dynamics Cambridge University Press2011

[4] V P Dymnikov Stability and Predictability of the Large ScaleAtmospheric Processes Institute of Numerical Mathematics ofthe Russian Academy of Sciences Moscow Russia 2007

[5] J G Charney ldquoThe dynamics of long waves in baroclinicwesterly currentrdquo Journal of Meteorology vol 4 pp 135ndash1621947

[6] E T Eady ldquoLong waves and cyclone wavesrdquo Tellus vol 1 pp33ndash52 1949

[7] R T Pierrehumbert and K L Swanson ldquoBaroclinic instabilityrdquoAnnual Review of Fluid Mechanics vol 27 no 1 pp 419ndash4671995

[8] R Gall ldquoA comparison of linear instability theory with theeddy statistics of a general circulation modelrdquo Journal of theAtmospheric Sciences vol 33 no 3 pp 349ndash373 1976

[9] I M Held ldquoProgress and problems in large-scale atmosphericdynamicsrdquo in The Global Circulation of the Atmosphere TSchneider and A Sobel Eds pp 1ndash21 Princeton UniversityPress Princeton NJ USA 2007

[10] B Farrell ldquoModal and non-modal baroclinic wavesrdquo Journal ofthe Atmospheric Sciences vol 41 no 4 pp 668ndash673 1984

[11] M K Tippett ldquoTransient moist baroclinic instabilityrdquo Tellus Avol 51 no 2 pp 273ndash288 1999

[12] D Hodyss and R Grotjahn ldquoNonmodal and unstable normalmode baroclinic growth as a function of horizontal scalerdquoDynamics of Atmospheres and Oceans vol 37 no 1 pp 1ndash242003

[13] J Pedlosky ldquoFinite amplitude baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 27 no 1 pp 15ndash30 1970

[14] R D Romea ldquoThe effects of friction and beta on finite ampli-tude baroclinic wavesrdquo Journal of the Atmospheric Sciences vol34 pp 1689ndash1695 1977

[15] J Pedlosky ldquoFinite-amplitude baroclinic waves at minimumcritical shearrdquo Journal of the Atmospheric Sciences vol 39 no3 pp 555ndash562 1982

[16] T Warn and P Gauthier ldquoPotential vorticity mixing bymarginally unstable baroclinic disturbancesrdquo Tellus A vol 41no 2 pp 115ndash131 1989

[17] A J Simmons and B J Hoskins ldquoThe lifecycles of some non-linear wavesrdquo Journal of the Atmospheric Sciences vol 35 pp414ndash432 1978

[18] A J Simmons and B J Hoskins ldquoBarotropic influences of thegrowth and decay of nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 37 pp 1679ndash1684 1980

[19] S B Feldstein and I M Held ldquoBarotropic decay of baroclinicwavesrdquo Journal of the Atmospheric Sciences vol 46 pp 1679ndash1684 1989

[20] C D Thorncroft B J Hoskins and M E McIntyre ldquoTwoparadigms of baroclinic-wave life-cycle behaviourrdquo QuaterlyJournal vol 119 no 509 pp 17ndash55 1993

[21] N A Phillips ldquoA simple three-dimensional model for the studyof largescale extratropical flow patternsrdquo Journal ofMeteorologyvol 8 pp 381ndash394 1951

[22] N A Phillips ldquoEnergy transformations and meridional circu-lations associated with simple baroclinic waves in a two-levelquasi-geostrophic modelrdquo Tellus vol 6 pp 273ndash286 1954

[23] L M Polvani and J Pedlosky ldquoThe effect of dissipation onspatially growing nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 45 no 14 pp 1977ndash1989 1988

[24] T G Shepherd ldquoNonlinear saturation of baroclinic instabilityPart I the two-layermodelrdquo Journal of the Atmospheric Sciencesvol 45 no 14 pp 2014ndash2025 1988

[25] G Balasubramanian and M K Yau ldquoBaroclinic instability ina two-layer model with parameterized slantwise convectionrdquoJournal of the Atmospheric Sciences vol 51 no 7 pp 971ndash9901994

[26] A Wiin-Nielsen ldquoOn the structure of atmospheric waves inmiddle latitudesrdquo Atmosfera vol 16 no 2 pp 83ndash102 2003

[27] D D Holm and B A Wingate ldquoBaroclinic instabilities of thetwo-layer quasigeostrophic alpha modelrdquo Journal of PhysicalOceanography vol 35 no 7 pp 1287ndash1296 2005

10 Advances in Meteorology

[28] J Egger ldquoBaroclinic instability in the two-layermodel interpre-tationsrdquo Meteorologische Zeitschrift vol 18 no 5 pp 559ndash5652009

[29] J Lamberts G Lapeyre and V Zeitlin ldquoMoist versus dry baro-clinic instability in a simplified two-layer atmospheric modelwith condensation and latent heat releaserdquo Journal of theAtmospheric Sciences vol 69 pp 1405ndash1426 2012

[30] D M W Frierson ldquoRobust increases in midlatitude static sta-bility in simulations of global warmingrdquo Geophysical ResearchLetters vol 33 no 24 Article ID L24816 2006

[31] N M J Hall B J Hoskins P J Valdes and C A Senior ldquoStormtracks in a high-resolution GCMwith doubled carbon dioxiderdquoQuarterly Journal vol 120 no 519 pp 1209ndash1230 1994

[32] S B Feldstein ldquoThe recent trend and variance increase of theannularmoderdquo Journal of Climate vol 15 no 3 pp 88ndash94 2002

[33] G J Marshall ldquoTrends in the southern annular mode fromobservations and reanalysesrdquo Journal of Climate vol 16 pp4134ndash4143 2003

[34] G M Ostermeier and J M Wallace ldquoTrends in the NorthAtlantic Oscillation-Northern Hemisphere annular mode dur-ing the twentieth centuryrdquo Journal of Climate vol 16 no 2 pp336ndash341 2003

[35] D W J Thompson and S Solomon ldquoInterpretation of recentSouthern Hemisphere climate changerdquo Science vol 296 no5569 pp 895ndash899 2002

[36] S J Lambert ldquoThe effect of enhanced greenhouse warming onwinter cyclone frequencies and strengthsrdquo Journal of Climatevol 8 no 5 pp 1447ndash1462 1995

[37] F Lunkeit K Fraedrich and S E Bauer ldquoStorm tracks ina warmer climate sensitivity studies with a simplified globalcirculation modelrdquo Climate Dynamics vol 14 no 11 pp 813ndash826 1998

[38] D J Lorenz and D L Hartmann ldquoEddy-zonal flow feedback inthe Southern Hemisphererdquo Journal of the Atmospheric Sciencesvol 58 no 21 pp 3312ndash3327 2001

[39] Q Geng and M Sugi ldquoPossible change of extratropical cycloneactivity due to enhanced greenhouse gases and sulphateaerosols study with a high-resolution AGCMrdquo Journal ofClimate vol 16 pp 2262ndash2274 2003

[40] J C Fyfe ldquoExtratropical Southern Hemisphere cyclonesharbingers of climate changerdquo Journal of Climate vol 16 pp2802ndash2805 2003

[41] J H Yin ldquoA consistent poleward shift of the storm tracksin simulations of 21st century climaterdquo Geophysical ResearchLetters vol 32 no 18 Article ID L18701 pp 1ndash4 2005

[42] S J Lambert and J C Fyfe ldquoChanges in winter cyclonefrequencies and strengths simulated in enhanced greenhousewarming experiments results from the models participating inthe IPCC diagnostic exerciserdquo Climate Dynamics vol 26 no7-8 pp 713ndash728 2006

[43] G Gastineau and B J Soden ldquoModel projected changes ofextreme wind events in response to global warmingrdquo Geophys-ical Research Letters vol 36 no 10 Article ID L10810 2009

[44] Y Wu M Ting R Seager H-P Huang and M A CaneldquoChanges in storm tracks and energy transports in a warmer cli-mate simulated by the GFDL CM21 modelrdquo Climate Dynamicsvol 37 no 1 pp 53ndash72 2011

[45] L Bengtsson K I Hodges and E Roeckner ldquoStorm tracks andclimate changerdquo Journal of Climate vol 19 no 15 pp 3518ndash35432006

[46] I Smith ldquoAn assessment of recent trends in Australian rainfallrdquoAustralian Meteorological Magazine vol 53 no 3 pp 163ndash1732004

[47] N Nicholls Detecting Understanding and Attributing ClimateChange Australian Greenhouse Office Publication 2007

[48] B C Bates P Hope B Ryan I Smith and S Charles ldquoKey find-ings from the Indian Ocean Climate Initiative and their impacton policy development in Australiardquo Climatic Change vol89 no 3-4 pp 339ndash354 2008

[49] J S Frederiksen and C S Frederiksen ldquoDecadal changes inSouthern Hemisphere winter cyclogenesisrdquo CSIROMarine andAtmospheric Research Paper 002 2005

[50] J S Frederiksen and C S Frederiksen ldquoInterdecadal changes insouthern hemisphere winter storm track modesrdquo Tellus A vol59 no 5 pp 599ndash617 2007

[51] J S Frederiksen C S Frederiksen and S L Osbrough ldquoMod-elling of changes in Southern Hemisphere weather systemsduring the 20th centuryrdquo in Proceedings of the 18th WorldIMACS Congress and International Congress on Modelling andSimulation pp 2562ndash2568 Cairns Australia July 2009

[52] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoCauses of changing Southern Hemispheric weathersystemsrdquo in Managing Climate Change I Jupp P Holper andW Cai Eds pp 85ndash98 CSIRO Publishing 2010

[53] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoChanges in Southern Hemisphere rainfall circulationand weather systemsrdquo in Proceedings of the 19th InternationalCongress on Modelling and Simulation (MODSIM rsquo11) pp 2712ndash2718 Perth Australia December 2011

[54] C S Frederiksen J S Frederiksen J M Sisson and S LOsbrough ldquoObserved and projected changes in the annual cycleof Southern Hemisphere baroclinicity for storm formationrdquoin Proceedings of the 19th International Congress on Modellingand Simulation (MODSIMrsquo 11) pp 2719ndash2725 Perth AustraliaDecember 2011

[55] N A Phillips ldquoGeostrophicmotionrdquoReviews of Geophysics vol1 no 2 pp 123ndash176 1963

[56] I Bordi K Fraedrich F Lunkeit and A Sutera ldquoTroposphericdouble jets meridional cells and eddies a case study andidealized simulationsrdquoMonthly Weather Review vol 135 no 9pp 3118ndash3133 2007

[57] I Bordi K Fraedrich M Ghil and A Sutera ldquoZonal flowregime changes in a GCM and in a simple quasigeostrophicmodel the role of stratospheric dynamicsrdquo Journal of theAtmospheric Sciences vol 66 no 5 pp 1366ndash1383 2009

[58] J G Charney ldquoOn the scale of atmosphericmotionsrdquoGeofysiskePublikasjoner vol 17 no 2 pp 3ndash17 1948

[59] A Burger ldquoScale consideration of planetary motions of theatmosphererdquo Tellus vol 10 pp 195ndash205 1958

[60] I A Pisnichenko ldquoUltralong-wave dynamics in a two-dimen-sional baroclinic atmosphere modelrdquo Atmospheric and OceanicPhysics vol 16 no 9 pp 883ndash892 1980

[61] I A Pisnichenko ldquoInfluence variable static stability on thedynamics of ultralong waves in two-dimensional baroclinicmodel of the atmosphererdquoAtmospheric andOceanic Physics vol19 no 11 pp 1223ndash1226 1983

[62] A Wiin-Nielsen ldquoOn barotropic and baroclinic models withspecial emphasis onultra-longwavesrdquoMonthlyWeather Reviewvol 87 pp 171ndash183 1959

[63] B W BinWang and A Barcilon ldquoMoist stability of a barocliniczonal flow with conditionally unstable stratificationrdquo Journal ofthe Atmospheric Sciences vol 43 no 7 pp 705ndash719 1986

Advances in Meteorology 11

[64] N A Phillips ldquoA coordinate system having some specialadvantages for numerical forecastingrdquo Journal of Meteorologyvol 14 pp 184ndash185 1957

[65] D M Alishaev ldquoDynamics of a two-dimensional baroclinicatmosphererdquoAtmospheric and Oceanic Physics vol 16 no 2 pp99ndash107 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

8 Advances in Meteorology

10

08

06

04

02

000 50 100 150 200 250 300

ΓΓd

u0 (m sminus1)

Figure 6 Domain of instability (filled) as a function of dimension-less temperature lapse rate ΓΓ

119889and vertically averaged zonal wind

velocity 1199060

(119888119894)max asymp 834msminus1 is reached at ΓΓ

119889asymp 055 Figure 6 shows

that increasing vertically averaged zonal wind 1199060is associated

with increasing 119888119894 This is further evident in Figure 8 which

shows 119888119894as a function of 119906

0for a range of ΓΓ

119889values The

lower ΓΓ119889

and the larger 119888119894 that is 119888

119894 increases with

decreasing static stability

4 Concluding Remarks

We have studied theoretically the impact of variations in thestatic stability parameter 120590

0and zonal wind shear Λ

120585on the

characteristics of baroclinically unstable waves of synopticscales using Eady-type model with the uniform Λ

120585between

upper and lower boundaries on an 119891-plane Quantitativeestimates of variations in 120590

0and Λ

120585on the growth rate 120594

119896

wavelength of maximum growth rate 119871120594max and short-wave

cutoff 119871min were obtainedAnalytical expressions are derived for sensitivity func-

tions for the growth rate 120594119896with respect to variations in static

stability parameter andwind shear velocityThese expressionsallow estimating to a first-order approximation the influenceof changes in 120590

0and Λ

120585on 120594119896 Analytical expressions for

relative sensitivity functions allow estimating the significanceof variations in 120590

0andΛ

120585on the growth rate of baroclinically

unstable waves with a given zonal wave numberTo study the impact of variations in atmospheric static

stability and zonal wind velocity on the instability of plan-etary scale waves the model with vertically averaged prim-itive equations with 120573-plane approximation was applied Ascontrol parameters we have used dimensionless temperature

15

10

5

0minus02 00 02 04 06 08 10

ΓΓd

u0 = 20u0 = 30u0 = 40

ci

(m sminus

1)

Figure 7 Imaginary part of phase speed 119888119894versus dimensionless

temperature lapse rate ΓΓ119889for different values of vertically averaged

zonal wind velocity 1199060

lapse rate ΓΓ119889and vertically averaged zonal wind velocity

1199060 We have estimated the influence of ΓΓ

119889and 119906

0on the

imaginary part of phase speed 119888119894 whichwas used as ameasure

of instabilityThe obtained results are qualitatively consistent with

changes in the essential weather patterns that occurred overthe last several decades in some areas of the SH and inparticular over Australia (eg [49 50 52ndash54]) Climatechange results suggest SH midlatitude static stability 120590

0may

increase and the MTG (the vertical wind shear Λ120585) may

decrease which according to our linear theoretical modelsleads to a slowing of the growth rate of baroclinic unstablewaves 120594

119896and an increasing wavelength of baroclinic unstable

wave with maximum growth rate 119871120594max that is a spectrum

shift of unstable waves towards longer wavelengths Thesemight affect the favourable conditions for the developmentof baroclinic instability and therefore the rate of cyclogenesisand a reduction in cyclone intensity The obtained sensitivityfunctions demonstrate that waves belonging to the short-wave part of the spectrum of unstable waves are moresensitive to changes in the static stability parameter thanwaves belonging to the long-wave part of the spectrum

To obtain more realistic estimates of the sensitivity of thegrowth rate of unstable waves with respect to static stabilityparameter and MTG numerical modeling based on a fullGCM is required It is hoped to carry out such work in thefuture

Advances in Meteorology 9

ΓΓd = 01ΓΓd = 04

ΓΓd = 23ΓΓd = 09

20

10

00 20 40 60

u0 (mmiddotsminus1)

ci(m

sminus1)

Figure 8 Imaginary part of phase speed 119888119894versus vertically

averaged zonal wind velocity 1199060for different values of dimensionless

temperature lapse rate ΓΓ119889

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors thank Dr I Pisnichenko for clarification ofcertain questions relevant to this paper

References

[1] J R Holton An Introduction to Dynamic Meteorology Aca-demic Press 3rd edition 1992

[2] G K Vallis Atmospheric and Oceanic Fluid Dynamics Cam-bridge University Press 2006

[3] M Mak Atmospheric Dynamics Cambridge University Press2011

[4] V P Dymnikov Stability and Predictability of the Large ScaleAtmospheric Processes Institute of Numerical Mathematics ofthe Russian Academy of Sciences Moscow Russia 2007

[5] J G Charney ldquoThe dynamics of long waves in baroclinicwesterly currentrdquo Journal of Meteorology vol 4 pp 135ndash1621947

[6] E T Eady ldquoLong waves and cyclone wavesrdquo Tellus vol 1 pp33ndash52 1949

[7] R T Pierrehumbert and K L Swanson ldquoBaroclinic instabilityrdquoAnnual Review of Fluid Mechanics vol 27 no 1 pp 419ndash4671995

[8] R Gall ldquoA comparison of linear instability theory with theeddy statistics of a general circulation modelrdquo Journal of theAtmospheric Sciences vol 33 no 3 pp 349ndash373 1976

[9] I M Held ldquoProgress and problems in large-scale atmosphericdynamicsrdquo in The Global Circulation of the Atmosphere TSchneider and A Sobel Eds pp 1ndash21 Princeton UniversityPress Princeton NJ USA 2007

[10] B Farrell ldquoModal and non-modal baroclinic wavesrdquo Journal ofthe Atmospheric Sciences vol 41 no 4 pp 668ndash673 1984

[11] M K Tippett ldquoTransient moist baroclinic instabilityrdquo Tellus Avol 51 no 2 pp 273ndash288 1999

[12] D Hodyss and R Grotjahn ldquoNonmodal and unstable normalmode baroclinic growth as a function of horizontal scalerdquoDynamics of Atmospheres and Oceans vol 37 no 1 pp 1ndash242003

[13] J Pedlosky ldquoFinite amplitude baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 27 no 1 pp 15ndash30 1970

[14] R D Romea ldquoThe effects of friction and beta on finite ampli-tude baroclinic wavesrdquo Journal of the Atmospheric Sciences vol34 pp 1689ndash1695 1977

[15] J Pedlosky ldquoFinite-amplitude baroclinic waves at minimumcritical shearrdquo Journal of the Atmospheric Sciences vol 39 no3 pp 555ndash562 1982

[16] T Warn and P Gauthier ldquoPotential vorticity mixing bymarginally unstable baroclinic disturbancesrdquo Tellus A vol 41no 2 pp 115ndash131 1989

[17] A J Simmons and B J Hoskins ldquoThe lifecycles of some non-linear wavesrdquo Journal of the Atmospheric Sciences vol 35 pp414ndash432 1978

[18] A J Simmons and B J Hoskins ldquoBarotropic influences of thegrowth and decay of nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 37 pp 1679ndash1684 1980

[19] S B Feldstein and I M Held ldquoBarotropic decay of baroclinicwavesrdquo Journal of the Atmospheric Sciences vol 46 pp 1679ndash1684 1989

[20] C D Thorncroft B J Hoskins and M E McIntyre ldquoTwoparadigms of baroclinic-wave life-cycle behaviourrdquo QuaterlyJournal vol 119 no 509 pp 17ndash55 1993

[21] N A Phillips ldquoA simple three-dimensional model for the studyof largescale extratropical flow patternsrdquo Journal ofMeteorologyvol 8 pp 381ndash394 1951

[22] N A Phillips ldquoEnergy transformations and meridional circu-lations associated with simple baroclinic waves in a two-levelquasi-geostrophic modelrdquo Tellus vol 6 pp 273ndash286 1954

[23] L M Polvani and J Pedlosky ldquoThe effect of dissipation onspatially growing nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 45 no 14 pp 1977ndash1989 1988

[24] T G Shepherd ldquoNonlinear saturation of baroclinic instabilityPart I the two-layermodelrdquo Journal of the Atmospheric Sciencesvol 45 no 14 pp 2014ndash2025 1988

[25] G Balasubramanian and M K Yau ldquoBaroclinic instability ina two-layer model with parameterized slantwise convectionrdquoJournal of the Atmospheric Sciences vol 51 no 7 pp 971ndash9901994

[26] A Wiin-Nielsen ldquoOn the structure of atmospheric waves inmiddle latitudesrdquo Atmosfera vol 16 no 2 pp 83ndash102 2003

[27] D D Holm and B A Wingate ldquoBaroclinic instabilities of thetwo-layer quasigeostrophic alpha modelrdquo Journal of PhysicalOceanography vol 35 no 7 pp 1287ndash1296 2005

10 Advances in Meteorology

[28] J Egger ldquoBaroclinic instability in the two-layermodel interpre-tationsrdquo Meteorologische Zeitschrift vol 18 no 5 pp 559ndash5652009

[29] J Lamberts G Lapeyre and V Zeitlin ldquoMoist versus dry baro-clinic instability in a simplified two-layer atmospheric modelwith condensation and latent heat releaserdquo Journal of theAtmospheric Sciences vol 69 pp 1405ndash1426 2012

[30] D M W Frierson ldquoRobust increases in midlatitude static sta-bility in simulations of global warmingrdquo Geophysical ResearchLetters vol 33 no 24 Article ID L24816 2006

[31] N M J Hall B J Hoskins P J Valdes and C A Senior ldquoStormtracks in a high-resolution GCMwith doubled carbon dioxiderdquoQuarterly Journal vol 120 no 519 pp 1209ndash1230 1994

[32] S B Feldstein ldquoThe recent trend and variance increase of theannularmoderdquo Journal of Climate vol 15 no 3 pp 88ndash94 2002

[33] G J Marshall ldquoTrends in the southern annular mode fromobservations and reanalysesrdquo Journal of Climate vol 16 pp4134ndash4143 2003

[34] G M Ostermeier and J M Wallace ldquoTrends in the NorthAtlantic Oscillation-Northern Hemisphere annular mode dur-ing the twentieth centuryrdquo Journal of Climate vol 16 no 2 pp336ndash341 2003

[35] D W J Thompson and S Solomon ldquoInterpretation of recentSouthern Hemisphere climate changerdquo Science vol 296 no5569 pp 895ndash899 2002

[36] S J Lambert ldquoThe effect of enhanced greenhouse warming onwinter cyclone frequencies and strengthsrdquo Journal of Climatevol 8 no 5 pp 1447ndash1462 1995

[37] F Lunkeit K Fraedrich and S E Bauer ldquoStorm tracks ina warmer climate sensitivity studies with a simplified globalcirculation modelrdquo Climate Dynamics vol 14 no 11 pp 813ndash826 1998

[38] D J Lorenz and D L Hartmann ldquoEddy-zonal flow feedback inthe Southern Hemisphererdquo Journal of the Atmospheric Sciencesvol 58 no 21 pp 3312ndash3327 2001

[39] Q Geng and M Sugi ldquoPossible change of extratropical cycloneactivity due to enhanced greenhouse gases and sulphateaerosols study with a high-resolution AGCMrdquo Journal ofClimate vol 16 pp 2262ndash2274 2003

[40] J C Fyfe ldquoExtratropical Southern Hemisphere cyclonesharbingers of climate changerdquo Journal of Climate vol 16 pp2802ndash2805 2003

[41] J H Yin ldquoA consistent poleward shift of the storm tracksin simulations of 21st century climaterdquo Geophysical ResearchLetters vol 32 no 18 Article ID L18701 pp 1ndash4 2005

[42] S J Lambert and J C Fyfe ldquoChanges in winter cyclonefrequencies and strengths simulated in enhanced greenhousewarming experiments results from the models participating inthe IPCC diagnostic exerciserdquo Climate Dynamics vol 26 no7-8 pp 713ndash728 2006

[43] G Gastineau and B J Soden ldquoModel projected changes ofextreme wind events in response to global warmingrdquo Geophys-ical Research Letters vol 36 no 10 Article ID L10810 2009

[44] Y Wu M Ting R Seager H-P Huang and M A CaneldquoChanges in storm tracks and energy transports in a warmer cli-mate simulated by the GFDL CM21 modelrdquo Climate Dynamicsvol 37 no 1 pp 53ndash72 2011

[45] L Bengtsson K I Hodges and E Roeckner ldquoStorm tracks andclimate changerdquo Journal of Climate vol 19 no 15 pp 3518ndash35432006

[46] I Smith ldquoAn assessment of recent trends in Australian rainfallrdquoAustralian Meteorological Magazine vol 53 no 3 pp 163ndash1732004

[47] N Nicholls Detecting Understanding and Attributing ClimateChange Australian Greenhouse Office Publication 2007

[48] B C Bates P Hope B Ryan I Smith and S Charles ldquoKey find-ings from the Indian Ocean Climate Initiative and their impacton policy development in Australiardquo Climatic Change vol89 no 3-4 pp 339ndash354 2008

[49] J S Frederiksen and C S Frederiksen ldquoDecadal changes inSouthern Hemisphere winter cyclogenesisrdquo CSIROMarine andAtmospheric Research Paper 002 2005

[50] J S Frederiksen and C S Frederiksen ldquoInterdecadal changes insouthern hemisphere winter storm track modesrdquo Tellus A vol59 no 5 pp 599ndash617 2007

[51] J S Frederiksen C S Frederiksen and S L Osbrough ldquoMod-elling of changes in Southern Hemisphere weather systemsduring the 20th centuryrdquo in Proceedings of the 18th WorldIMACS Congress and International Congress on Modelling andSimulation pp 2562ndash2568 Cairns Australia July 2009

[52] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoCauses of changing Southern Hemispheric weathersystemsrdquo in Managing Climate Change I Jupp P Holper andW Cai Eds pp 85ndash98 CSIRO Publishing 2010

[53] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoChanges in Southern Hemisphere rainfall circulationand weather systemsrdquo in Proceedings of the 19th InternationalCongress on Modelling and Simulation (MODSIM rsquo11) pp 2712ndash2718 Perth Australia December 2011

[54] C S Frederiksen J S Frederiksen J M Sisson and S LOsbrough ldquoObserved and projected changes in the annual cycleof Southern Hemisphere baroclinicity for storm formationrdquoin Proceedings of the 19th International Congress on Modellingand Simulation (MODSIMrsquo 11) pp 2719ndash2725 Perth AustraliaDecember 2011

[55] N A Phillips ldquoGeostrophicmotionrdquoReviews of Geophysics vol1 no 2 pp 123ndash176 1963

[56] I Bordi K Fraedrich F Lunkeit and A Sutera ldquoTroposphericdouble jets meridional cells and eddies a case study andidealized simulationsrdquoMonthly Weather Review vol 135 no 9pp 3118ndash3133 2007

[57] I Bordi K Fraedrich M Ghil and A Sutera ldquoZonal flowregime changes in a GCM and in a simple quasigeostrophicmodel the role of stratospheric dynamicsrdquo Journal of theAtmospheric Sciences vol 66 no 5 pp 1366ndash1383 2009

[58] J G Charney ldquoOn the scale of atmosphericmotionsrdquoGeofysiskePublikasjoner vol 17 no 2 pp 3ndash17 1948

[59] A Burger ldquoScale consideration of planetary motions of theatmosphererdquo Tellus vol 10 pp 195ndash205 1958

[60] I A Pisnichenko ldquoUltralong-wave dynamics in a two-dimen-sional baroclinic atmosphere modelrdquo Atmospheric and OceanicPhysics vol 16 no 9 pp 883ndash892 1980

[61] I A Pisnichenko ldquoInfluence variable static stability on thedynamics of ultralong waves in two-dimensional baroclinicmodel of the atmosphererdquoAtmospheric andOceanic Physics vol19 no 11 pp 1223ndash1226 1983

[62] A Wiin-Nielsen ldquoOn barotropic and baroclinic models withspecial emphasis onultra-longwavesrdquoMonthlyWeather Reviewvol 87 pp 171ndash183 1959

[63] B W BinWang and A Barcilon ldquoMoist stability of a barocliniczonal flow with conditionally unstable stratificationrdquo Journal ofthe Atmospheric Sciences vol 43 no 7 pp 705ndash719 1986

Advances in Meteorology 11

[64] N A Phillips ldquoA coordinate system having some specialadvantages for numerical forecastingrdquo Journal of Meteorologyvol 14 pp 184ndash185 1957

[65] D M Alishaev ldquoDynamics of a two-dimensional baroclinicatmosphererdquoAtmospheric and Oceanic Physics vol 16 no 2 pp99ndash107 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 9

ΓΓd = 01ΓΓd = 04

ΓΓd = 23ΓΓd = 09

20

10

00 20 40 60

u0 (mmiddotsminus1)

ci(m

sminus1)

Figure 8 Imaginary part of phase speed 119888119894versus vertically

averaged zonal wind velocity 1199060for different values of dimensionless

temperature lapse rate ΓΓ119889

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors thank Dr I Pisnichenko for clarification ofcertain questions relevant to this paper

References

[1] J R Holton An Introduction to Dynamic Meteorology Aca-demic Press 3rd edition 1992

[2] G K Vallis Atmospheric and Oceanic Fluid Dynamics Cam-bridge University Press 2006

[3] M Mak Atmospheric Dynamics Cambridge University Press2011

[4] V P Dymnikov Stability and Predictability of the Large ScaleAtmospheric Processes Institute of Numerical Mathematics ofthe Russian Academy of Sciences Moscow Russia 2007

[5] J G Charney ldquoThe dynamics of long waves in baroclinicwesterly currentrdquo Journal of Meteorology vol 4 pp 135ndash1621947

[6] E T Eady ldquoLong waves and cyclone wavesrdquo Tellus vol 1 pp33ndash52 1949

[7] R T Pierrehumbert and K L Swanson ldquoBaroclinic instabilityrdquoAnnual Review of Fluid Mechanics vol 27 no 1 pp 419ndash4671995

[8] R Gall ldquoA comparison of linear instability theory with theeddy statistics of a general circulation modelrdquo Journal of theAtmospheric Sciences vol 33 no 3 pp 349ndash373 1976

[9] I M Held ldquoProgress and problems in large-scale atmosphericdynamicsrdquo in The Global Circulation of the Atmosphere TSchneider and A Sobel Eds pp 1ndash21 Princeton UniversityPress Princeton NJ USA 2007

[10] B Farrell ldquoModal and non-modal baroclinic wavesrdquo Journal ofthe Atmospheric Sciences vol 41 no 4 pp 668ndash673 1984

[11] M K Tippett ldquoTransient moist baroclinic instabilityrdquo Tellus Avol 51 no 2 pp 273ndash288 1999

[12] D Hodyss and R Grotjahn ldquoNonmodal and unstable normalmode baroclinic growth as a function of horizontal scalerdquoDynamics of Atmospheres and Oceans vol 37 no 1 pp 1ndash242003

[13] J Pedlosky ldquoFinite amplitude baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 27 no 1 pp 15ndash30 1970

[14] R D Romea ldquoThe effects of friction and beta on finite ampli-tude baroclinic wavesrdquo Journal of the Atmospheric Sciences vol34 pp 1689ndash1695 1977

[15] J Pedlosky ldquoFinite-amplitude baroclinic waves at minimumcritical shearrdquo Journal of the Atmospheric Sciences vol 39 no3 pp 555ndash562 1982

[16] T Warn and P Gauthier ldquoPotential vorticity mixing bymarginally unstable baroclinic disturbancesrdquo Tellus A vol 41no 2 pp 115ndash131 1989

[17] A J Simmons and B J Hoskins ldquoThe lifecycles of some non-linear wavesrdquo Journal of the Atmospheric Sciences vol 35 pp414ndash432 1978

[18] A J Simmons and B J Hoskins ldquoBarotropic influences of thegrowth and decay of nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 37 pp 1679ndash1684 1980

[19] S B Feldstein and I M Held ldquoBarotropic decay of baroclinicwavesrdquo Journal of the Atmospheric Sciences vol 46 pp 1679ndash1684 1989

[20] C D Thorncroft B J Hoskins and M E McIntyre ldquoTwoparadigms of baroclinic-wave life-cycle behaviourrdquo QuaterlyJournal vol 119 no 509 pp 17ndash55 1993

[21] N A Phillips ldquoA simple three-dimensional model for the studyof largescale extratropical flow patternsrdquo Journal ofMeteorologyvol 8 pp 381ndash394 1951

[22] N A Phillips ldquoEnergy transformations and meridional circu-lations associated with simple baroclinic waves in a two-levelquasi-geostrophic modelrdquo Tellus vol 6 pp 273ndash286 1954

[23] L M Polvani and J Pedlosky ldquoThe effect of dissipation onspatially growing nonlinear baroclinic wavesrdquo Journal of theAtmospheric Sciences vol 45 no 14 pp 1977ndash1989 1988

[24] T G Shepherd ldquoNonlinear saturation of baroclinic instabilityPart I the two-layermodelrdquo Journal of the Atmospheric Sciencesvol 45 no 14 pp 2014ndash2025 1988

[25] G Balasubramanian and M K Yau ldquoBaroclinic instability ina two-layer model with parameterized slantwise convectionrdquoJournal of the Atmospheric Sciences vol 51 no 7 pp 971ndash9901994

[26] A Wiin-Nielsen ldquoOn the structure of atmospheric waves inmiddle latitudesrdquo Atmosfera vol 16 no 2 pp 83ndash102 2003

[27] D D Holm and B A Wingate ldquoBaroclinic instabilities of thetwo-layer quasigeostrophic alpha modelrdquo Journal of PhysicalOceanography vol 35 no 7 pp 1287ndash1296 2005

10 Advances in Meteorology

[28] J Egger ldquoBaroclinic instability in the two-layermodel interpre-tationsrdquo Meteorologische Zeitschrift vol 18 no 5 pp 559ndash5652009

[29] J Lamberts G Lapeyre and V Zeitlin ldquoMoist versus dry baro-clinic instability in a simplified two-layer atmospheric modelwith condensation and latent heat releaserdquo Journal of theAtmospheric Sciences vol 69 pp 1405ndash1426 2012

[30] D M W Frierson ldquoRobust increases in midlatitude static sta-bility in simulations of global warmingrdquo Geophysical ResearchLetters vol 33 no 24 Article ID L24816 2006

[31] N M J Hall B J Hoskins P J Valdes and C A Senior ldquoStormtracks in a high-resolution GCMwith doubled carbon dioxiderdquoQuarterly Journal vol 120 no 519 pp 1209ndash1230 1994

[32] S B Feldstein ldquoThe recent trend and variance increase of theannularmoderdquo Journal of Climate vol 15 no 3 pp 88ndash94 2002

[33] G J Marshall ldquoTrends in the southern annular mode fromobservations and reanalysesrdquo Journal of Climate vol 16 pp4134ndash4143 2003

[34] G M Ostermeier and J M Wallace ldquoTrends in the NorthAtlantic Oscillation-Northern Hemisphere annular mode dur-ing the twentieth centuryrdquo Journal of Climate vol 16 no 2 pp336ndash341 2003

[35] D W J Thompson and S Solomon ldquoInterpretation of recentSouthern Hemisphere climate changerdquo Science vol 296 no5569 pp 895ndash899 2002

[36] S J Lambert ldquoThe effect of enhanced greenhouse warming onwinter cyclone frequencies and strengthsrdquo Journal of Climatevol 8 no 5 pp 1447ndash1462 1995

[37] F Lunkeit K Fraedrich and S E Bauer ldquoStorm tracks ina warmer climate sensitivity studies with a simplified globalcirculation modelrdquo Climate Dynamics vol 14 no 11 pp 813ndash826 1998

[38] D J Lorenz and D L Hartmann ldquoEddy-zonal flow feedback inthe Southern Hemisphererdquo Journal of the Atmospheric Sciencesvol 58 no 21 pp 3312ndash3327 2001

[39] Q Geng and M Sugi ldquoPossible change of extratropical cycloneactivity due to enhanced greenhouse gases and sulphateaerosols study with a high-resolution AGCMrdquo Journal ofClimate vol 16 pp 2262ndash2274 2003

[40] J C Fyfe ldquoExtratropical Southern Hemisphere cyclonesharbingers of climate changerdquo Journal of Climate vol 16 pp2802ndash2805 2003

[41] J H Yin ldquoA consistent poleward shift of the storm tracksin simulations of 21st century climaterdquo Geophysical ResearchLetters vol 32 no 18 Article ID L18701 pp 1ndash4 2005

[42] S J Lambert and J C Fyfe ldquoChanges in winter cyclonefrequencies and strengths simulated in enhanced greenhousewarming experiments results from the models participating inthe IPCC diagnostic exerciserdquo Climate Dynamics vol 26 no7-8 pp 713ndash728 2006

[43] G Gastineau and B J Soden ldquoModel projected changes ofextreme wind events in response to global warmingrdquo Geophys-ical Research Letters vol 36 no 10 Article ID L10810 2009

[44] Y Wu M Ting R Seager H-P Huang and M A CaneldquoChanges in storm tracks and energy transports in a warmer cli-mate simulated by the GFDL CM21 modelrdquo Climate Dynamicsvol 37 no 1 pp 53ndash72 2011

[45] L Bengtsson K I Hodges and E Roeckner ldquoStorm tracks andclimate changerdquo Journal of Climate vol 19 no 15 pp 3518ndash35432006

[46] I Smith ldquoAn assessment of recent trends in Australian rainfallrdquoAustralian Meteorological Magazine vol 53 no 3 pp 163ndash1732004

[47] N Nicholls Detecting Understanding and Attributing ClimateChange Australian Greenhouse Office Publication 2007

[48] B C Bates P Hope B Ryan I Smith and S Charles ldquoKey find-ings from the Indian Ocean Climate Initiative and their impacton policy development in Australiardquo Climatic Change vol89 no 3-4 pp 339ndash354 2008

[49] J S Frederiksen and C S Frederiksen ldquoDecadal changes inSouthern Hemisphere winter cyclogenesisrdquo CSIROMarine andAtmospheric Research Paper 002 2005

[50] J S Frederiksen and C S Frederiksen ldquoInterdecadal changes insouthern hemisphere winter storm track modesrdquo Tellus A vol59 no 5 pp 599ndash617 2007

[51] J S Frederiksen C S Frederiksen and S L Osbrough ldquoMod-elling of changes in Southern Hemisphere weather systemsduring the 20th centuryrdquo in Proceedings of the 18th WorldIMACS Congress and International Congress on Modelling andSimulation pp 2562ndash2568 Cairns Australia July 2009

[52] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoCauses of changing Southern Hemispheric weathersystemsrdquo in Managing Climate Change I Jupp P Holper andW Cai Eds pp 85ndash98 CSIRO Publishing 2010

[53] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoChanges in Southern Hemisphere rainfall circulationand weather systemsrdquo in Proceedings of the 19th InternationalCongress on Modelling and Simulation (MODSIM rsquo11) pp 2712ndash2718 Perth Australia December 2011

[54] C S Frederiksen J S Frederiksen J M Sisson and S LOsbrough ldquoObserved and projected changes in the annual cycleof Southern Hemisphere baroclinicity for storm formationrdquoin Proceedings of the 19th International Congress on Modellingand Simulation (MODSIMrsquo 11) pp 2719ndash2725 Perth AustraliaDecember 2011

[55] N A Phillips ldquoGeostrophicmotionrdquoReviews of Geophysics vol1 no 2 pp 123ndash176 1963

[56] I Bordi K Fraedrich F Lunkeit and A Sutera ldquoTroposphericdouble jets meridional cells and eddies a case study andidealized simulationsrdquoMonthly Weather Review vol 135 no 9pp 3118ndash3133 2007

[57] I Bordi K Fraedrich M Ghil and A Sutera ldquoZonal flowregime changes in a GCM and in a simple quasigeostrophicmodel the role of stratospheric dynamicsrdquo Journal of theAtmospheric Sciences vol 66 no 5 pp 1366ndash1383 2009

[58] J G Charney ldquoOn the scale of atmosphericmotionsrdquoGeofysiskePublikasjoner vol 17 no 2 pp 3ndash17 1948

[59] A Burger ldquoScale consideration of planetary motions of theatmosphererdquo Tellus vol 10 pp 195ndash205 1958

[60] I A Pisnichenko ldquoUltralong-wave dynamics in a two-dimen-sional baroclinic atmosphere modelrdquo Atmospheric and OceanicPhysics vol 16 no 9 pp 883ndash892 1980

[61] I A Pisnichenko ldquoInfluence variable static stability on thedynamics of ultralong waves in two-dimensional baroclinicmodel of the atmosphererdquoAtmospheric andOceanic Physics vol19 no 11 pp 1223ndash1226 1983

[62] A Wiin-Nielsen ldquoOn barotropic and baroclinic models withspecial emphasis onultra-longwavesrdquoMonthlyWeather Reviewvol 87 pp 171ndash183 1959

[63] B W BinWang and A Barcilon ldquoMoist stability of a barocliniczonal flow with conditionally unstable stratificationrdquo Journal ofthe Atmospheric Sciences vol 43 no 7 pp 705ndash719 1986

Advances in Meteorology 11

[64] N A Phillips ldquoA coordinate system having some specialadvantages for numerical forecastingrdquo Journal of Meteorologyvol 14 pp 184ndash185 1957

[65] D M Alishaev ldquoDynamics of a two-dimensional baroclinicatmosphererdquoAtmospheric and Oceanic Physics vol 16 no 2 pp99ndash107 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

10 Advances in Meteorology

[28] J Egger ldquoBaroclinic instability in the two-layermodel interpre-tationsrdquo Meteorologische Zeitschrift vol 18 no 5 pp 559ndash5652009

[29] J Lamberts G Lapeyre and V Zeitlin ldquoMoist versus dry baro-clinic instability in a simplified two-layer atmospheric modelwith condensation and latent heat releaserdquo Journal of theAtmospheric Sciences vol 69 pp 1405ndash1426 2012

[30] D M W Frierson ldquoRobust increases in midlatitude static sta-bility in simulations of global warmingrdquo Geophysical ResearchLetters vol 33 no 24 Article ID L24816 2006

[31] N M J Hall B J Hoskins P J Valdes and C A Senior ldquoStormtracks in a high-resolution GCMwith doubled carbon dioxiderdquoQuarterly Journal vol 120 no 519 pp 1209ndash1230 1994

[32] S B Feldstein ldquoThe recent trend and variance increase of theannularmoderdquo Journal of Climate vol 15 no 3 pp 88ndash94 2002

[33] G J Marshall ldquoTrends in the southern annular mode fromobservations and reanalysesrdquo Journal of Climate vol 16 pp4134ndash4143 2003

[34] G M Ostermeier and J M Wallace ldquoTrends in the NorthAtlantic Oscillation-Northern Hemisphere annular mode dur-ing the twentieth centuryrdquo Journal of Climate vol 16 no 2 pp336ndash341 2003

[35] D W J Thompson and S Solomon ldquoInterpretation of recentSouthern Hemisphere climate changerdquo Science vol 296 no5569 pp 895ndash899 2002

[36] S J Lambert ldquoThe effect of enhanced greenhouse warming onwinter cyclone frequencies and strengthsrdquo Journal of Climatevol 8 no 5 pp 1447ndash1462 1995

[37] F Lunkeit K Fraedrich and S E Bauer ldquoStorm tracks ina warmer climate sensitivity studies with a simplified globalcirculation modelrdquo Climate Dynamics vol 14 no 11 pp 813ndash826 1998

[38] D J Lorenz and D L Hartmann ldquoEddy-zonal flow feedback inthe Southern Hemisphererdquo Journal of the Atmospheric Sciencesvol 58 no 21 pp 3312ndash3327 2001

[39] Q Geng and M Sugi ldquoPossible change of extratropical cycloneactivity due to enhanced greenhouse gases and sulphateaerosols study with a high-resolution AGCMrdquo Journal ofClimate vol 16 pp 2262ndash2274 2003

[40] J C Fyfe ldquoExtratropical Southern Hemisphere cyclonesharbingers of climate changerdquo Journal of Climate vol 16 pp2802ndash2805 2003

[41] J H Yin ldquoA consistent poleward shift of the storm tracksin simulations of 21st century climaterdquo Geophysical ResearchLetters vol 32 no 18 Article ID L18701 pp 1ndash4 2005

[42] S J Lambert and J C Fyfe ldquoChanges in winter cyclonefrequencies and strengths simulated in enhanced greenhousewarming experiments results from the models participating inthe IPCC diagnostic exerciserdquo Climate Dynamics vol 26 no7-8 pp 713ndash728 2006

[43] G Gastineau and B J Soden ldquoModel projected changes ofextreme wind events in response to global warmingrdquo Geophys-ical Research Letters vol 36 no 10 Article ID L10810 2009

[44] Y Wu M Ting R Seager H-P Huang and M A CaneldquoChanges in storm tracks and energy transports in a warmer cli-mate simulated by the GFDL CM21 modelrdquo Climate Dynamicsvol 37 no 1 pp 53ndash72 2011

[45] L Bengtsson K I Hodges and E Roeckner ldquoStorm tracks andclimate changerdquo Journal of Climate vol 19 no 15 pp 3518ndash35432006

[46] I Smith ldquoAn assessment of recent trends in Australian rainfallrdquoAustralian Meteorological Magazine vol 53 no 3 pp 163ndash1732004

[47] N Nicholls Detecting Understanding and Attributing ClimateChange Australian Greenhouse Office Publication 2007

[48] B C Bates P Hope B Ryan I Smith and S Charles ldquoKey find-ings from the Indian Ocean Climate Initiative and their impacton policy development in Australiardquo Climatic Change vol89 no 3-4 pp 339ndash354 2008

[49] J S Frederiksen and C S Frederiksen ldquoDecadal changes inSouthern Hemisphere winter cyclogenesisrdquo CSIROMarine andAtmospheric Research Paper 002 2005

[50] J S Frederiksen and C S Frederiksen ldquoInterdecadal changes insouthern hemisphere winter storm track modesrdquo Tellus A vol59 no 5 pp 599ndash617 2007

[51] J S Frederiksen C S Frederiksen and S L Osbrough ldquoMod-elling of changes in Southern Hemisphere weather systemsduring the 20th centuryrdquo in Proceedings of the 18th WorldIMACS Congress and International Congress on Modelling andSimulation pp 2562ndash2568 Cairns Australia July 2009

[52] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoCauses of changing Southern Hemispheric weathersystemsrdquo in Managing Climate Change I Jupp P Holper andW Cai Eds pp 85ndash98 CSIRO Publishing 2010

[53] J S Frederiksen C S Frederiksen S L Osbrough and J MSisson ldquoChanges in Southern Hemisphere rainfall circulationand weather systemsrdquo in Proceedings of the 19th InternationalCongress on Modelling and Simulation (MODSIM rsquo11) pp 2712ndash2718 Perth Australia December 2011

[54] C S Frederiksen J S Frederiksen J M Sisson and S LOsbrough ldquoObserved and projected changes in the annual cycleof Southern Hemisphere baroclinicity for storm formationrdquoin Proceedings of the 19th International Congress on Modellingand Simulation (MODSIMrsquo 11) pp 2719ndash2725 Perth AustraliaDecember 2011

[55] N A Phillips ldquoGeostrophicmotionrdquoReviews of Geophysics vol1 no 2 pp 123ndash176 1963

[56] I Bordi K Fraedrich F Lunkeit and A Sutera ldquoTroposphericdouble jets meridional cells and eddies a case study andidealized simulationsrdquoMonthly Weather Review vol 135 no 9pp 3118ndash3133 2007

[57] I Bordi K Fraedrich M Ghil and A Sutera ldquoZonal flowregime changes in a GCM and in a simple quasigeostrophicmodel the role of stratospheric dynamicsrdquo Journal of theAtmospheric Sciences vol 66 no 5 pp 1366ndash1383 2009

[58] J G Charney ldquoOn the scale of atmosphericmotionsrdquoGeofysiskePublikasjoner vol 17 no 2 pp 3ndash17 1948

[59] A Burger ldquoScale consideration of planetary motions of theatmosphererdquo Tellus vol 10 pp 195ndash205 1958

[60] I A Pisnichenko ldquoUltralong-wave dynamics in a two-dimen-sional baroclinic atmosphere modelrdquo Atmospheric and OceanicPhysics vol 16 no 9 pp 883ndash892 1980

[61] I A Pisnichenko ldquoInfluence variable static stability on thedynamics of ultralong waves in two-dimensional baroclinicmodel of the atmosphererdquoAtmospheric andOceanic Physics vol19 no 11 pp 1223ndash1226 1983

[62] A Wiin-Nielsen ldquoOn barotropic and baroclinic models withspecial emphasis onultra-longwavesrdquoMonthlyWeather Reviewvol 87 pp 171ndash183 1959

[63] B W BinWang and A Barcilon ldquoMoist stability of a barocliniczonal flow with conditionally unstable stratificationrdquo Journal ofthe Atmospheric Sciences vol 43 no 7 pp 705ndash719 1986

Advances in Meteorology 11

[64] N A Phillips ldquoA coordinate system having some specialadvantages for numerical forecastingrdquo Journal of Meteorologyvol 14 pp 184ndash185 1957

[65] D M Alishaev ldquoDynamics of a two-dimensional baroclinicatmosphererdquoAtmospheric and Oceanic Physics vol 16 no 2 pp99ndash107 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 11

[64] N A Phillips ldquoA coordinate system having some specialadvantages for numerical forecastingrdquo Journal of Meteorologyvol 14 pp 184ndash185 1957

[65] D M Alishaev ldquoDynamics of a two-dimensional baroclinicatmosphererdquoAtmospheric and Oceanic Physics vol 16 no 2 pp99ndash107 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in