geofisical fluid dinam 2d1

8/13/2019 Geofisical Fluid Dinam 2d1

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490 Geophysieal Huid Dynamics 6. Ekman Layer a F ree Surlaee 491

TO 1 - i)2pvy

u

z/O(b) Profiles of and v

8 re aS deplh

(a) o d o g r ~ p h

31tg

F ig . 13.7 Ekman lay er al a free surface. The left panel shows velocily vectors alvarious depths;values of -z / a re ind ic al ed a lo ng Ihe c urve I ra ce d out by Ihe I Ip of Ihe velocllY veclors. Therighlpanel shows vertical distributions of u and v

v

then the net (depth-integrated) Coriolis force, directed to the right of thedepth-integrated transport, can balance the wind stress. .

The horizontal uniformity assumed in the solution is not a senous hmJlat ion. Since Ekman layers near the ocean surface have a thickness -5 0 mmuch smaIler than the scale of horizontal variation L 100 km), the solutionis still 10caIly applicable. The absence of horizontal pressure gradient assumedhere can also be relaxed easily. Because o f t he thinness of the layer, anyimposed horizontal pressure gradient remains constant across th.e layer. Thepresence of a horizontal pressure gradient merely adds a depth-mdependentgeostrophic velocity to the Ekman solution. Suppose t he sea surface slopesdown tothe north, so that there is a pressure force acting northward throughoutthe Ekman layer and below (Figure 13.8). This means tha t a t the b o t ~ o m ofthe Ekman layer z/o -00) there is a geostrophic velocity U to the ng?t.ofthe pressure force. The surface Ekman spiral forced by t he wmd stress Jomssmoothly to this geostrophic velocity as z/ o -oo.

Pure Ekman spira ls are not observed in the surface layer of the ocean,mainly because the assumptions of constant eddy viscosity and steadiness areparticularly restrictive. When the f10w is v ~ r g e d ayer a few d y ~ howe. er,several instances have been found in which the current does look IIke a splral.One such example is shown in Figure 13.9.Explanation in Terms of Vortex TiltingWe have seen in previous chapters that the thickness of a viscous layer usualIygrows in a nonrotating flow, either in t ime or in the direction of fiow. The

>

(28)

(29)

30J Tv d z= - pi

v = A e l+i z /8 +B e- I+ ; z /8where we have defined

The solution of (27) is

We shall see shordy that o is the thickness of the Ekman layer. The constantBis zero because the field must remain finite as z -oo. Thesurface boundaryconditions (24) and (25) can be combined as p/ly dVldz =r at z=O, fromwhich (28) gives

Substitution of this into (28) gives the velocity componentsTI P 8 z )= I]V: e Z / cos - 8 4

T p z/ 8 ( Z )v = - J]V e SIn - 8 4The Swedish oceanographer Ekman worked out this solution in 1905. The

solution is shown in Figure 13 .7 for the case of the northern hemisphere, inwhich I is positive. The velocities at various depths are plotted in Figure 13.7a,where each arrow represents the velocity vector at a certain depth. Such a plo tof v versus u is sometimes called a hodograph plot. The vertical distributionsof u and vare shown in Figure 13.7b. The hodograph shows (hat the surfacevelocity is deflected 45 to th e right ofthe applied wind stress. [In thesouthernhemisphere the def lect ion is to the left of the surface stress.] The velocityvector rotates c lockwise (1ooking down) with depth, and the magnitudeexponentially decays with an e-folding scale of which is called the Ekmanlayer thickness. The tips of the velocity vector at various depths form a spiral,called the Ekman spiral.

The components of the volume transport in the Ekman layer areroo udz=OThis shows that the net transporl is 10 the right 01 the applied stress and isindependent 01 Vy. I n fac t, th e result Jv dz = -T/lp fol lows directly f rom avertical integration of the equation of motion in the form -plv = d stress / dzso that the result does not depend on the eddy viscosity assumption. The factthat the transport is to th e right of the applied stress makes sense, because


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492 Geophysica/ Huid Dynamics I 7 Ekman Layeron a Rigid SurJace 493

pressure force

:

NWS 1: = 1 1 dyn/cm2

40m

80 m

o 10 cm s2040{0

-z/8 = O

z 8t ~ = = = ~ _ r ~ r _ l I

Fig. 13.8 Ekman layer al a f ree surface in the presence of a pressure gradient. The geostrophicvelocity forced by the pressure gradient is UEkman solution, in contrast, results in a viscous layer that does notgrow eitherin time or space. This can be explained by examining the vorticity equationPedlosky, 1987). The vorticity.components in the x and y directions are

i w iJv v xdy iJz dz

iJu w wy = iJz x = dzwhere we have used w = Using these, the z-derivative of the equations ofmotion 22) and 23) gives

= 1 d2 Wydz v dz z

-20

3

u cm/s) 20

31)

the solid earth or the boundary layer over the ocean bottom. We assume tha ta t large dis tances f rom the sur face the velocity is toward the X direction andhas a magnitude U Far from the wall viscous forces are negligible, so thatthe Coriolis force can only be balanced by a pressure gradient:

Fig. 13.9 An observed velocitydistribution near the coast of Oregon. Velocity isaveraged over7 days. Wind stress had a magnitude of 1 1 dyn/cm 2 and was directed nearly soulhward, asindicated at lhe top of the figure. The upper panel shows venical dislribulions of u and D andlhe lower panel shows the hodograph in which depths are indcated in meters. The hodographis similar to lhal of a surface Ekman layer of deplh 16 m Iying over the bot tom Ekman layerextending from a deplh of 16 n lo lhe ocean bottom). [After Kundu 1977).]

d 2wx dz = v dZ

2

The r ight side of these equations represent diffusion of vorticity. WithoutCoriolis forees this diffusion would cause a thickening of the viscous layer .The presence of planetary rotation, however, means that vertical f luid linescoincide with the planetary vor tex l ines . The t il ting of vertical f luid lines,represented by terms on the lef t sides of 31), then causes arate of change ofhorizontal component of vortieity Ihat just cancels the di fusion termo

7 Ekman Layer on a Rigid SurfaceConsider now a horizontally independent and steady viscous layer on a solidsurface in a rotat ing flow. This can be the a tmospher ic boundary layer over

:>

cm/s)

jU=_ ' dpp dy 32)


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494 Geophysical Fluid Dynamics 7 Ekman Layer on a Rig id Surjace 495

where we have defined the complex velocity v== u + iv The boundary condit ions 35 and 36 in terms of th e complex velocity are

b Profiles of ti aod v

2/0

uC

tpressureforce

a HodographI U

v

on a semi-infinite plate the Blasius solution of Section lOA in which thethickness is proportional to 1/ {[J

Figure 13.10b shows the vertical distribution of the velocity components.Fa r from the wall the velocity is entirely in the x direction, and the Coriolisforce balances the pressure gradient . As the wal l is approached, retardingefiects decrease u and the associated Coriolis force, so that the pressuregradient which is independent of z forces a component v in the directionof the pressure force. Using 41 the net transport in the Ekman layer normalt o t he uniform stream outside the layer is

Fig.13.10 Ekman layer at a rigid surface. Theleft panel shows velocity vectors at various depths;values o f z / {j are indicated along thecurve traced out by the tip ofthe velocity vectors. The rightpanel shows vertical distributions of u and

3839

37

34

3536

as Z Hxat Z =0

at z = Oas z ? ro

v=Ov=O

V uv=o

u= Uu=O

d 2 vfu = l /v 2 fUdzwhere we have replaced -p- l dp dy by fU in accordance with 32 . Theboundary conditions ar e

The particular solution of 37 is V = U The total solution is therefore

This simply says t ha t t he flow O utside the viscous layer is in geostrophicbalance, U being the geostrophic velocity. Fo r our assumed case of positiveU and 1 we must have dp / dy < O so t hat t he pressure falls with y That is,the pressure force is directed ala ng th e positive y direction, r es ul ti ng i n ageostrophic flow U to the right ofthe pressure force in the northern hemisphere.The horizontal pressure gradient remains constant within the thin boundarylayer.

Nearthe salid surface the viscous forces ar e important, s o t ha t t he balancewithin the boundary layer isd 2ufv = l/v dZ 2 33

where z is taken vertically upward from the solid surface. Multiplying 34by i and adding 33 , the equatons of motion become

d 2 V if = v Udz l/v

where == /2vv/f. To satisfy 3 8 , we must have B = O Condition 39 givesA = - U The velocity components then become

According to 41 , t he t ip of the velocity vector describes a spiral for variousvalues of z Figure 13.10a . As w it h t he Ekman layer at a fr ee surface, thefrictional efiects are confined within a layer of thickness 0= ../2I1v/f whichincreases with l/v and decreases with the rotation rate j Interestingly the ayerthickness is independent of the magnitude of the free-stream velocity U; thisbehavior is quite difierent f rom th2t of a steady nonrotating boundary layer

u = U[ l e- z / cos z/ o ]v = U e- z / 5 sin zj8

40

41

r vdz= U[;;rZ=Wowhich is direc ted to the left of the free-stream velocity, in the direction of thepressure force.

If the atmosphere were i n laminar motion, Jlv wou ld b e equal to itsmolecular value for a ir , a nd th e Ekman ayer thickness a t a l at it ud e of 45where f = 10 -4 S I would be about S - 0.4 m. The observed thickness of theatmospheric boundary layer is of order 1 km, which implies an eddy viscosityof order l/v - 50 m2/s . G. I. Taylor 1915 , in fact, tried to estimate the eddyviscosity by matchingthe predicted velocity distributions 41 with the observedwind at various heights.

The Ekman layer solution on a solid surface demonstrates that the threeway balance between th e Coriolis force, th e pressure force, and the frictional


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496 Geophysical Fluid Dynamics 8. Shallow Water Equarions 497

Fig. 13.11 Balance of forces wilhin an Ekman layer, showingthat velocity u has a componenloward low pressure.

Shallow Water Equations80th surface and internal gravi ty waves were discussed in Chapter 7. Theeffect of planetary rotation was assumed to be smail , which is valid i f t hefrequency W of the wave is much larger than the Coriol is parameter f In this

42)

43)

ap aYpg-ay ay apg-ax ax

au avH y - + H 71 w 71 - w O =ax ay

Fig. 13.12 Layer of fluid on aRa bollom.

chapter we are considering phenomena slow enough for w to be comparableto f Consider surface gravi ty waves in a shallow layer of homogeneous fluidwhose mean depth is H I f we restrictourselves to wavelengths A much largerthan R then the vertical velocities are much smal ler than the horizontalvelocities. In Section 7.6we saw that the acce eration aw al is then negligiblein the vert ical momentum equation, so that the pressure distribution is hydrostatic. We also demonstrated that th e fluid pa rt ic1e s execu te a horizontalrectilinear motion that is independent of z. When the effects of pianetaryrotation are included, the horizontal velocity is still depth-independent, butthe particle orbits are no longer rectilinear but elliptic on a horizontal plane,as we shall see in the following sect ion.

Consider a layer of fluid over a t lat horizontal bottom Figure 13.12). LetZ be measured upward from the bottom surface, and be the displacementofthe free surface. The pressure at height z fram the bollom, being hydrostatic,is gived by

p =pg H 71 -zThe horizontal pressure gradients are therefore

Since these are independent of z t he resul ting hor i zontal mot ion IS alsodepth-independent.

Now consider the continuity equationau av aw =0ax ay az

Since au ax and av ay are independent of z the continuity equation requiresthat w varies l inearIy with z fram zero a t t he bottom t o th e maximum valuea t t he free surface. Integrat ing vert ical ly across the water column from z Oto z R 17 and not ing that u and v r depth-independent, we get

-S.

Coriolisforce

highp

friction

low P

pressureforce

force within the boundary layer results in a component of flow directed towardthe lower pressure.Thebalanceof forces within the boundary layer is illustratedin Figure 13.1 J. The net frict ional force on an element is oriented approximatelyopposite to the veloci ty v ~ o r u. t is c1ear that a balance of forces is possibleonl y i f t he v el oc it y vector has a component f rom high to low pressure , asshown. Frict ional forces therefore cause the t low around low pressure centerto spiral inward Mass conservation requires that the inward converging flowshould rise over a low pressure system, resul t ing in cloud formation andrainfal . This is what happens i n a cyc lone , which is a low pressure sys tem.In contrast, over a h igh pressure system the air sinks as i t spirals outward dueto frict ional effects. The arrival of high pressure systems therefore brings inc1ear skies and fair weather, because he sinking air does not resul t in c10udformation.

Frictional effects, in particular the Ekman transport by surface winds, playa fundament al r ol e in th e theory of wind-driven ocean circulation. Possiblythe mostimportant result of such theories was given byHemy Stommel in 1948.He showedthat the northward increase of the Coriolis parameter ji s responsiblefor makingthecurrents along the western boundary ofthe ocean for example, theGulf Stream in the Atlantic and the Kuroshio in the Pacific) much stronger thanthe currents on the eastern side. These are discussed in books on physicaloceanography and will no t be presented here. Instead, we shal l now tum ourat lention to the influence of Coriolis forces on inviscid wave morions.


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498 GeophysicalFluid Dynamics 9. Normal Modes in a Continuously Stratified Layer 499

which c an b e w ri tt en as

(52)

(53)

(54)

51)

(48)

(47)

(50)

(49)ap0= gpaz

au 1 ap fv=at Po axav 1 ap fu= -at Po ay

ro f=,,;; W H l/Jn z) dzal dl Jp Pn ndz

p PON 2 w Oat g

ro[u, v, pi Po] = [u,,, Vn, p,,]l/Jn(Z)

n=O

fluid layer. No w consider a continuously stratified medium and assume tha tthe horizontal scale of motion is much larger than t he ver ti cal s ca le . The .pressure distribution is therefore hydrostatic, an d the equations of motion are

au av aw =0ax ay az

where p and p represent perturbations of pressure an d densi ty from the s la teof rest. The advective term in the density equation is written in the inearizedform w dpj dz) = -P oN 2 wl g where N z) is the buoyancy frequency. In thisform th e rate of change of density at a point is assumed to be only du e to thevertical advection of the backgr ound density distribution pez), as discussedin Section 7.18.

In a continuously stratified medium, it is convenient to u se t he rne thodof separation of variables an d write q = qn(x, y, t)l/Jn z for sorne variable q.The solution is thus written as the su m of various vertical modes, which arecalled normal modes because t he y t ur n o ut to be o rt h og o na l t o e ac h other.The vertical structure of a mode i s descr ibed by l , and q describes thehorizontal propagation of the mode . Although eacn mode propagates onlyhorizontally, the su m of a number of rnodes can also propagate vertically ifthe various q are out of phase.We assume separable solutions of the form

(46)

(45)

(44)

av dT- + f u = - g at ay

aT+H au av ) =0ato ax ay

dU dT fv= g-dt ax

In the m or nentum equations of (45), th e pressure gradient terrns ar e wr ittenin the forrn (42) an d the n o n l i n e ~ advective terms have been neglected underthe srnall amplitude assumption. Equations (45), c al l ed t h e shallow waterequations, govern the motion of a layer of fluid i n w hi ch the horizontal scaleis much larger than the depth of the layer. These equations will be used inth e following sections fo r studying various types of gravity waves.

Although the aboye analysis has been forrnulated for a layer ofhomogeneous fluid, Equation (45) is applicable to internal waves in a stratifiedrnedium, if we replaced H by the equivalent deplh He, defined by

where w 7/) is th e vertical velocity at the surface, and w O) = Ois th e verticalvelocity a t t h e b o tt o m. The surface velocity is given by

r ar a7 rw 7 / )= - = - + u - + v -Dt at ax ayTh e continuity equation (43) then becomes

au av a 7/ ih a H r ) - + H + 7 / ) - + - + u - + v - = Oax ay iN ax ay

a7 a a- +- [ u H + 7 / ) ] +- [ v H + 7 / ) ] = oat ax ayThis simply says that the divergence of th e horizontal transport depresses thefree surface. Fo r small amplitude waves, t h e q u ad ra t ic n o nl i ne a r t e rm s c an b eneglected in c om pa ri so n t o t he l in ea r t erms, so t ha t t he divergence term in(44) simplifies to HV . u.

The linearized continuity and m om entum equations are then

whe re e is t h e s p ee d of long nonr otating internal gravity waves. This will bedemonstrated in th e following seclion.

9. Normal Modes in a Continuously Stratified LayerI n t h e p r e ce di n g s e ct io n we considered a homogeneous medium, an d derivedthe governing equations f or waves of wavelength larger than the depth of the

where the amplitudes u,,, Vn, Pn, w, and p are functions of (x,) , t). The z-axisis mea su re d f rom the upper free surface of t he f luid l ayer , and Z = Hrepresents the bol tom wal l. The reasons ror assuming the various forrns ofz-dependence in (52)-(54) a re the following: The variables u, v, an d p havet h e s a me vertical structure in order to be consistent with (48) and (49). Thecontinuity equation (47) requires that the vertical structure of w s h ou l d b e the


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500 Geophysical Fluid Dynami


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502 Geophysical Fluid Dynamics 9. NormalModesin a Continuous/y St atijied Layer 503

which gives

NHtan = OCn

The vertical modal structure is found from 69 . Since the magnitude of aneigenfunction is arbitrary, we can divide l o by Ao, obtaining

Nz coN Nz N 2z /to=cos sin = 1Co g Co g

where we have used N1zl/co 1 [since NH/co 1], and N Z/g 1 [sinceN 2 H/g= NH/co coN/g 1 both sides of (70) being much less than 1].Fo r this mode the vertical structure of u v and p is therefore nearly depthindependent. The corresponding structure for w [given by J/to dz as indicatedin ( 53 )] is linear in z with zero a t t he bolloro and a maximum at the upperfree surface. A stratified medium therefore has a mode of motion that behaveslike that in an unstratified medium; this mode doe s not feel the stratification.The n = Omode is called the barotropic mode.

The remaining modes n ;; 1 ar e baroclinic. For these modes cnNI g 1 butNH I Cn is no t small, as can be s een in Figure 13.13, so that the baroclinicroots of (70 ) are nearly given by

(68)

(67)

(69)

(70)

at z =0

at z= -H

with the boundary condionslj N 2_n - f J =0dz g n

dfJn=OdzThe set 66 - 68 defines an eigenvalue problem, with fn as the eigenfunctionand Cn as the eigtOnvalue. The solution of (66 ) is

Application of the surface boundary condition (67) givescnNBn= - - -Ag

The bottom boundary condition (68) then givesNH cnNtanCn g

Taking a typical depth-average oceanic value of N 1 0- 3 s -1 and H - 5 km,the eigenvalue for the first baroclinic mode is c, - 2 mis. The correspondingequivalent depth is He = di g - 0.4m.

An examination ofthe algebraic steps leading to(70) shows that neglecting(he right side is equivalent to replacing the upper boundary condition (65)by w = O a t Z = O. This is called the r ig id l id approximation. The baroclinicmodes are negligibly distorted by the rigid lid approximation. In contrast, therigid lid approximation applied to t he barotropic mode would yield Co=


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504 Geophysical Fluid Dynamics 10. High and Lo , Frequency Regimes in Shallow Water Equations 505

v = ; e j k x + I . I - w l )where k is the eastward wavenumber and I is the northward wavenumber.Then (75) gives

where V = i lax 2+a lay is the horizontal Laplacian operator.Equation (75) is Boussinesq, linear, and hydrostatic , but otherwise quitegeneral in the sense that it is applicable to both high and low frequencies .

Consider wave solutions of the form

73

(74)

75

(72)

a v a , ,av av gH V v + f - - - g H , B - = Oa J al H o al ax

a u av a au av) f gH - -al 2 al ax ax aya2v au a au av)2 f =gH - al al ay ax ay

Now take alal of (73) and use (72), obtainingaJv f[fav a au aV)J i a ll av) gH - - =gH - -al al ax ax ay ayal ax ay

To eliminate u, we first obtain a vorticity equation by cross differentiating andsubtracting the momentum equations in (45):a au av) au av)ai ay - ax -fo ax +ay -{3v=OHere we have made the customary {3-plane approximation, valid if the y-scaleis small enough so that t>flf 1 Accordingly, we ha ve treated f as constantan d replaced it by an average value fo) excepl when di l dy appears ; thi s iswhy we have wri tten fo in the second term of the aboye equation. Taking thex-derivative, multiplying by gH, and adding to (74), we final ly get a vortici tyequation in terms of v only:

ing inlernal gravity waves. The j3-effect will be considered in this section. Sincef varies only northward, horizontal isotropy is lost whenever the ,B-effect isincluded, and i t becomes necessa ry to dis tinguish between the diffe renthorizontal directions. We shall follow the usual geophysical convention thatthe x-axis is direc ted eas tward and the y-axis is directed northward, with uand v the corresponding velocity components.

The simplest way to perform the analysis is to examine the v-equat ion. Asingle equation for v c an b e derived by first taking the time derivatives of themomentum equat ions in (45) and using the continuity equation to eliminateaTllal. This gives

Fig.13.t4 Vertical distribution of afew normalmodesin a stratified medium of uniformbuoyancyfrequency.

because it satisfies do/ni dz =O at z = - H. The n th mode O has n zerocrossings within the layer (Figure 13.14).A decomposition into normal modes is only possible in the absence oftopographic variations an d mean currents with shear. It is valid with or withoutCoriolis forces an d with or without the (3-effect. However, the hydrostaticapproximation here means that t he f re quenci es a re much smaller than Under this condition the eigenfunctions are independent of the fre quency, as(56) shows. Without the hydrostatic approximation the eigenfunctions l Jbecome dependent on the frequency w. This is discussed, for example, inLeBlond and Mysak (1978).

Summary: Small amplitude motion in a frict ionless continuouslystratified ocean can be e o m p o s e ~ in terms of noninteracting vertical normalmodes. The vertical structure of each mode is del ined by an eigenfunction1jI,, z . If the horizontal scale of the waves is much larger t han the verticalsca le , then the equations governing the horizontal propagation of each modeare identical to those of a shallow homogeneous layer, with the (ayer depth Hreplaced by an equivalentdepth defined by = gHe For a medium of constantN the baroc inic (n; ) eigenvalues are g iven by c =NH/7T n while thebarotropic eigenvalue is Co = JgIl. The rigid l id approximation is quite goodfor the baroc inic modes.

10. High an d Lo w FrelJuency Regim es in Shallow WaterEquations

We shall now examinewhat terms are negligiblein the shal ow waterequationsfor the various frequency ranges. Our analysis isvalid fora singlehomogeneouslayer or for a stratified medium. In the lalter case H has to be interpreted asthe equivalenl depth, and e has t o be interpreted as the speed of long nonrotat-

w J - c wK - fw -c 2{3k = O (76)where K 2 = e + 12 and e ~ t can be shown that al roots of (76) arereal, two ofthe roots being superinertial > f and thethird being subinertial j Equat ion (76) i s the complete dispe rs ion relat ion for l inea r sha llowwater equations. In various parametricranges it takes simpler forms, representing simpler waves.


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506 Geophysical F1uid Dynamics 11. Gravity Waves with Ra.a.ian 507

l

This is the dispersion relation of gravity wayes in the presenee of Coriolisforces. [The relat ion can be most simply deriyed by setting the determinantofthe set of linear homogeneous equations (77)-(79) to zero.) t can be writtenas

where K = e 1 is the magnitude ofthe horizontal wavenumber. The dispersion relation shows that the waves can propagate in any horizontal directionand have w > f Gravity waves affeeted by Coriolis forces are called Poincarwaves, Sverdrup waves, orsimply rotational gravily waves. [Sometimes the namePoincar wave is used to describe those rotational gravity waves that satisfythe boundary condi tions in a channel.] In spite of their name, the solutionwas first worked out by Kelvin (Gill, 1982, page 197). A plot of (82) is shownin Figure 13.15. tis seen that the waves are dispersive exeept for w when(82) gives w 2=gHK 2, so that the propagation speed is w/K=,gH. The highfrequency imit agrees with our previous discussion of surface gravity wavesunaffected by Coriolis forces.

(82)

81 )

(80)

Substituting these in (79), we getw 2 - j = gH k 2+ 2

Solving for and obetween (77) and (78), we get = 2 wk + if7w V= 2gi 2 -ifk wfw

First consider high frequency waves w f. Then the third term of (76) isnegligible compared to t he first termo Moreo ver , the fo urt h term is alsonegligible in this range. Compare, for example, the fourth and second terms:

c2n k - _1 3c2wK 2 wKwher e we have a ss umed t ypi cal values of ,I3=2xlO-1I m- l s- , w=3 j 3 x 10-4 S I and 27Tj K -100 km. For w 1, therefore, the balance is betweenthe first and second t erms in (76) , and t he roo ts a re w = K gH, whichcorrespond to a propagation speed of w j K = gH. The eftects of both f and 3 are therefore negligible for high frequency wayes, as is expected since theyare t oo f as t t o be affected by the Coriolis efteets.Next consider w > 1, but w - f Then the third term in (76) is not negligible,bu t the {3-effect is. These are grayity waves infiuenced by Coriolis forces;gravity waves are discussed in the next seetion. Howeyer, the time seales arestill too short for the motion to be affected by the ,I3effect.Lastly, considervery slow waves for which w fThen the ,I3-efleetbeeomesimportant, and the first term in (76) becomes negligible.Compare, forexample,the first and the last terms:

w3 1c2 3kTyp ical v alue s f or the oeean a re e-200m/s for the barotropic mode,c 2 mis for the baroclil)ic mode, { = 2 X 10 - m-, S I 271 /k lOO km, andw - 10- 5 S l. Thi s makes the aboye r at io about 0.2 x 10-4 for the barotropicmode and 0.2 for the baroc lini c mode. The first t erm in (76) is thereforenegligible for w f.

Equation (75) governs the dynamics of a variety of wave motions in theocean and the atmosphere, and the discussion inthis section shows what termscan be dropped under various limiting conditions. An understanding of theselimiting conditions will be useful in the following sections.

KFig. 13.15 Dispersion relalions fOI Poincar and Kelvin waves.

Gravi ty Waves with RotationIn this chapter we shall examine several f ree wave solutions of the shallowwater equat ions. In the present section we shall study gravity waves withfrequencies in the range w >1, for which the {3-effect is negligible, as deroanst ra ted in the preceding section . Consequently, the Cor io lis f requency f isregarded as constant here. Consider progressive waves of the form

u, v 71 = o 7] ei kx ly- wIv:here V and i/ are the complex ampli tudes, and the real part of the rightslde \s meant. Then (45) gives

-iw - jo = -ikgi 77- iwv j = - ilg7] 78

-iwi+ iH k+ IV =0 (79)

f

geofisical fluid dinam 2d1

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