some aspects of the theory of fronts and frontal analysis

551.509.311 : 551.515.81

Some aspects of the theory of fronts and frontal analysis

By T. H. KIRK Meteorological Ofice, Brucknell

(Manuscript received 6 December 1965; in revised form 28 March 1966; communicated by F. H. Bushby)

SUMMARY

Fronts are considered as aspects of development and it is shown that their characteristics can be deduced without recourse to the traditional concept of air masses. The approach interprets the current practice of synoptic analysis in terms of dynamical theory and, as a corollary, affords the possibility of utilising automatic computational methods for the objective depiction of frontal zones and their associated weather.

1. INTRODUCTION

For many years meteorologists have been aware of the many limitations of the frontal theory and of the practice of frontal analysis. Critical comment is to be found in the work of Bleeker (1958a), Flohn (1958), Reed (1955) and Sutcliffe (1952). A recent comprehensive account of some of the difficulties inherent in the concept of fronts and in their practical interpretation has been given by Taljaard, Schmitt and Van Loon (1961).

Fronts have been discussed from many aspects without agreement either on a suitable definition or on what constitutes their real essence. For the most part, the discussion has proceeded from a prior consideration of the temperature field and the concept of air mass has been adopted, either explicitly or implicitly.

Recently, Bleeker (1958b), Eliassen (1962) and Sawyer (1956) have stressed the role of vertical circulation as an essential characteristic of a front and their approach has been essentially dynamical in character, although limited to simple models.

In this paper, the fundamental approach is that a front must be regarded as an aspect of the developmental field in the same way that a depression is viewed as a manifestation of development. Accepting this point of view, it is then possible to show that known frontal characteristics can be interpreted in terms of the general equations of motion.

The immediate advantages accruing from this standpoint are :

(a) there is no appeal to the unsatisfactory concept of air mass;

(b) fronts are necessary products of the cyclonic development ;

(c) fronts must be closely related of necessity to weather and to cyclogenesis;

(d) the dynamical characteristics of fronts can be interpreted directly.

Throughout, fronts are considered as upper-air features and the influence of the friction layer is negjected.

2. TEMPERATURE DISTRIBUTION IN RELATION TO VORTICITY

Fundamental to this new approach is the relation between the temperature field and the

The equations of motion may be written in vector form, in isobaric coordinates :

vorticity. This may be derived as follows :

dV d t - + f f k x V = - V 4 *

374

FRONTS AND FRONTAL ANALYSIS 375

where V is the velocity vector, # the geopotential, Fz a unit vertical vector andf the Coriolis parameter. Taking the ' divergence ' of each side

dV Div- dt = f( - ~ 2 4

where 5 is the vorticity and where variations of latitude have been neglected for simplicity.

Differentiating with respect to p , and inserting the relation h#p = - a where tc is the specific volume :

For quasi-geostrophic flow

Thus V2 x determines the vertical lapse rate of geostrophic vorticity and vice versa (N.B., p increases downwards).

Since GI = RT/p, Eq. ( 2 ) becomes

To elucidate Vz T Div GT, in Eq. (3), set VT = lOTl a, where (VT( is the magnitude of the temperature gradient, and zz is a unit vector in the direction of the ascendant. Then,

V 2 T = a . V (VTI + lVTl Diva. ' (4)

The first r.h.s. term in Eq. (4) is a measure of the degree of crowding of the isotherms. The second term determines the degree of turning of the isotherms. Both effects are significant for the quantity V2 T.

The crowding of the isotherms has been accepted in the past as a characteristic of a frontal zone. The frontogenetic effect due to the turning of the isotherms has hitherto received little attention although there is no a priori reason for neglecting it.

The quantity Vz T therefore appears to be a useful and significant parameter for the definition of a front in the field of temperature. The forward and rear edges of the frontal zone are specified by critical values of Vz T while the zone itself is characterized by a gradient of V2 T, i.e., by V (V2 T) .

It is now apparent that ' discontinuities ' of temperature gradient arise from the differential concentration of geostrophic vorticity, in the vertical, whether by advection, heating, vertical motion, mixing processes or other means. It must follow that these discontinuities are natural products of the circulation in a depression - one might say that they are inevitable products of the circulation.

3. THE FACTORS SIGNIFICANT FOR FRONTOGENESIS

Eliassen (1959) has examined the concentration of thermal gradient in horizontal deformation fields and has concluded that the mechanism cannot provide for the degree of sharpness of frontal zones found in nature. He argues that the completion of the frontogenetic process must be due to some other mechanism and suggests that vertical circulation is probably most significant. This vertical circulation is envisaged as a mechanism whereby approximate geostrophic balance is maintained in the atmosphere.

376 T. 13. KIRK

The argument may be given more precise form by the use of the vorticity equation which may be written in the form

311 - + Div (qV) + Div at

where q is the vertical component of absolute vorticity. The last term of Eq. (5) may be written

The first of these terms is the twisting term; the second term is the vertical advection of vorticity. Jones and Graystone (1962) have demonstrated that these two terms tend to balance each other in atmospheric motions and that their sum is an order of magnitude less than either of them. Expressing this mathematically, with geostrophic values, gives

A gradient of vertical velocity (0 dp ld t ) in a thermal gradient (Va) therefore entails a non-zero value for V2 M and this term is a measure of the ' discontinuity ' of the thermal gradient such as occurs at a frontal zone.

If a value of Vz a is achieved by means other than the action of the twisting term, the quasi-geostrophic control ensures the creation of a gradient of vertical velocity through the mechanism expressed by Eq. ( G ) , i.e., the occurrence of frontal activity. Thus Eq. (6) expresses a valve-like action whereby gradients of vertical motion and the temperature discontinuities find mutual adjustment.

It is important to note that this mechanism supplements the concentration of thermal gradient in horizontal deformation fields and provides for the completion of the frontogenetic process as envisaged by Eliassen. The recognition of the significance of Vz u (proportional to Vz T on an isobaric surface and hence expressible as the divergence of the temperature gradient) leads to a view of frontal activity inherently associated not only with thermal gradients but also with gradients of vertical motion.

4. THE ROLES OF CURVATURE VORTICITY AND SHEAR VORTICITY

The quantity V2 u may be written - f d(,lBp. The geostrophic vorticity may be expressed by

where B/as and 3/3n represent differentiation along and across the contour, V, is the geostrophic wind speed and 9, is the direction of the contour measured in a direction anticlockwise from east. The curvature vorticity 5, = V, 3#,/ds and the shear vorticity 1, = - bVg/3n.

which may be written in the form


The term inside the brackets is usually of minor importance compared with the first term in regions of strong curvature. It follows that the frontogenetic effect of the curvature vorticity term is dependent on the rate of change along the contour of the vertical shear of wind direction.

Also,

In zones of pronounced speed shear the last term is of minor importance compared with the first term. It follows that the frontogenetic effect of the shear vorticity term is dependent on the cross-contour gradient of the vertical shear of wind speed. Both these criteria can be used on upper-air charts for the identification of frontal zones.

5 . FRONTAL ACTIVITY IN RELATION TO DEVELOPMENT

The next step is to see how frontal activity can be conceived in terms of development. From Eq. (1)

3V 3p d t 3P

x E + f - = v u x K . 3 dV --

For geostrophic flow, the vertical shear of the geostrophic wind balances the solenoidal field, represented by V u x K . When the motion is not completely geostrophic the unbalanced solenoids available for a change of circulation must be represented by the disparity between the vertical shear termfJV/bp and the solenoidal term Vu x K . They must therefore be represented by 3/3p dV/dt x E. Representing the unbalanced solenoidal vector by R

[( :: 1 1 3 V z m = - f V - D i v v + Curl

3P f- + V 2 u 6 .

If the flow is quasi-geostrophic f d[/bp = - Vz 0: and Vz N = - f V 3/3p Div 7. Thus the Laplacian of the unbalanced solenoidal vector N is proportional to the gradient of the developmental field. Unbalanced solenoids, available for changes of circulation, are therefore concentrated where there is a gradient of the development.

6. THE ROLE OF THE DEFORMATION

The importance of the deformation in large-scale development has already been recognized and Petterssen (1953) has shown that approximately

f 1 - V2$ = + [Fz - 5'1 where F is the magnitude of the deformation.

The assumption of quasi-geostrophic flow thus implies an essential balance between the magnitudes of the vorticity and the deformation. If this balance is not achieved, then

and the influence of the deformation in the determination of V2 u is apparent.

378 T. H. KIRK

If dV/dt be regarded as an index of dynamical activity, the role of the deformation

The deformation is usually defined in terms of the quantities A and B where

can be elucidated to some extent as follows :

3u 3v 3v 3u A = - - - - . B = - + - . 3x by ' 3x ay

The deformation F is then given by F2 = A2 + B'. Now consider the vector

F = rv-vr* x k:

where V is the wind speed and # is the direction of the streamline measured in a direction anticlockwise from east. Then

F = A' 5 + B' A where t and f i are unit vectors along and

/FI2 = (A')' + (B')' = F2. across the flow and

The quantity V F appears to be more tractable than F ; for convenience it will be referred to as the ' deformation transport.' After considerable algebra it can be shown that

dV -

dt VC2 - = t f [k. Curl WF) - V . Vt;,] + ri f [ F 2 - Div (VF) + K . (YC, x V)].

This equation shows how the deformation is related to the separate terms V * VC, and K . ( V l , x V).

If u and v are Cartesian components of the velocity then it is easily shown that

and K * Curl (VF) = u'i2 v - vC2 u

Div (VF) - F' = uV2 u + vV2 u,

which enable the appropriate deformation terms to be evaluated from the wind field.

7. THE DIAGNOSIS OF FRONTS

The results of the previous discussion suggest that variations of vorticity and divergence are closely related to many physical characteristics of fronts. It is useful therefore to examine the various criteria that are adopted in current practice for the identification of fronts.

(a) Temperature field

A discontinuity of temperature gradient is determined by the critical values of - V2 a- A front may be regarded as bounded on either side by these critical values and characterized by a maximum gradient of V2 a, i.e. by a maximum of V (V2 a). Since - V2 a = f 3 [ , / 3 p it follows that the critical values of V2 a define the upper and lower boundaries of a zone of maximum gradient of geostrophic vorticity.

( b ) Barometric tendency field

At a constant pressure surface, say at 1,000 mb approximating to the surface chart, the tendency may be represented by &$/at. A discontinuity of the tendency gradient is given by a critical value of - V2 34/3t , i.e., by a critical value of - f b[,/bt. A discontinuity in the field of tendency gradient therefore identifies the maximum local rate of change of geostrophic vorticity.


(c) W i n d field

V 2 V = V (Div 0) - 05 x E, Thus V2V, defining the discontinuity of wind, depends on both the gradient of

For the geostrophic wind vorticity and the gradient of the divergence.

The discontinuity in the geostrophic wind therefore occurs where the gradient of geostrophic vorticity is a maximum.

( d ) Vertical motion

For the criteria based on the effects of vertical motion, such dewpoint, the parameter dpldt may be considered as appropriate.

Since the equation of continuity imposes the relationship

as rainfall, clouds and

--- d p - Div V, bp d t

it follows that * d t = S D i v V d p and V2- d p d t = Vz DivVdp.

In other words, the relevant factor is the distribution of the integrated divergence. This is not strictly related to events at any particular level.

It will be seen that the various criteria used in practice for locating a front are, in effect, examining different aspects of the vorticity and divergence fields and hence some discrepancies between them are inevitable. These discrepancies are minimized in the ideal case when the vorticity is concentrated to a vortex line. This concentration occurs in practice more effectively and frequently at the surface.

8. THE PHYSICAL ASPECTS OF LOCAL FRONTOGENESIS

In the foregoing account the greatest emphasis has been placed on the kinematical aspects of fronts and frontogenesis. If, as suggested, frontogenesis is not confined to the concentration of thermal gradient in a deformation field but is a complex process, it is not readily apparent what index would be appropriate for frontogenesis or even whether such an index could be a valid representation. However, the concept of local frontogenesis, by which is meant the local rate of increase of V2 a, is useful in an examination of the physical factors of importance in the process.

Since ha d a ba d p - = - _ V . V a - - - b t d t 3p d t

3 da 3 t d t - v ~ ~ = V Z - - V ~ ( V . V ~ ) - V ~

The 1st law of thermodynamics may be written

d T d a dQ -. - “ z + P z ’ d t

where dQ/dt is the rate of heat entry to the system and c, is the specific heat at constant

da 1 d v2-= --v2-

d t P d t

volume, whence [cU T - Q1.

380 T. H. KIRK

The term V2 daldt therefore measures the frontogenetic effect of differential heating, including processes such as condensation and evaporation which are accounted for in the term dQ/dt.

The term - V2 (V . V a ) measures the effect of differential advection. One of the most important characteristics of a front is that its passage in general changes the advection process, i.e., at the front there is a ‘ discontinuity ’ in the gradient of the thermal advection. The magnitude or sharpness of this discontinuity is expressed by - V2 (V . Va) . From Eq. (7) therefore, it follows that the factors making for local frontogenesis are :

(a) differential heating, including the effects of condensation and evaporation, radiation

(b) differential advection;

(c) differential vertical motion, acting through the term V2 (balbp dpldt).

Since all these processes contribute to 3/3t V2 a, i.e., to - f >/at 3<,/3p, it follows that the change of the vorticity lapse can be regarded as the common factor by which all frontogenetic processes are made effective.

etcetera;

9. A RE-APPRAISAL

It is now possible to re-appraise the position as :

(a) The airmass concept is not essential to the theory of fronts and can well be discarded or rather re-interpreted.

(b) Fronts may be conceived as products of the developmental process. Their characteristics can then be deduced from the dynamical equations.

(c) Variations of the gradients of temperature and other elements (‘ discontinuities,’ if sufficiently sharp) can be interpreted by the distribution of Div VT etcetera, i.e., by the use of the Vz operator. This affords a method of investigating frontal structure at any level.

(d) Frontogenesis is associated with the concentration of vorticity and it is reasonable to assume that this concentration of vorticity is the fundamental mechanism in terms of which the frontogenetic processes must operate.

(e) The airmass theory can have a validity only if re-interpreted in terms of vorticity transfer.

ACKNOWLEDGMENTS The writer is grateful to F. H. Bushby for communicating this paper and to R. Dixon

for discussions on some mathematical difficulties. This paper is published by permission of the Director-General of the Meteorological Office.

Bleeker, W.

Eliassen, A.

REFERENCES 1958a ‘ Fronts and the jet-stream; historical remarks and objections

to the frontal theory,’ Met. Abhandlungen Band IX, Heft 1, pp. 77-83.

1958b ‘ Fronts and the jet-stream; the front as a circulation system,’

1959 ‘On the formation of fronts in the atmosphere,’ Rossby Memorial Volume, Rockefeller Institute Press, pp. 277- 287.

‘ On the vertical circulation in frontaI zones,’ Oslo Geofys.

Ibid., pp. 85-93.

1962 Publ., 24, pp. 147-160.


Flohn, H. 1958

Jones, D. E. and Graystone, P. 1962

Petterssen, S. 1953

Reed, R. J.

Sawyer, J. S.

1955

1956

Sutcliffe, R. C. 1952

Taljaard, J. J,, Schmitt, W. and 1961 Van Loon, H.

' Luftmassen, Fronten and Strahlstrome,' Met. Rund., 11,

' A note on the equation of mean motion as used in numerical prediction,' Quart. 1. R. Met. Soc., 88, pp. 250-255.

' On the relation between vorticity, deformation and divergence and the configuration of the pressure field,' Tellus, 5, pp. 231-237.

' A study of a characteristic type of upper level frontogenesis.'

' The vertical circulation at meteorological fronts and its relation to frontogenesis,' Proc. Roy. Soc., A, 234,

' Principles of synoptic weather forecasting,' Quart. J. R.

' Frontal analysis with application to the Southern Hemis-

pp. 7-13.

J . Met., 12, pp. 226-237.

pp. 346-362.

Met. SOC., 78, pp. 291-320.

phere,' Notos, 10, 1/4, pp. 25-58.

some aspects of the theory of fronts and frontal analysis

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