A note on t~e physical interpretation o] the balance equation

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1-locrynnzo t . 4. i957

A N O T E O N T H E P H Y S I C A L I N T E R P R E T A T I O N O F T H E B A L A N C E

E Q U A T I O N

v o JT~.CH VITEK Metearologlcal Institute, Charles University, Prague*)

If the changes in the Coriolis parameter are neglected the relation between the fields of pressure and motion in the horizontal non-divergent flow is determined by the equation

v +2x-lf0' ( (1> lO~ a y ' --~0----~-~ / J

In Eq. (13 ~ denotes the stream function, q~ the geopotential, ~. is the Coriolis parameter, x and y are horizontal coordinates of the standard coordinate system. Relation (1) is the so-called balance equation, the properties of which were dealt with in detail for example by Bolin [1].

The balance equation reduces to the relation for the geostrophic vorticity if the non-linear erms are equal to zero, i. e. if

a x, a y , ~o-;-~-y] = o. (2)

This relation can thus be considered as a sufficient condition for the vorticity of the horizontal non-divergent flow to be equal to the geostrophic vorticity. Equation (2) is then the differential equation defining the geometric properties of such a non-divergent flow, the vorticity of which fully satisfies the geostrophic approximation.

Equation (2) is satisfied by the solution

( x , y ) = A x + By + f ( a x + by + c), (3)

*) Address: Praha II , Ke Karlovu 3.

Studla geoph, et geod. t (1957) 385

V. Vitelc

where f is an arbitrary function having continuous partial derivatives of the second order, A,. B, a, b and c are constmats. Let us first consider the simple case when

f ( a x -F b y + c ) = a x + by + c, so that

~ ( x , y ) = axx.-~-bly + c. (4)"

Solution (4) corresponds to the ,,degenerate" non-divergent flow, the relative vorticity ~ = V[ y" of which is everywhere zero. In this case the constant a~ has the meaning of a meridional compo- nent of velocity, the constant b~ is the negative zonal component of velocity. The streamlines. are straight lines. Solution (4) is of no importance from the synoptic point of view.

I f A = B = 0, it holds that ~ ( x , y ) = f ( a x + by W c). (5).

The relative vorticity is then determined by the equation

d ' f (z) ( x , y ) = ( a 2 + b 2) d z ~ - - g ( z ) " (6),

In Eq. (6), for the sake of simplicity, we put z = a x + by + c. The slope of the tangent o f the streamline ~ = const, is then at every point equal to

0 ~v (x,y) O x a

0 ~o (x, y) b O y

The same result obviously holds for the isoline of relative vorticity ~ = const. Both the stream- lines and isoLines of relative vorticity are thus parallels. This also follows directly from relations (5) and (6). For the left sides of these equations to attain constant values, it suffices to put a x + by = ~-~ const. The field of motion described by Eqs. (5) and (6) is translationally ineffective with respect to the relative vorticity, i. e. there is no horizontal advection of the relative vorticity.

In the general case described by relation (3) the relative vorticity is again defined by Eq. (6); the isolines of the relative vorticity are thus again straight lines a x + by = const. The geometric properties of the field of the streamlines, on the other hand, depends on the choice of the function f ( z ) . If, for example, we put f ( z ) = ( a x + by + c)~ the streamline is a parabola. In the field o~ motion given by relation (3) horizontal advection of the relative vorticity generally takes place.

Solution (3) is not, of course, the only solution of Eq. (2). If, for example, we assume a solution of the type ~o (x, y) = F (x) G (.y), we obtain

1 C ~o = (A~ x + B1) 1 - c ( A , y + B2) c - ~ . (7)

The stxeamlines are then generally higher parabolas (hyperbolas) according to whether the constant- C is greater or smaller than zero. In this paper we shall consider only solution (3), which obviously holds more widely than solution (7).

The considerations dealt with so far can be summed up as follows: Eq. (2) is a sufficient condi- tion for the vorticity of the horizontal non-divergent flow to be equal to the geostrophic vorticity. The solution of this equation prescribe~ the geometric properties of the corresponding field off motion. I f we limit ourselves to solution (3) then the geostrophic vorficity describes such a field where the isohnes of the relative vorticity are parallels. The geostrophic vorticity thus need not correspond to the real situation particularly in the nei~hbourhood of the baric centres and out- standing ridges (troughs), where there are usually closed centres of vortieity. This probably partly ~xplains the fact that the models working with geostrophic hypothesis relatively rarely fully predict the position and intensity' of the centres of cyclones and anticyclones.

Received 7. 1. 1957 Reviewer: M . Brdi~kcc

Rcjr~rc/,/ce.

[1] B. Bo l in : Numerical Forecasting with the Barotropic Model, Tellus, 7 (1955), 27.

~88 Studia geoph, et geod. 1 (1957~

Remarks on the mechanism ol surface-pressure changes in baroelinic a tmosphere

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HPHMEHAHHE K ~HSHHECKOH HHTEPHPETARHH YPABHEHH~ BAhAHCA

VOJTI~CH VITEK Memeopo.~oz~qeck~ ~cm~myrn l(ap~o~a ynnsepc~mema, Hpam.

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I I o c r y n n ~ o 7. t . t957

R E M A R K S O N T H E M E C H A N I S M O F S U R F A C E - P R E S S U R E C H A N G E S

I N B A R O C L I N I C A T M O S P H E R E

STANISLAV BRANDEJS Metearological Institute, Charles University. Pragut*)

Let us consider an isobari~ layer Po---'Pi, identify the isobaric surface Po with the level 100 cb, the isobaric surface Px with the level 20 cb, and assume that the wind vector tJ in each level of the chosen layer is given by the relation

13 (x, y , p; t) = 13o ( x , y ; t) + a (p) tJ" (x, y ; t), (I)

where 13o is the wind vector in the isobaric surface Po, 19' = 19 s - - Do; tJs denotes the wind vector in the isobaric surface

p,

f ' Ps = (Po - - P l ) - x p dp = -~- (Po + Px) = 60 cb

Pt

and the proportionality factor a (p) = ( P 0 - Ps) - x ( P c - P). Relation (1) holds analogically for the relative vorticity r = ]~ �9 ~7p X D, where

( 0 0 ) v p = - 5 - ; i + - - ~ i p

and ~ is the uni t vector of the Cartezian coordinate system i, I, ~ pointing to the local zenith. F rom the foregoing definition of relative vorticity i t is obvious that

r ( x , y , p ; t) = •0 ( 'x ,y ; t) + a ( p ) r ( x , y ; t), (2)

where r denotes the relative vorticity in the isobaric surface Po and r = ~. Vp • l~" is the so-called thermal vordcity.

Let us substitute relations (1) and (2) into the simplified vorticity equation in the form

0r 2" 20~ 0--7- + ~" vp (r + ) - - ~ = o, (3)

dp where :t denotes the Coriolis parameter and co = ~ is the generalized vertical velocity in the

*) Address: Praha II , Ke Kaxlovu 3.

StudJa 9eoph. et geod. I (1967) 387