D

— <Pti H--<Pi> cos (p I--<Pco — (oa + (uFk + vF

Let us substitute Equations (2.5.13) into this equation and average it over the longitude. As a result we have

(luY + M2 + [u*2] + [>*2]\ + l d_ |M2 + M2 M dt \ 2 2 / a cos (p d(p [ 2

[t^l + Ct^2] „ I d JM2 + M2r +---[o] + ([u][i/*u*] + M[r*2])j cos (p + — j---[<o]

ru*2-| , r *2-| + --iyt-_ [ft,] + ([M][U*0>*] + |>]|>*C0*])

I jA

([®]M + [®*t>*]) cos (p--([®][a>] + [0)*CO*])

([>][a] + |>*a*]) + (M[FJ + M[F„]) + flWJ] + [WJ]). (2.5.26)

Subtracting (2.5.21) from (2.5.26) we have the desired equation for KE:

+ l>*2]M > cos q> + — ^-—± [o>] + ([u*fi)*][u]

+ |>*co*][>])} + - ({u*t?*] ^ + [>*2] ^ J a \ d(p dip J

+ I [u*(o*] ^ + [i?*co*] ^T) + tan^ ([>*«*] [u] - [u*2]M)

Integrating (2.5.27) over the whole mass of the atmosphere and taking into account the boundary conditions we finally have d

where

KEa2 cos <p dX dip dp/g = C(AE, KE) - C(KE, Ku) - D(KE), (2.5.28)

([u*FJ] + [t;*F*])a2 cos <p dX dip dp/g. (2.5.30)

In Equations (2.5.22) and (2.5.28) the constituents C(KE, KM) describe the mutual transformations of kinetic energy of zonal mean (KM) and eddy (KE) motions. It is obvious from Equation (2.5.24) that these constituents represent the work of Reynolds stresses on the gradients of zonal velocity. In addition, the first and third terms in the integrand describe the so-called barotropic instability of zonal circulation. The remaining terms in (2.5.22) and (2.5.27) describe the dissipation of kinetic energy of zonal mean and eddy motions (components D(KM) and D(KE)) and mutual transformations of kinetic and available potential energy (components C{AM, KM) and C(AE, KE)).

The budget equations for the available potential energy can be obtained in much the same manner as we have just shown for KM and KE. Therefore, without repeating the derivation, one might restrict oneself to a discussion of the final expressions. But to avoid misunderstanding, which often occurs when presenting this subject, we will proceed in the opposite direction. So, we will start from the heat budget equation, written in terms of potential temperature. This equation, after transition to the isobaric system of coordinates, takes the form

Let us define 0 as the sum of the average (over the sphere surface) value 9 and of its deviation 6'. We define velocity components as well as heat sources and sinks in the atmosphere similarly. Substitution of these relationships into (2.5.31) and subsequent subtraction of the equation for 9

from the resulting equality yields

Here we have assumed that the multiplier 9/T on the right-hand sides of Equations (2.5.31) and (2.5.32) does not depend on horizontal coordinates, and have also taken into consideration the fact that in accordance with the continuity equation integrated over the surface of a sphere the isobaric vertical velocity co obeys the equation 8a>/8p = 0; from here, with co equal to zero at the upper atmosphere boundary (p = 0), it follows that <5 = 0. We introduce definitions

F'T = [M + F%, where, as before, the square brackets signify an average over the longitude, the asterisk is a deviation from the zonal mean, and then we substitute (2.5.34) into (2.5.33), rewritten in divergent form, and average the equation obtained over the longitude. As a result we have

~ + —— ^ ((MM) + [W]) cos <p) dt a cos (p dip dp T cp dp or, after multiplying by [0], ot 2 a cos (p o<p\ 2 /

+ — (WL [Q,] + [0][0*co*] ) - ([0*i>*] — + [0*co*] — )[0]

T cp dp

We transform the expression for the last term on the right-hand side of (2.5.35). But first we note that

where, as before, ya = g/cv and y = — 8 T/dz are adiabatic and real vertical temperature gradients. Thus, dp g\P J

0 0

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