WjP uiP 1r ir T

+ (fiu'jT' + 0.61 gufflSu + v — uju} -2v-±-±. (3.10.4)

OXk i cxk

Here the first term on the left-hand side describes time change in the covariance u[u'j, the second term describes its transfer by mean motion, the third term describes the rate of generation due to the interaction of mean and fluctuating motions, the fourth term describes the divergence of the transfer by velocity fluctuations. The first four terms on the right-hand side describe the changes created by the interaction of pressure with the velocity field, and by buoyancy forces. And, finally, the last two terms describe the intensity of molecular diffusion and the rate of dissipation.

From (3.10.4) with i = j the budget equation for the velocity dispersion S71 follows:

5—tt _ d —py du, d , ,, — U: + Uk - U: + 2U:Uk--1--UkU:

at dxk oxk oxk

and from this, after division of all terms by 2, the budget equation for turbulent kinetic energy b = u'2/2 follows:

where the last term on the right-hand side describing the rate of turbulent energy dissipation is usually designated as s.

The budget equations for temperature (T,2) and specific humidity (q'2) dispersions are derived similarly. In this case the heat and humidity budget equations are initial equations. These equations, after being transformed to divergence form and rejecting small terms describing the radiative heat influx and water vapour phase conversions, take the form

where Xt and yq are coefficients of the heat and water vapour molecular diffusion.

Decomposing uk, Tand q again into mean and fluctuating components, and averaging the equations obtained we find df/dt + d(ukT+ u[T)/dxk = xT d2T/dx2k, (3.10.9)

Next, subtracting (3.10.9) and (3.10.10) from (3.10.7) and (3.10.8) and multiplying the resulting equations, dT'/dt + d(u'kT+uk r + u'kT' - u[T)/dxk = xT d2T'/dxl; (3.10.11)

dq'/dt + d(u'kq + ukq' + u'kq' - u'kq')/dxk = xq d2cf/dx\, (3.10.12)

by T' and q', respectively, we have after averaging d T'2 d T'2 —- dT 1 d —^ 82 T'2 (dT'\2

----1- uk---1- ukq--1---ukq = xq , dt 2 dxk 2 dxk 2 dxk dx£ 2 \dxk

In Equations (3.10.13) and (3.10.14) the first terms on the left-hand side describe the change in dispersion over time; the second terms describe the ordered transfer; the third terms describe the generation due to interaction of the mean and fluctuating fields; the fourth terms describe the divergence of transfer by velocity fluctuations; the terms on the right-hand side describe, respectively, the molecular diffusion and the rate of equalization (degeneration) of temperature and specific humidity inhomogeneities.

If we adopt conditions of quasi-stationarity and horizontal homogeneity, ignore the divergence of the transfer by fluctuating motion and molecular diffusion in the vertical direction, and direct the xt axis along the tangential wind stress, then Equations (3.10.6), (3.10.13) and (3.10.14) are rewritten in the form u[u'3 = ul = [_s- (pu'3r + 0.6\gu'iq')~]{dujdx3)

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