## Nn Tlg and nm

Let nn « 1, ng « 1 and nm « 1. In this case, according to Golitsyn, the function \j/F can be replaced by its limiting value (equal to unity) corresponding to zero values of IIn, Il9 and Hm. The assumption about the existence of the finite limit of the function ipF (in other words, complete self-similarity by parameters nn, II9 and nm) represents one more hypothesis forming the basis of the theory. An application of this hypothesis allows us to obtain a simple formula for the total kinetic energy E of the atmospheric circulation. Indeed, for the given assumptions,

where the factor 2n is introduced for the sake of convenience.

On the other hand, E = \(Ana2mA)V2, where V is the root-mean-square velocity of atmospheric motion. Hence, in accordance with (5.2.1)

V = (27[)1/2ma" 1/2(/<t)1/8cp- V/8a2- (5-2.2)

Next, if the advective heat transport in the atmosphere is balanced by the long-wave radiation emitted into space, that is, mAcpV^-^(fa)T4, (5.2.3)

na/2

where 5T is the characteristic equator-to-pole temperature difference, then after substitution of the expressions for V and Tr into (5.2.3) we have

ST=i(2n)ll2mAll2(fayilsc-ll2qll8a-1. (5.2.4)

We note now that for the Earth's conditions the inequality nn « 1 is not valid (for the Earth, IIQ = 1.43). Nevertheless, an agreement between estimates obtained with the help of (5.2.1), (5.2.2) and (5.2.4) and observed data turned out to be satisfactory. This demonstrates that the similarity theory may be used in this case, too.

In order to take into account the existence of two media - the atmosphere and ocean - and the processes of interaction between them Zilitinkevich and Monin proposed modification of the similarity theory as follows. Firstly, they replaced Equation (5.2.3) by the heat budget equation for the ocean-atmosphere system averaged over time throughout the annual cycle and over an area of the sphere surface between the equator and some fixed latitude (say, cp = 7r/4). This equation, on the assumption that there is no heat transport through the equator, is written in the form

where vT

MHTA = c mA — a is the meridional heat transport in the atmosphere at latitude (p = n/4, referred to a unit area of the sphere belt 0 < <p < n/4; MHT0 is the meridional heat transport in the ocean normalized in the same way; RA is the average (in the examined belt) net radiation flux at the upper boundary of the atmosphere; v is the meridional component of wind velocity; an overbar means averaging over the circle of latitude q> = n/4.

Second, the net radiation flux RA(cp) = FH<p) — F\q>) at the upper boundary of the atmosphere is set proportional to the difference in the temperature Ts at the underlying surface for (p = n/4 and its global average value T0. Substitution of the expressions for the absorbed short-wave solar radiation flux Fl(cp) = (4/n)q cos <p and for the outgoing long-wave radiation flux FT(<p) = q(l + b0(Ts — T0)/T0) into the relationship RA(q>) ~ (Ts — T0) yields

Here ST0 = (TE — TP) is the annual mean difference of the underlying surface temperatures at the equator (TE) and the pole ( TP); b0 = 2.6 is the numerical constant.

At (p = 0 and cp = n/2 from (5.2.6) it follows that

and from (5.2.7), after averaging over the domain 0 < <p < n/4 and using the ratio

connecting the average temperature Tr of the outgoing emission with the global average temperature T0 of the underlying surface , it appears that

where b1 = 0.16 and b2 = 1.8 are the new numerical constants; values of these and all other constants mentioned below are borrowed from Zilitinkevich and Monin (1977).

Thirdly, when describing the meridional heat transport MHTA in the atmosphere Zilitinkevich and Monin suggested that the main contribution to the average product vT is made by the correlation of synoptic velocity and temperature disturbances, and that the basic mechanism of these disturbances in the stratified atmosphere is the baroclinic instability of the zonal flow. In this case vT ~ aTav, where crv ~ Lu/a, erx ~ LST0/a are the root mean-square disturbances of the velocity and temperature; L ~ nj/2cr/Q is the horizontal scale of baroclinic disturbances proportional to Rossby's deformation radius; IIT = ((/c — l)//c)(l — y/ya) «0.1 is the non-dimensional combination describing the vertical atmospheric stratification; ya and y are the adiabatic and actual vertical temperature gradients in the atmosphere.

Substituting vT into MHTA and using the definitions of IIm and nn we obtain

T\T\2rT

where b3 « 12 is one more numerical constant.

The mass-weighted vertical average zonal wind velocity u appears in (5.2.11). To find it, the difference analogue of the thermal wind formula

{l = b*c15T}> nn Tr is used. It follows from the geostrophic relation with an allowance for the hydrostatic equation, the Clapeyron equation and the approximate equality dp/p x 5T/T; here p is the atmospheric pressure, b4 = 0.35 is the numerical constant.

Finally, in estimating the meridional heat transport MHT0 in the ocean it is assumed that on average throughout the year the total amount of heat entering the ocean area being examined has to be balanced by the meridional transport, that is, (2/3)Q0 = MHT0 where Q0 is the resulting heat flux at the ocean-atmosphere interface referred to the unit area 2na2 sin 7t/4 of the zonal belt of 0 < (p < 7i/4; the factor 2/3 is the fraction of the ocean in the zonal belt (the relative ocean area in the domain being examined).

It is also anticipated that the seasonal and latitude variations of water temperature encompass not the entire ocean thickness but only the upper active layer, and that the vertical difference in temperatures at the surface and at the low boundary of the ocean active layer in the zonal belt 0 < <p < ii/A is proportional to the horizontal temperature difference <5TW = (Te— 7j), where Tt is the sea water freezing temperature. Considering these assumptions the vertical heat flux is defined as Q0/poco = kn^T^/h, where h is the thickness of the active layer; kH is the effective coefficient of the vertical heat diffusion including the proportionality factor between the vertical and horizontal temperature differences in the ocean; p0c0 is the volume heat capacity of sea water.

The expression for Q0/poco written above contains two new unknowns kH and h. The first one is found from the condition of the existence of strongly stable stratification within the active layer. In this case the flux Richardson's number

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