where Y is the meridional extension of the domain examined. It is clear that Equation (5.7.8) has to be fulfilled at any instant of time and at any height in the atmosphere, and its rejection is equivalent to the introduction of false sources and sinks of momentum.
One more method for the parametrization of second moments on the basis of the theory of baroclinic instability was proposed by Stone (1972). He defined second moments by formulae of the form
where r'U = —(gH0/fT)(86/dy) is the zonal component of the thermal wind; Ri = (gHl/<%2T)(8Q/8z) is the Richardson number; ku...,k5 are numerical factors.
It has been shown that the relations (5.7.9) connecting momentum and heat fluxes with the temperature field are valid for forced disturbances and invalid for free disturbances. According to Lorenz (1979) the forced disturbances are disturbances for which changes in external heat sources and sinks determine the changes in the temperature gradient, and they, in turn, determine changes in the heat flux. In such disturbances the variations of the heat flux and temperature gradient turn out to be positively correlated. As for the case of free disturbances, the opposite situation occurs: the changes in heat flux arising from the internal instability of the atmospheric circulation and changes in the temperature gradient generated by them turn out to be negatively correlated. The disturbances in the heat flux with spatial scales less than the planetary scale, and also planetary oscillations with time scales more or less than the seasonal time scale are free; the seasonal oscillations of the planetary scale are forced.
A similar method of parametrization of second moments was developed by Petuhov (1989). This method, as well as all the others mentioned above, contains many hypotheses and empirical constants obtained as applied to the present-day conditions and this, naturally, restricts the implementation of such parametrizations. An approach based on closing the atmospheric hydro-thermodynamic equations with the help of equations for second moments (so-called Friedman-Keller equations), and on simplifying the latter by exclusion of third moments, has higher universality. Such simplification also represents rather restrictive hypotheses but it has the advantage that it has much less influence on the characteristics of the zonal average circulation. This approach, proposed by Monin (1965), has still not gained wide acceptance and was realized only for the case of the barotropic atmosphere approximated in the form of an incompressible, non-dissipative, two-dimensional spherical film.
Finally, one more, still not accepted, method of parametrization of second moments reduces to the application of the following relations (see Chalikov, 1982):
which were obtained from experimental data handling processing with an allowance for dimensional considerations. Here Ri = (jf2(d6/8(r)/R(d9/dy)2 is the Richardson number; Ki = is the Kybel number; Ma =
<%(gR(89/8a)~il2 is the Mach number; Re = %a/K is the Reynolds number; K is the macroviscosity coefficient describing motions of a subgrid scale; other designations are the same.
The functions Fu ..., F4 appearing in (5.7.10) contain too many dimension-
less arguments to determine them from empirical data. But the number Re for K = 105 m2/s is of order 103, and the numbers Ma and Ki are of order 10"2 and 10"1 respectively. Therefore, it is expected that Equations (5.7.10) will have the property of self-similarity on Re, Ma and Ki and, hence, the question of their use reduces to finding functions Fu ..., F4 of two arguments cr and Ri.
As for the second moments created by the stationary non-zonal motions, there are no justifiable parametrization schemes for these at present. But, as has been shown by Stone and Miller (1980), the net meridional heat transport corresponding to stationary and transient disturbances of the zonal circulation is correlated with the meridional temperature gradient more than with any separate constituent of this transport. This circumstance, as well as the fact that both these components complement each other, point to the existence of negative feedback between them and to one source of their origin - the baroclinic instability of the zonal circulation. This lifts some of the burden from researchers, allowing them to use (for the meridional heat transport resulting from stationary disturbances at least) the same parametrization as for the meridional heat transport initiated by transient disturbances, or even to consider the joint effect of non-zonal motions without separating them into stationary and transient motions.
We make some brief remarks here about the potential of zonal models by reference to one of them, developed by Dymnikov et al. (1979). This model successfully simulates the observed three-cell structure of the meridional circulation in the troposphere of both hemispheres. It also provides good reproduction of the easterly winds in tropical and polar latitudes and the jet flow in the vicinity of the 200 hPa isobaric surface in temperate latitudes. Meanwhile, the velocity of the zonal winds in temperate latitudes turns out to be slightly excessive due to rejecting the effects of the Earth's surface topography and overestimating of the meridional temperature gradient. This last circumstance is caused by underestimation (by about 10 K) of the air temperature in polar latitudes, and this underestimation is caused, in its turn, by inaccurate specification of the underlying surface temperature in polar regions, by underestimation of the meridional heat transport due to eddy disturbances and by the coarse vertical resolution.
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