## Twodimensional zonal models

Imagine that the underlying surface is homogeneous along a latitude circle, and non-zonal motions are determined only by the instability of the zonal circulation. We average the initial hydrothermodynamic equations along the latitude circle. The resulting equations will contain statistical moments. To close the equations it is necessary to find the connection between these moments and the characteristics of the zonal circulation whose existence at the homogeneous (in a zonal direction) Earth is ensured by the fact that all statistical moments generated by zonal averaging have to be invariant to a rotation relative to the Earth's axis. Thus, in the case discussed it is possible in principle to construct a zonal atmospheric model.

When taking into account inhomogeneities of the underlying surface (say, continents and oceans) the property of invariance of statistical moments is not valid. Moreover, owing to the change in the thermophysical properties of the underlying surface in a zonal direction, steady areas of high pressure over the land and depressions over the ocean are formed. These areas are not described by zonal models. But if for the land and ocean such longitudinal distinctions are decisive due to the boundedness of their extensions in a zonal direction, for the atmosphere they do not play such an important role, and because of this zonal models of the atmosphere turn out to be representative, from the viewpoint of the reproduction of the meridional structure of the atmospheric circulation, in this case as well.

The distinctive property of the atmosphere forming the basis for the derivation of zonal model equations is the approximate symmetry of the thermodynamic regime relative to the axis of the Earth's rotation. A high degree of zonality in the distribution of many characteristics of the atmospheric circulation due to the zonality of the diurnal insolation points to this fact. We represent any dependent variable appearing in the initial hydrothermo-dynamic equations as a — [a] + a* + a', where [a] is the zonal average stationary component; a* and a' are stationary and time-dependent deviations from the zonal average defined by the equalities [a*] = 0, a' = 0; the overbar means time averaging. Then after averaging over time (with the period of averaging being a fortiori less than the annual cycle in order to take seasonal variability into account) and over longitude the initial system of atmospheric hydrothermodynamic equations reduces to the form d_ It

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