To the theory of convective flows in a rotating stratified medium over a thermally inhomogeneous surface

A theoretical model of circulations over a thermally inhomogeneous horizontal surface in a gravity field is considered. The model is more general than that used in a number of previous works. It is free from the assumption about the relative thinness of the Ekman boundary layer, which (although not always justified) significantly simplified the calculations, since it was associated with the presence of a small parameter in the problem. On the basis of the proposed model, an analytical solution is found for a linear stationary two-dimensional convective flow problem in a semi-infinite stably stratified medium rotating around a vertical axis. Constitutive parameters are introduced - analogs of the Rayleigh and Taylor numbers, in which a given horizontal scale of thermal inhomogeneities appears as a spatial scale. For mesoscale atmospheric currents, which are characterized by very large values of these numbers, the consideration is limited to the case when the values of the Rayleigh numbers are much larger than the values of the Taylor numbers, but less than the latter to the 3/2 power (a situation typical for such atmospheric currents). Relationships for analyzing the dependences of the components of velocity and helicity on the parameters of the problem are obtained. A number of general statements about the ratios of different helicity "components" in the discussed thermal circulations, in particular, in atmospheric currents with characteristic horizontal scales of the order of hundreds of kilometers, have been proved. Examples of numerical calculations of the vertical distribution of these components are given.