Approximate analytical solution of the nonlinear problem of a flow over a thermally inhomogeneous underlying surface

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The problem of the thermal circulation over the underlying surface, has been studied analytically for the case when the temperature of the underlying surface depends linearly on one of the horizontal coordinates. A horizontal pressure gradient is specified at the upper boundary of the medium horizontal layer (that has been rotating around the vertical axis) being under consideration, and this fact provides the existence of the background horizontal flow. The problem is essentially nonlinear, since, first, the heat advection, second, the square friction and the heat exchange at the underlying surface are taken into account. The solution depends on three non-dimensional parameters that are determined by the absolute values of the specified horizontal temperature and pressure gradients and by the angle between these vectors. In dependence on the values of the above mentioned parameters the solution properties may be very different. When the horizontal temperature gradient is absent, the solution is a generalization of the Ekman boundary layer classical theory for a case of the nonlinear friction against the underlying surface. The temperature and pressure fields essentially depend on the existence or absence of the background motion velocity component in the direction of the temperature horizontal gradient.