Application of the Lorenz equations for studying thermal convection in the lithosphere

Stationary solutions including wave solutions with constant amplitudes are found for nonlinear equations of thermal convection in a layer with nonlinear rheology. The solution is based on the Fourier expansion of unknown velocities and temperatures with only the first and first two terms retained in the velocity and temperature series, respectively. This method, which can be regarded as the Lorenz method, yields the Lorenz equations that fairly well describe the thermal convection in a layer with Newtonian rheology if the Rayleigh number is not very large. The obtained generalization of the Lorenz equations to the case of an integral (having a memory) nonlinear rheology implies that only the first term is retained in the Fourier series for the stress components, i.e., the nonlinear rheological equation is harmonically linearized. However, in the Fourier series of temperature, it is essential to keep the second term: this term, which is independent of the horizontal coordinate, models the thermal boundary layer that characterizes the developed convection. We constructed the bifurcation curves that describe the stationary convection in the nonlinear integral medium simulating the rheology of the mantle, and analyzed the stability of stationary convective flows. The Lorenz method is applied to study small-scale thermal convection in the lithosphere of the Earth.