Influences of Thermal Boundary Layer on Atmospheric Cellular Convection

Abstract

A theory is developed which appears to explain the flatness of the convection cells that has been observed from satellites. Rayleigh's thermal convection theory, which describes the transition from conduction to convection of a horizontal layer of fluid heated from below, is studied in conjunction with the thermal boundary condition which is more applicable to the atmosphere than the boundary condition of constant temperature. The thermal boundary condition is expressed by a linear combination of the vertical temperature gradient and the temperature itself. For hydrodynamical boundary condition, two cases are studied, one is “slip” and the other is “non-slip”. The eigenvalues of the governing differential equation are obtained numerically at the marginal state. The results show that the eigenvalues are largely influenced by a nondimensional number γ which is the ratio of the vertical temperature gradient term to the temperature term at the boundaries. The critical Rayleigh number Ray decreases as the ratio decreases and Rαy approaches a value of approximately 120 as γ approaches zero. Dependency of the eigenvalues upon γ is large for the lowest eigenvalue, but γ exerts small effect on the other eigenvalues. A non-dimensional number a, defined by kd where d is the depth of convection layer and k is the horizontal wave number, decreases also as γ decreases. Since a represents a ratio of the vertical to horizontal dimensions of a cell, the result indicates that the flat convection cells common in the atmosphere are developed when γ is small at the boundaries. Furthermore, the Rayleigh number becomes nearly uniform at its lowest value over a wide range of a, showing that cellular convection can set in with a rather broad variety of cell dimensions. The dynamical boundary conditions, slip or non-slip, however, are found to exert only small influences upon a as compared with the thermal boundary condition.