On turbulence caused by thermal instability

Abstract

In this paper a statistical theory of turbulence in an incompressible fluid caused by the joint effects of gravity, and thermal instability, is developed. The mathematical theory is based on the equations of continuity and heat conduction and the Boussinesq form of the equations of motion in which the variations of density (resulting from the variations in temperature) are taken into account only in so far as they modify the action of gravity. By restricting oneself to a portion of the fluid far from the bounding surfaces one can treat the turbulence as approximately homogeneous and axisymmetric and use the theory of axisymmetric vectors and tensors recently developed by the writer (Chandrasekhar 1950a). A number of correlations between the various field quantities (such as the velocity components, fluctuations in temperature, etc.) at two different points in the medium are defined; and a closed system of equations for the defining scalars are derived for the case when the non-linear terms in the equations of motion and heat conduction can be neglected and a constant mean adverse temperature gradient is maintained. Under stationary conditions when the time derivatives of the various correlations are zero, there is an exact balance between the dissipation of kinetic energy by viscosity and the liberation of potential energy by gravity. A fundamental set of solutions of the equations governing stationary turbulence is obtained; these solutions, varying periodically in the vertical direction, enable a generalized Fourier analysis of the various correlation functions. According to these solutions, a Fourier analysis of correlations such as u_\|(0)u'_\|(z)̅ of the vertical velocities at two points directly above one another and separated by a distance z, cannot include wave-lengths less than a certain minimum value depending on the physical parameters and on the temperature gradient maintained. We may thus speak of a smallest size for the eddies. Further, it appears that the field of turbulence can be analyzed into two modes characterized by the kinetic energy being confined, principally, to the vertical or to the horizontal direction.