One-Dimensional Analog of the Saltzman-Lorenz Problem for Thermal Convection

Abstract

In a previous paper PANCHEV /1/ proposed the following system of equations

$${w_t} + w{w_z} = v{w_{zz}} + \beta \theta {\text{ , }}{\theta _t} + w{\theta _z} = \ae {\theta _{zz}} + zw$$

(1.1)

as one-dimensional analog to the classical system of Rayleigh-Benard or Saltzman-Lorenz, governing the thermal convection in horizontal fluid layer heated from bellow. The fundamental properties of the latter — nonlinearity, diffusivity (ν,k>0) and forcing (β, γ>0) are preserved in (1.1). Moreover, similarly to the Burgers’ single equation, it is matematically much more tractable than the complete system.