Convection problems in a half space

It is often useful to define a convection problem on a half space. For example, (Hurle et al., 1982) use the velocity in a phase change problem to transform their stability analysis to one on a half-space; also (Hurle et al., 1967) have heat conducting half-spaces bounding a fluid layer to investigate the effects of finite conductivity at the boundary. While it may offer some simplicity to deal with a half-space configuration, from the mathematical point of view it does introduce new complications. In particular, (Galdi and Rionero, 1985) derive a very sharp result on the asymptotic behaviour of the base solution for which the energy maximum problem for R _E admits a maximizing solution. Roughly speaking, either the base solution must decay at least linearly at infinity, or the gradient of the base solution must decay at least like 1/z2 (if z > 0 is the half-space.) To describe this result and related ones in geophysics it is convenient to return to the general equations for a heat conducting linearly viscous fluid.