The Effect of Long-Wavelength Thermal Modulations on the Onset of Convection in an Infinite Porous Layer Heated From Below

The onset of finite-amplitude convection in a horizontal porous layer of infinite extent is considered. Attention is focused on the case of spatially periodic heating and cooling on the lower and upper boundaries, respectively. In particular, we analyse the effects of small-amplitude, symmetric, thermal modulations with a wavelength which is large compared with the layer depth. Weakly nonlinear theory is used to derive Landau-Ginzburg equations for the amplitude of convection in the form of transverse and longitudinal rolls. It is found that these patterns do not necessarily have the same spatial periodicity as the thermal forcing and may even be spatially quasiperiodic. The most unstable transverse roll, however, always has the same wavelength as the thermal modulations. We show also that for certain ranges of values of the modulation wavenumber the first mode to appear as the Rayleigh number is increased is, somewhat surprisingly, a rectangular cell of large-aspect-ratio planform. This mode is a linear superposition of two rolls equally aligned at a small angle away from the direction of the longitudinal roll.