Nonlinear compressible magnetoconvection. Part 3. Travelling waves in a horizontal field

We present results of numerical experiments on two-dimensional compressible convection in a polytropic layer with an imposed horizontal magnetic field. Our aim is to determine how far this geometry favours the occurrence of travelling waves. We therefore delineate the region of parameter space where travelling waves are stable, explore the ways in which they lose stability and investigate the physical mechanisms that are involved. In the magnetically dominated regime (with the plasma beta,$\hat{\beta}$= 8), convection sets in at an oscillatory bifurcation and travelling waves are preferred to standing waves. Standing waves are stable in the strong-field regime ($\hat{\beta}$= 32) but travelling waves are again preferred in the intermediate region ($\hat{\beta}$= 128), as suggested by weakly nonlinear Boussinesq results. In the weak-field regime ($\hat{\beta}$≥ 512) the steady nonlinear solution undergoes symmetry-breaking bifurcations that lead to travelling waves and to pulsating waves as the Rayleigh number, $\circ{R}$, is increased. The numerical experiments are interpreted by reference to the bifurcation structure in the ($\hat{\beta}$, $\circ{R}$)-plane, which is dominated by the presence of two multiple (Takens-Bogdanov) bifurcations. Physically, the travelling waves correspond to slow magnetoacoustic modes, which travel along the magnetic field and are convectively excited. We conclude that they are indeed more prevalent when the field is horizontal than when it is vertical.