Rayleigh–Bénard convection in the presence of spatial temperature modulations

Motivated by recent experiments, we study a rich variation of the familiar Rayleigh–Benard convection (RBC), where the temperature at the lower boundary varies sinusoidally about a mean value. As usual the Rayleigh number R measures the average temperature gradient, while the additional spatial modulation is characterized by a (small) amplitude δ m and a wavevector q m . Our analysis relies on precise numerical solutions of suitably adapted Oberbeck–Boussinesq equations (OBE). In the absence of forcing (δ m = 0), convection rolls with wavenumber q c bifurcate only for R above the critical Rayleigh number R c . In contrast, for δ m ≠0, convection is unavoidable for any finite R ; in the most simple case in the form of ‘forced rolls’ with wavevector q m . According to our first comprehensive stability diagram of these forced rolls in the q m – R plane, they develop instabilities against resonant oblique modes at R ≲ R c in a wide range of q m / q c . Only for q m in the vicinity of q c , the forced rolls remain stable up to fairly large R > R c . Direct numerical simulations of the OBE support and extend the findings of the stability analysis. Moreover, we are in line with the experimental results and also with some earlier theoretical results on this problem, based on asymptotic expansions in the limit δ m → 0 and R → R c . It is satisfying that in many cases the numerical results can be directly interpreted in terms of suitably constructed amplitude and generalized Swift–Hohenberg equations.