The onset of zonal modes in two-dimensional Rayleigh–Bénard convection

We study the stability of steady convection rolls in two-dimensional Rayleigh–Bénard with free-slip boundaries and horizontal periodicity over 12 orders magnitude Prandtl number $(10^{-6} \leq Pr 10^6)$ 6 Rayleigh $(8{\rm \pi} ^4 < Ra 10^8)$ . The analysis is facilitated by partitioning our modal expansion into so-called even odd modes. With aspect ratio $\varGamma = 2$ , we observe that zonal modes (with wavenumber equal to zero) can emerge only once roll state consisting becomes unstable perturbations. determine boundary $(Pr,Ra)$ plane remarkably intricate features corresponding qualitative changes solution, as well three regions where lose subsequently regain increased. asymptotic limit $Pr \to 0$ find become almost instantaneously, eventually leading nonlinear relaxation osculations bursts, which explain a weakly analysis. In complementary large- $Pr$ limit, at instability switch off abruptly large, but finite, number.