Hopf/steady-state mode interaction in a vertical slot—Effect of 1:4 resonance

Abstract

Hopf/steady-state mode interaction in a vertical slot with side-wall heating is revisited. Linear stability analysis reveals that a pair of Hopf modes and a steady mode share the critical Grashof number at a crossover Prandtl number P*=12.454 256 44. About a quarter century ago, the non-resonant mode interaction between the critical modes was investigated for a Prandtl number close to P*. In the present paper, we claim that the critical Hopf modes and the steady mode have wavenumbers approximately in a ratio of 1:4. We then examine the effect of 1:4 resonance on the Hopf/steady mode interaction although the 1:4 resonance arises at the quartic-order approximation. This implies that the resonance is so weak that its effect of phase coupling is expected to be negligibly small compared with the effect of coupling acting through the moduli of amplitudes arising at the cubic order. We find, however, that the resonance is indispensable for the bifurcation of symmetric mixed modes with phase differences of 0 and π, from which an asymmetric mixed mode bifurcates.