Oscillatory convection

oscillatory convection is time-dependent convection emerging through nonzero-frequency overstability or a Hopf bifurcation. At onset a complex-conjugate growth-rate pair crosses the imaginary axis. Travelling and standing convection rolls are distinct nonlinear possibilities, and their stability cannot be decided from the linear angular frequency alone.

At fixed orientation, opposite travelling amplitudes can obey ${\dot{Z}}_{\pm }=(r+i{\omega }_0)Z_{\pm }-(g|Z_{\pm }{|}^2+h|Z_{\mp }{|}^2)Z_{\pm }$. For $r>0$ and $g_r=\mathrm{Re}g>0$, a travelling branch has one nonzero intensity $r/g_r$ and is stable to its counter-propagating amplitude when $h_r>g_r$. A standing branch has both intensities $r/(g_r+h_r)$ and is stable within the pair when $g_r>h_r$ and $g_r+h_r>0$. These follow by linearizing the real intensity equations. Spatial modulation and other orientations remain separate stability tests.