Weakly nonlinear rotating-convection roll stability

A saturated convection roll is not stable merely because its single-amplitude cubic coefficient is positive. Perturbations include phase sidebands, transverse bending, differently oriented convection rolls, oscillatory modes and large-scale Eulerian mean flow. Their growth rates follow from the corresponding enlarged amplitude equations or the full linearization about the convection roll. Küppers–Lortz instability and small-angle instability of rotating convection rolls are distinct orientation-dependent mechanisms.

For finite Prandtl number and stress-free boundaries, nearly parallel convection rolls can interact with a weakly damped large-scale Eulerian mean flow. The ordinary regular two-roll amplitude expansion becomes nonuniform as the angle tends to zero, so the mean-flow mode must be retained. This mechanism is not automatically the finite-angle Küppers–Lortz instability, and thresholds depend on the stated boundary and parameter regime.

For an existing convection roll of intensity $r/g_0$, the amplitude of a new orientation at relative angle $\vartheta$ grows at rate $r[1-g(\vartheta )/g_0]$. Rotation permits $g(\vartheta )\ne g(-\vartheta )$. If one oblique orientation has $g(\vartheta )<g_0$, the convection roll is unstable immediately above a supercritical stationary threshold. This is the Küppers–Lortz mechanism. Successive orientation replacements can produce time-dependent convection roll switching; a universal attractor does not follow from the two-mode calculation. The perturbing orientation requires three-dimensional disturbances even if the initial convection roll is two-dimensional.