= Wavenumber selection in rotating convection
For the free-slip plane layer, onset selects the smallest <Rayleigh number> across all admissible stationary and oscillatory <normal modes>. The first vertical mode wins because increasing vertical index raises viscous cost and reduces oscillatory admissibility. Oscillatory minimization replaces the right-hand side of the stationary wave-number equation by $P^2\mathrm{Ta}\pi^2/(1+P)^2$. For the oscillatory minimum also require positive <angular frequency> squared. Where both selected curves are admissible, at rapid rotation the ratio of the minimized thresholds is $R_{o,c}/R_{s,c}\sim2P^{4/3}/(1+P)^{1/3}$. Equality gives $8P^4-P-1=0$, with positive root about $0.67660$; below it sufficiently rapid rotation can select oscillatory primary onset.
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