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Convective overstability

{title2=$\sigma=-\frac{N^2\beta}{2(\beta^2+\Omega^2)}$}

For $|N^2|\ll\Omega^2$, this leading amplitude-growth rate is largest at <thermal diffusion rate of a Fourier mode> $\beta=\Omega$, giving $\sigma_{\rm max}=-N^2/(4\Omega)$ when $N^2<0$.

Convective overstability growth rate

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 321 / 3 / d / Solution

Convective overstability growth rate ( $σ = - \frac{N ^{2} β}{2 ( β ^{2} + Ω ^{2} )}$ )

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Convective overstability growth rate (σ=−2(β2+Ω2)N2β​)

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Convective overstability growth rate ( $σ = - \frac{N ^{2} β}{2 ( β ^{2} + Ω ^{2} )}$ )

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