Infinite-Prandtl-number convection ($\mathrm{Pr}\to \infty$)

In thermal-time units, taking the Prandtl number large removes momentum inertia at leading order. The Stokes flow momentum equation then determines velocity instantaneously from the temperature perturbation, while the nonlinear heat equation retains its time derivative. Small Prandtl number does not generally give this model. An exact contrasting shear in a free-slip layer is $u=U_0{e}^{-\mathrm{Pr}{\pi }^{2}t}\cos \pi z$, $w=\theta =0$: it decays slowly as the Prandtl number tends to zero but is excluded by the instantaneous Stokes equation.

For a stress-free, fixed-temperature layer, the infinite-Prandtl number growth rate of the fundamental vertical normal mode is ${\lambda }_{\infty }=R{k}^2/(k^2+{\pi }^2{)}^2-(k^2+{\pi }^2)$. For large Rayleigh number, its maximum has $k^2={\pi }^2-8{\pi }^6/R+O(R^{-2})$ and ${\lambda }_{\max }=R/(4{\pi }^2)-2{\pi }^2+O(R^{-1})$. The infinite-Prandtl number limit must be taken first, or inertia must remain small on the resulting growth-time scale. This optimum differs from the free-slip convection neutral curve minimum, which selects onset rather than fastest growth far above it.

For a nonzero horizontal Fourier mode $k$, solve $(D^2-k^2{)}^{2}w=R{k}^2\theta$, $w=w^{\prime \prime }=0$, to obtain $w={\mathcal{W}}_R\theta$. Under homogeneous temperature Dirichlet boundary conditions, the sine modes diagonalize this self-adjoint operator, with eigenvalues $R{k}^2/(k^2+n^2{\pi }^2{)}^2$. The temperature linear operator is ${\mathcal{L}}_R=D^2-k^2+{\mathcal{W}}_R$. At a marginal mode $g=\sin (n\pi z)$ it has ${\mathcal{W}}_{R}g=(k^2+n^2{\pi }^2)g$.