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 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 ${\mathcal{W}}_{R}g=(k^2+n^2{\pi }^2)g$.