{disambiguate=double-diffusive convection}
{title2=$k_z=0$}

An <elevator mode (double-diffusive convection)> is a <normal mode> whose fluid motion is purely vertical while its velocity, <temperature> and composition <perturbations> are independent of height. Neighbouring columns can move upward and downward without a vertical return-flow structure, hence the name. Such modes describe the fastest-growing disturbances in the ideal local <salt fingering> problem.<Double-diffusive processes in stellar astrophysics by Pascale Garaud>[Section 2.2.1, paragraph following equation 16: vertically invariant elevator modes]{ref}

Take $z$ upward and write the <velocity field> and scalar <perturbations> as
$$
\mathbf u=(0,0,w(x,y,t)),\qquad \theta=\theta(x,y,t),\qquad s=s(x,y,t).
$$
The <incompressibility> condition is automatic because $\nabla\cdot\mathbf u=\partial_z w=0$. More surprisingly, the perturbation's own <advection> of every one of these fields vanishes:
$$
\boxed{(\mathbf u\cdot\nabla)f=w\partial_z f=0.}
$$
The background need not be uniform: if its <temperature> and <salinity> gradients are constants $T_z,S_z$, vertical motion still advects those backgrounds. This supplies the terms $T_z w$ and $S_z w$ below.

In a nonrotating local model with the <Boussinesq approximation>, constant <kinematic viscosity> $\nu$ and constant <thermal diffusivity> $\kappa_T$ and <solutal diffusivity> $\kappa_S$, the ansatz reduces the equations to
$$
\begin{aligned}
\partial_t w&=g(\alpha\theta-\beta s)+\nu\nabla_h^2w,\\
\partial_t\theta+T_z w&=\kappa_T\nabla_h^2\theta,\\
\partial_t s+S_z w&=\kappa_S\nabla_h^2s.
\end{aligned}
$$
Here $g>0$ is <gravitational acceleration>, $\alpha$ the <thermal expansion coefficient>, $\beta$ the <haline contraction coefficient>, and $\nabla_h^2=\partial_x^2+\partial_y^2$ the horizontal <Laplace operator>. A constant pressure perturbation is compatible with the absent horizontal motion.

For example, let all three fields be proportional to $e^{\sigma t}\cos(kx)$, with $k>0$. The upward and downward columns alternate with <wavelength> $2\pi/k$. Their amplitudes satisfy a three-variable <linear differential equation>; eliminating the scalar amplitudes gives the $q=1$ <characteristic polynomial> used in the <salt-finger instability criterion>. Although that criterion is a <linear stability> result, the vanishing self-<advection> means that this particular ansatz also solves the corresponding nonlinear local equations exactly within the stated ideal model.

\Image[/elevator-mode-vertical-columns.png]
{title=An <elevator mode (double-diffusive convection)> across one horizontal <wavelength>}
{description=Original schematic with vertical <velocity field> $w=W\cos(kx)$. Each height has the same pattern of upward and downward motion; the lower plot shows its horizontal profile. The top and bottom of the drawn region are not impermeable walls.}

That exactness does not imply that unlimited growth describes a laboratory tank. An impermeable horizontal wall requires the vertical velocity to vanish there, excluding a nonzero height-independent pattern. Other <perturbations> can also grow on the rising and falling columns and disrupt them. Elevator modes are therefore useful ideal building blocks, with boundary conditions and additional disturbances determining how faithfully a real flow follows them.


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