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In double-diffusive convection, a density ratio compares the buoyancy contributions from two stratifying quantities. A convention must be stated. For vertical coordinate $z$ increasing upward and warm salty water above cooler fresher water, take $R_{\rho }=\alpha T_z/(\beta S_z)$, with positive thermal expansion coefficient $\alpha$ and haline contraction coefficient $\beta$.

The linear equation of state gives ${\rho }_z={\rho }_0(-\alpha T_z+\beta S_z)$. Thus $R_{\rho }>1$ makes the combined density stratification stable, even though the salt contribution alone is destabilizing. Faster heat diffusion can still enable salt fingering. The reciprocal ratio is also used in the literature, so a numerical density ratio is meaningless without its convention.^[ref]

Let $T_i,S_i$ be the temperature and salinity of the original supplied liquid, and $T_{\infty },S_{\infty }$ those of the surroundings. For reservoir values $T,S$, define the dimensionless cooling and freshening measureswith positive reference differences. Here $\alpha$ is the thermal expansion coefficient, $\beta$ the haline contraction coefficient, $A$ the interface area and $R_{\rho }=\beta (S-S_{\infty })/[\alpha (T-T_{\infty })]$ the density ratio. Increasing $\theta$ describes cooling toward the surroundings; increasing $\eta$ describes freshening. The flux law below is an assumed double-diffusive convection closure.

The steady subscript denotes balance between supply, overflow and interfacial transfer. A model of this kind separates the stabilizing bulk salinity contrast from the faster transfer of heat across the interface.

A steady well-mixed hot salty reservoir receives volume flux $Q$ and loses the same flux over a weir. Heat and salt balances are $Q(T_i-T)=A{F}_T$ and $Q(S_i-S)=A{F}_S$. If $\beta F_S/(\alpha F_T)=R_F$ and $\alpha F_T=b(\alpha \Delta T{)}^{4/3}/R_{\rho }^2$, define $R_0=\beta (S_i-S_{\infty })/[\alpha (T_i-T_{\infty })]$. Then ${\eta }_s=(R_F/R_0){\theta }_s$ and $C=Ab[\alpha (T_i-T_{\infty }){]}^{1/3}/(Q{R}_0^2)$. For $R_0>1$ and $0\le R_F<1$ the interface is statically stable while preferential heat transfer can drive double-diffusive convection.

An elevator mode 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.^[ref]

Take $z$ upward and write the velocity field and scalar perturbations asThe 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: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 toHere $g>0$ 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.

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.

The haline contraction coefficient measures the fractional increase of mass density with salinity at fixed temperature and pressure:Its units depend on the salinity variable $S$: a dimensionless mass fraction and a value in grams per kilogram require different numerical coefficients.

For small changes, $\delta \rho /\rho \simeq -\alpha \delta T+\beta \delta S$, with thermal expansion coefficient $\alpha$. This formula separates two competing buoyancy contributions. In thermosolutal convection, their combination determines the static density gradient, while unequal diffusion rates determine whether that apparently stable gradient can nevertheless become unstable.

Let $T_i,S_i$ be the temperature and salinity of the original supplied liquid, and $T_{\infty },S_{\infty }$ those of the surroundings. For reservoir values $T,S$, define the dimensionless cooling and freshening measureswith positive reference differences. Here $\alpha$ is the thermal expansion coefficient, $\beta$ the haline contraction coefficient, $A$ the interface area and $R_{\rho }=\beta (S-S_{\infty })/[\alpha (T-T_{\infty })]$ the density ratio. Increasing $\theta$ describes cooling toward the surroundings; increasing $\eta$ describes freshening. The flux law below is an assumed double-diffusive convection closure.

With a constant buoyancy-flux ratio and the preceding steady initial state, the heat and salt balance equations give $d\eta /d\theta =q$ and ${\eta }_s=q{\theta }_s$, yielding $\eta =q\theta$ after shutdown.

After inflow and overflow cease, a fixed-volume reservoir initially in the preceding steady state obeys $\eta =q\theta$ and $\dot{\theta }=k(1-\theta {)}^{10/3}/(1-q\theta {)}^2$, with $k=Ab[\alpha (T_i-T_{\infty }){]}^{1/3}/(V{R}_0^2)$. Integrating gives $k(t-t_0)=G(\theta )-G({\theta }_s)$, where $G=3(1-q{)}^2(1-\theta {)}^{-7/3}/7+3q(1-q)(1-\theta {)}^{-4/3}/2+3q^2(1-\theta {)}^{-1/3}$. For $q<1$, heat contrast tends to zero only at infinite time, while a nonzero salt contrast remains.

In an ideal unbounded fluid governed by the Boussinesq approximation, take uniform background temperature and salinity gradients $T_z,S_z>0$, with $z$ upward. Let $\alpha ,\beta >0$ be the thermal expansion coefficient and haline contraction coefficient, and let the thermal diffusivity ${\kappa }_T$ exceed the solutal diffusivity ${\kappa }_S>0$. With the conventiona strictly stable total mass density gradient has $R_{\rho }>1$. The band admitting exponentially growing salt fingering modes isThe upper equality is the marginal long-wavelength limit. This local linear stability result assumes no rotation and constant positive kinematic viscosity; finite boundaries can restrict the available wavelengths.^[ref]

A dimensional derivation makes the signs explicit. Put $A=g\alpha T_z$ and $B=g\beta S_z$, where $g>0$ is the gravitational acceleration. Consider a normal mode with growth rate $\sigma$, squared wavenumber $K=|\mathbf{k}{|}^2>0$ and horizontal fraction $q=k_h^2/|\mathbf{k}{|}^2$, where $\mathbf{k}$ is the wave vector and $k_h$ its horizontal magnitude. Eliminating the temperature, salinity and pressure perturbations from the linearized equations gives the characteristic polynomialwhere $\nu >0$ is the kinematic viscosity.

For an elevator mode, $q=1$ and the perturbation is independent of $z$. Its constant term isIf $R_{\rho }=A/B<{\kappa }_T/{\kappa }_S$, choose a sufficiently small positive $K$ to make $P(0)<0$. Since $P(\sigma )$ tends to positive infinity along the positive real axis, the intermediate value theorem supplies a positive real growth rate.

For the converse, write $P(\sigma )={\sigma }^3+a_1{\sigma }^2+a_2\sigma +a_3$. When $R_{\rho }\ge {\kappa }_T/{\kappa }_S$, all three coefficients are positive, andThe Routh-Hurwitz criterion therefore puts every root in the left half-plane. This covers oblique modes too, since $0\le q\le 1$. If $R_{\rho }<1$, ordinary density-driven overturning is already available; that is a different regime from fingering within a stably stratified background.