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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]

Fingering convection is a form of double-diffusive convection in which a rapidly diffusing stabilizing contribution to buoyancy competes with a more slowly diffusing destabilizing contribution. The combined mass density stratification can resist an ordinary vertical displacement, yet unequal diffusion permits narrow rising and sinking structures to grow.

A displaced parcel initially carries both contributions. If the restoring anomaly diffuses away faster than the destabilizing anomaly, the remaining buoyancy reinforces its motion. In the common thermal-compositional case, thermal diffusivity exceeds solutal diffusivity; the initial stability of the combined mass density profile is therefore insufficient to decide stability once diffusion acts.^[ref]

In seawater, salt fingering realizes this competition with temperature and salinity: the thermal contribution stabilizes while salt-rich material above fresher material supplies the destabilizing contribution. Its laboratory examples make the different transport rates visible.^[ref]

In a star, chemical composition can play the slower role. A larger mean molecular weight tends to increase mass density at fixed pressure and temperature, so a molecular-weight inversion can oppose the thermal restoring effect. Rapid thermal transport can then permit fingering even when the total local stratification is stable.^[ref]

The density ratio measures the competition, and the salt-finger instability criterion shows why there is also an upper limit to the stabilizing contribution: if it is too strong, unequal diffusion no longer produces growing modes. The elevator mode gives a particularly simple local pattern with vertical motion and no height dependence.

Melvin E. Stern was a physical oceanographer who developed a mathematical account of the instability now called salt fingering. His 1960 paper The Salt-Fountain and Thermohaline Convection by Melvin E. Stern explained how warm, saline water above cooler, fresher water can become unstable even when the combined density stratification is gravitationally stable.^[ref]

The decisive ingredient is that heat spreads by heat conduction much faster than salt spreads by diffusion. A displaced parcel can lose its temperature anomaly while retaining its salinity anomaly, changing the restoring or overturning density difference. Stern's paper made this into a stability problem and asked whether the resulting motions could matter for vertical ocean mixing. This connected small-scale molecular transport with a possible large-scale oceanographic consequence, while leaving its environmental importance as an observational question. He taught at Florida State University from 1987 to 2006.^[ref]

Portrait: Melvin Stern in the Florida State Times remembrance, page 11.

Salt fingering is double-diffusive convection in which narrow rising and sinking fluid structures develop because heat and dissolved salt diffuse at different rates. Warm, salty water above colder, fresher water can have a stable total mass density stratification: the stabilizing temperature difference outweighs the destabilizing salinity difference.^[ref]

A small upper parcel displaced downward enters colder surroundings. It loses heat relatively quickly while retaining more of its salt, becoming denser and continuing downward. A fresher parcel displaced upward can gain heat while remaining relatively fresh and light. The distinction between thermal diffusivity and solutal diffusivity supplies the feedback; stability of the initial combined mass density profile does not exclude this mechanism. The density ratio records the competing contributions, while the salt-finger instability criterion specifies the parameter regime.

A laboratory analog can replace heat and salt by two dissolved substances. David Osmond describes a 1820-by-80-by-120 millimetre tank with salt and sugar solutions entering at opposite ends and an overflow at the centre. The supplied solutions had matching mass density, but salt diffused faster than sugar. The salt-rich plume could lose salt faster than it gained sugar, producing a lighter core and denser surroundings. Fingering and diffusive layers appeared in different regions, showing that identical source densities need not remain dynamically equivalent when composition diffuses.^[ref]

In this analog, salt plays the faster-diffusing role played by heat in the ocean, while sugar plays the slower-diffusing role of oceanic salt. Keeping that correspondence explicit prevents reversing the physical explanation.