Salt-finger instability criterion ($1<R_{\rho }<{\kappa }_T/{\kappa }_S$)

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.