{title2=$1<R_\rho<\kappa_T/\kappa_S$}

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 convention
$$
R_\rho=\frac{\alpha T_z}{\beta S_z},\qquad \tau=\frac{\kappa_S}{\kappa_T},
$$
a strictly stable total <mass density> gradient has $R_\rho>1$. The band admitting exponentially growing <salt fingering> modes is
$$
\boxed{1<R_\rho<\tau^{-1}.}
$$
The 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.<Double-diffusive processes in stellar astrophysics by Pascale Garaud>[Section 2.2.1, equations15–16 and the definition of the density ratio]{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 polynomial>
$$
P(\sigma)=(\sigma+\nu K)(\sigma+\kappa_TK)(\sigma+\kappa_SK)+qA(\sigma+\kappa_SK)-qB(\sigma+\kappa_TK),
$$
where $\nu>0$ is the <kinematic viscosity>.

For an <elevator mode (double-diffusive convection)>, $q=1$ and the perturbation is independent of $z$. Its constant term is
$$
P(0)=K\left[\nu\kappa_T\kappa_SK^2+A\kappa_S-B\kappa_T\right].
$$
If $R_\rho=A/B<\kappa_T/\kappa_S$, 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, and
$$
a_1a_2-a_3=K^3\left[(\nu+\kappa_T+\kappa_S)(\nu\kappa_T+\nu\kappa_S+\kappa_T\kappa_S)-\nu\kappa_T\kappa_S\right]+qK\left[(\nu+\kappa_T)A-(\nu+\kappa_S)B\right]>0.
$$
The <Routh-Hurwitz criterion> therefore puts every root in the left half-plane. This covers oblique modes too, since $0\le q\le1$. If $R_\rho<1$, ordinary density-driven overturning is already available; that is a different regime from fingering within a stably stratified background.


 Salt-finger instability criterion (source code)