= Frozen-temperature solutal interface dispersion
{title2=$sC_I=(Dq-V)[VC_I/D-(G+\Gamma k^2)/m]$}
For an interface perturbation $\eta e^{ikx+st}$ with <instantaneous interface kinetics>, take the liquid concentration perturbation proportional to $e^{-qz}$. Here $\Gamma>0$ is the coefficient of <curvature-induced melting-temperature depression>. The <advection-diffusion equation> and interfacial <mass conservation> give
$$
s=Dq^2-Vq-Dk^2,\qquad sC_I=(Dq-V)\left[\frac{VC_I}{D}-\frac{G+\Gamma k^2}{m}\right].
$$
For a growing <normal mode> the unique decaying liquid branch has $q=(V+\sqrt{V^2+4D(s+Dk^2)})/(2D)$. Writing $b=(G+\Gamma k^2)/(mC_I)$, its rate is $s=(D/2)(V/D-b)(\sqrt{b^2+4k^2}-b)$. Instability requires $G+\Gamma k^2<mVC_I/D$. The <Gibbs-Thomson effect> suppresses sufficiently short waves; a <quasistatic solute diffusion> to solute diffusion changes the rate and must be stated separately.
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