= Solution
At times long compared with the encounter time at depth $H$, the fixed depth $H$ is negligible relative to the growing diffusion lengths. Salt obeys the <diffusion equation> with diffusivity $D$. With
$$
\eta_s=\frac{z}{2\sqrt{Dt}},
\qquad
\eta_s(h)=\frac{\lambda}{\epsilon},
\qquad
\epsilon=\sqrt{\frac D\kappa},
$$
the self-similar concentration profile in the liquid is
$$
\boxed{
C(z,t)=C_0+(C_i-C_0)
\frac{\operatorname{erfc}\eta_s}
{\operatorname{erfc}(\lambda/\epsilon)}}.
$$
It has $C(h,t)=C_i$ and tends to $C_0$ in the far field.
Because the solid contains no salt, conservation of solute at the moving boundary requires the rejected solute flux from <Fick's first law> to equal the rate at which the interface sweeps up salt:
$$
\boxed{C_i\dot h=-D C_z(h^+,t)}.
$$
Substituting the similarity profile gives
$$
\lambda C_i
=
(C_i-C_0)
\frac{\epsilon e^{-(\lambda/\epsilon)^2}}
{\sqrt{\pi}\operatorname{erfc}(\lambda/\epsilon)}.
$$
This relation determines $C_i$ if $\lambda$ is already known. More generally it must be solved together with the thermal <Stefan condition> and the interfacial phase-equilibrium relation $T_i=-mC_i$ from the <liquidus>. For $\epsilon\ll1$, salt occupies a much thinner boundary layer than heat and $C_i$ can be much larger than $C_0$.
Solved by gpt-5.6-sol high.
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