= Solution
Write $H=L/c_p$, $k=\rho c_p\kappa$ and $\epsilon=\sqrt{D/\kappa}$. The requested model neglects density differences and uses the same thermal transport coefficients in the interfacial balances. Let $C_b=C(b(t),t)$ and $T_b=T_m-mC_b$. With no imposed heat gradient in the fresh liquid, its <temperature> remains $T_m$. The prescribed linear <temperature> in the <ice> is
$$
T_i(z,t)=T_m+(T_b-T_m)\frac{z-a}{b-a}.
$$
The salt and <temperature> fields in the lower liquid satisfy <diffusion equations>, $C_t=DC_{zz}$ and $T_t=\kappa T_{zz}$. For constant interfacial values and the proposed <similarity solution>, their decaying solutions are
$$
\begin{aligned}
C(z,t)&=C_0+(C_b-C_0)
\frac{\operatorname{erfc}[z/(2\sqrt{Dt})]}{\operatorname{erfc}(\lambda_b)},\\
T_l(z,t)&=T_\infty+(T_b-T_\infty)
\frac{\operatorname{erfc}[z/(2\sqrt{\kappa t})]}{\operatorname{erfc}(\epsilon\lambda_b)}.
\end{aligned}
$$
These follow by substituting a variable $z/(2\sqrt{\chi t})$ into the <diffusion equation>, which gives $U''+2\eta U'=0$; integrating gives the <complementary error function>. Their values at $b$ and at infinity give the required boundary data.
For the upward-moving fresh-water/<ice> boundary, the <Stefan condition> is $\rho L\dot a=kT_{i,z}$: negative $\dot a$ means fresh water freezes. At the lower boundary the signed heat balance is
$$
\rho L\dot b=k[T_{i,z}-T_{l,z}]_{z=b}.
$$
Salt cannot enter the pure <ice>. With no bulk liquid motion its interfacial conservation law is
$$
DC_z(b,t)=-C_b\dot b.
$$
The derivatives and interface velocities are
$$
\dot a=\lambda_a\sqrt{D/t},\quad\dot b=\lambda_b\sqrt{D/t},\quad
T_{i,z}=-\frac{T_m-T_b}{2(\lambda_b-\lambda_a)\sqrt{Dt}},
$$
$$
T_{l,z}(b)=-\frac{(T_b-T_\infty)e^{-(\epsilon\lambda_b)^2}}
{\sqrt{\pi\kappa t}\operatorname{erfc}(\epsilon\lambda_b)},\qquad
C_z(b)=-\frac{(C_b-C_0)e^{-\lambda_b^2}}
{\sqrt{\pi Dt}\operatorname{erfc}(\lambda_b)}.
$$
Define $F(x)=\sqrt\pi\,x e^{x^2}\operatorname{erfc}x$. Substitution into all three balances gives
$$
\boxed{2H\epsilon^2\lambda_a=-\frac{T_m-T_b}{\lambda_b-\lambda_a},}
$$
$$
\boxed{H=\frac{T_b-T_\infty}{F(\epsilon\lambda_b)}
-\frac{T_m-T_b}{2\epsilon^2\lambda_b(\lambda_b-\lambda_a)},\qquad
F(\lambda_b)=\frac{C_b-C_0}{C_b}.}
$$
Together with $T_b=T_m-mC_b$, these determine the interfacial constants. In particular $C_b=C_0/[1-F(\lambda_b)]$.
The physical branch has $\lambda_a<\lambda_b<0$. Since $F$ is negative at negative arguments, the salt relation gives $0<C_b<C_0$: lower-surface <melting> dilutes the brine. Combining the two heat balances eliminates the <ice> gradient and gives
$$
T_b-T_\infty=H\frac{\lambda_b-\lambda_a}{\lambda_b}F(\epsilon\lambda_b)>0.
$$
Thus $T_m>T_b>T_\infty>T_L(C_0)$. The interface is at local <liquidus> equilibrium with diluted brine, while the undiluted far-field brine remains above its own <liquidus>. The similarity model describes a layer starting with zero thickness, or the long-time limit when an initial finite thickness no longer sets the dominant scale.
For the small-diffusivity-ratio limit, put $A=\epsilon\lambda_a$ and $\Delta=T_m-T_\infty$, keeping $S=H/\Delta$ and the imposed concentrations fixed. Seek $A=O(1)$ and $\lambda_b=O(1)$. Then $F(\epsilon\lambda_b)\sim\sqrt\pi\epsilon\lambda_b$ and $\lambda_b-\lambda_a\sim-\lambda_a$. The upper heat balance gives
$$
T_m-T_b\sim2HA^2.
$$
In the lower balance the two terms of order $\epsilon^{-1}$ must cancel, giving
$$
\frac{T_b-T_\infty}{\sqrt\pi}+\frac{T_m-T_b}{2A}=0.
$$
Consequently
$$
\boxed{T_b-T_\infty\sim\frac{\Delta}{1-2A/\sqrt\pi},}
$$
and $T_b-T_\infty\sim-\sqrt\pi HA$. Adding the two <temperature> differences yields $2A^2-\sqrt\pi A=1/S$. The negative root is
$$
\boxed{4\epsilon\lambda_a\sim\sqrt\pi-\sqrt{\pi+8/S}.}
$$
The resulting $C_b=(T_m-T_b)/m$ lies strictly between zero and $C_0$. The salt relation therefore determines a finite negative $\lambda_b$, confirming its $O(1)$ scaling. The top interface travels on the thermal scale $\sqrt{\kappa t}$, whereas the lower <melting> interface travels on the shorter solutal scale $\sqrt{Dt}$. This is the <brine-ice sandwich with disparate diffusion scales>; $S$ is the <latent-to-sensible heat ratio>, the reciprocal of the usual sensible-to-latent <Stefan number> convention.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-55-ice-brine.png]
{title=Similarity temperature and concentration profiles, local liquidus and ice-brine phase-diagram path}
The sketches show constant fresh-water <temperature>, a declining linear <ice> <temperature>, and a much broader thermal than solutal boundary layer in the brine. <Concentration> is zero in freshwater and pure <ice>, jumps to $C_b$ across the lower interface, and rises to $C_0$. Accordingly the local <liquidus> is $T_m$ on the salt-free side, jumps to $T_b$ and falls rapidly to $T_L(C_0)$. In the <phase diagram>, the brine starts on the <liquidus> at $(C_b,T_b)$ and ends above it at $(C_0,T_\infty)$; the pure <ice> has <concentration> zero, and the interface tie-line joins it to the diluted brine.
Physically, the cold saline liquid permits freshwater to freeze above, but salt lowers the <liquidus> and allows the existing <ice> to melt into brine below even though that brine is colder than $T_m$. Dilution raises the lower interfacial <freezing> <temperature>. Heat released by upper <freezing> supplies the lower <melting> and the conductive loss to the colder deep brine. With $|\lambda_a|\gg|\lambda_b|$, upper <freezing> exceeds lower <melting> and the <ice> layer thickens as $b-a\propto\sqrt t$.
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