= Solution
Before the ice reaches the salty layer, the state moves vertically in a temperature--concentration <phase diagram> along $C=0$: cooling crosses the pure-water melting point and produces pure ice. After the encounter, the liquid state starts at $(T_\infty,C_0)$ and approaches the interface state
$$
(T_i,C_i)=(-mC_i,C_i)
$$
on the <liquidus> $T_L=-mC$. The solid remains on the $C=0$ axis.
When $\epsilon\ll1$, the solutal diffusion length $\sqrt{Dt}$ is much smaller than the thermal diffusion length $\sqrt{\kappa t}$. Immediately ahead of the interface, rejected salt makes $C$ fall steeply from $C_i$ to $C_0$, so the local liquidus $T_L=-mC$ rises steeply with depth while the actual temperature changes comparatively slowly. <Constitutional supercooling> occurs wherever
$$
\boxed{T(z)<T_L(C(z))=-mC(z)}.
$$
Its local onset criterion at the interface is
$$
\boxed{T_z(h^+)<-mC_z(h^+)}.
$$
The supercooled zone lies in the thin salty boundary layer immediately ahead of the planar front. A small forward protrusion then enters liquid that is already below its local freezing point, so the planar interface is susceptible to a morphological instability.
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