= Solution
Let $E(\lambda)=\sqrt\pi\lambda e^{\lambda^2}\operatorname{erfc}(\lambda)$. The salt equation in the <saline Stefan problem> gives $C_i-C_0=E(\lambda)C_i$. Since the <complementary error function> is positive on the real axis and $C_i>0$, the signs of $C_i-C_0$ and $\lambda$ agree. Thus freezing is equivalent to $C_i>C_0$, or $T_i<T_0=-mC_0$. Using the leading thermal balance,
$$
T_i-T_0=\frac{\theta_+-\theta_-}{2}+O(\epsilon),\qquad \boxed{\lambda>0\quad\Longleftrightarrow\quad\theta_->\theta_+}
$$
at leading order away from a vanishing <temperature> difference. In fact the exact onset is also $\theta_-=\theta_+$: at $\lambda=0$ the salt balance forces $C_i=C_0$, and the <Stefan condition> then forces $T_{-\infty}+T_\infty=2T_0$. The solution with $\lambda=0$ has a stationary interface and balancing conductive <heat fluxes>, despite nontrivial diffusion of the initial <temperature> discontinuity. For completeness, the exact freezing criterion follows without relying on that local crossing. By the <salt-rejection function for a saline Stefan front>, $T_i(\lambda)$ decreases continuously from $T_0$ to $-\infty$ as positive $\lambda$ increases. If $T_{-\infty}<T_0$, let $\lambda_s$ be where $T_i=T_{-\infty}$. On $[0,\lambda_s]$, the solid-side thermal term decreases, the liquid-side term increases, and the latent-heat term increases: the factors $e^{-\epsilon^2\lambda^2}/\operatorname{erfc}(-\epsilon\lambda)$ and $e^{-\epsilon^2\lambda^2}/\operatorname{erfc}(\epsilon\lambda)$ decrease and increase respectively. These monotonicities follow from the positive integral representation of $e^{v^2}\operatorname{erfc}v$. The thermal residual starts at $(\theta_--\theta_+)/\sqrt\pi$ and is negative at $\lambda_s$, so a positive root exists uniquely precisely when $\theta_->\theta_+$. Beyond $\lambda_s$ both thermal contributions to freezing are negative, so no further root is possible. If $T_{-\infty}\ge T_0$, the same sign argument excludes every positive root.
Back to article page
