= Conservation of salt with salt-retaining ice
{title2=$S/S_0=(M/M_0)^{r-1}$}
Let liquid mass $M$ freeze incrementally, and let each new ice increment retain <salinity> $rS$ with constant $0\leq r\leq1$, where $S$ is the instantaneous well-mixed liquid salinity. Conservation gives $d(MS)=rS\,dM$, hence $dS/S=(r-1)dM/M$. Integrating yields $S/S_0=(M/M_0)^{r-1}$. For frozen mass fraction $f$, $\Delta S=S_0[(1-f)^{-(1-r)}-1]\simeq(1-r)S_0f$. Specifying a fixed bulk ice salinity instead produces a different, though first-order compatible, finite-mass balance.
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