Ice-bearing wall-plume thermodynamic invariant ($\varphi Q\simeq (-E-c_{p}mS)/(L+c_{p}m{C}_0)$)

An ice-bearing plume at local liquidus equilibrium conserves excess enthalpy $E$ and salt $S$ relative to the ambient liquidus water. Eliminating its temperature and brine composition gives $\varphi Q[L+c_{p}m{C}_0-L\varphi ]=-E-c_{p}mS+E\varphi$. To dilute-crystal order this fixes its advected ice volume, hence its leading buoyancy flux. A positive rising branch needs a positive source invariant.

For one exposed entraining edge and conserved positive dilute ice flux $I_0$, the line-plume equations give the displayed pure far-field solution. Its width grows linearly, speed is constant and ice fraction decays inversely with distance from a virtual origin. Two exposed edges replace $e$ by $2e$. The dilute solution is invalid near the formal origin, where its ice fraction diverges.