 Ice-bearing wall-plume thermodynamic invariant

Consider an equilibrium ice-water wall line plume with volume flux per unit span $Q$, ice volume fraction $\varphi$, liquid salinity $C$, and temperature $T=-mC$. The positive coefficient $m$ is the slope magnitude of a linear liquidus; ambient liquid has salinity $C_0$ and temperature $T_0=-m{C}_0$. Use equal reference densities and equal specific heat capacities $c_p$ in the thermodynamic bookkeeping, and let $L$ be the latent heat of fusion per unit mass.

When entrainment only adds that reference liquid and no crystals or heat escape, the excess enthalpy and salt fluxes are conserved:Combining them gives $E+c_{p}mS=-\varphi Q(L+c_{p}mC)$. Eliminating $C$ with the salt relation yieldsThus the leading dilute-crystal balance is $\varphi Q\simeq (-E-c_{p}mS)/(L+c_{p}m{C}_0)$. More precisely, this limit requires both $L\varphi \ll L+c_{p}m{C}_0$ and $|E|\varphi \ll |E+c_{p}mS|$; cancellation in the source combination can invalidate it even at small $\varphi$.

This conserved leading ice flux can drive a self-similar ice-bearing wall plume. Suspended frazil ice also changes buoyancy in ice-shelf plume models, whose additional crystal-growth, deposition and pressure effects need their own balances.^[ref]