= Ice-bearing wall-plume thermodynamic invariant
{title2=$\phi Q\simeq(-E-c_pmS)/(L+c_pmC_0)$}
Consider an equilibrium <ice>-water <wall line plume> with <volume flux> per unit span $Q$, ice volume fraction $\phi$, 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 $T_0=-mC_0$. 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:
$$
E=Q[c_p(T-T_0)-L\phi],\qquad S=Q[(1-\phi)C-C_0].
$$
Combining them gives $E+c_pmS=-\phi Q(L+c_pmC)$. Eliminating $C$ with the salt relation yields
$$
\phi Q[L+c_pmC_0-L\phi]=-E-c_pmS+E\phi.
$$
Thus the leading dilute-crystal balance is $\phi Q\simeq(-E-c_pmS)/(L+c_pmC_0)$. More precisely, this limit requires both $L\phi\ll L+c_pmC_0$ and $|E|\phi\ll|E+c_pmS|$; cancellation in the source combination can invalidate it even at small $\phi$.
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.<Frazil ice formation in an ice shelf water plume by Lars H. Smedsrud and Adrian Jenkins>[Sections 1–2, suspended-ice buoyancy and coupled mixture/heat/salt balances; the equilibrium invariant here uses the explicitly simpler assumptions above]{ref}
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