 Self-similar ice-bearing wall plume

In a dilute, well-mixed wall line plume, let $b$ be its width, $w$ its upward velocity, $\varphi \ll 1$ its ice volume fraction, and $e>0$ its one-edge entrainment coefficient. Define $g_i=g({\rho }_l-{\rho }_i)/{\rho }_l$ from the liquid and ice mass densities. The mixture's reduced gravity is approximately $g_i\varphi$, so $g_i$ itself is a material density-contrast scale, not the changing mixture reduced gravity.

Neglect crystal slip, deposition and other buoyancy contributions. If the ice-bearing wall-plume thermodynamic invariant supplies a constant positive ice flux $I_0=\varphi bw$, then volume flux $Q=bw$ and kinematic momentum flux $M=b{w}^2$ obeyA constant-velocity solution has $b=eZ$ with $Z$ measured from its plume virtual origin. Substitution givesIts width grows linearly and ice fraction falls inversely with height. Two exposed entraining edges replace $e$ by $2e$. The formal origin lies outside the dilute model because the ice fraction diverges there; finite-source conditions require matching away from that origin.