= Self-similar ice-bearing wall plume
{title2=$b=eZ,\quad w=(g_iI_0/e)^{1/3},\quad\phi=w^2/(g_iZ)$}
In a dilute, well-mixed <wall line plume>, let $b$ be its width, $w$ its upward <velocity>, $\phi\ll1$ 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 $g_i\phi$, 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 $I_0=\phi bw$, then <volume flux> $Q=bw$ and kinematic <momentum flux> $M=bw^2$ obey
$$
Q'=ew,\qquad M'=g_i\phi b=\frac{g_iI_0}{w}.
$$
A constant-<velocity> solution has $b=eZ$ with $Z$ measured from its <plume virtual origin>. Substitution gives
$$
w=\left(\frac{g_iI_0}{e}\right)^{1/3},\qquad \phi=\frac{w^2}{g_iZ}.
$$
Its 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.
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