 Melting-driven saline wall plume

A glacier supplies fresh meltwater at speed $U>0$ to a rising wall line plume whose buoyancy is dominated by salinity. With one entraining edge, entrainment coefficient $\epsilon$, volume flux $Q=bw$ and negligible wall stress, $Q^{\prime }=\epsilon w+U$ and $(b{w}^2{)}^{\prime }=g{\beta }_C(C_{\infty }-C)b$. Enthalpy and salt balances give $[Q(T_{\infty }-T){]}^{\prime }=U[L/c_p+T_{\infty }-T_s]$ and $[Q(C_{\infty }-C){]}^{\prime }=U(C_{\infty }-C_s)$. If latent heat dominates and entrainment dominates the volume supply, integration from a virtual origin $Z=0$ and power-law balance give the displayed width and speed. The temperature and salinity deficits are $ULZ/(c_{p}Q)$ and $U(C_{\infty }-C_s)Z/Q$, both proportional to $Z^{-1/3}$. The approximation is a far-field plume solution, not a solution at the virtual origin.