Wall line plume

A wall line plume is a one-sided line plume adjoining a solid wall. Under a top-hat pure-plume model, $d(bW)/ds=\alpha W$ because only its outer edge entrains ambient fluid.

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.

An ice-bearing plume at local liquidus equilibrium conserves excess enthalpy $E$ and salt $S$ relative to the ambient liquidus water. Eliminating its temperature and brine composition gives $\varphi Q[L+c_{p}m{C}_0-L\varphi ]=-E-c_{p}mS+E\varphi$. To dilute-crystal order this fixes its advected ice volume, hence its leading buoyancy flux. A positive rising branch needs a positive source invariant.

For one exposed entraining edge and conserved positive dilute ice flux $I_0$, the line-plume equations give the displayed pure far-field solution. Its width grows linearly, speed is constant and ice fraction decays inversely with distance from a virtual origin. Two exposed edges replace $e$ by $2e$. The dilute solution is invalid near the formal origin, where its ice fraction diverges.

In a triangular-profile wall line plume, axial velocity and reduced gravity decrease linearly from their wall values to zero across the plume width $b$. If the wall speed is $W$, its volume and kinematic momentum fluxes per unit span are $V=bW/2$ and $J=b{W}^2/3$.