Constant-flux porous gravity current ($t_c=\varphi K_{b}Q/U^3$)

Constant actual volume input $Q$ into a porous gravity current with background flow gives early symmetric scales $h\sim [(Q/\varphi {)}^{2}t/a{]}^{1/3}$ and $x\sim [a(Q/\varphi )t^2{]}^{1/3}$. For $U>0$, drift dominates after $t_c=\varphi K_{b}Q/U^3$. The late interior depth is $Q/U$ and the steady upstream edge is $x_{-}=-K_{b}Q/U^2$, with upstream linear profile $h=U(x-x_{-})/K_b$. Upstream volume flux is zero because background drift balances gravity spreading.

The downstream edge of a continuously fed porous gravity current with background flow has a transition width proportional to $\sqrt{a{h}_{\mathrm{int}}t}$ around drift position $ut$. Its similarity solution satisfies $(FF\prime )\prime +\eta F\prime /2=0$, $F(-\infty )=1$, and $F({\eta }_N)=0$. A finite front has $F\sim ({\eta }_N/2)({\eta }_N-\eta )$. A clipped linear ramp is a useful mass-preserving approximation, but not an exact solution of this nonlinear transition equation.