Downstream similarity of a shear-driven gravity current ($y_N=(3\mu \Delta \rho gQx/{\tau }^2{)}^{1/3}$)

For a steady point-source shear-driven viscous gravity current far downstream, $h\ll y_N\ll x$ permits dropping the downstream gravity flux. With $w=h^2$, the equation is $w_x=D(w^2{)}_{yy}/(4A)$, a quadratic porous medium equation. Conserving $A\int wdy=Q$ yields $y_N=(3\mu \Delta \rho gQx/{\tau }^2{)}^{1/3}$ and $h^2=\tau (y_N^2-y^2)/(2\Delta \rho gx)$ inside the current. The central height decays as $x^{-1/6}$. Near the source, balancing shear and gravity gives an upstream reach of order $\sqrt{\mu \Delta \rho gQ}/\tau$, whose numerical coefficient is not determined by scaling.