= Downstream similarity of a shear-driven gravity current
{title2=$y_N=(3\mu\Delta\rho gQx/\tau^2)^{1/3}$}
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 $A\int w\,dy=Q$ 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.
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