Far-downslope width of a constant-flux porous current ($Y_N\propto X^{\beta /(2\beta +1)}$)

In a slender steady porous gravity current, the dominant balance is $(h^{\beta }{)}_X=(h^{\beta }{h}_Y{)}_Y$. If its width and height scales are $W,H$, fixed downslope volume flux gives $H^{\beta }W=\text{constant}$, and transverse spreading gives $W^2\sim HX$. Eliminating $H$ yields the displayed width exponent. The porosity exponent affects transient storage but not this steady balance.