 Solution

Seek a steady downslope solution with $u>0$ constant. The volume conservation equation reduces to $h_x=E$, so the constant-speed entraining gravity current on a slope hasThe conserved buoyancy flux per unit width is $B=u{g}^{\prime }h$, givingSubstitute these expressions into the momentum equation. Its advective term is $E{u}^2$ and its hydrostatic term is $\frac{1}{2}E(B/u)\cos \theta$. The balance is thereforewhich givesThe assumed slope inequality makes ${\beta }_{\ast }>0$. The increasing thickness dilutes reduced gravity while leaving $g^{\prime }h$ and the buoyancy flux constant. The denominator includes both bottom drag and the momentum needed to accelerate newly entrained ambient fluid.