 Solution

Because $\hat{W}<0$, species 1 now settles more slowly and species 2 more rapidly than in the equal-speed case. The supplied inequality says that species 2 provides the larger initial particle-density contribution to the reduced gravity; with exact Boussinesq density contrasts, the corresponding condition is ${\tilde{\varphi }}_1({\rho }_1-{\rho }_a)<{\tilde{\varphi }}_2({\rho }_2-{\rho }_a)$. The dominant buoyancy contribution is therefore removed earlier, so $g^{\prime }(t)$ and the gravity-current front condition fall below their equal-settling values at first order in $ϵ$. The runout length decreases. The slower loss of the weaker species-1 contribution only partly compensates for this effect.