 Solution

Take an arbitrary reference depth $h_0>0$, and select $u_0=(gS{h}_0/f{)}^{1/2}$ so the uniform reference flow balances drag and slope. Work with positive downstream velocity, so $u_{\ast }=\sqrt{f}u$ in the dimensional constitutive laws. Introduce the erosion and bedload scalesMatching erosion with deposition, suspended advection with erosion, and bed storage with erosion gives respectivelyUse $h_0$ for the scales of $\eta$ and $s$ too, so $h=\eta -s$ survives unchanged. After dividing the balances by their matched scales, the parameters areFor example the momentum balance divided by $g{h}_0/x_0$ has inertial prefactor $F^2$, surface-gradient term $-{\eta }_x$, and equal slope/drag prefactors $S{x}_0/h_0=\delta$, because $f{u}_0^2=gS{h}_0$. The suspended balance has source $u^3-c/u$, and the bed balance has its negative with unit bed-time coefficient by the definition of $t_0$. These substitutions give the entire stated dimensionless system; $q_b=u^3$ is dimensionless bedload, with its dimensional scale already absorbed into $\beta$.