 Solution

Bedload transport carries sediment by rolling, sliding and short hops near the bed. Suspended sediment is held in the water column by turbulent mixing. The competition between particle settling velocity and the turbulent vertical velocity scale chiefly determines which description applies; particle size and density influence that settling velocity.

Scale the depths and bed/free-surface elevations by $d$, both velocities by $U$, distance by $L$, and time by $T$. A choice producing exactly the requested coefficients isThe sediment continuity equation then has unit coefficients, while both material-acceleration time derivatives acquire the factor $L/(UT)=a/d$. Dividing water momentum by $gd/L$ and sediment momentum by $U^2/L$ givesFor example $d=1\mathrm{m}$, $a=10^{-3}\mathrm{m}$, $S=10^{-3}$, $f=10^{-2}$ and ${\rho }_s/{\rho }_w=2.65$ give $\epsilon =10^{-3}$ and $\delta =2.65\times 10^{-4}$. These are illustrative values: small effective mobile-layer thickness and small slope explain the two small parameters.

Set $\epsilon =\delta =0$ at leading order. Water continuity gives $hu=q$, with spatially constant flux; take a prescribed constant $q$. Water momentum integrates to $\eta +F^2{u}^2/2=C$. Since $s=\eta -h$,The bedload transport model with sediment inertia has remaining equations $s(u{)}_t+v_x=0$ and $v{v}_x=(u-v{)}^2-\kappa v$.

Choose a uniform state of this reduced model with $h_0=u_0=q=1$, constant $s_0=C-F^2/2-1$, andIt satisfies $(1-v_0{)}^2=\kappa v_0$. For small perturbations $U,V$ of the two velocities, put $a_1=2(1-v_0)>0$ and $b_1=a_1+\kappa >0$. Linearization givesFor a Fourier mode $e^{ikx+\lambda t}$ this yieldsEvery nonzero mode decays for $F<1$ and grows for $F>1$. The spatially uniform mode is neutral, and $F=1$ is a singular threshold where the neglected terms matter. In a general uniform state the controlling sign is $q/u_0^2-F^2{u}_0$: equivalently the actual mean-flow Froude number is less or greater than one. The displayed $F=1$ criterion uses the stated unit-depth, unit-velocity normalization of the reduced steady state.