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Bedload transport is motion of sediment grains along or close to the bed, represented in a depth-averaged model by a solid-volume flux per unit width. The Exner equation balances this flux divergence and changes in porous-bed solid storage against exchange with suspended sediment. Consistent mass units are necessary when adding this equation to a suspended mass-concentration balance.

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.

Here $h$ is water depth, $u$ the depth-averaged downstream speed, $s$ bed elevation relative to the reference slope, and $\eta =s+h$ free-surface elevation relative to that reference. The constants ${\rho }_w,{\rho }_s$ are water and grain densities, $S$ is the reference bed slope, $\tau$ bed shear stress, $\varphi$ bed porosity, $c$ suspended sediment mass concentration, $q_b$ bedload solid volume flux per unit width, and $v_E,v_D$ erosion and deposition velocities. The settling speed is denoted by $v_s$ in the constitutive laws.

Depth integration of incompressible water conservation gives the depth-storage plus discharge-divergence balance. Depth-averaged momentum has local and advective acceleration on the left, and gravity driven by the actual surface gradient together with bed drag on the right. Hydrostatic pressure and a shallow-water approximation underlie this Saint-Venant description.

Suspended sediment conservation first has the conservative form $(hc{)}_t+(huc{)}_x={\rho }_s(v_E-v_D)$; the water balance reduces it to the material-derivative form printed in the source. Solid bed volume per unit horizontal area is $(1-\varphi )s$. Its storage plus bedload divergence equals deposition minus erosion, giving the Exner bed balance. Adding the suspended and bed balances after converting bed volume to grain mass cancels exchange terms: erosion and deposition transfer sediment between the two reservoirs rather than create it.

Suspended sediment is particulate material carried within a fluid rather than solely along the bed. Its concentration balance includes advection, erosion into suspension and deposition from suspension. For water depth $h$ and suspended mass concentration $c$, the conservative balance $(hc{)}_t+(huc{)}_x={\rho }_s(v_E-v_D)$ reduces using water continuity to $h(c_t+u{c}_x)={\rho }_s(v_E-v_D)$.