= Solution
Particles settle toward the bed under their submerged weight. A concentration increasing downward gives an upward turbulent diffusive flux, which can oppose that settling. If <turbulence> cannot sustain the particles, they enter the weakly mixed near-wall region and can deposit. <Sediment resuspension> requires bed forces strong enough to entrain grains, not merely a specified diffusivity far above the bed.
For noncohesive grains, an appropriate criterion uses the <Shields parameter>
$$
\boxed{\Theta=\frac{\tau_d}{(\rho_s-\rho)gd}>\Theta_c,}
$$
where $d$ is grain size, $\rho_s$ its <density> and $\Theta_c$ the <sediment entrainment threshold> expressed dimensionlessly. The threshold depends on grain geometry, contacts and viscous effects; cohesion changes the criterion. Sustained suspension additionally requires sufficiently vigorous upward turbulent motions relative to $W_s$. Bed motion and well-mixed suspension are therefore not identical conditions.
At sedimentation-resuspension equilibrium, the total vertical particle flux is constant. Set $\kappa_T=Kz$ with $K=C/2$. The stationary equation gives
$$
J=-W_s\phi-Kz\phi_z=\text{constant},
\qquad \phi(z)=A z^{-W_s/K}-\frac{J}{W_s}.
$$
Since $W_s>0$ and $\phi\to0$ at infinity, necessarily $J=0$. Settling and upward diffusion then balance pointwise. Taking a reference height $z_1=1$ in the specified length units gives the <equilibrium settling-diffusion profile>
$$
\boxed{\phi(z)=\phi_1\left(\frac z{z_1}\right)^{-P},\qquad
P=\frac{W_s}{K}=\frac{2W_s}{\alpha\hat u},\qquad z_1=1.}
$$
It is nonnegative and decays to zero. The profile describes the region $z\gg\delta$; it does not assert that this <power law> continues through the viscous layer down to the physical bed. The supplied reference concentration summarizes the bed exchange and near-wall physics that the outer diffusion closure does not resolve.
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