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The integrated budget is ${\partial }_t(g^{\prime }h)+B_x=S_B-v_s{g}^{\prime }$. A steady spatial increase needs a source $S_B$ exceeding deposition, for example sediment resuspension or lateral particle input. In unsteady flow, local depletion can produce $B_x>0$ even without particle addition.

Vertical turbulence carries particles upward as well as downward. Its mixing time $h/u^{\prime }$ should be short relative to the settling time $h/V_s$; hence $u^{\prime }\gg V_s$ is a useful well-mixed criterion, subject also to the eddy correlation time and particle response time. A more resolved description balances turbulent diffusion against downward settling velocity. A nearly uniform interior concentration does not eliminate the downward drift.

At an absorbing bed, particles crossing the boundary settle and do not return unless sediment resuspension is separately supplied. With well-mixed concentration $\varphi$, the particle volume deposition flux is $V_s\varphi$ per unit horizontal area. Thus stirring can preserve an almost uniform suspension while its total particle content decreases. The present model neglects sediment resuspension and any significant change of bulk fluid volume due to the dilute particles.

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 parameterwhere $d$ 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 givesSince $W_s>0$ and $\varphi \to 0$ 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 profileIt 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.

For a steady no-source particle-laden gravity current, $B_x=-v_s{g}^{\prime }<0$. Simply entraining particle-free ambient fluid increases volume and dilutes particle volume fraction; it cannot create particle buoyancy flux.

A steady increase can occur if sediment resuspension or bed erosion adds more suspended particle buoyancy than settling removes, if a tributary or lateral particle source supplies a positive flux, or if the buoyancy per unit particle amount increases through a change of the surrounding fluid's density. Changes of particle properties or another buoyancy source can also alter the flux, but they go beyond the fixed-particle, fixed-reference model. In a model with a source $S_B$,Thus the buoyancy production in a particle-laden current requires $S_B>v_s{g}^{\prime }$ in a steady current. In an unsteady current, $B_x$ can be positive even without external addition: local depletion has ${\partial }_t(g^{\prime }h)<-v_s{g}^{\prime }$, so more buoyancy leaves a section than enters it while stored particles are being released. An instantaneous spatial flux increase is not by itself proof of particle production.