= Solution
For a steady no-source <particle-laden gravity current>, $B_x=-v_sg'<0$. 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 $S_B$,
$$
\partial_t(g'h)+B_x=S_B-v_sg'.
$$
Thus the <buoyancy production in a particle-laden current> requires $S_B>v_sg'$ in a steady current. In an unsteady current, $B_x$ can be positive even without external addition: local depletion has $\partial_t(g'h)<-v_sg'$, so more buoyancy leaves a section than enters it while stored particles are being released. \b[An instantaneous spatial flux increase is not by itself proof of particle production.]
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