= Weak-settling attenuation of a sloping gravity current
{title2=$B^{1/3}\sim B_o^{1/3}-\frac{v_s}{3\beta_*E}\ln\left(1+\frac{E(x-x_0)}{h_0}\right)$}
For slowly depositing particles, $B_x=-v_sB/(uh)$ and local adjustment gives $u\sim\beta_*B^{1/3}$ with the <constant-speed entraining gravity current on a slope> coefficient. At leading order $h\sim h_0+E(x-x_0)$ and $B^{1/3}=B_o^{1/3}-[v_s/(3\beta_*E)]\ln[1+E(x-x_0)/h_0]$. Its validity requires settling slow compared with the current and, for small $E$, compared with entrainment.
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