Deposit profile of a finite triangular-valley particle current ($m_d(x)=\frac{M_0}{R}[\frac{14}{3}(1-{\eta }^{3/4})-\frac{7}{5}(1-{\eta }^{5/2})],\eta =x/R$)

Under the same box-model assumptions, $M_0={\rho }_p{\varphi }_{0}W$ is initial particle mass and $m_d$ is final deposited mass per downstream length. Integrate the vertical particle deposition flux across the instantaneous triangular-valley width $2\lambda h$ from arrival of the front at $x$ until settling is complete. This gives the displayed profile on $0\le x\le R$ and zero beyond the runout. Its integral is $M_0$, verifying particle mass conservation. Areal deposit density also depends on the lateral coordinate because the wetted valley width changes as the current thins.

// Target: geophysics.bigb