MPM BL

Meyer-Peter Müeller Bedload Formulation

The Meyer-Peter Müeller (1948) formulation is

$$\Phi =\max \left[{8\left({\theta _{sf}-\theta _{c}}\right)^{1.5},\quad 0}\right]$$

where $\Phi$ is the magnitude of the non-dimensional transport rate for each sediment class, $\theta _{sf}$ is the non-dimensional Shields parameter for skin stress

$$\theta _{sf}={\frac {\tau _{sf}}{\left({s-1}\right)gD_{50}}}$$

$\theta _{c}=0.047$ is the critical Shields parameter, and $\tau _{sf}$ is the magnitude of total skin-friction component of bottom stress computed from

$$\tau _{sf}=\left({\tau _{bx}^{2}+\tau _{by}^{2}}\right)^{0.5}$$

where ${\tau _{bx}}^{2}$ and ${\tau _{bx}}^{2}$ are the skin-friction components of bed stress, from currents alone or the maximum wave-current combined stress, in the $x$ and $y$ directions. These are computed at cell faces ($u$ and $v$ locations) and then interpolated to cell centers ($\rho$ points). The bedload transport vectors are partitioned into $x$ and $y$ components based on the magnitude of the bed shear stress as

$$q_{blx}=q_{bl}{\frac {\tau _{bx}}{\tau _{sf}}};\quad \quad q_{bly}=q_{bl}{\frac {\tau _{by}}{\tau _{sf}}}$$