MPM BL

Meyer-Peter Müeller Bedload Formulation

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