SD BL

Soulsby & Damgaard Bedload Formulation

The Soulsby and Damgaard (2005) formulae account for the combined effects of mean currents and asymmetrical waves on bedload flux. Their formulations are based on numerical integration, over a wave cycle, of the non-dimensional transport equation

$${\overrightarrow {\Phi }}=\max \left[{A_{2}\theta ^{0.5}\left({\theta _{sf}-\theta _{c}}\right){\frac {\overrightarrow {\theta _{sf}}}{\theta _{sf}}},\quad 0}\right]$$

where ${\overrightarrow {\Phi }}$ and ${\overrightarrow {\theta _{sf}}}$ are vectors with components in the direction of the mean current and in the direction perpendicular to the current, e.g., ${\overrightarrow {\Phi }}=\left({\Phi _{\parallel },\,\Phi _{\bot }}\right)$, ${\overrightarrow {\theta _{sf}}}=\left({\theta _{sf\parallel },\,\theta _{sf\bot }}\right)$, $\theta _{sf}=\left|{\overrightarrow {\theta _{sf}}}\right|$, $\theta _{c}$ is the critical Shields parameter, and $A2=12$ is a semi-empirical coefficient. The implementation of this method requires computation of transport rates in the directions parallel and perpendicular to the currents as:

$$\Phi _{\parallel }=\max \left[{\Phi _{\parallel 1},\quad \Phi _{\parallel 2}}\right]$$

where

$$\Phi _{\parallel 1}=A_{2}\theta _{m}^{0.5}\left({\theta _{m}-\theta _{c}}\right)$$

$$\Phi _{\parallel 2}=A_{2}\left({0.9534+0.1907\cos 2\varphi }\right)\theta _{w}^{0.5}\theta _{m}+A_{2}\left({0.229\gamma _{w}\theta _{w}^{1.5}\cos \varphi }\right)$$

$$\Phi _{\bot }=A_{2}{\frac {0.1907\theta _{w}^{2}}{\theta _{w}^{3/2}+1.5\theta _{m}^{1.5}}}\left({\theta _{m}\sin 2\varphi +1.2\gamma _{w}\theta _{w}\sin \varphi }\right)$$

where $\theta _{m}$ is the mean Shields parameter and $\tau _{m}$ is

$$\tau _{m}=\tau _{c}\left({1+1.2\left({\frac {\tau _{w}}{\tau _{w}+\tau _{c}}}\right)^{1.5}}\right)$$

and $\tau _{c}$ is the bottom stress from the currents only, $\tau _{w}$ is the bottom stress from the waves only calculated in the bottom-boundary layer routines (see below). The asymmetry factor $\gamma _{w}$ is the ratio between the amplitude of the second harmonic and the amplitude of the first harmonic of the oscillatory wave stress. Following the suggestion of Soulsby and Damgaard (2005), we estimate the asymmetry factor using Stokes second-order theory (e.g., Fredsøe and Deigaard, 1992) and constrain it to be less than 0.2. The non-dimensional fluxes (Eqns. (30) and (33)) are rotated into $x$ and $y$ directions using the directions for mean current and waves and dimensionalized with Eqn. (24) to yield values for $q_{blx}$ and $q_{bly}$ for each sediment class.