SSW BBL

SSW bottom boundary layer formulation

Wave-orbital calculations

Near-bed wave-orbital characteristics, including representative orbital velocity $u_{br}$, representative period $T_{r}$, and average direction of wave propagation $\theta _{w}$ (degrees, nautical convention, which is positive clockwise from north) are defined according to Madsen (1994). When SWAN results are used, these correspond to UBOT, PWAVE, and DWAVE. If surface-wave statistics (e.g., $H_{s}$, $T_{d}$, and $\theta _{w}$) are provided, they can be converted to bottom orbital velocity externally (using, for example, the routines suggested in Wiberg and Sherwood (2008) and provided as UBOT in a SWAN input file. Alternatively, if SSW_CALC_UB is defined, orbital velocity $u_{br}$ is calculated according to linear wave theory as follows:

$$u_{br}={\frac {H_{s}}{2\sinh(kh)}}$$

where $kh$ is wavenumber x depth, and $k$ can be approximated using one of the methods described here.

Ripple Geometry

Ripple height $\eta _{r}$ and wavelength $\lambda _{r}$ are calculated using information from the previous time step and the Malarkey and Davies (2003) implementation of the Wiberg and Harris (1994) formulation, which is valid for wave-dominated conditions. They approximate ripple wavelength as $535D_{50}$ and ripple steepness as:

$${\frac {\eta _{r}}{\lambda _{r}}}=\exp \left[{-0.095\left({\ln \left({\frac {d_{0}}{\eta _{r}}}\right)}\right)^{2}+0.442\left({\ln \left({\frac {d_{0}}{\eta _{r}}}\right)}\right)-2.28}\right]$$

where $d_{0}=u_{br}T/\pi$ is the wave-orbital diameter. When transport stage is below the threshold for sediment transport ($T_{*}=frac{\tau _{wc}}(\tau _{ce}<1$), ripple dimensions from the previous time step are retained.

An alternative formulation for time-dependent ripple roughness is in development branches and is described here.

Bottom Roughness

Roughness lengths associated with grain roughness $z_{0N}$, sediment transport roughness $z_{0ST}$, and bedform roughness (ripples) $z_{0BF}$ are estimated as:

$$z_{0N}=2.5D_{50}/30$$

$$z_{0ST}=\alpha D_{50}\cdot a_{1}{\frac {T_{*}}{1+a_{2}T_{*}}}$$

$$z_{0BF}=a_{r}\eta _{r}^{2}/\lambda _{r}$$

where the sediment transport coefficients are $\alpha =0.056$, $a1=0.068$, and $a_{2}=0.0204\ln(100D_{50}^{2})+0.0709\ln(100D_{50})$ (Wiberg and Rubin, 1989) with the bedform roughness $D_{5}0$ expressed in meters, and where $a_{r}$ is a coefficient that may range from 0.3 to 3 (Soulsby, 1997). Grant and Madsen (1982) proposed $a_{r}=27.7/30$, but we use a default value of $a_{r}=0.267$, as suggested by Nielsen (1992). The roughness lengths are additive, so subsequent BBL calculations use $z_{0}=\max \left[{z_{0N}+z_{0ST}+z_{0BF},\quad z_{0MIN}}\right]$, where $z_{0MIN}$ allows setting a lower limit on bottom drag (default $z_{0MIN}=5e^{-5}$m).

Wave-current combined stress and roughness

Initial estimates of (kinematic) bottom stresses based on pure currents $\tau _{c}$ ( = $\tau _{b}$ ) and pure waves $\tau _{w}$ ($\tau _{b}=0$) are made as follows.

$$\tau _{c}={\frac {\left({u^{2}+v^{2}}\right)\kappa ^{2}}{\ln ^{2}(z/z_{0})}}$$

and $\tau _{w}=0.5f_{w}u_{b}^{2}$, where $f_{w}$ is the Madsen (1994) wave-friction factor, which depends on the ratio of the wave-orbital excursion amplitude to the bottom roughness length $A_{b}/k_{b}$, where $A_{b}=u_{br}T/(2\pi )$ and $k_{b}=30z_{0}$. $f_{w}=$

$$0.3,A_{b}/k_{b}\leq 0.2$$

$$\exp(-8.82+7.02(A_{b}/k)^{-0.078}),0.2<A_{b}/k_{b}\leq 100$$

$$\exp(-7.30+5.61(A_{b}/k)^{-0.109}),A_{b}/k_{b}>100$$

The pure-current and pure-wave limits are used as initial estimates for calculations towards consistent profiles for eddy viscosity and velocity between $z_{0}$ and $z_{r}$, using either the model of Madsen (1994) or Styles and Glenn (2000). Both of these models assume eddy viscosity profiles scaled by $u_{*wc}={\sqrt {\tau _{wc}}}$ in the wave boundary layer (WBL) and $u_{*c}={\sqrt {\tau _{b}}}$ in the current boundary layer, calculated as $K_{M}=$

$$\kappa u_{*wc}z,z<\delta _{wbl}$$

$$\kappa u_{*c}z,z>\delta _{wbl}$$

where $\delta _{wbl}$ is the thickness of the WBL, which scales as $u_{*wc}T/(2\pi )$. $\tau _{wc}$ represents the maximum vector sum of wave- and current-induced stress, but the $\tau _{b}$ is influenced by the elevated eddy viscosity in the WBL, and must be determined through an iterative process. The shape and elevation of the transition between these profiles and other details differ among the two models, but both the models of Madsen (1994) or Styles and Glenn (2000) return values for the horizontal vectors $\tau _{b}$, $\tau _{w}$, and $\tau _{wc}$. The parameter $\tau _{b}$ is the mean (over many wave periods) stress used as the bottom boundary condition in the momentum equations, and $\tau _{wc}$ is the maximum instantaneous stress exerted over the bottom by representative waves and currents.

Skin friction - form drag partitioning

When ripples are present, $\tau _{wc}$ is a combination of form drag, which does not directly contribute to sediment transport, and skin friction, which does. The next step in the BBL calculations is to estimate the skin-friction component of $\tau _{wc}$ using the ripple dimensions and a bedform drag-coefficient approach (Smith and McLean, 1977; Wiberg and Nelson, 1992), as follows.

$$\tau _{sfm}=\tau _{wc}\left[{1+0.5C_{dBF}{\frac {\eta _{r}}{\lambda _{r}\kappa ^{2}}}\left({\ln {\frac {\eta _{r}}{\left({z_{0N}+z_{0ST}}\right)}}-1}\right)^{2}}\right]^{-1}$$

where $C_{dBF}\approx 0.5$ is a bedform drag coefficient for unseparated flow (Smith and McLean, 1977).

Maximum shear stress

Because shear stress varies between ripple crests and troughs, an estimate of the maximum shear stress at the crests $\tau _{sfm}$ is calculated for use in sediment-transport algorithms as:

$$\tau _{sf}=\tau _{sfm}(1+8{\frac {\eta _{r}}{\lambda _{r}}})$$