# 12.1. Bottom Boundary Layer model¶

Related CPP options:

 BBL Activate bottom boundary layer parametrization ANA_WWAVE Set analytical (constant) wave forcing (hs,Tp,Dir). ANA_BSEDIM Set analytical bed parameters (if SEDIMENT is undefined) Z0_BL Compute bedload roughness for ripple predictor and sediment purposes Z0_RIP Determine bedform roughness ripple height and ripple length for sandy bed Z0_BIO Determine (biogenic) bedform roughness ripple height and ripple length for silty beds

Preselected options:

#ifdef BBL
# ifdef OW_COUPLING
# elif defined WAVE_OFFLINE
# elif defined WKB_WWAVE
# else
#  define ANA_WWAVE
# endif
# ifdef SEDIMENT
#  undef  ANA_BSEDIM
# else
#  define ANA_BSEDIM
# endif
# ifdef SEDIMENT
#  define Z0_BL
# else
#  undef  Z0_BL
# endif
# ifdef Z0_BL
#  define Z0_RIP
# endif
# undef  Z0_BIO
#endif


DESCRIPTION

Reynolds stresses, production and dissipation of turbulent kinetic energy, and gradients in velocity and suspended-sediment concentrations vary over short vertical distances, especially near the bed, and can be difficult to resolve with the vertical grid spacing used in regional-scale applications. CROCO provides algorithms to parameterize some of these subgrid-scale processes in the water column and in the bottom boundary layer (BBL). Treatment of the BBL is important for the circulation model solution because it determines the stress exerted on the flow by the bottom, which enters the Reynolds-averaged Navier-Stokes equations as a boundary conditions for momentum in the x and y directions:

$\begin{split}K_m \frac{\partial u}{\partial s} = \tau_{bx} \\ K_m \frac{\partial v}{\partial s} = \tau_{by}\end{split}$

Determination of the BBL is even more important for the sediment-transport formulations because bottom stress determines the transport rate for bedload and the resuspension rate for suspended sediment.

CROCO implements either of two methods for representing BBL processes: (1) simple drag-coefficient expressions or (2) more complex formulations that represent the interactions of wave and currents over a moveable bed. The drag-coefficient methods implement formulae for linear bottom friction, quadratic bottom friction, or a logarithmic profile. The other, more complex wave-current BBL model is described by Blaas et al. (2007) with an example of its use on the Southern California continental shelf. The method uses efficient wave-current BBL computations developed by Soulsby (1995) in combination with sediment and bedform roughness estimates of Grant and Madsen (1982), Nielsen (1986) and Li and Amos (2001).

The linear and/or quadratic drag-coefficient methods depend only on velocity components u and v in the bottom grid cell and constant, spatially-uniform coefficients $$\gamma_1$$ and $$\gamma_2$$ specified as input:

$\begin{split}\tau_{bx} = (\gamma_1 + \gamma_2 \sqrt{u^2+v^2}) ~u \\ \tau_{by} = (\gamma_1 + \gamma_2 \sqrt{u^2+v^2}) ~v\end{split}$

where $$\gamma_1$$ is the linear drag coefficient and $$\gamma_2$$ is the quadratic drag coefficient. The user can choose between linear or quadratic drag by setting one of these coefficients to zero. The bottom stresses computed from these formulae depend on the elevation of u and v (computed at the vertical mid-elevation of the bottom computational cell). Therefore, in this s-coordinate model, the same drag coefficient will be imposed throughout the domain even though the vertical location of the velocity is different.

Logarithmic drag (with roughness length $$z_0$$)

To prevent this problem, the quadratic drag $$\gamma_2$$ can be computed assuming that flow in the BBL has the classic vertical logarithmic profile defined by a shear velocity $$u_*$$ and bottom roughness length $$z_0$$ (m) as:

$\left | u \right | = \frac{u_*}{\kappa} \ln \left ( \frac{z}{z_0} \right )$

where $$\left | u \right | =\sqrt{u^2+v^2}$$, friction velocity $$u_*=\sqrt{\tau_b}$$, z is the elevation above the bottom (vertical mid-elevation point of the bottom cell), $$\kappa=0.41$$ is von Kármán’s constant. $$z_0$$ is an empirical parameter. It can be constant (default) or spatially varying. Kinematic stresses are calculated as`

$\begin{split}\tau_{bx} = \frac{\kappa^2}{ \ln^2 \left ( z/z_0 \right )} \sqrt{u^2+v^2} u \\ \tau_{by} = \frac{\kappa^2}{ \ln^2 \left ( z/z_0 \right )} \sqrt{u^2+v^2} v\end{split}$

The advantage of this approach is that the velocity and the vertical elevation of that velocity are used in the equation. Because the vertical elevation of the velocity in the bottom computational cell will vary spatially and temporally, the inclusion of the elevation provides a more consistent formulation.

Combined wave-current drag (BBL)

To provide a more physically relevant value of $$z_0$$, especially when considering waves and mobile sediments, a more complex formulation is available (BBL).

The short (order 10-s) oscillatory shear of wave-induced motions in a thin (a few cm) wave-boundary layer produces turbulence and generates large instantaneous shear stresses. The turbulence enhances momentum transfer, effectively increasing the bottom-flow coupling and the frictional drag exerted on the wave-averaged flow. The large instantaneous shear stresses often dominate sediment resuspension and enhance bedload transport. Sediment transport can remold the bed into ripples and other bedforms, which present roughness elements to the flow. Bedload transport can also induce drag on the flow, because momentum is transferred to particles as they are removed from the bed and accelerated by the flow. Resuspended sediments can cause sediment-induced stratification and, at high concentrations, change the effective viscosity of the fluid.

The BBL parameterization implemented in CROCO requires inputs of velocities u and v at reference elevation z, representative wave-orbital velocity amplitude $$u_b$$, wave period T, and wave propagation direction $$\theta$$ (degrees, clockwise from north). The wave parameters may be the output of a wave model such as WKB or WW3 or simpler calculations based on specified surface wave parameters. Additionally the BBL models require bottom sediment characteristics (median grain diameter $$D_{50}$$, mean sediment density $$\rho_s$$, and representative settling velocity $$w_s$$); these are constant (ANA_BSEDIM) or based on the composition of the uppermost active layer of the bed sediment during the previous time step if the sediment model is used.

The wave-averaged, combined wave–current bottom stress is expressed as function of $$\tau_w$$ and $$\tau_c$$ (i.e., the stress due to waves in the absence of currents and due to currents in the absence of waves, respectively) according to Soulsby (1995):

$\bar{\tau}_{wc}=\tau_c \left ( 1+1.2 \left ( \frac{\tau_w}{\tau_w + \tau_c} \right )^{3.2} \right )$

The maximum wave–current shear stress within a wave cycle is obtained by adding $$\bar{\tau}_{wc}$$ and $$\tau_w$$ (with $$\phi$$ the angle between current and waves):

$\tau_{wc}= \left ( (\bar{\tau}_{wc}+\tau_w \cos{\phi})^2 + (\tau_w \sin{\phi})^2 \right )^{1/2}$

The stresses $$\tau_c$$ and $$\tau_w$$ are determined using:

$\begin{split}\tau_{c} = \frac{\kappa^2}{ \ln^2 \left ( z/z_0 \right )} { \left | u \right | }^2 \\ \tau_{w} = 0.5 \rho f_w u_b^2\end{split}$

$$u_b$$, the bottom orbital velocity, is determined from the significant wave height $$H_s$$ and peak frequency $$\omega_p$$ using the Airy wave theory:

$u_b=\omega_p \frac{H_s}{2 \sinh{kh}}$

with h the local depth and k the local wave number from the dispersion relation. The wave-friction factor $$f_w$$ is, according to Soulsby (1995):

$f_w=1.39(u_b/\omega_p z_0)^{-0.52}$

The wave–current interaction in the BBL is taken into account only if $$u_b>1$$ cm/s; otherwise, current-only conditions apply.

Shear stress for sediment resuspension and roughness length due to bed form

To determine the shear stress relevant for sediment resuspension and the roughness length due to bed forms, we follow the concept of Li and Amos (2001) briefly summarized here. First, the maximum wave–current skin friction $$\tau_{s}$$ is computed from the equations above, using the Nikuradse roughness $$z_0= D_{50}/12$$.

A bed-load layer develops as soon as the maximum wave–current skin friction $$\tau_{s}$$ exceeds the critical stress $$\tau_{cr}$$. This layer affects the stress effective for ripple formation and sediment resuspension. Subsequently, for sandy locations, ripple height and length are computed, leading to a space- and time-dependent ripple roughness length $$z_0= z_{rip}$$, which is used to compute the drag on the flow (instead of a constant value when BBL is not activated). This drag provides boundary conditions to the momentum and turbulence equations (KPP or GLS).