Bed friction

As already mentioned earlier in section 3.1.3, caution is needed when discretising the bottom boundary conditions for momentum, (8). They are an example for a physical condition which has to be modified for the numerical discretisation, since the discrete velocity point nearest to the bottom is half a grid box away from the point where the boundary condition is defined. Furthermore, due to the logarithmic law, high velocity gradients are typical near the bed. Simply setting the discrete bottom velocity to zero, would therefore lead to large discretisation errors. Instead, a flux condition using bottom stresses is derived from the law of the wall.

For the determination of the normalised bottom stresses

$$\frac{\tau^x_b}{\rho_0}=u_*^{bx}u_*^b,$$ (49)

$$\frac{\tau^y_b}{\rho_0}=u_*^{by}u_*^b$$ (50)

with the friction velocities $u_*^b=\sqrt{\tau_b/\rho_0}$ with $\tau_b=\sqrt{(\tau^x_{b})^2+(\tau^y_{b})^2}$, assumptions about the structure of velocity inside the discrete bottom layer have to be made. We use here the logarithmic profile

$$\frac{u(z')}{u_*} =\frac{1}{\kappa} \left(\frac{z'+z_0^b}{z_0^b}\right),$$ (51)

with the bottom roughness length $z_0^b$, the von Kármán constant $\kappa=0.4$ and the distance from the bed, $z'$. Therefore, estimates for the velocities in the centre of the bottom layer can be achieved by:

$$u_b = \frac{u_*^{bx}}{\kappa} \left(\frac{0.5h_1+z_0^b}{z_0^b}\right),$$ (52)

$$v_b = \frac{u_*^{by}}{\kappa} \left(\frac{0.5h_1+z_0^b}{z_0^b}\right).$$ (53)

For $h_1\rightarrow 0$, the original Dirichlet-type no-slip boundary conditions (8) are retained. Another possibility would be to specify the bottom velocities $u_b$ and $v_b$ such that they are equal to the layer-averaged log-law velocities (see Baumert and Radach (1992)). The calculation of this is however slightly more time consuming and does not lead to a higher accuracy.