Wave-induced set-up

In a (geographic) 1D case the computation of the wave-induced set-up is based on the vertically integrated momentum balance equation which is a balance between the wave force (gradient of the wave radiation stress normal to the coast) and the hydrostatic pressure gradient (note that the component parallel to the coast causes wave-induced currents but no set-up).

  $$\frac{dS_{xx}}{dx} + \rho g H \frac{d \overline{\eta}}{dx} = 0$$ (2.145)

where $d$ is the total water depth (including the wave-induced set-up) and $\eta$ is the mean surface elevation (including the wave-induced set-up) and

  $$S_{xx} = \rho g \int [n \cos^2 \theta + n - \frac{1}{2}]E d \sigma d\theta$$ (2.146)

is the radiation stress tensor.


Observation and computations based on the vertically integrated momentum balance equation of Dingemans et al. (1987) show that the wave-induced currents are mainly driven by the divergence-free part of the wave forces whereas the set-up is mainly due to the rotation-free part of these forces. To compute the set-up in 2D, it would then be sufficient to consider the divergence of the momentum balance equation. If the divergence of the acceleration in the resulting equation is ignored, the result is:

  $$\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y}... ...artial}{\partial y} (\rho g H \frac{\partial \overline{\eta}}{\partial y}) = 0$$ (2.147)

This approximation can only be applied to open coast (unlimited supply of water from outside the domain, e.g. nearshore coasts and estuaries) in contrast to closed basin, e.g. lakes, where this approach should not be used.