Introduction

Wave setup is usually confined to narrow zones in the immediate vicinity of the shoreline. The size of such areas is small enough that the setup process can be considered to be quasi-stationary. Wave-induced currents are usually weak compared to tidal currents. This implies an equilibrium between the wave-induced force and gradient of the wave setup,

  $$gd \left ( \frac{\partial \zeta}{\partial x} + \frac{\partial \zeta}{\partial y} \right ) + F_x + F_y = 0$$ (5.1)

where $\zeta$ is the setup, $d$ the water depth and $F_{i}$ is the wave-induced force in $x_i$-direction per unit mass. In order to reduce the number of equations to one, we use the observation by Dingemans (1997) that wave-driven currents are mainly due to the divergence-free part of the wave forces whereas the setup is mainly due to the rotation-free part of the force field. We therefore take the divergence of eq. (5.1) to obtain the following elliptic partial differential equation for $\zeta$,

  $$\frac{\partial}{\partial x} (gd \frac{\partial \zeta}{\partial x}... ...al y}) + \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} = 0$$ (5.2)

This Poisson equation needs one boundary condition in each point of the boundary of the computational domain. Two types of boundary conditions are foreseen; the first one is used on the open boundaries and on the shoreline where the shoreline is defined as the line where the depth is zero:

  $$F_{n} + gd \frac{\partial \zeta}{\partial n} = 0$$ (5.3)

with $n$ the outward direct normal. It is not possible to use this boundary condition on all boundary points because then there remains an unknown constant. So some point for which we take the boundary point with the largest depth, the setup is assumed to be $\zeta=0$.


The second type of boundary condition with given value of $\zeta$ is also used in nested models. The setup computed in the larger model is used as boundary condition in the nested model. In the nested model the setup is given in all points of the outer boundary. On the shoreline inside the area again eq. (5.3) is used.


The Poisson equation (5.2) together with its boundary conditions will be solved numerically on a curvilinear grid. The next section discusses the details of the method. After each iteration performed in SWAN new values of the setup are being calculated and added to the depth, so that the SWAN model incorporates the effect of setup on the wave field. An output quantity SETUP is added so that the user can be informed about the magnitude and distribution of the wave setup.