Discretization

Discretization of Eq. (2.19) is carried out using the finite difference method. The homogeneous part of Eq. (2.19) is given by

  $$\frac{\partial N}{\partial t} + \frac{\partial c_x N}{\partial x... ..._\sigma N}{\partial \sigma} + \frac{\partial c_\theta N}{\partial \theta}\, .$$ (3.1)

We choose a rectangular grid with constant mesh sizes $\Delta x$ and $\Delta y$ in $x-$ and $y-$direction, respectively. The spectral space is divided into elementary bins with a constant directional resolution $\Delta \theta$ and a constant relative frequency resolution $\Delta \sigma/\sigma$ (resulting in a logarithmic frequency distribution). We denote the grid counters as $1 \leq i \leq M_x$, $1 \leq j \leq M_y$, $1 \leq l \leq M_{\sigma}$ and $1 \leq m \leq M_{\theta}$ in $x-$, $y-$, $\sigma-$ and $\theta-$spaces, respectively. All variables, including e.g. wave number, group velocity, ambient current and propagation velocities, are located at points $(i,j,l,m)$. Time discretization takes place with the implicit Euler technique. We obtain the following approximation of Eq. (3.1):

$$\frac{N^{n}-N^{n-1}}{\Delta t}\vert _{i,j,l,m} + \frac{[c_x N]_{i+1/2}-[c_x N]_{i-1/2}}{\Delta x}\vert^{n}_{j,l,m} +$$

$$\frac{[c_y N]_{j+1/2}-[c_y N]_{j-1/2}}{\Delta y}\vert^{n}_{i,l,m} +$$

$$\frac{[c_\sigma N]_{l+1/2}-[c_\sigma N]_{l-1/2}}{\Delta \sigma}\vert^{n}_{i,j,m} +$$

$$\frac{[c_\theta N]_{m+1/2}-[c_\theta N]_{m-1/2}}{\Delta \theta}\vert^{n}_{i,j,l}\, ,$$ (3.2)

where $n$ is a time-level with $\Delta t$ a time step. In case of a stationary computation, the first term of Eq. (3.2) is removed and $n$ denotes an iteration level. Note that locations in between consecutive counters are reflected with the half-indices.