Governing equations in curvilinear co-ordinates

A curvilinear grid is characterized by the co-ordinates of the grid points, i.e.

  $$x_{i,j}\, , \qquad i=1,\cdots,M \, , j=1,\cdots,N$$ (3.89)

  $$y_{i,j}\, , \qquad i=1,\cdots,M \, , j=1,\cdots,N$$ (3.90)

The four-sweep method is unchanged, so in the first sweep action densities in the points $(i-1,j)$ and $(i,j-1)$ are used to compute the action densities in the point $(i,j)$. Numerical approximations are obtained by a two-dimensional Taylor expansion with respect to the point $(x_{i,j},y_{i,j})$.


The differences in quantities between neighbouring grid points in the curvilinear grid are denoted as follows

  $$\Delta x_1 = x_{i,j} - x_{i-1,j} \, , \qquad \Delta y_1 = y_{i,j} - y_{i-1,j}\, , \qquad \Delta F_1 = F_{i,j} - F_{i-1,j}$$ (3.91)

and

  $$\Delta x_2 = x_{i,j} - x_{i,j-1} \, , \qquad \Delta y_2 = y_{i,j} - y_{i,j-1}\, , \qquad \Delta F_2 = F_{i,j} - F_{i,j-1}$$ (3.92)

The partial derivatives can be found from the two-dimensional Taylor expansions

  $$\Delta F_1 = \frac{\partial F}{\partial x} \Delta x_1 + \frac{\partial F}{\partial y} \Delta y_1$$ (3.93)

and

  $$\Delta F_2 = \frac{\partial F}{\partial x} \Delta x_2 + \frac{\partial F}{\partial y} \Delta y_2$$ (3.94)

It follows that the partial derivatives can be approximated by

  $$\frac{\partial F}{\partial x} \approx \frac{\Delta y_2 \Delta F_1 - \Delta y_1 \Delta F_2}{[D]}$$ (3.95)

and

  $$\frac{\partial F}{\partial y} \approx \frac{\Delta x_1 \Delta F_2 - \Delta x_2 \Delta F_1}{[D]}$$ (3.96)

where

  $$[D] = \Delta y_2 \Delta x_1 - \Delta y_1 \Delta x_2$$ (3.97)

Thus, in curvilinear co-ordinates the complete propagation terms (including time-derivative, but ignoring dependence on $\sigma$ and $\theta$ temporarily) read

$$\left ( \frac{1}{\Delta t} + (D_{x,1} + D_{x,2})c_{x,i,j}^+ + (D_{y,1}+D_{y,2}) c_{y,i,j}^+ \right ) N_{i,j}^+$$

$$- \frac{N_{i,j}^-}{\Delta t} - D_{x,1} (c_x N)_{i-1,j}^+ - D_{y,1} (c_y N)_{i-1,j}^+$$

$$- D_{x,2} (c_x N)_{i,j-1}^+ - D_{y,2} (c_y N)_{i,j-1}^+ = S_{i,j}^+$$ (3.98)

where

  $$D_{x,1} = \frac{\Delta y_2}{[D]} \, , \quad D_{y,1} = -\frac{\De... ...D_{x,2} = -\frac{\Delta y_1}{[D]} \, , \quad D_{y,2} = \frac{\Delta x_1}{[D]}$$ (3.99)

Here, the superscript $+$ denotes the new time level $t$, and $-$ the old time level $t-\Delta t$. The equation for a stationary computation is found by putting $1/\Delta t$ to 0.


Again, the marching method is stable as long as the propagation direction towards the point $(i,j)$ is enclosed between the lines connecting this point with its neighbours $(i-1,j)$ and $(i,j-1)$. It can be shown that this is the case if

  $$D_{x,1}c_x + D_{y,1}c_y \geq 0\, \quad \mbox{and} \quad D_{x,2}c_x + D_{y,2}c_y \geq 0$$ (3.100)

This set of criterions enables the SWAN program to decide whether a certain spectral direction does belong in the sweep which is being processed (in this the first sweep).


In the second sweep, $\Delta x_1 = x_{i,j} - x_{i,j-1}$, etc. and $\Delta x_2 = x_{i,j} - x_{i+1,j}$, etc. In the third sweep, $\Delta x_1 = x_{i,j} - x_{i+1,j}$, etc. and $\Delta x_2 = x_{i,j} - x_{i,j+1}$, etc. In the fourth sweep, $\Delta x_1 = x_{i,j} - x_{i,j+1}$, etc. and $\Delta x_2 = x_{i,j} - x_{i-1,j}$, etc. Otherwise, all of the above equations and conditions remain the same.


Conservation of action in the numerical approximation can be demonstrated for the triangle of which the corners are the three points $(i,j)$, $(i-1,j)$ and $(i,j-1)$. If for each side of this triangle the energy flux is computed as the inner product of the average of $cN$ and an inward-pointing normal of the side itself, then the three energy fluxes are exactly in balance assuming that the situation is stationary, and the source term is zero. In this case it is found that

$$\left [ c_x N \right ]_{i,j}^{+} (\Delta y_2 - \Delta y_1) + \lef... ...]_{i-1,j}^{+} (-\Delta y_2) + \left [ c_x N \right ]_{i,j-1}^{+} (\Delta y_1) +$$

$$\left [ c_y N \right ]_{i,j}^{+} (\Delta x_1 - \Delta x_2) + \lef... ...{i-1,j}^{+} (\Delta x_2) + \left [ c_y N \right ]_{i,j-1}^{+} (-\Delta x_1) = 0$$ (3.101)