An illustrative explanation of the sweeping approach

In the absence of a current, the direction of propagation of the wave crest is equal to that of the wave energy. For this case, the propagation velocity of energy ($c_x$, $c_y$) is equal to the group velocity ($c_{g,x}$, $c_{g,y}$). In presence of a current this is not the case, since the propagation velocities $c_x$ and $c_y$ of energy are changed by the current. Considering the applied numerical procedure in SWAN, it is initially more convenient to explain the basic principles of the numerical procedure in the absence of a current than in the situation where a current is present. So, first, we shall focus on the sweeping technique in the absence of a current. After this, we shall discuss the numerical procedure in case a current is present.


The computational region is a rectangle covered with a rectangular grid. One of the axes (say the $x-$axis) is chosen arbitrary, for instance perpendicular to the coast. The state in a grid point ($x_i$,$y_j$) in an upwind stencil is determined by its up-wave grid points ($x_{i-1}$,$y_j$) and ($x_i$,$y_{j-1}$). This stencil covers the propagation of action density within a sector of 0$^o -$90$^o$, in the entire geographic space; see Figure 3.2.

Figure 3.2: Numerical scheme for wave propagation in geographic space with below the first quadrant for which the waves are propagated.

Hence, this procedure is called sweep 1 and encloses all wave energy propagation over the first quadrant in spectral space. This quandrant is the corresponding domain of dependence in the directional space. By rotating the stencil over 90$^o$, the next quadrant 90$^o -$180$^o$ is propagated. Rotating the stencil twice more ensures propagation over all four quadrants (see Figure 3.1). This allows waves to propagate from all directions. Hence, the method is characterized as a four-sweep technique.


The gain of such a stencil is that the propagation is unconditionally stable because the wave characteristics lie within the concerning quadrant. Thus, propagation is not subjected to a CFL criterion. In addition, the principle of causality requires that the numerical domains of dependence of both geographic and spectral spaces must be identical.


In cases with bottom refraction or current refraction, action density can shift from one quadrant to another. This is taken into account in the model by repeating the computations with converging results (iterative four-sweep technique). Typically, we choose a change of less then 1% or so in significant wave height and mean wave period in all geographic grid points to terminate the iteration (see Section 3.4).


Note, however, we may even choose more sweeps than the proposed 4 ones^3.3. The number of sweeps is denoted by $M$ and the directional interval of each sweep equals 360$^o/M$. It is expected that the number of iterations may reduced, since the solution is guaranteed to be updated for all wave directions at all grid points in one series of sweeps, provided the directional interval is sufficiently small (e.g. 8 sweeps with an interval of 45$^o$ each, or 12 sweeps of each 30$^o$). This is certainly the case at deep water where wave rays are just straight lines. However, at shallow water, this becomes less obvious because of the presence of refraction. In this case the wave energy may jump multiple directional bins in one (large) time or distance step. It may leave a sweep sector too early or it may even skip this sweep, especially when the sweep interval is relatively small ($<$ 30$^o$) and depth changes per grid cell are relatively large. The result is that wave rays erroneously cross and that a number of wave components in one sweep within one time/distance step overtakes some other bins in another sweep ahead, which implies that causality is violated, resulting in a possible model instability. See also Section 3.8.3. Hence, for such shallow water cases, choosing a relative large number of sweeps, $M>4$, is more likely to prove counter-productive.


The numerical procedure as described above remains in principle the same when a current ($U_x$,$U_y$) is present. The main difference is that the propagation velocities of energy are no longer equal to the group velocity of the waves but become equal to $c_x = c_{g,x}+U_x$ and $c_y = c_{g,y}+U_y$. To ensure an unconditionally stable propagation of action in geographical space in the presence of any current, it is first determined which spectral wave components of the spectrum can be propagated in one sweep. This implies that all wave components with $c_x > 0$ and $c_y > 0$ are propagated in the first sweep, components with $c_x<0$ and $c_y > 0$ in the second sweep, components with $c_x<0$ and $c_y<0$ in the third sweep, and finally, components $c_x > 0$ and $c_y<0$ in the fourth sweep. Since the group velocity of the waves decreases with increasing frequency, the higher frequencies are more influenced by the current. As a result, the sector boundaries in directional space for these higher frequencies change more compared to the sector boundaries for the lower frequencies. In general, four possible configurations do occur (see Figure 3.3). Consider, for instance, one fixed frequency propagating on a uniform current. The current propagates at an angle of 45$^o$ with the $x-$axis. The sign of the current vector and strength of the current are arbitrary. The shaded sectors in Figure 3.3 indicate that all the wave components that are propagating in the direction within the shaded sector, are propagated in the first sweep ($c_x > 0$, $c_y > 0$).

Figure 3.3: Four possible configurations and propagation velocities $c_x$, $c_y$ for a fixed frequency in the presence of a current propagating at an angle of 45$^o$ with the $x-$axis.

The top-left panel (A) represents a situation in which both $c_x$ and $c_y$ are negative due to a strong opposing current, i.e. wave blocking occurs. None of the wave components is propagated within the first sweep. The top-right panel (B) represents a situation in which the current velocity is rather small. The sector boundaries in directional space are hardly changed by the current such that the sector boundaries are approximately the same as in the absence of a current. The bottom-left panel (C) reflects a following current that causes the propagation velocities of the wave components in two sectors to be larger than zero. In this specific case, all the waves of the shaded sectors are propagated within the first sweep. The bottom-right panel (D) represents a case with a strong following current for which all the action is take along with the current. For this case the fully 360$^o$ sector is propagated in the first sweep.


After it has been determined which wave components are propagated in one sweep, i.e., the sector boundaries in directional space have been determined for each frequency, the integration in frequency and directional space can be carried out for those wave components.