Implementation of DIA within the four-sweep technique

In SWAN, the quadruplets are integrated using the DIA, see Section 2.3.4. As a consequence of the four-sweep technique, two different types of methods can be used to calculate the four wave-wave interactions.

1. The first method implies that the interactions are calculated in every iteration prior to the first sweep. This method ensures conservation of energy density, but has the disadvantage that the spectral source term $S_{\rm nl4}(\sigma,\theta)$ for every grid point in geographical space has to be stored in internal memory. Such an integration method increases the amount of required memory with a factor about 2. The source term is stored in memory and is then explicitly integrated for a particular sweep. This method is particularly useful when ambient current is included.
2. The second method is slightly different, in which the interactions are calculated and integrated for every sweep separately. The calculation of the energy transfer for a specific quadrant requires also the calculation of the transfer within a sector of about 33$^o$ in the adjacent two quadrants. Calculating the interactions for one sweep increases the computation time for the quadruplets with a factor of about 1.66 (2 adjacent sectors of 33$^o$ times 4 sweeps divided by 360$^o$). Contrary to the first method, the total rate of this energy shift is not stored in memory so that energy density is not conserved per sweep (and per iteration). However, this does not influence the converging results. Within this second method, two different numerical schemes are available, namely, a semi-implicit scheme and a fully explicit scheme. The use of a fully explicit scheme is recommended because of the computational efficiency. A semi-implicit scheme increases the computation time of the quadruplets with a factor of about 2. Note that this method does not work when ambient current is included in the model. The reason is that the bounds of the directional sector within a sweep may not be the same for every frequency bin. So there may be some overlap of frequency bins between the quadrants, and so this method does not conserve energy anymore. To prevent this the user is advised to choose the first method.