Conservative elimination of negative energy densities

The numerical solution obtained with the hybrid central/upwind scheme Eqs. (3.20) and (3.21) is, in principle, not free from spurious oscillations or wiggles, unless $\mu~=~\nu~=~0$. However, these wiggles will not result in negative densities in the energy-containing part of the two-dimensional wave spectrum, as they occur against a background level of energy. On the other hand, the flanks of the spectrum are immediately adjacent to bins of zero energy. Hence, negative energy prevails especially in these flanks, either at very low frequencies ($\sim 0.03$ Hz) or in the edges of spread waves, which is likely to be generated by the hybrid scheme exhibiting numerical dispersion. Since broad banded wave energy field is relatively smooth away from blocking frequency (low gradients), these negative energy densities tend to be rather small. They can be effectively removed through the so-called conservative elimination (Tolman, 1991). In short, all negative energy density for each frequency within a sweep is removed, and the energy densities for all directions within this directional sector at a given frequency are multiplied by a constant factor to conserve energy.


The conservative elimination algorithm is outlined as follows. For a given frequency $f$ and a given directional range $[\theta_1, \theta_2]$ of one of the four sweeps, we differentiate between positive and negative contributions of the energy density, as follows

  $$E\,(f,\theta) = E^{+}\,(f,\theta) + E^{-}\,(f,\theta)$$ (3.27)

with $E^{+} > 0$ and $E^{-} < 0$. Next, we compute the total energy within the sweep at frequency $f$

  $$E_{{\rm tot}}\,(f) = \int_{\theta_1}^{\theta_2}\,E\,(f,\theta)\,d\theta$$ (3.28)

and we integrate the positive densities over the considered directions

  $$E^p_{{\rm tot}}\,(f) = \int_{\theta_1}^{\theta_2}\,E^{+}\,(f,\theta)\,d\theta$$ (3.29)

Based on these two quantities, we compute the following factor

  $$\alpha\,(f) = \frac{E_{{\rm tot}}\,(f)}{E^p_{{\rm tot}}\,(f)} \leq 1$$ (3.30)

Finally, we remove the negative contributions by setting $E^{-}\,(f,\theta)$ = 0, while the positive densities are multiplied by factor $\alpha\,(f)$ to preserve the total energy at given frequency $f$.


The effectiveness of this conservative elimination algorithm can, however, be poor for a number of reasons. One of the reasons is that the directional resolution is too coarse for the scale of directional spreading, so that energy is spread over a few directional bins. For instance, a wave spectrum with small directional spreading ($< 10^o$) distributed over a number of directional bins of each $10^o$ within a sweep. As a result, within this directional sector the total amount of negative densities can be larger than the amount of positive ones. This implies $\alpha < 0$. In such a case, conserving energy within this sweep makes no sense, and we will then eliminate the negative energy densities and to leave the positive densities as they are. This is called strict elimination. It must be noted that in that case the hybrid scheme Eq. (3.21) is inaccurate anyway, and strict elimination will most likely not worsen this case.


Another reason is the frequent occurrence in the exchange of wave energy between directional sweep sectors (see Section 3.4 for details), which may enhance the gradient in the energy density locally and thereby generates more negative energy densities. This is particular the case if the directional discretization is too coarse for the scale of spreading. Finally, a relatively strong turning rate tends to provoke strict elimination.


As a rule, for a typical field case over 95% of the number of occurrences of elimination of negative energy densities concerns conservative elimination. Hence, less than 5% of this number is related to (non-conserved) strict elimination, which is acceptable.