Triads

In this section two methods are described for the computation of nonlinear interactions at shallow water. The first method is called the LTA method and is based on a Boussinesq theory. It generates second (and possibly fourth and eighth) higher harmonics, which are, however, persistent over large distances away from the surf zone. The second one, called the DCTA method, heuristically captures the $k^{-4/3}$ spectral tail at shallow water depths, while generating all transient sub and super harmonics.


LTA


The Lumped Triad Approximation (LTA) of Eldeberky (1996), which is a slightly adapted version of the Discrete Triad Approximation (DTA) of Eldeberky and Battjes (1995), is used in SWAN in each spectral direction:

  $$S_{\rm nl3} (\sigma) = S^-_{\rm nl3} (\sigma) + S^+_{\rm nl3} (\sigma)$$ (2.91)

with

  $$S^+_{\rm nl3} (\sigma) = \max \Bigl[0,\alpha\, c_\sigma\, c_{g,\s... ...sin(-\beta) \left \{ E^2(\sigma/2) - 2 E(\sigma/2) E(\sigma) \right \} \Bigr]$$ (2.92)

and

  $$S^-_{\rm nl3} (\sigma) = -2 S^+_{\rm nl3} (2\sigma)$$ (2.93)

in which $\alpha$ is a tunable scaling factor that controls the strength of triad interactions, $c_\sigma$ and $c_{g,\sigma}$ are the phase and group velocity, respectively, at $\sigma$, $\beta$ is the biphase of self-self interaction, and finally, $J$ is the interaction coefficient based on the Boussinesq theory of Madsen and Sørensen (1993), as follows

  $$J = \frac{k^2_{\sigma/2} (gd + 2 c^2_{\sigma/2})} {k_\sigma d (gd + \frac{2}{15} gd^3 k^2_\sigma - \frac{2}{5} \sigma^2 d^2)}$$ (2.94)

As proposed by Eldeberky (1996), the biphase $\beta$ is approximated as

  $$\beta = -\frac{\pi}{2} + \frac{\pi}{2} \tanh (\frac{m}{Ur})$$ (2.95)

with Ursell number $Ur$ given by

  $$Ur = \frac{g\,H_{m0}}{8\sqrt{2}} \left( \frac{T_{m01}}{\pi\,d} \right)^2$$ (2.96)

and $m$ a tunable coefficient.


Eldeberky and Battjes (1995) proposed $m = 0.2$ based on a laboratory experiment. However, our recent experience shows that this relatively low value triggers some instability that artificially amplifies higher harmonics in the triad computation. Yet it will be less prone to error if the value of $m$ is increased. As suggested by Doering and Bowen (1995), the optimal agreement of Eq. (2.95) with the data of some field measurements is obtained with a value of $m=0.63$, which also reflects a robust numerical performance.


Recently, De Wit (2022) proposed a biphase parametrization that is derived from SWASH experiments demonstrating the dependence of biphase on local bed slope and local peak wave period. This parametrization is implemented in SWAN version 41.45. Note that this parametrization allows the biphase values to be positive, which potentially makes it possible to include the effect of recurrence (i.e. to transfer wave energy back to the primary peak).


DCTA


Booij et al. (2009) proposed a heuristic triad formulation, in analogy with the quadruplet interaction, that accounts for both the generation of all super harmonics and the transition to a universal tail of $k^{-4/3}$. The original expression for the Distributed Collinear Triad Approximation (DCTA) is given by

  $$S_{\rm nl3} (\sigma_1) = \lambda \, \frac{\sin(-\beta)\,\tilde{k}... ...,2}k_2^p\,N(\sigma_2) - \sigma_1\,c_{g,1}k_1^p\,N(\sigma_1) \Bigr] \,d\sigma_2$$ (2.97)

for the quasi-resonance conditions $\sigma_3 = \vert\sigma_2 - \sigma_1\vert$ (note that the frequencies match but the wave numbers not). Here, $\lambda$ is a calibration coefficient that controls the magnitude of triad interactions, $\tilde{k} = \tilde{\sigma}/\sqrt{gd}$ with $\tilde{\sigma}$ the mean frequency as given by Eq. (2.67), $p = 4/3$ is a shape coefficient to force the high-frequency tail, and $\overline{k} = (k_1+k_2+k_3)/3$ is a characteristic wave number of the triad. Note that the factor $\tanh(\overline{k}d)/\overline{k}d$ in Eq. (2.97) accounts for the increasing resonance mismatch with increasing wave number (Booij et al., 2009).


However, recent study (Zijlema, 2022) has demonstrated that the following energy-flux conservative expression

  $$S_{\rm nl3} (\sigma_1) = \lambda \, c_{g,1}\frac{\sin(-\beta)\,\t... ...Bigl[ c_{g,2}k_2^p\,E(\sigma_2) - c_{g,1}k_1^p\,E(\sigma_1) \Bigr] \,d\sigma_2$$ (2.98)

yields improved prediction accuracy over the original one (2.97). This revised formulation is implemented in version 41.45.


In the above DCTA formulations, a collinear approximation is applied by which it is assumed that the primary contribution to the triad interactions arises from collinear interactions. Effectively, the triad source term for each directional bin is taken independently of other directional bins. An extension to noncollinear triad interactions is proposed by Benit and Reniers (2022) which includes a transfer reduction scaling based on the angle difference between two noncollinear interacting components. The final expression yields

$$S_{\rm nl3} (\sigma_1,\theta_1) = \lambda \, c_{g,1}\frac{\sin(-\beta)\,\tilde{k}^{2-p}}{\tilde{\sigma}^2\,d^2} \,\, \times$$

$$\int_{0}^{2\pi} \int_{0}^{\infty} \biggl (\frac{\tanh\,\overline{... ..._2,\theta_2) - c_{g,1}k_1^p\,E(\sigma_1,\theta_1) \Bigr] \,d\sigma_2\,d\theta_2$$

with $G(\Delta \theta_{nm})$ the transfer function of Sand (1982) and $\Delta \theta_{nm} = \theta_n - \theta_m$.


As a final note, the triad wave-wave interactions due to either LTA or DCTA are calculated only for $Ur \geq 0.1$.