Page updated: March 22, 2021
Author: Curtis Mobley
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The last ﬁgure on the previous page shows a contour plot of a two-dimensional, one-sided variance spectrum ${\Psi }_{1s}\left({k}_{x},{k}_{y}\right)$ and a contour plot of a random surface generated from that variance spectrum. A particular spreading function is implicitly contained in that two-dimensional variance spectrum. The eﬀect on the generated sea surface of the spreading function contained within ${\Psi }_{1s}\left({k}_{x},{k}_{y}\right)$ warrants discussion.

As we have seen (e.g. Eq. 4 of the Wave Variance Spectra: Examples page), a 2-D variance spectrum is usually partitioned as

 $\Psi \left({k}_{x},{k}_{y}\right)=\frac{1}{k}\mathsc{𝒮}\left(k\right)\Phi \left(k,\phi \right)\equiv \Psi \left(k,\phi \right)\phantom{\rule{0.3em}{0ex}}.$

Here $\mathsc{𝒮}\left(k\right)$ is the omnidirectional spectrum, and $\Phi \left(k,\phi \right)$ is the nondimensional spreading function, which shows how waves of diﬀerent frequencies propagate (or “spread out”) relative to the downwind direction at $\phi =0$.

One commonly used family of spreading functions is given by the “cosine-2S” functions of Longuet-Higgins et al. (1963), which have the form

 $\Phi \left(k,\phi \right)={C}_{S}{cos}^{2S}\left(\phi ∕2\right)\phantom{\rule{0.3em}{0ex}},$ (1)

where $S$ is a spreading parameter that in general depends on $k$, wind speed, and wave age. ${C}_{S}$ is a normalizing coeﬃcient that gives

 ${\int }_{0}^{2\pi }\Phi \left(k,\phi \right)\phantom{\rule{0.3em}{0ex}}d\phi =1$ (2)

for all $k$.

Figure 1 shows the cosine-2S spreading functions for values of $S=2$ and 20. These spreading functions are strongly asymmetric in $\phi$, so that more variance (wave energy) is associated with downwind directions ($|\phi |<90\phantom{\rule{2.6108pt}{0ex}}deg$) than upwind ($|\phi |>90\phantom{\rule{2.6108pt}{0ex}}deg$). The larger the value of $S$, the more the waves propagate almost directly downwind ($\phi =0$), rather than at large angles relative to the downwind direction. However, the cosine-2S spreading functions always have a least a tiny bit of energy propagating in upwind directions, as can be seen for the $S=2$ curves. This is crucial for the generation of time-dependent surfaces, as will be discussed on the next page.

Figure 2 shows a surface generated with the omnidirectional variance spectrum of Elfouhaily et al. (1997) (ECKV) as used on the previous page, combined with a cosine-2S spreading function (1) with $S=2$ for all $k$ values. The wind speed is $10\phantom{\rule{2.6108pt}{0ex}}m\phantom{\rule{2.6108pt}{0ex}}{s}^{-1}$. The simulated region is $100×100\phantom{\rule{2.6108pt}{0ex}}m$ using $512×512$ grid points. Note in this ﬁgure that the mean square slopes (mss) compare well with the corresponding Cox-Munk values shown in Table . The mss values for the generated surface are computed from ﬁnite diﬀerences, e.g.

 $ms{s}_{x}\left(r,s\right)=\frac{z\left(r+1,s\right)-z\left(r,s\right)}{x\left(r+1\right)-x\left(r\right)}$

averaged over all points of the 2-D surface grid. The $⟨{𝜃}_{x}⟩$ and $⟨{𝜃}_{y}⟩$ values shown in the ﬁgure are the average angles of the surface from the horizontal in the $x$ and $y$ directions). These are computed from

 ${𝜃}_{x}\left(r,s\right)=tan\left(ms{s}_{x}\left(r,s\right)\right)\phantom{\rule{0.3em}{0ex}},$

etc., averaged over all points of the surface.

 slope variable DFT value Cox-Munk formula Cox-Munk value $ms{s}_{x}$ 0.031 $0.0316U$ 0.032 $ms{s}_{y}$ 0.021 $0.0192U$ 0.019 $mss$ 0.052 $0.001\left(3+5.12U\right)$ 0.054

Table 1: Comparison of Cox-Munk mean square slopes and values for the DFT-generated 2-D surface of Fig. 2.

The spreading function used in Fig 2 was chosen (with a bit of trial and error) to give a distribution of along-wind and cross-wind slopes in close agreement with the Cox-Munk values (except for a small amount of Monte-Carlo noise). Figure 3 shows a surface generated with a cosine-2S spreading function with $S=20$; all other parameters were the same as for Fig. 2. This $S$ value gives wave propagation that is much more strongly in the downwind direction $\phi =0$, as would be expected for long-wavelength gravity waves in a mature wave ﬁeld. The surface waves thus have a visually more “linear” pattern, whereas the waves of Fig. 2 appear more “lumpy” because waves are propagating in a wider range of angles $\phi$ from the downwind.

As shown in on the Wave Variance Spectra: Theory page , the total mean square slope depends only on the omnidirectional spectrum $\mathsc{𝒮}\left(k\right)$. Thus the total mss is the same (except for Monte Carlo noise) in both ﬁgures 2 and 3, but most of the total slope is in the along-wind direction in Fig. 3.

Real spreading functions are more complicated than the cosine-2S functions used here. In particular, some observations Heron (2006) of long-wave gravity waves tend to show a bimodal spreading about the downwind direction, which transitions to a more isotropic, unimodal spreading at shorter wavelengths. Although omnidirectional wave spectra are well grounded in observations, the choice of a spreading function is still something of a black art. You are free to choose any $\Phi \left(k,\phi \right)$ so long as it satisﬁes the normalization condition (2).