**Page updated:**
March 22, 2021 **Author:** Curtis Mobley

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# Spreading Function Effects

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

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

Here $\mathcal{\mathcal{S}}\left(k\right)$ 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

$$\Phi \left(k,\phi \right)={C}_{S}{cos}^{2S}\left(\phi \u22152\right)\phantom{\rule{0.3em}{0ex}},$$ | (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 $, so that more variance (wave energy) is associated with downwind directions ($\left|\phi \right|<90\phantom{\rule{2.6108pt}{0ex}}deg$) than upwind ($\left|\phi \right|>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 $100\times 100\phantom{\rule{2.6108pt}{0ex}}m$ using $512\times 512$ 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 $\u27e8{\mathit{\theta}}_{x}\u27e9$ and $\u27e8{\mathit{\theta}}_{y}\u27e9$ 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

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

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 |

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 $\mathcal{\mathcal{S}}\left(k\right)$. 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).