Page updated: April 16, 2020
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 Ψ1s(kx,ky) 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 Ψ1s(kx,ky) 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

Ψ(kx,ky) = 1 k𝒮(k)Φ(k,φ) Ψ(k,φ).

Here 𝒮(k) is the omnidirectional spectrum, and Φ(k,φ) is the nondimensional spreading function, which shows how waves of different frequencies propagate (or ”spread out”) relative to the downwind direction at φ = 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

Φ(k,φ) = CS cos2S(φ2), (1)

where S is a spreading parameter that in general depends on k, wind speed, and wave age. CS is a normalizing coefficient that gives

02πΦ(k,φ)dφ = 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 φ, so that more variance (wave energy) is associated with downwind directions (|φ| < 90deg) than upwind (|φ| > 90deg). The larger the value of S, the more the waves propagate almost directly downwind (φ = 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 1: The cosine-2S spreading functions for S = 2 and 20. Top panel: polar plot in φ; bottom panel: linear in φ.

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 10ms1. The simulated region is 100 × 100m 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.

mssx(r,s) = z(r + 1,s) z(r,s) x(r + 1) x(r)

averaged over all points of the 2-D surface grid. The 𝜃x and 𝜃y 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

𝜃x(r,s) = tan(mssx(r,s)),

etc., averaged over all points of the surface.


Figure 2: A sea surface generated with the ECKV omnidirectional spectrum and a cosine-2S spreading function with S = 2. Compare with Fig. 3. Generated by IDL routine

slope variable DFT value Cox-Munk formula Cox-Munk value

mssx 0.031 0.0316U 0.032
mssy 0.021 0.0192U 0.019
mss 0.052 0.001(3 + 5.12U) 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 φ = 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 φ from the downwind.


Figure 3: A sea surface generated with the ECKV omnidirectional spectrum and a cosine-2S spreading function with S = 20. Compare with Fig. 2.

As shown in on the Wave Variance Spectra: Theory page , the total mean square slope depends only on the omnidirectional spectrum 𝒮(k). 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 Φ(k,φ) so long as it satisfies the normalization condition (2).

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