Wave spectral shapes

A contribution from Michel Olagnon (Ifremer)

Current engineering practice favours the JONSWAP spectrum to represent the frequency distribution of the incoming waves when computing the action of these waves. In addition, the JONSWAP spectrum is conveniently implemented in every hydrodynamic code. Yet, assessment of the JONSWAP spectrum was almost exclusively carried out for the North Sea and within a small frequency interval about the peak frequency, where the bulk of the structural responses could be found at the time the model was developed.

In many cases, the lack of a better validated model stretches the application domain for the JONSWAP spectrum far beyond the range of conditions that it was designed for. It is noteworthy that cos^2s power is commonly used to model directional spreading without much more justification, and that floating systems sensitive to crossed-seas might thus suffer from non-conservative design.

It does not mean either that current practices would be invalid or unsafe, but this shows clearly that they would strongly benefit from a reassessment of the way spectral shapes are used in the computations and of the actual validity of these methods for frequencies significantly away from the spectral peak, as well in the high frequency domain for ringing excitation as in the low frequency domain for the responses of moored systems.

It should also be investigated whether the current models sufficiently take into account directional spreading, multiple peaks, realistic energy contents for all frequency bands of interest and shape characteristics of spectra at locations where the climate is very different from that of the North Sea. Suggestions for improved or alternative models, yet simple and easy to use by engineers, might also be made.

The high frequency contents of the spectrum can be split between bound and free waves. Since one cannot think of any generating mechanism for free high frequency waves that could compete with the non linear composition of lower frequency waves into bound waves, it is a sensible assumption that the linear spectrum will exhibit a high rate of decay, and that most energy in any high frequency band will be the result of wave-wave interaction. Moreover, the results of Forristall and Prevosto in the WACSIS JIP have shown that as far as water surface elevation is concerned, second-order reconstruction of the high frequency energy contents starting from a linear spectrum is quite satisfactory. Linear spectra can be obtained with standard buoys, but may still differ significantly from the JONSWAP model, especially by featuring several peaks. Yet, it is very likely that using a correct shape of the free waves spectrum in its most energetic frequency range, and computing the second-order energy contents would accurately enough represent the required input to high frequency hydrodynamic responses computations.

Further investigations might be more useful if they consider the full scale verification of the whole chain of computations, from the linear spectrum to say ringing responses.

Similar to the high frequency end, the low frequency contents may be split between bound and free waves. However, swell, wave groups or seiches can generate low frequency energy that is independent of that of the bound waves resulting from the non-linear composition of higher frequency waves. The two can thus not be easily distinguished. One may also wonder if the low frequency content of the wave trains is stationary on the time-scale of a sea state, or if that energy comes in bursts that would require dedicated processing methods. In any case, special care should be exercised in de-trending and smoothing/windowing the data, to keep actual low frequency wave energy distinct from the effects of frequencies leaking from tidal components or from computational artefacts.

One might also assume that low frequency responses are mostly due to fluid-structure interaction non-linearities, and that the excitation energy stemming from pure wave-wave interaction can be safely neglected. Even if this were the case, if computations can provide a good estimation of the low-frequency content of the wave spectrum, that would be a strong assessment of the ability of the chosen representation to lead to the true low-frequency response.

For both cases, the low- and high-frequency tails of the spectrum may be obtained through wave-wave interaction calculations from the main body of the spectrum, and are a first stage that may already give some sensible information, provided the main body is accurately represented.

Not very many measurements are reliable when it comes to high and low frequency contents of the spectrum. For the high frequency part, it is a well-known feature of buoys that they "linearize" the shape of the free surface, by staying longer than they should on the crests as they are carried over by the wave, and remaining for shorter durations in the troughs. Sensors mounted on a fixed platform are thus almost mandatory to carry out some study of the high frequency tail of the wave spectrum. The sampling frequency and the cut-off frequency of the measurement systems should also be sufficiently high to keep observations with full high frequency contents. Yet, some carefully designed experiments such as WACSIS have allowed us to establish the high frequency tail of the spectrum from numerical computation of the wave-wave interactions, in such a manner that most of the problems of high frequency wave input for structural response calculation are now solved.

Determining the low frequency content requires long time-series and special processing in order to be correctly assessed. Unfortunately, very few long time-histories have ever been recorded, mostly because of storage limitations. To process these time-series, one has to face the problem of the stationarity of the sea state, which may be too short to allow valid analysis of phenomena that may not appear a sufficient number of times in a single sea state.

Filtering problems may also occur, since acceleration integration in buoys would amplify errors in the low frequency, and for the Datawell sensor it is reported that if the instrument platform oscillates, this is registered as additional heave (Krogstad).

The upper frequency tail of the spectrum seems to require rather practical implementation and dissemination of existing results and methods than dedicated studies. Yet, it is still dependent on a correct model for the main body of the spectral energy. Investigations should thus focus on

1. the improvement of the shape of the main body of the spectrum, especially in order to deal with multi-peak spectra, and
2. on the fit of shapes in different regional climates, especially in swell dominated areas.

For the low-frequency part of the spectrum, the following tasks are suggested:

1. inventory of existing long time-series of wave records;
2. analyses of the capacity of processing methods to extract the actual low-frequency excitation energy and its "gustiness";
3. sensitivity studies of the calculated (difference terms) wave-wave interaction low-frequency energy with respect to spectral shape in the main body of the spectrum (bandwidth parameters, multiple peaks) and to the discretization/time duration characteristics of the representation of the sea state;
4. analyses of field data, and comparison of their low-frequency contents to the calculations.