Dispersion Relation

Substitution of the solutions for <p(x, z, t) and n(x, t) into the Bernoulli equation (Eq. 8.11) will give the following relationship between wave frequency and wavenumber:

This is the dispersion relation (so-called for reasons which will become apparent later). A number of useful properties of the motion can now be derived. We will firstly examine the differences between waves in deep water and waves in shallow water.

Waves in deep water are defined to be those for which the depth of the water is large compared to the wavelength of the wave. We may consider the wavelength to be the inverse of the wave number (dropping the factor of 2n since we are dealing with orders of magnitude), thus deep water is defined as kH » 1 (8.17)

We now consider what happens to the dispersion relation in this case. Looking at Fig. 8.3, we see that for large x, tanh x asymptotes towards 1 so this means that for large kH, tanh kH approaches 1 and the dispersion relation reduces to m2 = gk (8.18)

On the other hand, shallow water waves are those for which the wavelength is long compared to the water depth, i.e. k » H and thus kH « 1 (8.19)

Again, we consider the behaviour of tanh x, this time for small x and see that it approaches the line y=x. So for small kH, tanh kH approaches kH and the dispersion relation reduces to

One interesting question to ask is "How deep is deep water?" or "How shallow is shallow water?". Consider the approximation that we have made for deep water, i.e.

The point at which deep water becomes "deep" is the point at which we claim that this approximation is true, so it really depends on how far along the asymptote you want to go. A glance at the plot of y=tanh x shows that tanh is already quite close to 1 at a value of x = 2, and in fact tanh (2.0) = 0.96 This factor of 4% could well be "small enough", so if we take this to be the point beyond which the approximation holds, then deep water can be defined as that for which kH > 2. This means that or the water depth needs to be greater than about a third of the wavelength for the deep water approximation to apply. Typical swell waves in the ocean with periods of about 8 sec, have wavelengths of about 100 m, so these will be considered deep water waves right up until a depth of about 30 m, i.e. the waves will only start to feel the bottom when they are in water of less than 30 m depth. With the typical coarse spatial resolutions of global wave forecasting models (see Sect. 8.5) there are very few grid points that are in depths of 30 m or less, so it is often a reasonable approach to run global systems with deep water physics only.

Now considering the shallow water approximation, we see that y=tanh x is very close to they=x line for values of x less than about 0.5, in fact tanh (0.45) « 0.422. Again, if we think that this is a tolerable approximation, then we can say that our shallow water approximation holds when kH < 0.45, or

In other words, for the waves to behave as purely shallow water waves, the water depth needs to be less than 7% of the wavelength. Our 100 m long swell waves will thus only become purely shallow water waves when the water depth is less than 7 m. On top of this, wavelengths become shorter in shallow water, moving the shallow water limit for swell with deep water wavelengths of 100 m to even shallower water.

An important point to note here is that the definitions for "deep water" and "shallow water" are actually defined as relationships between the wave and the water depth, rather than as an absolute value of the water depth, so there is no specific depth at which the water can be called either "deep" or "shallow". For example, the wavelength of a tsunami is related to the width of the rupture of the earthquake that generated it. This is typically of order 100 km wide. Therefore, tsunamis will act as shallow water waves when the water depth is less than 7% of 100 km which is 7000 m. Almost all of the global ocean is shallower than this, so this is why tsunamis are considered to be shallow water waves.

8.3.1 Phase Velocity and Group Velocity

Some interesting features of wave propagation can be easily derived from the deep and shallow water approximations to the dispersion relation. The phase speed of a wave is simply the speed of propagation of the wave crest. The definition of the period (T) of a wave is the time taken for successive crests of a wave to pass a fixed point, thus a wave will move a distance X in time T and so the phase speed (cp) is

For a disturbance represented by a number of different sinusoidal waves, the group velocity describes the velocity at which the energy of the group of waves is propagating. This can be shown to be (e.g. Holthuijsen 2007; Young 1999)

Equation (8.26) says that in deep water, the individual waves are propagating at twice the speed of the energy that they are carrying. This is an intriguing concept and it can be seen quite easily in nature. If you throw a small stone in a puddle, providing the puddle is deep enough, you will see a group of ripples propagating outwards obeying the deep water dispersion relation. As the ripples propagate away from the disturbance, you will see that individual waves appear at the back of the group, move forwards through the group and then disappear as they get to the front of the group. Equation (8.26) also shows that the speed of propagation of the waves is related to the wavenumber, so waves of different wavelengths will propagate at different speeds. For a disturbance composed of waves of a number of different frequencies (or wavelengths), as they propagate away from the area of disturbance the longer waves will travel faster than the shorter waves and thus the wave energy will disperse. This is where the term dispersion relation comes from.

Equation (8.27) says that in shallow water, the individual waves propagate at the same speed as the wave energy and this speed is dependent only on the water depth. Thus waves of all wavelengths will travel at the same speed and shallow water waves are therefore non-dispersive.

In addition to these interesting features of wave propagation, further useful properties of the motion can be derived from Eq. (8.14). For example, it can be shown that the trajectories of the fluid particles (defined by u and w) describe circles in deep water and ellipses in shallow water. These are often referred to as the orbital velocities of the waves. The derivation is not shown here, but details can be found in Young (1999), Holthuijsen (2007) or Kundu (1990).

8.4 Basic Definitions

The analysis above is mainly concerned with the very simple situation where we consider just one sinusoidal wave component. We have seen that it is possible to derive some readily seen characteristics of the ocean surface with the various assumptions, however, it is clear that this is not a valid description of the actual ocean surface. A more appropriate description is that the sea-surface is characterised as the superposition of a large number of sinusoidal components, with each of these sinusoidal components behaving as described in the previous section. Figure 8.4 shows an example with five sinusoidal components. Each of these components has a different frequency and a different amplitude and they sum together to produce the more complex sea-surface elevation depicted at the bottom. This is again just in one dimension but it can easily be extended to two dimensions by considering a range of different wave directions as well.

Thus, the sea-surface elevation in general can be described by

i=i where ai, Mi and fa represent the amplitude, frequency and phase of the ith wave component, respectively.

Fig. 8.4 Representation of a 1-D ocean surface as a sum of 5 sinusoidal components

8.4.1 The Wave Spectrum

Consider the variance of the sea-surface elevation. This is, by definition, the mean of the square of the surface elevation, and so, assuming the mean of n is zero:

2N 1

We can also consider how this variance is distributed over the different frequencies present in the wave fields, i.e., over the frequency interval Aft. This gives us the variance density spectrum:

J 2Af which becomes, in the limit

This is the frequency spectrum. It can be generalised to the directional case as

So to summarise, the directional frequency spectrum F (f, 0) can be used to describe the variability of the sea-surface elevation. Note that there is no phase information in this description, so the actual surface elevation as depicted in Fig. 8.4 could not be reconstructed from the spectrum, but instead, it describes the distribution of the energy in the wave field according to wave frequency and direction.

The wave spectrum is a very useful construct and is the prognostic variable for current state-of-the-art wave models. A couple of examples of directional wave spectra are shown in Fig. 8.5.

The top panel of this figure shows both a full directional wave spectrum and its directionally integrated one-dimensional equivalent. This depicts a relatively simple sea-state in which most of the wave energy is propagating towards the west, with a fairly large spread around this direction. The peak energy occurs at a frequency of around 0.15 Hz, i.e. most of the energy is being carried by waves with period of about 6.7 sec (this is the peak period, Tp). For the spectrum shown in the bottom

Fig. 8.5 Examples of directional wave spectra

Fig. 8.5 Examples of directional wave spectra panel, there are a number of different components to the sea state, with wave energy clearly propagating in a number of different directions. You can imagine that the sea-state described by this wave spectrum would look quite complicated and very different to the wave field represented by the spectrum in the top panel.

8.4.2 Significant Wave Height

Significant Wave Height (H) is another very important concept that is used frequently to describe the sea state. The idea of wave height for a simple sinusoidal wave is trivial—the wave height is defined to be twice the amplitude, so for each of the 5 wave components depicted in Fig. 8.4, it is straightforward to determine the wave height. But what is the wave height of the resulting wave field?

Hs has come to be used to describe a number of different "wave heights" that can be derived from a wave field. These are all typically very close in value, but given their different methods of derivation, there are some subtle differences of which it is important to be aware.

The original definition is that based on visual observations. Someone out on a boat in the open ocean can observe the waves and estimate what the "average" wave height is. Clearly this will be a subjective estimate and different observers may well produce different wave height estimates. This is called the Significant Wave Height.

A second definition is that obtained through direct observations of the sea-surface elevation. In this case, the Significant Wave Height is defined to be the average of the one-third highest waves in a sample, where a "wave" is defined through the upward or downward crossing definition (see, for example Holthuijsen (2007) for definitions of these). In this case, the resulting wave height should more accurately be referred to as H1/3, but Significant Wave Height is more often used. It has been shown that the visually observed wave height is closely correlated to this definition of wave height (Jardine 1979). It implies that an observer only sees the higher waves, and automatically ignores smaller waves riding on the dominant waves.

Hs can also be derived from the wave spectrum. Using the definition that it is the mean value of the highest one-third of the waves in a given record, and assuming that the wave heights (or more specifically the crest heights) are Rayleigh-distribut-ed, then H133 can be shown to be equal to (Holthuijsen 2007):

where m0 is the zeroth-order moment of the wave spectrum given by

This is equivalent to the volume enclosed by the two-dimensional spectrum (the one-dimensional version would be the area under the curve of the one-dimensional spectrum). The value of 4.004... is typically rounded to 4 and so the spectrally-derived definition of H1/3, which more formally should be referred to as Hm0 can be written as

Again, this is almost always referred to as Hs. In order to determine this from a modelled wave spectrum, the integral needs to be expressed as a sum over the discrete frequency and directional range of the modelled spectrum. Given that the model has a limited range of frequencies that it can resolve, a high-frequency tail is usually included, with a slope of f-n, where n is usually 4 or 5, so it is straightforward to determine the area under this part of the spectrum and it can be added to the H. (See the one-dimensional spectrum in Fig. 8.5—the spectral values stop abruptly at the highest frequency that the model is able to resolve).

The Significant Wave Height is a statistical measure for the wave height. Clearly, individual waves can be both lower and higher. It can be shown that in a simple spectrum describing a single coherent wave system, the probability distribution of the height of individual waves closely follows the Rayleigh distribution (e.g., Holthuijsen 2007). This distribution implies that 1 in 100 waves is expected to be as large as 1.51 Hm0, and 1 in 1000 waves is expected to be as large as 1.86Hm0. Higher waves rapidly become less likely, which is why waves higher than approximately 2.0Hm0 are typically called "freak" or "rogue" waves.

We have seen here that there are a number of different ways of describing the "wave height" of a particular wave field and these are typically all referred to as Significant Wave Height, or H. Clearly, this one value used for describing the sea-state is a gross simplification. It would be reasonable to use this to describe a simple sea-state in which there is only one dominant component to the wave field, but consider the two sea-states in Fig. 8.5. The Hs is similar in each panel (Hs = 1.36 m in the top panel compared to Hs = 1.03 m in the bottom panel) even though the sea-states depicted by the spectra are very different. Simply using Hs to describe a sea-state means that you lose a lot of information about the structure of the wave field. This is similar to giving a weather forecast with a simple maximum temperature value. It doesn't tell you whether you need to take your umbrella or not!

8.5 Operational Wave Modelling 8.5.1 Background and Basics

This section focuses on operational wave modelling in the context of wave forecasting. As mentioned previously, most current state of the art wave forecast models

are phase-averaged third generation models, which have the wave spectrum as their prognostic variable. The most common models in usage at international forecasting centres are WAM (WAMDIG 1988; Komen et al. 1994) and WAVEWATCH III® (Tolman et al. 2002, 2009). These are computationally efficient models that can be used for large scale global forecasting. Also the SWAN model (Booij et al. 1999; Ris et al. 1999) is extensively used, but more for near-shore engineering applications. A review of the state of the art of operational (and research) wave modelling can be found in Cavaleri et al. (2007).

The basis of virtually all wind wave models used in operational forecasting is some form of the balance equation for the wave energy spectrum F (f, 0) as discussed in Sect. 8.4.1. In its most simple form, it is given as dF

at where the left hand side represents the effects of linear propagation, and the right hand side represents sources and sinks for spectral wave energy. Propagation, in its simplest form, only considers wave components in the spectrum to propagate along great circles, until the wave energy gets absorbed at the coast (either as part of the propagation algorithm, or due to the dissipation source terms). More advanced versions of this equation, as used in prevalent models, also consider refraction (changing of wave direction due to interaction with the bottom in shallow water) and shoaling (changing of wave height and length due to changing water depths), and some consider similar effects due to the presence of mean currents. So far, all operational wave models consider linear propagation only. Many operational models now address the effects of unresolved islands and reefs as sub-grid obstructions.

Traditionally, three source terms have been considered; Sn describing the input of wave energy due to the action of the wind, Snl describing the effects of nonlinear interactions between waves, and Ss describing the loss of wave energy due to wave breaking or "whitecapping". Many early models for shallow water applications added a wave-bottom interaction source term, Sbot, which was typically concerned with wave energy loss due to friction in the bottom boundary layer. Of these source terms the nonlinear interactions have a special relevance. Effects of nonlinear interactions occur as source terms in this equation, because the propagation description in the equation is strictly linear. Furthermore, the interactions are essential for wave growth, and not for propagation. They represent the lowest order process known to effectively lengthen waves during growth, and they have been shown to stabilize the spectral shape at frequencies higher than the spectral peak (e.g., Komen et al. 1994). Nonlinear interactions consider resonant exchanges of energy, action and momentum between four interacting wave components, governed by a six-dimensional integration over spectral space. The SWAMP study in the 1980's (SWAMP group 1985) identified the explicit computations of these interactions as essential for practical wave models. The development of the Discrete Interaction Approximation (DIA) (Hasselmann et al. 1985) made this economically feasible. Models that explicitly compute nonlinear four-wave interactions are identified as third-generation wave models.

Present operational wave models address source terms in a much more detailed fashion. Wind input is turning into wind-wave interactions, and can include feedback of energy and momentum to the atmosphere ("negative input"). Further to this, wave breaking is seen as impacting on atmospheric turbulence, and hence influencing atmospheric stresses and wave growth. Nonlinear interactions now regularly include both four-wave interactions in deep water and three-wave (triad) interactions in shallow water. Wave dissipation now regularly addresses traditional whitecap-ping in the deep ocean, and separate mechanisms for depth-induced ("surf") breaking, and much slower dissipation mechanisms that influence swell travelling across basins with decay time scales of days to weeks. Many additional wave-bottom interactions are also considered in shallow water. Most prevalent are bottom friction source terms, but other processes such as wave-sediment interactions associated with bottom friction, percolation and scattering of waves due to bottom irregularities have been proposed and are available in some wave models. Of special recent interest is the interaction of waves with muddy bottoms, which both adds a source term and may modify the dispersion relation and hence wave propagation. Source terms for other processes such as wave-ice interactions and effects of rain on waves have been proposed, but are presently not used in any practical wave models.

8.5.2 Operational Centres

Many operational weather forecast centres run operational wind wave models. This is not done by accident. During the 1974 Safety of Life at Sea (SOLAS) conference, international agreement was reached to consider wind waves as part of the weather, explicitly giving weather forecast centres the responsibility to do wave forecasting for the public. The first numerical wave predictions, however, far precede this date, and in the U.S.A. can be traced back to 1956 (see historical overview in Tolman et al. 2002).

Many of the larger weather forecast centres such as the European Centre for Medium Range Weather Forecasts (ECMWF1, Europe), The National Centers for Environmental Prediction (NCEP2, USA) and the Bureau of Meteorology (Bureau3, Australia) produce wave forecasts for up to 10 days ahead, on 6-12 h forecast cycles. Most of these centres use a global wave model, with one or more higherresolution nested regional models for areas of special interest. For example, the configuration of WAM at the Bureau (as at end of 2009) is shown in Fig. 8.6. The highest resolution model (blue boundary) is run at 0.125° resolution in latitude and longitude, and is nested inside a model at 0.5° spatial resolution (red boundary) which is in turn nested inside the global model at 1°. Typically, the higher reso

1 Web site at http://www.ecmwf.int.

2 Wave data at http://polar.ncep.noaa.gov/waves.

3 Wave data at http://www.bom.gov.au/marine/waves.shtml.

20 E 40 E 60 E 80 E 100 E 120 E 140 E 160 E 160W 140 W 120W 100 W 80 W 60 W 40W 20 W

20 E 40 E 60 E 80 E 100 E 120 E 140 E 160 E 160W 140 W 120W 100 W 80 W 60 W 40W 20 W

135 E 150 E 165 E 180 165W 150 W 135 W 120 W105 W 90 W 75 W 60 W 30x30 15x10 10x10 8x4 4x4

Fig. 8.6 Examples of configurations of some operational wave model systems. Top panel shows the Bureau and bottom panel shows NCEP

135 E 150 E 165 E 180 165W 150 W 135 W 120 W105 W 90 W 75 W 60 W 30x30 15x10 10x10 8x4 4x4

Fig. 8.6 Examples of configurations of some operational wave model systems. Top panel shows the Bureau and bottom panel shows NCEP

lution models obtain data from the lower resolution models without feeding any information back, but full two-way nesting of such models is now used at NCEP (Tolman 2008). Configuration of the NCEP system (as at end of 2009) is also shown in Fig. 8.6. This incorporates a range of different spatial resolutions ranging from global at 0.5° down to the highest resolution models at 4 arc minutes (1/15th of a degree) around the coastlines. The spatial resolutions of the wave models are typically dictated by the resolutions of the atmospheric models from which the wave models obtain their wind forcing and additionally, by the availability of computing resources. In an operational forecasting environment, a major consideration is the time taken for the model to complete a forecast and the speed with which the results can be disseminated.

Some centres also run specialised wave models for specific conditions; for example NCEP run wave models specifically for hurricanes, with specialized forcing from hurricane weather models. Finally, several centres run wind wave ensembles, to provide probabilistic information on the expected reliability of the forecast. While such ensembles have been generated for up to a decade, they have not been scrutinized as much as corresponding atmospheric ensembles, and may not have reached the same level of maturity. Further details of operational wave forecast systems can generally be found at the websites for the forecast centres as given in the footnotes.

In addition to differences in the spatial resolutions of the models, there is considerable variety in other aspects of the operational implementations of wave forecast systems at each forecasting centre. For example, the wind forcing used to force the wave model will typically be provided by the centre's Numerical Weather Prediction (NWP) model, and these can vary considerably in detail. Whether the wave model incorporates data assimilation or not can also contribute to differences in the forecasts. The most widely used data source that is assimilated in wave models is Hs from satellite altimeters. This can significantly improve the skill of wave forecasts (Greenslade and Young 2005), particularly in cases where the surface winds are known to have deficiencies. One limitation to the assimilation of Hs data is that it can not provide any direct information on the observed wave spectrum, so a number of assumptions need to be made in adjusting the modelled spectrum (Greenslade 2001). This issue can be somewhat overcome by incorporating the assimilation of wave spectra from Synthetic Aperture Radar (SAR), such as is performed at the ECMWF (ECMWF 2008). In situ wave buoys could also provide wave spectra for assimilation. However, the limitation of these is that compared to satellite data, they are very sparsely distributed and they tend to be located near the coast, for logistical reasons. The fact that they are typically not used in wave data assimilation schemes means that they can be used as a valuable independent data source for model verification.

Many of the operational forecast centres share their model results through a wave model intercomparison study supported by the Joint Commission for Oceanography and Marine Meteorology (JCOMM) (Bidlot et al. 2007). Model forecasts are also compared to observations from in situ buoys around the globe. This project provides a mechanism for benchmarking and the quality assurance of wave forecast products. The results are available each month to all participants and published on the web.4 An example of the intercomparison at one location is shown in Fig. 8.7. This shows 24-hour forecasts of Hs and Tp at buoy 44005 (located 78 nautical miles off the coast of New Hampshire, in the northwest Atlantic) for the month of November 2009.

In the top panel, it can be seen that all wave models are able to forecast the Hs reasonably well, with the synoptic scale variability being captured very well. There is some spread around the observed Hs, and for this example, most of the models have overpredicted the peak H s occurring around the 15th of November. The Tp is also quite well captured this month, particularly the dominance of long waves (high

4 Web site at http://www/jcomm.info

Forecast (t = t+24) wave height and averaged buoy data at buoy 44005

-augobs^-ECMWF -b-UKMO FNMOC AES 5 - «CEP METFR DWD ■■« AUSBM-«- SHOM • JMA -&■ KMA a PRTOS

NOVEMBER 2009

Forecast (t=t+24) peak period and averaged buoy data at buoy 44005

avg obs —ECMWF UKMO -«- FNMOC -— AES -*■• NCEP METFR DWD ' AUSBM "«" SHOM « JMA -o- KMA PRTOS

Fig. 8.7 An example of results from the wave intercomparison activity

Fig. 8.7 An example of results from the wave intercomparison activity wave period) during the middle of the month and the trend towards shorter period waves at the end of the month. The high variability in Tp seen in both the observations and the models from the 3rd to the 13th suggests that there were a number of different wave systems present during this period.

There are also a number of summary results from this intercomparison activity produced each month. An example is shown in Fig. 8.8. This shows the root-mean-

Fig. 8.8 Summary statistics for one month from the wave model intercomparison project. Each coloured line represents forecasts from a different operational centre. Top panel: H, middle panel, u10 and bottom panel, T . The x-axis in each panel represents forecast period, in days

square (rms) error amongst the forecast models averaged over all buoy data available for the three parameters, Hs, Tp and u10 (wind speed at 10 m above the surface). The error is defined as the difference between the modelled and observed parameters. The rms error can be seen as a measure of the skill of a model. Figure 8.8 shows that the rms error for a 24-hour forecast (1 day) is approximately 0.5 m, although it varies from about 0.4 m to about 0.7 m. Errors of wave models normalized with mean conditions for the better models are of the order of 15% for hindcasts and short term forecasts (results not shown). Another feature obvious from this figure is the growth in error with forecast period. It can also be seen that there is a strong correlation between the rms error in the surface winds and the rms error of the wave forecasts, i.e. those centres that have accurate surface winds also have high skill for the wave forecasts. With continuously improving weather models at all centres, differences in wave models and in the selection of numerical and physical options in these models is becoming more and more apparent and important. After a decade of relatively small changes to wind wave modelling approaches, this has recently lead to an increased interest in improved physics approaches in the corresponding wave models.

8.5.3 Outlook

As mentioned above, a renewed interest in wave model development has surfaced in the last few years. This is particularly clear with the recently started National Oceanographic Partnership Program (NOPP) project which aims to provide the next generation of source term formulations for operational wind wave models. Literally all source terms in the wave models will be addressed in this study, with a focus on deep water and continental shelf physics. A greater focus of operational centres on coastal wave modelling is emerging, Partly due to increased requirements from users of the service and also due to the increasing ability of wave models to address this, given advances in computing power. With this, alternate modelling approaches such as curvilinear and unstructured grids are becoming more prevalent and more important.

Furthermore, the mode of operation of many forecast centres is slowly changing. Traditionally, operational centres have focused on isolated topical forecast problems such as weather and waves. More and more, such centres are moving toward an integrated earth-system modelling approach, where the links between models are seen as essential to improve the quality of the individual models. Wind waves literally are the interface between the atmosphere and the ocean. In a systems design approach, a wind wave model could become an advanced boundary layer module for an integrated atmosphere-ocean modelling system. At ECMWF, a first step into this direction was made more than a decade ago, when their wind wave model started providing real time surface roughness information (including wave-induced roughness) to the weather model. At NCEP, coupled atmosphere-ocean models are used for climate and hurricane prediction. Experimental versions of the hurricane model now include a three-way-coupled system, consisting of full weather model (HWRF), a full ocean model (HYCOM) and a full wave model (WAVEWATCH III). A similar system is under development at the Bureau. In such a model the wind waves play a key role; they modify surface roughness and therefore stresses; they may temporarily store momentum extracted from the atmosphere, and release this to the ocean in a geographically distant location; spray generated by waves influences (and links) momentum, heat and mass fluxes between the ocean and atmosphere. Indeed, the most complete estimates of spray production are directly related to the wave spectrum, and hence require a full wave model. Another important forecast problem in which wind waves become important is coastal inundation, where many coastal inundation problems are directly linked to momentum produced by incoming swell rather than by wind pushing up water in a tradition storm surge situation. Several decades of experience with wave-driven coastal circulation and inundation problems can be found in the civil engineering literature, but these experiences have not yet been used in operational forecasting procedures.

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