Notes on Cavaleri (2009)

Cavaleri, L. (2009). Wave modeling-missing the peaks. Journal of Physical Oceanography, 39(11), 2757-2778.


  1. General
  2. What to Worry About
  3. The State of the Art
  4. Reasons for the Errors
    1. Physics: Wind
    2. Physics: Wave Generation
    3. Physics: Whitecapping
    4. Physics: Nonlinear Interactions
    5. Numerics
    6. Modeling
  5. General Comments
  6. Where to Act


This is a very interesting paper, which although focussed on extreme conditions, “analyzes the capability of the present wave models of properly reproducing the conditions during and at the peak of severe and extreme storms”, it identifies a lot of potential issues with spectral wave models. Even in the abstract there are some very interesting quotes, such as:

“wind accuracy is still a relevant factor at the peak of the storms.”

“a sensitivity study is suggested to identify the most critical areas in a wave model to determine where to invest for further improvements.”

“The limits on the description of the physics of the processes when using the spectral approach, particularly in extreme conditions, are considered.”

What to Worry About (pp. 2757—2758)

Basically the introduction, this section presents some of the apparent issues and what to do about them. The following quotes are self explanatory:

Issues with parameterisation:

“The strong demand for practical results has also led to solutions where the complexity of the problem and the difficulties of getting enough data in the right conditions have required partially empirical solutions.”

Issues with calibration over time scales:

“tuning is generally completed on the bulk of the data. The more rare a special case is, the more likely it is to be poorly represented by the “tuned” rule, especially if in these “different” conditions the physics of the process does change.”

Issues with validation:

“Of course, a low bias does not exclude positive and negative errors by the model, for whatever reason. This is why we also use correlations and scatter indices. However, a straight-forward study of several scatter diagrams or time series comparisons and related statistics will quickly reveal that the models have a marked tendency to under-estimate the largest wave heights, and in particular the peaks, more in heavy storm conditions.”

Identification of errors:

“Obviously the first step is to identify the reasons for our errors. This paper is an attempt in this direction for what concerns spectral wave modeling.”

“This leads to a rather long, possibly discouraging, list of potential sources of errors.”

Solutions to the problems:

“we try to discuss where it is possible to act, where work can be done, and which actions could be taken to overcome at least part of the present limitations.”

“there is a wide spectrum of opinions of where exactly we are and which are the right ways to go.”

The State of the Art (pp. 2758—2761)

General discussion that normal conditions of wave models perform quite well, particularly local models which can be calibrated specifically

“Nowadays, the average error of an advanced wave model is easily down to a few, typically negative, percents, with a bias of the order of 10 cm or fewer.”

“Compared to the quoted general performance of operational global models, even better results are achieved by special studies, typically concerning some specific storms.”

The focus on the poor performance in high wave heights is emphasized:

“when we analyze the statistics with respect to wave height… it is immediately evident that the average error, both as bias and rms, is strongly dependent on $H_{s}$, the largest wave heights suffering underestimates up to 1m or more.”

The ability for local models to perform better is generally attributed to improved description of wind fields:

“Indeed, the improvements in the description of the surface wind fields, derived from the model’s higher resolutions, and an improved description of the physical processes at work, particularly at the air-sea interface, have been key elements in the improvement of the wave results.”

This relationship of improved wind fields leading to multiplicative improvements in the wave field, has led to a belief that errors in the wind field is the main source of error in wave modelling. Yet, this (lengthy) quote and perhaps our own investigations in Hebmarine may indicate this is not entirely the case:

“[There was an] assumption that the wind field inaccuracy was the main reason for the, typically negative, errors in wave modeling. However, … this attitude is rapidly approaching an end. Although the wind errors should certainly still be considered, wave modelers also have to look into their own machines if they want to decrease further the differences between wave model results and the measured truth.”

There is a lot of evidence of missing peaks and troughs in wave modelling. The time series in Figure 2 (p. 2760) shows that, even for non-extreme conditions, models perform better in the average than in the extremes

“The one-month plot (March 2008) shows very clearly the repeated tendency of the model to miss the peaks.”

Figures 3 and 4 show an interesting relationship that shows that the average wind speeds become asymptotic with resolution quickly, with a similar (but not complete) convergence for average wave heights. Figure 4 shows that no such convergence is apparent for peak wind speeds or wave heights. There is an argument that problems in the wave field are all related to the wind field, however, this quote aims to dispel that:

“with increasing resolution the area [influenced] by the wind underestimate is getting smaller and smaller. Because the waves are an integrated effect, in space and time, of the driving wind fields, we should expect to see a progressively reduced effect of these “wind misses” on the waves - something obviously not true from the right panel of Fig. 4. Therefore, although the wind quality has obviously an effect, we must conclude that the problem lies also within the wave model itself.”

Reasons for the Errors (p. 2761—2771)

This section (actually called “The search for a good reason (where we find we have to deal with a whole plethora of processes)”) tries to identify the problems leading to the underestimation of extremes and splits the discussion into the physics within models, the numerics of the models and a more general modelling discussion.

Physics: Wind (p. 2762)

I think I’ll just quote the summary here:

Summary. Underestimates of the surface wind speeds are more frequent in the high value range. Model resolution is critical. Smoothing of the fields, a frequent practice in meteorological modeling, leads to lower peak values. Gustiness is not properly considered. Air density can vary substantially, directly affecting wave generation.”

The interesting part about resolution is capturing the gradients, so that a small area storm can be more difficult to model that a large one and the gradients will be more substantial. The term Limited-Area Model (LAM) is defined, which might describe the Hebmarine model, although it may be larger, being defined, roughly, as not global.

Physics: Wave Generation (p. 2764)

Miles (1957) is the method used for the generation of waves from wind in numerical models. It was improved by Snyder (1981), but the measurements were taken for wind speeds of about 7-10 ms${}^{-1}$. This has issues:

“It is natural to wonder if we are allowed, as we presently do, to use the same findings in extreme storms, with winds up to 30 or 40 ms${}^{-1}$, much higher in hurricanes. The nice orderly picture that the Miles process implies is likely not to correspond to the truth once we move to high winds.”

The physics in high wave heights is different…

“Banner and Melville (1976) had shown that the transfer of energy to waves is characterized by a series of bursts that happen when the surface air boundary layer detaches from the sea surface soon after the crest of breaking waves. In severe, and more so in extreme, conditions this can be expected to be the case for practically each single wave.”

Apparently, as waves get bigger the physics changes somewhat. Water droplets being blown off the waves can have an impact as their acceleration is felt by the atmosphere and creates higher surface drag, whilst they also kill the short waves which enable momentum transfer.

At the very high wind speeds, so much foam is generated that there is no troughs in the wave, which invalidates the Miles (1975) approach apparently.

There are also issues with linear theory, even for deep water problems, when large waves are considered:

“All the spectral wave models use linear wave kinematics; that is, the phase speed $c$ of a given spectral component is evaluated on the basis of linear theory. However, the infinitesimal approximation implied and the finite dimension of real waves may lead to appreciable differences.”

The phase speed being higher for steeper waves decreases the difference between the wind speed, $U$, and $c$ (which would reduce transfer) but longer waves also mean that the waves could be larger.

Interestingly gustiness causes waves to grow faster than the equivalent uniform wind speed. There is also a phenomena that happens when the phase speed of the waves is close to the wind speed. Something called the “diode” effect (from Abdalla and Calaleri (2002)).

The differences can be significant: straightforward numerical experiments showing that a 30% wind variability can lead, in a longer time scale, to a 30% increase of the maximum significant wave height.”

Here is the summary for this section:

Summary. Doubts exist on the validity of the Miles (1957) theory (although modified) in extreme conditions. The physics does change in such a situation. At present we do not have a good physical, hence numerical, model of what is going on at very high wind speeds.

The finite, large amplitude of stormy waves, particularly the dominant ones, implies a phase speed greater than dictated by linear theory. In the end this may lead to larger wave heights.

The level of gustiness experienced in the field is often underestimated in meteorological models. Therefore, its consideration in wave models does not lead to the related enhancements found in the measured results. Besides, models do not reproduce the possible longer-term oscillations of both the winds and wave fields. In certain conditions, but typically in the very severe storms, this leads to a strong underestimate of the possible maximum values.

Cold, hence frequently gusty, winds are also characterized by a higher air density. Neglecting its variations often leads to an underestimate of the resulting wave heights.”

Physics: Whitecapping (p. 2765)

There are two main whitecapping models in use, but they seem to vary in outputs, which is particularly interesting in terms of uncertainty quantification as this is being “made up” somewhere in the models:

“If used in their correct software environment, both of these approaches provide reasonable and often good results in most practical applications; however, it is stunning that their quantifications of the process differ by 100% or more, which tells a lot about the difference between sound physics and correct operational results.”

Better approaches have been found recently, which has improved the bias significantly for low wave heights (see Figure 1). This Ardhuin et al. (2008) formulation still requires parameterisation, however, and does still not work well in the high ranges.

Here is the section summary:

Summary. During the present decade, progress has been made in understanding the physics of breaking waves. These results have only recently found their way into one operational model. Notwithstanding these advances, there is still a good level of empiricism in the way we attack the problem, and most of the operational models still cling to very empiric, often totally inconsistent, approaches. All the proposed and used solutions require some tunable constants. As such, they are tuned to the bulk of the results and may fail in extreme conditions, when the physics of the process is likely to be substantially different.”

Physics: Nonlinear Interactions (p. 2766)

So non-linear interactions are solved with exact solutions, but under some strong assumptions. Nonetheless, the computing power required is very high for the original formulation of Hasselmann (1962 etc.) so a less intensive version called the discrete interaction approximation (DIA) has been used instead.

There are some issues with DIA, particularly with the transfer of too much energy into the low frequencies or to spread the energy directionally, will cause less growth through wind inputs.

Efforts are being made to provide solutions closer to the original Hasselmann formulation as more computing power becomes available, but the strong assumptions of the original formulation are still “not exact”, for reasons such as

“the model is a large time limit closure and neglects near-resonant modes that can change the spectrum on much shorter scales… and in a region of rapid changes.”

“the Hasselmann model presumes a homogeneous and Gaussian sea state, which is of course an idealization, especially in heavy and extreme conditions.”

“it is a truncated model of a weakly nonlinear system. In other words, it includes the lowest-order resonances, whereas on longer time and space scales, other nonlinear contributions may become important.”

Finally, this is the summary:

Summary. Beside its sometime incorrect spectral distribution, the present widely used DIA approximation leads to too-wide energy distributions in the spectra, both in frequency and in direction. This decreases the wind input to waves. New, better solutions are on the way. It is expected that these solutions will be implemented in operational models in the near future. However, all these solutions are still approximations, based on some strong hypotheses that are likely not to be satisfied in heavy and extreme sea conditions. The related implications for wave modeling have not yet been explored.”

Numerics (p. 2767)

The role of the numerics are described as:

“to import into the spectral energies the inputs-outputs derived from the physical processes, and then to advect the resulting energy in the direction of the speciļ¬c components.”

The main issue with the numerics, particularly for extremes in the diffusion of energy throughout the spatial and frequency domains. Caverli says that “the problem is unavoidably linked to the discontinuity associated to a grid”, although there are some arguments that grid refinement is perhaps not always ideal. However, I think this relates more to the time-step size, so without a fixed CFL number.

“In some numerical schemes, the spreading due to diffusion can be reduced by using a higher-resolution grid or, where possible, using larger integration time steps, that is, reducing their number in a given time interval.”

“a careful analysis is required to achieve the best balance between diffusion and overall integration time.”

Other fixes include higher order numerical schemes, although these have some risk from the “garden sprinkler effect”. Also higher frequency resolution can be used but this has a double hit in terms of computing time as more non-linear interaction calculations must be done as well.

Generally these issues effect small area storms with large gradients, but the correct advection of swell is also important.

“A large-scale storm will be marginally affected by diffusion. However, a strong isolated peak will be substantially decreased during advection, its energy being redeployed at the neighboring grid points.”

“a lower Hs also implies a lower peak period and in turn an underestimate of the consequent swell period, hence group speed.”

“the correct advection of swell may happen to be of relevance for our present discussion.”

Here is the summary:

Summary. Diffusion, unavoidably associated to the use of discrete grids, leads to a smoothing of the fields, hence to an underestimate of the extremes. Although higher-order schemes may improve the situation, they often do so at the expenses of other details. Engineering solutions have been devised for practical applications.

Higher-resolution grids or, Courant number permitting, larger time steps may help. However, in each case a careful analysis is required to be sure that the introduced changes act in the right direction.

A higher spectral resolution would also be beneficial, especially for long-distance swell. This could be relevant in cross-sea conditions.”

Modeling (p. 2768)

This section considers other modelling considerations. These include combined effects, spectrum impacts and sensitive modelling phenomena such as “dynamic generation”.

In terms of wave-current interactions, the paper say this:

“Most of the currents we find in the sea are not strong enough to affect at a significant level the waves that characterize a storm. However, in certain areas where the current speed is not negligible with respect to the group speed of the relevant wave system - typically the Gulf Stream, the Kuroshio and the Agulhas Current - the interaction with currents may substantially enhance the height of the waves.”

Currents can reduce the wavelength of waves and enhance their wave height, with increased steepness. This increased steepness can lead to more energy transfer from wind. Getting accurate current information (ocean circulation, rather than tidal), is very difficult.

There are some issues with the classical Pierson Moskowitz spectrum, in that models don’t tend to converge to this spectrum in all cases. This may be due to the model, or the development of the spectrum.

“the very concept of fully developed conditions is still a matter of debate (see, e.g., Glazman and Pilorz 1990; Glazman 1994; Hwang and Wang 2004). As Alves et al. (2003) point out, the present convenient use of P-M64 is more a matter of a lack of conclusive evidence in either direction than scientific truth.”

There are also issues with model quirks, such as the use of limiters in some models to control energy exchange. The model parts do not interchange either, for the same physical processes, which is not easily explained:

“[We cannot take] part of the physics of WAM, for example, the whitecapping subroutines, and plunge it into WAVEWATCH. Surprisingly, at least the first time, we find that the $H_{s}$ results may change up to 40%.”

Dynamic generation, “the case when the heart of the storm, that is, its more energetic waves, moves with the weather system that generates it”, is a particular test for wave modelling as many of the talked about issues (particularly the correct phase speed) come into play. This is why, for some cases, “all the wave models substantially underestimated the measured peak values (Cardone et al 1996)”.

Finally, the analytical modelling of the tail of the energy spectrum in models may cause some problems. A tail is required because the number of frequency bins is finite [although a lot of the high frequency bins tend to be empty in Hebmarine data]. The spectrum has a cut-off frequency and the selection of this frequency, and the power of the analytical tail leads to differences:

“it implies possible $H_{s}$ differences of some tens of centimeters, particularly in heavy and extreme conditions, when the cutoff can be at a relatively low frequency.”

The tail can also impact on the “level of input” and the losses due to whitecapping. There is also a general issue of models frequency components and real waves which is illucidated by this quote:

“For advection a spectral model considers separately all the spectral components, the motion of each one independent of the other components. This is not the case for a buoy, especially in the high-frequency range, where the single components ride on top of the larger and longer ones, with a potential nonnegligible influence on their spectral representation.”

Finally, this is the summary:

Summary. A highly detailed knowledge of the distribution of currents in the oceans is required for a proper evaluation of the wave-current interactions, hence the possibility of enhanced or focused wave fields. Such knowledge is not yet available.

Even when fetch and duration allow, wave models do not necessarily converge to the classical Pierson-Moskowitz spectrum. On the other hand, given the present level of accuracy of wave models and measurements, one could wonder about the accuracy of the data that led to this spectrum. Also the very concept of fully developed conditions is still a matter of debate.

The use of limiters is in itself a strong indicator of the approximation with which the physics is represented in wave models, especially in extreme conditions; similarly so for the impossibility of exchanging sections of the programs representing the same physical process between two different models.

The importance of accurately representing the phase and group speeds in extreme conditions becomes apparent in the case of dynamical generation.

“The frequency after which to apply a tail and the slope of the tail is still a matter of debate. This may imply a nonnegligible contribution to the overall energy of the system, hence to the resulting wave height.”

General Comments (pp. 2771—2773)

This section considers more general wave modelling issues, rather than focussing on the extremes. A discussion of other sources of error is given:

“Easy examples are the theoretical approximations implied in modeling, and the convergence and the accuracy of the numerical procedure.”

A mention of nearshore processes and there issues is also given, with reference to the WISE group paper for further details.

There is an interesting discussion on the balance of wave models and how they have been tweaked to get the errors from each process to cancel in some effect. [This, obviously, stands in the way of software process. You almost have deliberate failure in the unit testing]. Discussing the main drivers, which are wind and wave breaking, the author says:

“We succeed, within limits, because the system is self-regulating. For instance, if at a certain time breaking is underestimated, the wave conditions rapidly grow to a level where breaking becomes impelling again, bringing the system under control.”

This appears to be a hindrance to further development in wave models as “the present results of the operational wave models are in general surprisingly good.” This means that there is a reluctance to change the operational models for more exact physics, because of these special balances.

“The dominant rule in practical applications is the following: a change to a model becomes permanent if and only if its effect is positive, that is, the score of prolonged applications improves.”

“that the present models, good as they are, represent a careful balance among the different processes, each one with its larger or smaller inaccuracies. Having one process correct and all the other ones with their own previous approximations does not necessarily imply better results.”

Issues with modelling and measurement for extreme conditions are mentioned. There is also some discussion about the validity of the spectral approach, although an alternative method seems somewhat daunting. There is an interesting quote from another paper, Komen et al. (1994) about trying to improve the quality of existing models:

“one should not be too optimistic about the effect of further refinements…”

Where to Act (pp. 2773—2776)

This section considers what can be done to improve the present situation and has some good quotes for justifying work on uncertainty quantification. It considers the itemised issues for large amplitude waves and then the general case for modelling.

Model resolution is discussed for both wind:

“Starting as before with the wind, the most obvious possible improvement, particularly for the most intense areas of severe and extreme storms, is an increased resolution of the meteorological models… each additional step ahead $\alpha$ in resolution implying a much larger increase, $\alpha^{3}$ or $\alpha^{4}$, of the computer capabilities.”

and wave:

“Here, too, an $\alpha$ increase implies an $\alpha^{3}$ increase in computer time.”

A highlighted issue with adding additional computer power, is that in the large centers wave modelling is not as important as other climate factors,

“there are larger scientific and social interests driving the advance of the meteorological models, interests for which the wave modeling community is probably a minority.”

Also, although meteorological models are advancing, “doubling the resolution of the meteorological models every five years or so”, issues with a lack of understanding of gustiness and a lack of proper modelling of the atmospheric spectrum.

Regarding computing power for waves, adapted meshes are proposed:

“what we really need is a flexible, most likely finite element, dynamical grid whose density in space is able to adapt continuously to the local gradients while the wave system moves from one area to another.”

Not only refinement of spatial grid, but also refinement of the spectral representation should be of benefit:

“a better spectral resolution, both in frequency and direction, and improved advection algorithms are key elements to reducing the smoothing of the fields, the consequent lowering of the maxima, and the error in the swell arrival time in a certain area. Although the former is only a matter of computer power, the latter one is a subtle compromise with the need to avoid unnaturally patchy fields.”

The need for more computing power also extends into the underlying physical processes, particularly the nonlinear interactions.

“The nonlinear interactions are probably the single item where the problem we face is the lack of sufficient computer power.”

In the broader context of current interaction, again, a lack of computation power comes to the fore, although understanding and a lack of data also play a part:

“The problem for the circulation modelers has its roots both in the physics of the processes at work, in the computer power for higher resolution, and in the lack of data.”

This raises an issue of whether some of these problems can be modelling deterministically. In terms of ocean currents, given the lack of data, this is extremely difficult, thus “It seems that, at least for certain aspects, we will have to live more and more in a probabilistic world.”

Other, not often considered phenomena and also mentioned, such as Stokes drift (non-linear transfer of mass by Stokes waves) and current-wind interactions.

Most excitingly, a short discussion of uncertainty quantification and model testing is hiding in the back of the paper. For instance, when considering how best to distribute future resources:

“a more rational way could be to analyze the sensitivity of the final results to a refinement in the various processes and decide to invest along the most sensitive lines”.

“it is not immediately obvious which elements the model is more sensitive to.”

“It seems that a sensitivity study would be a useful exercise.”

I believe that a general uncertainty analysis process ideally lends itself to the above issues.

Interestingly, there is mention of Bayesian techniques for uncertainty quantification (is this how I found this paper?), although the difficulties with them are acknowledged

“Sophisticated methods for model reproduction based on a Bayesian approach have recently been proposed… However, the sheer complexity and variety of the possible situations suggest that this is not a practical way, especially when we expect, as we presently do, to go beyond the classically integrated parameters $H_{s}$, $T_{m}$ and $\Theta_{m}$ (mean period and direction, respectively) and on to argue about the structure of the spectra.”

Finally, there is some discussion of using cutting edge simulation to move away from, but also maybe inform the processes used in spectral modelling, “I believe a possibility is offered by the direct numerical simulation of what is going on in the sea”. Thus, some higher resolution CFD type studies of large and breaking waves would be beneficial.

“Even a two-dimensional realization (i.e., a vertical section of the atmosphere and the sea surface), although with its limitations, would be a valuable start. Given these high waves, we could model the wind flow over it and evaluate from basic principles how much energy is passing from the atmosphere to the waves.”

The last mention is of the ultimate mega total atmosphere model, which we are working towards, but might never reach.

“Obviously, the full simulation of air and sea, possibly starting as before from already developed conditions and modeling their physical evaluation with all the processes at work: wind, boundary layer, vortex shedding, waves, breaking, limited crest length, foam, turbulence, shear currents, and so on.”

© 2013 Mathew B. R. Topper