Numerical Wind-Wave Models

Estimates of wave conditions produced from numerical wind-wave models are an important complement to in-situ and remotely sensed observations. The usefulness of

in-situ observations is limited by the sparseness of their deployment, whilst altimeter observations are limited by spatial and temporal sampling issues, lack of spectral and directional information and problems measuring close to the coast. Model data can be produced with dense resolution in space and time and can provide long histories for assessing inter-annual and climatic variability.

Understanding the errors in model data is vital for the calculation of uncertainty of derived wave energy statistics. In this section we start by giving an introduction to numerical wind-wave models and briefly discuss the factors which can affect the accuracy of their predictions. We then give a qualitative description of the error structure of model data.

2.3.1. Brief introduction to numerical wind-wave models

Wave models attempt to replicate the growth, propagation and decay of ocean waves based on the winds over the area in question. The fundamental concept underpinning wind-wave modelling is the energy balance equation. This states that the evolution of the wave spectrum is the sum of three source terms describing the input of energy from the wind, non-linear transfer of energy within the spectrum and the dissipation of energy from breaking or shallow water processes. It can be written as (e.g. Tolman et al, 2002) interactions and S is the dissipation source term. In slowly varying conditions such as _ds non-steady currents or water depth, wave action density (defined as the variance

spectrum divided by the intrinsic angular frequency - the frequency measured in a frame of reference moving with the current) is conserved rather than wave energy. When equation 2.3.1 is rewritten in terms of wave action it is known as the action balance equation. A detailed derivation of the energy and action balance equations is given by Komen et al (1994).

In the first generation of wave models non-linear interactions were not computed. The

transfer processes, but due to limitations in computer power at the time, nonlinear interactions were computed in a simple parameterised form. The third generation of wave models, used at present, compute an explicit representation of the all three source terms and the step-by-step evolution of the wave spectrum, without a priori

assumptions about the spectral shape. There are currently two third generation wave models run operationally at meteorological agencies: the WAM model (WAMDI Group, 1988; Komen et al., 1994) and the WaveWatch III model (Tolman et al, 2002).

As well as using information on the wind field, wave observations from in-situ measurements, satellite altimeters and SAR can be routinely assimilated into both analysed and hindcast wave fields (see e.g. Lionello et al, 1992; Hasselmann et al, 1997;

Voorrips, 1999; Abdalla et al, 2006). Model runs with assimilated data have been shown to significantly reduce the errors in modelled wave parameters.

2.3.2 Sources of error in wave models

The error sources in the estimates produced by wave models can be viewed as either internal or external to the model. The internal sources of error are the formulation of sources terms or ‘model physics’ and the numerical scheme, while external errors refer to errors in the input data, primarily the wind field. An in-depth review of the present state of the art and limiting factors in the physics and numerics of wave modelling is given by Cavaleri et al (2007). In the following we give a brief overview of the error sources.

2.3.2.1 Input data

In validation studies it is common practice to assess quality of wind input at the same time as the wave estimates, in order to estimate the relative importance of internal and external errors. It is not straightforward to separate internal from external errors. Janssen (1998) has presented a simple model for the error in Hs resulting from wind speed errors and shows that they are proportional to the square of the error in wind speed. Rogers et al (2005) have shown that, in contrast to previous studies, the errors in the wind fields used at Fleet Numerical Meteorology and Oceanography Center (FNMOC) are no longer the dominant source of errors in wave estimates from WaveWatch III. However, this is not to say that the quality of the wind forcing is no longer important to accuracy.

Recently, Feng et al (2006) have tested the sensitivity of WaveWatch III to four

different wind input fields and shown that the accuracy is critically sensitive to choice of the wind field product. Other input fields such as currents, bathymetry, or bottom conditions are less important in the open ocean but become significant in shallower coastal waters.

2.3.2.2 Numerical resolution

Describing a continuous physical process such as wave growth, propagation and dissipation with a discrete model can lead to significant errors. The resolution of the geographic grid, the time step for integration and the spectral resolution (number of frequency and direction bins) all affect accuracy. The propagation of swell on a grid with discrete directional resolution can lead to the disintegration of a continuous swell field into discrete packets. This process is known as the Garden Sprinkler Effect and is discussed by Tolman (2002b). The accuracy of swell propagation is also affected by blockage by small islands and ice which are not resolved in the spatial grid (Tolman, 2003; Ponce de Leon and Guedes Soares, 2005). Coarse geographic and temporal resolution can also lead to small intense systems being subject to some smoothing, resulting in systematic underestimation of peak wind speeds and hence peak wave heights (Tolman 2002a).

2.3.2.3 Model physics

There remain many open questions about the formulation of source terms in spectral wave models. Amongst the most important ones are the method used to estimate non-linear interactions, spectral dissipation in deep water, and air–sea momentum transfer at high wind speeds. At high wind speeds many of the assumptions about the processes involved are stretched to their limit. Moreover observation of wave growth in extreme conditions are, by their nature, limited. Rogers et al (2005) stress that “given the

necessary reliance on approximations in today's state-of-the-art wave models, it may be especially difficult for these models to have ‘universal’ tuning. In particular, tuning for applications at one scale may inevitably degrade performance at another scale. For example, tuning to short-fetch empirical growth curves probably will not produce a skilful global model.”

2002; Cavaleri, 2006). Liu et al (2002) show that even when working with accurate, carefully evaluated wind fields, the wave model results show a scatter not justified by the known uncertainties in the input information. Nevertheless, some improvements can be expected to be made by improving the physics and numerics in wave models.

2.3.3. Qualitative description of model errors

The performance of models in terms of integral parameters such as Hs and Te is, on the whole, fairly good. However, Cavaleri (2006) notes that the comparison between modelled and measured spectra is often unsatisfactory, not only in the details, but sometimes also in the general structure. In this study we will only consider the accuracy of integral parameters.

As noted before, modelled wave spectra can be considered an estimate of the average conditions over the grid spacing and time step used in the model. Typically, global or oceanic scale wave models will be run with a grid spacing somewhere between 0.5° and 3° (about 50-300km) with a time step of 3 or 6 hours. Measured data is usually obtained over a smaller scale, with buoys representing a point average over time (between 20 minutes and 1 hour) and altimeter data representing an instantaneous spatial average over an area of 5-10km in diameter. The spatial and temporal variability of wave

conditions will therefore result in differences between measurements and modelled data.

These differences are sometimes referred to as ‘representativeness errors’ and the error is assigned to the measured data (Janssen et al, 2007).

The errors in modelled parameters exhibit short term temporal correlation. That is, an over- or under-estimate in Hs or other parameters will typically persist for a number of hours. For instance models will tend to over or under predict the intensity of an entire storm, which leads to correlation of errors for up to a few days.

Additionally errors in different parameters can be correlated. At high sea states, since wave spectra tend toward standard Bretschneider or JONSWAP type forms, an overestimate in model Hs will result in an overestimate of period as well. This correlation of errors in different parameters means that one needs to be careful when calibrating model data, since adjusting model parameters independently may lead to changes in the shape of the joint distribution.

Errors in modelled parameters can be thought of as having a mean or bias and also a random component. Both the mean and bias component will have a complex

dependence on the actual wave conditions. For instance the bias of a model estimate of Hs may have a dependence on the actual Hs, period, spectral shape, swell age, etc.

Moreover, it has been shown by numerous authors that biases change both with location and with time. This is due to the way that errors occur in models and their propagation through the model domain.

Janssen (2008) presents a particularly clear illustration of the non-stationary biases in ECMWF WAM model spectra. A plot of the bias in spectral energy binned by

frequency shows that the model tends to over-predict energy at lower frequencies in the (Northern Hemisphere) summer and much less in the winter time. Moreover, the magnitude of this bias and its dependence on both frequency and time of year changes from year to year. He notes that the main reasons for the changing biases are that large swells generated in the Southern Ocean in the Southern Hemisphere winter time are not well modelled due to unresolved islands and atolls (mainly in the Pacific) and the formulation of the dissipation source term.

This goes to show that it is difficult to define and adjust for a ‘mean error component’

since varying conditions lead to varying amounts of internal and external errors occurring and aggregating over the model domain. Therefore errors in wind seas and young swells can be expected to have different characteristics to older swells that have propagated further, increasing uncertainties.

A further reason for non-stationary biases in model data is changes made to the models themselves. This is more of an issue for archived data from operational models than for hindcasts. However, despite the fact that hindcasts are run with a constant model setup, the quality of the input wind fields and assimilated wave data may be varying.

Finally, we note that modelled data may be subject to temporal offsets, with the model predicting that a storm arrives slightly earlier or later than it actually does. This type of error is sometimes referred to as a ‘jitter error’. Jitter errors are not so important when

To summarise, the main features of the errors in model data are:

• The bias and variance of modelled parameters may depend on multiple factors such as Hs, Te, swell age, etc.

• Errors in parameters will exhibit short-term autocorrelation.

• There may also be correlation of errors between parameters, e.g. errors in Hs

and Te may be correlated.

• The bias and variance of the modelled parameters may be non-stationary in both time and space.

• There may be temporal offsets or ‘jitter errors’ in modelled parameters.