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VII Waves

5.5 Parameterisation of subgrid bathymetry

5.5.1 Introduction

By looking at monthly mean analysis wave height increments, especially during the Northern Hemisphere summer, it appears that there are areas where the wave model first guess is systematically too high or too low. The underestimation in wave heights tends to be located in the active storm track areas or in areas affected by the Indian sub-continent monsoon. It is known that this underestimation is likely caused by too weak model winds. On the other hand, the overestimation for most of the tropical and northern Pacific cannot be explained in terms of local winds. After further scrutiny, it appears that these systematic overestimations are often present in areas where small island chains exist (French Polynesia and Micronesia in the Pacific Ocean, Maldives Islands and Andaman Islands in the Indian Ocean and Azores and Cape Verde Islands in the Atlantic Ocean, ...).

Hence, it appears that small islands and submerged bathymetric features that are not at all resolved by the coarse wave model grid (55 km) may have a larger impact on the wave climate than it is usually assumed. Although, in the operational grid up to CY28R1, representation of some islands were artificially enhanced to produce the necessary blocking to wave propagation, the results were not very satisfactory. A more appropriate and automatic procedure has been designed to deal with small and not so small islands and reefs. This simple parameterisation was successfully implemented in CY28R1.

5.5.2 Treatment of unresolved bathymetry

Up to CY28R1, the operational 55 km wave model grid was based on the ETOPO5 data set that represents the land and sea-bottom elevation on a 5-minute latitude/longitude grid. This data set is available from the National Geophysical Data Center (http://www.ngdc.noaa.gov/mgg/). Recently, NGDC produced a finer data set, namely ETOPO2 that has a 2-minute resolution. Fig. 5.3 shows the bathymetry for an area centred on the Tuamotu Archipelago in the South Pacific as derived from the ETOPO2 data (only sea points with water depth less than 300m are shown). The complexity of the bathymetry is clearly visible. This data set can be used to produce the wave model grid by averaging the depths of all ETOPO2 sea points within a model grid box and vice-versa for land points. A model grid box is considered to be over sea if 50% or more ETOPO2 points are sea points and a small area 4 by 4 minutes centred on the model grid point is not land. Fig. 5.4 shows the resulting mean depth for the 55 km grid. Much of the shallow features of the archipelago are gone. It is therefore not surprising that when model swell propagates across this area, very little attenuation is experienced (even with the model shallow water physics switched on).

Based on a similar idea as in Tolman (2003) and Hardy et al. (2001), we have modified the wave propagation scheme to limit the amount of wave energy that can be advected through these sub grid bathymetric features. The WAM model uses a simple first order upwind scheme that requires the knowledge of the wave spectral flux entering a given grid box in the upwind direction. These incoming fluxes are specified by the product of the wave spectral component and the corresponding mean group velocity perpendicular to the upwind grid box facet. However, in reality, if small islands or shallow water features are present, only part of incoming energy will reach the central grid point. With the availability of a finer resolution topographic data set such as ETOPO2, it is possible to estimate how much obstruction these features would produce.

For each model grid point, the 2-minute data are analysed line by line in all four cardinal directions from the model grid point up to the neighbouring grid points. If land or shallow water features are present, then the proportion of how much energy would have propagated from the neighbouring points to the grid point along each line is reduced accordingly. Land and very shallow features are entirely blocking the flux along the respective line, provided deeper data points on both sides surround them (in order to be an obstruction and not a coastline). On the other hand, points, which are shallow enough to affect the incoming waves but deep enough that they do not block the waves, are only reducing the flux in proportion to the total number of points in a line. How relatively shallow the water is depends on the frequency (wavelength) of the spectral component under consideration. Very shallow features are defined such that their respective depth is less than 0.1/k, where k is the deep water wave number () of the incoming waves. If waves are not entirely obstructed by the small bathymetric features, they are however assumed to be partially blocked if the depth is of the order 2/k or less and the mean grid point depth is at least 10 times larger than that threshold.

The total obstruction for each upwind flux is then obtained by summing over all lines that are intersecting the corresponding grid box facet. High frequency waves are less affected by the bathymetry than low frequency components. Thus, at each grid point there is a transmission factor for each discretised frequency bin corresponding to all four cardinal directions. Fig. 5.5 shows how much energy is allowed to propagate towards the north for the first frequency bin of the model (wavelength ~ 1360 m) for the same area as in Fig. 5.3 . These long waves will indeed be quite attenuated as they cross the Archipelago. On the other hands, the short waves should be a lot less affected by the unresolved bathymetry. Fig. 5.6 displays the corresponding transmission coefficient for the very short waves in the model (wavelength ~ 6 m). The impact of the unresolved bathymetry is indeed much reduced.

The net benefit of using this new parameterisation was illustrated in Bidlot and Janssen (2003) (http://w3ec2.ecmwf.int/wave/documents/memo_CY28R1.pdf ) and in the e-suite for cycle 28R1.

Figure 5.3 ETOPO2 bathymetry obtained from the National Geophysical Data Center (only sea points shallower than 300 m are shown).

Figure 5.4 WAM bathymetry (only sea points shallower than 300 m are shown) for the 55 km grid.

Figure 5.5 Percentage of the wave energy that is allowed propagates northwards for the lowest frequency bin (~ 0.035 Hz).

Figure 5.6 Percentage of the wave energy that is allowed propagates northwards for the highest frequency bin (~ 0.55 Hz).