Pierson Scale Models
Toney Talbot
Amalgam are recognised worldwide as makers of the finest hand-made large scale models, particularly at 1:8 scale. Our work is unique in its attention to detail together with a focus on creating models that truly capture the style and spirit of each car. Since 1995 we have dedicated our energy and passion to achieving a level of accuracy, precision and excellence that raises our finished 1:8 replicas, far above anything previously created at any scale. Following many hours of hand working, these models precisely capture the entirety of the original and are almost impossible to discern from a real car in photographs.
SandyDuck '97 wind conditions at NDBC buoys in the region. Wind speed has been converted from 5- to 10-m altitude using a power-law relation. Direction is measured clockwise from north (NDBC convention)
SandyDuck '97 time series of frequency spectra at NDBC buoy 44014. Buoy data are compared with the SWAN model. Note that scaling of ordinate axis is not constant, so the comparison is essentially normalized
Lake Michigan time series of frequency spectra at two NDBC buoy locations. The SWAN models are compared with measurements. The n1.0PM,NDS model is not included in this comparison, because it is practically identical to the n1.0PM model in this simulation: (a) NDBC buoy 45002 and (b) NDBC buoy 45007
Mississippi Bight wind data measured at the two NDBC buoys in the region. Wind speed has been converted from 5- to 10-m altitude using a power-law relation. Direction is measured clockwise from north (NDBC notation)
Mississippi Bight time series of frequency spectra at location of NDBC buoy 42040. The SWAN models are compared with measurements. The n1.0PM,NDS model is not included in this comparison, because it is practically identical to the n1.0PM model in this simulation.
It is credible that both of the problems with the SandyDuck simulation are due to low-frequency energy being overdissipated by the model. In section 5, we propose two general types of modifications of the sink term that would tend to reduce dissipation of lower frequencies: 1) alteration of two of the free parameters of the dissipation term used by the model, that of Komen et al. (1984), and 2) by disallowing the breaking of swell. (The definition of swell that we use is given in section 5.) With the second modification, the problem of swell being dissipated during the wind event is corrected; however, neither modification consistently corrects the general underprediction of wind sea energy at and below the spectral peak.
Whitecapping is probably the least understood deep water source/sink mechanism. This dissipation is not easily measured, so prevailing theories provide only vague guidance and the formulas used in wave models tend to be quite empirical. Donelan and Yuan (1994, denoted as DY hereinafter) provide a concise explanation of the whitecapping term used by SWAN, which is based on Hasselmann (1974), KHH, and WAMDI Group (1988).
Van Vledder (1999) experiments with tuning the free parameters of the dissipation term of (4) using the SWAN model, along with two DIA parameters (five tuning parameters total). He tuned to the Kahma and Calkoen (1992) growth curve using a relatively short (25 km) fetch case. His tuned source term has only weak dependence of dissipation on the (k/km) term. Van Vledder demonstrates a shortcoming with the whitecapping formulation of SWAN [also WAM and early versions of WAVEWATCH (e.g., Tolman 1991)]: an artificial impact of swell energy on wind wave growth. This erroneous behavior of the model is due to the dissipation term being strongly weighted by the spectral mean frequency (or wavenumber), which is in turn very sensitive to the presence of swell. Solutions to this particular problem are under development (e.g., Holthuijsen and Booij 2000).
Comparisons with the lidar data by Rogers et al. (2000) indicate that, while total energy is generally well predicted, mean wave period is significantly underpredicted by the model. In order to investigate this simulation error further, we compare time series of energy at several frequencies (buoy vs model). These are shown in Fig. 3. Here, model spectra are interpolated to the buoy frequencies. Two discrepancies are immediately obvious: 1) energy at and below the spectral peak (approximately 0.18 Hz during the wind event) is underpredicted during the wind event, while energy above the spectral peak is slightly overpredicted, and 2) during the wind event, there is a sudden drop in the swell energy that is not observed in the data.
The latter discrepancy is easily explained: because of the dependence of the dissipation term on the integrated wave steepness, the dissipation of the swell frequencies, previously insignificant, becomes significant in the presence of wind sea. The cause of the first problem is less clear; most likely it is a result of combined inaccuracies of the three deep-water source/sink terms and wind forcing. (Wind measurements during this period of SandyDuck indicate that conditions are quite complex. Thus, there is a higher than normal level of uncertainty with regard to the wind forcing.) To resolve the first problem, we focus our attention on the dissipation term. This is justified by two facts: (i) the dissipation term is the least accurate of the three terms and (ii) the dissipation term is, by design, a closure term and is therefore a means to compensate for inaccuracies in the other two terms until more accurate (or, in the case of Snl computationally expedient) formulations for those two terms are developed.
Below, we detail two modifications that are intended to reduce the two types of problems mentioned above. We have two specific objectives in designing these modifications: 1) that the details of their design should be independent of the motivating simulation (SandyDuck) and 2) that consideration is given to conditions of general nature.
Both events are characterized by a growth phase (during which wind speed and wave energy grow rapidly), a brief transition phase (when wind speed is highest and source, sink, and propagation terms roughly balance), and a decay phase (when wind speed is decreasing, waves decay via whitecapping and propagation). The decay of the second event is caused by a simultaneous decrease in wind speed and fetch length. Note that for buoy 45007, the second, northerly event is of significantly longer fetch than the first, southerly event, while for buoy 45002, the reverse is true. Thus, considering the buoys separately, we are modeling two short fetch events and two longer fetch events with this simulation. Bulk wave parameters measured by the two buoys are summarized in Table 1.
The two events are modeled in one simulation, initialized with zero energy state at 1600 local time on 8 November. Wind data from buoys 45002 and 45007 are used. Wind data from the two coastal data locations are not used because of the sheltering apparent in data from those locations (wave comparisons are made at the two open-water locations, where sheltering effects are less important). Wind is simply assumed uniform in x (longitude), with variability in y (latitude) determined by interpolation between the two data points. Model results are compared at the locations of buoys 45002 and 45007. See the appendix for additional details on model setup.
With the n1.5PM model, the impact is seen mostly in the higher frequencies (which are dissipated more because of higher n and higher Cds). In the lower frequencies, the higher n and higher Cds have an opposing effect on dissipation and largely balance each other.
The n2.0 model produces a clear improvement in results, especially at buoy 45007. At buoy 45002, there is a moderate overprediction of some energies for the short-fetch case, but we expect that an exaggeration of wind speeds near the coastline is at least partially to blame for this. Otherwise, the improvement is quite significant.
The SWAN model is being developed as a near-real-time wave model for the Mississippi Bight region as a component of the Northern Gulf of Mexico Littoral Initiative. Figure 8 shows the computational grid and the location of two NDBC buoys. The test case used herein was designed by Hsu et al. (2000) as a preliminary simulation for the nowcast model, covering the time period of 0000 UTC 19 October through 0100 UTC 22 October 1999. Wind data from both buoys are shown in Fig. 9. It is an ideal test case: winds range from weak to moderately high strength and are directed offshore (fetch limited). Very little wave energy enters the computational domain through the open ocean boundary, minimizing concerns of error in specification of the boundary conditions. Wind and wave data are available within the computational grid from two NDBC buoys. The wind field appears to be fairly uniform between the two buoys during this period and is generally from a northerly direction. Therefore, the model is forced with uniform wind input, based on data from buoy 42040. The bottom friction formulation from the Joint North Sea Wave Project (JONSWAP: Hasselmann et al. 1973) is used. (See the appendix for additional details on model setup.)
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