SEARCH

SEARCH BY CITATION

Keywords:

  • Gulf of Mexico;
  • bi-modal spectra;
  • directional spectra;
  • hurricane-generated waves;
  • spectral wave model

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Study Area and Data Description
  5. 3. Dominant Wave Directions
  6. 4. Mono-modal Frequency Spectra
  7. 5. Bi-modal Spectra
  8. 6. Summary and Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[1] Hurricane-induced directional wave spectra in the Gulf of Mexico are investigated based on the measurements collected at 12 buoys during 7 hurricane events in recent years. Focusing on hurricane-generated wave spectra, we only consider the wave measurements at the buoys within eight times the radius of the hurricane maximum wind speed (Rmax) from the hurricane center. A series of numerical experiments using a third-generation spectral wave prediction model were carried out to gain insight into the mechanism controlling the directional and frequency distributions of hurricane wave energy. It is found that hurricane wave spectra are almost swell-dominated except for the right-rear quadrant of a hurricane with respect to the forward direction, where the local strong winds control the spectra. Despite the complexity of a hurricane wind field, most of the spectra are mono-modal, similar to those under fetch-limited, unidirectional winds. However, bi-modal spectra were also found in both measurements and model results. Four types of bi-modal spectra have been observed. Type I happens far away (>6 × Rmax) from a hurricane. Type II is bi-modal in frequency with significant differences in direction. It happens in the two left quadrants when the direction of hurricane winds deviates considerably from the swell direction. Type III is bi-modal in frequency in almost the same wave direction with two close peaks. It occurs when the energy of locally-generated wind-sea is only partially transferred to the swell energy by non-linear wave-wave interactions. Type IV was observed in shallow waters owing to coastal effects.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Study Area and Data Description
  5. 3. Dominant Wave Directions
  6. 4. Mono-modal Frequency Spectra
  7. 5. Bi-modal Spectra
  8. 6. Summary and Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[2] The Gulf of Mexico is extremely susceptible to the impact of frequent tropical cyclones. Hurricane Katrina (2005) caused the death of at least 1,836 people and total property damage estimated at $81 billion [Knabb et al., 2006]. This kind of extreme meteorological events can generate enormous waves [Wang et al., 2005; Holliday et al., 2006; Liu et al., 2008]. The highest recorded significant wave height reached 16.91 m during Hurricane Katrina, though the sparse buoys were likely to miss the maximum. Many ocean and coastal engineering applications require the information about the directional and frequency distributions of wave energy beyond significant wave heights and peak wave periods. The directional wave spectral data can be obtained from both in situ measurements (e.g., using directional wave buoys) and remote sensing (e.g., the Scanning Radar Altimeter, SRA [Walsh et al., 1996], and the Synthetic Aperture Radar, SAR [Elachi et al., 1977; Young and Burchell, 1996]). Young [2006] analyzed the directional spectrum data recorded from directional buoys during the passage of 9 tropical cyclones off Australia's North-West coast. The airborne SRA can measure the directional spectra in all quadrants of a hurricane's inner core [Wright et al., 2001]. In addition to measurements, numerical modeling is a useful and convenient way to obtain the spatial and temporal variations of directional spectra and to gain insight into the mechanism controlling the directional and frequency distributions of hurricane wave energy. Moon et al. [2003] used the third-generation spectral wave model, WAVEWATCH III®, to simulate the directional wave spectra generated by Hurricane Bonnie (1998).

[3] In the Gulf of Mexico, the U.S. National Oceanic and Atmospheric Administration's (NOAA) National Data Buoy Center (NDBC) has maintained a wave buoy network including 12 buoys, which provides the directional wave spectrum data every one hour. Although the region is big and the number of buoys is limited, there still are a fairly large number of hurricane-generated directional wave spectra acquired in the past several years. No detailed analysis of the data, however, exists in the literature. In this paper, the characteristics of directional spectra of hurricane-generated waves in the Gulf of Mexico are studied based on the buoy measurements. A series of numerical experiments using a third-generation spectral wave prediction model have been carried out to probe for the mechanism that controls the directional and frequency distributions of hurricane-generated wave energy.

2. Study Area and Data Description

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Study Area and Data Description
  5. 3. Dominant Wave Directions
  6. 4. Mono-modal Frequency Spectra
  7. 5. Bi-modal Spectra
  8. 6. Summary and Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[4] As shown in Figure 1, the directional wave data in the Gulf of Mexico were collected at 12 buoys during 7 hurricane events in recent years. Focusing on hurricane-generated wave spectra, we only consider the wave measurements at the buoys within eight times the radius of the hurricane maximum wind speed from the hurricane center. The average values of central pressure (Pc), maximum wind speed (Vmax) and its radius (Rmax) as well as the average hurricane forward speed (Vt) were calculated based on the best track data from the U.S. National Hurricane Center.

image

Figure 1. The Gulf of Mexico, 12 buoy locations (stars) and 7 hurricane tracks (lines with dots).

Download figure to PowerPoint

[5] The source data measured at these buoys were analyzed by NOAA using the Fast Fourier Transform to obtain the 2D directional spectra (see Steele et al. [1998] for details). Compared to the Maximum Likelihood Method, the results should be similar but may give slightly boarder directional spreading [Young, 2006]. Following the approach of Young [2006], the direction of forward movement of a hurricane was used to transform (rotate) the spectral data into a frame of reference moving north at the hurricane forward speed. In this moving frame of reference, the location of the measurements moves relative to the hurricane center that is fixed, and the spatial dimensions were normalized by the radius of maximum winds.

3. Dominant Wave Directions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Study Area and Data Description
  5. 3. Dominant Wave Directions
  6. 4. Mono-modal Frequency Spectra
  7. 5. Bi-modal Spectra
  8. 6. Summary and Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[6] The observed and modeled dominant wave directions are showed in Figure 2. The observed directions were averaged within every 1Rmax × 1Rmax square. Areas with no vectors indicate where there were no measurements. The third-generation spectral wave model, SWAN [Booij et al., 1999] (version 40.85), is utilized to simulate wave fields driven by an idealized hurricane that moves north at a constant forward speed. Thirty-five frequencies are exponentially spaced from 0.02135 Hz to 0.6 Hz with 72 evenly spaced directions (5° resolution), and a time step of 15 min is used. Frequency resolution meets the requirement (Δf = 0.1 f) of the Discrete Interaction Approximation (DIA) method [Hasselmann et al., 1985] for the approximation of nonlinear 4-wave (quadruplets) interactions. The whitecapping expression of van der Westhuysen [2007] is chosen, which features the non-linear, saturation-based expression of van der Westhuysen et al. [2007] for the dissipation of components with saturation levels above a predefined threshold, and that of Komen et al. [1984] for those below it. This is combined with the wind input expression of Yan [1987]. Default values are used for other settings. The hurricane parameters are taken from the average values (Case 1) for 7 hurricanes, that is, 944 mb, 104 knot, 22 nm and 5.9 m/s for Pc, Vmax, Rmax and Vt, respectively (see Table S1 in the auxiliary material for details). By using these parameters, the hurricane wind fields can be easily generated using the Holland wind vortex [Holland, 1980] with the consideration of hurricane forward speed. This Holland wind field is used to drive the wave model.

image

Figure 2. (left) Observed and (right) modeled distributions of wind directions (black arrows), dominant wave directions (red arrows) and significant wave heights (the contour interval is 0.5 m).

Download figure to PowerPoint

[7] As shown in Figure 2, the observed wind directions are consistent with Young's results [see Young, 2006, Figure 4a]. The modeled results are similar to the observations. Wind directions are counterclockwise relative to the center for a north-hemisphere hurricane. Dominant wave directions show that waves in two front quadrants radiate out from a region to the right of the hurricane center where the winds are very strong. In the right-rear quadrant, the angle between winds and waves is relatively small (less than 60°) at most locations, which means the waves are mainly generated locally. In the left-rear quadrant, waves at some locations radiate out from the same region as waves in two front quadrants, and at other locations waves are locally generated with small directional deviation to winds. It can be found from the numerical experiments that the hurricane forward speed affects greatly on the dominate wave direction in the left-rear quadrant. The spatial distribution of significant wave heights shows that the maximum height occurs near Rmax to the right of the hurricane center.

4. Mono-modal Frequency Spectra

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Study Area and Data Description
  5. 3. Dominant Wave Directions
  6. 4. Mono-modal Frequency Spectra
  7. 5. Bi-modal Spectra
  8. 6. Summary and Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[8] The locations of observed spectra cover all four quadrants relative to the hurricane center (shown in Figure 3 with black points). In order to identify the number of spectral peaks in each energy-frequency (1D) spectrum, we establish two criteria. One is that the trough-peak differences on both sides of a peak should be larger than 15% of the maximum energy density of this spectrum; the other is that the frequency difference between two successive peaks should not be smaller than 0.04 Hz. If both conditions are met, the peak is identified. By this way, 448 out of 530, which is 85% of the total measured frequency spectra, are mono-modal. Thus, most observed hurricane-induced spectra (see the black points without colored symbols) are mono-modal, consistent with Young's [2006] finding. Note that the bi-modality with two peaks at the same frequency but different directions cannot be identified by using the frequency spectrum only. Moreover, the criteria may exclude the bi-modality contained within the high-frequency tail [e.g., Hwang et al., 2000]. Both bi-modal features will not be considered in Section 5.

image

Figure 3. Relative locations of observed spectra. Bi-modal features are denoted in different symbols. The locations of observed and modeled spectra used in this paper are indicated as well.

Download figure to PowerPoint

[9] Young [2006] found that the spectral parameters defining the hurricane spectrum are similar to those obtained for simple fetch-limited cases. In this paper, the relationship between the non-dimensional wave energy (ɛ) and the non-dimensional peak frequency (v) was studied using both the observed data and the model results of Case 1. The observed data shows a good linear correlation between log10 (ɛ) and log10 (v), and can be expressed as

  • equation image

where ɛ = g2Etot/U104 and v = fpU10/g; g is the gravitational acceleration; U10 is the wind speed at 10m elevation; Etot is the total energy of a spectrum and can be estimated by the significant wave height Hs using (Hs/4)2; fp is the peak wave frequency. The model result agrees very well with this observed form (see Figure S1 in the auxiliary material for details). However, this relationship differs (e.g., for the swell part) from the form by Donelan et al. [1985] which is consistent with Young's [2006] results. Numerical experiments suggest that both hurricane wind intensities and forward speeds would affect this relationship.

[10] The modeled frequency spectra at different locations (M1-M4 in Figure 3) in four quadrants are shown in Figure 4a. All four spectra are mono-modal. The energy densities in the two right quadrants are larger than those in the two left quadrants because of the hurricane wind asymmetry mainly caused by the hurricane forward movement. In order to estimate how each quadrant's winds contribute to those spectra, a series of numerical experiments were carried out. The whole hurricane winds were divided into four parts by each quadrant. The results driven by each quadrant winds separately are shown in Figure 4a. It can be seen that local winds contribute most to the spectra in the two right quadrants. Especially, the spectra in the right-rear quadrant are mainly locally generated. In the right-front quadrant, the spectra are the combination of the locally generated waves and the swell originated from the right-rear quadrant. The spectra in the two left quadrants are swell-dominated, remotely generated from the right quadrants, especially from the front right quadrant. The peak frequencies using the four individual-quadrant winds are higher than those using the whole hurricane winds. This means that the effect of nonlinear interactions is underestimated in the wave field driven by the separated, experimental winds, since some energy cannot transfer from higher frequencies to lower frequencies.

image

Figure 4. (a) Modeled 1-D spectra under different wind conditions. (b) Three modeled source terms at four locations (M1-M4).

Download figure to PowerPoint

[11] Figure 4b shows the source terms at those locations using the whole hurricane winds, including the combination of wind input and whitecapping, the nonlinear quadruplet interactions, and the combination of all these terms, respectively. The nonlinear interaction term always transfers wave energy from high frequencies to low frequencies, while this trend is different from the trend of the total source term. This means that all source terms contribute to the final results, not mainly from the nonlinear interactions or the wind input and whitecapping alone. This result is consistent with the numerical tests by Banner and Young [1994]. Young [2006] also mentioned that the wind input and dissipation terms may still be important in the stage of wave growth, although he emphasized that the non-linear term plays a dominant role in controlling the spectral shape for “mature” hurricane waves. Notice that these are numerical results. Further studies are needed.

5. Bi-modal Spectra

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Study Area and Data Description
  5. 3. Dominant Wave Directions
  6. 4. Mono-modal Frequency Spectra
  7. 5. Bi-modal Spectra
  8. 6. Summary and Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[12] Although most of the hurricane-generated wave spectra are mono-modal, there are still about 15% of spectra which are bi-modal. It can be seen from Figure 3 that most bi-modal locations are in the two left quadrants. If only considering the observations in these two quadrants, the bi-modal percentage increases to 23%.

[13] Four types (I–IV) of bi-modal spectra are found in the buoy measurements. Type I (Figure 5, O2) is bi-modal mainly due to the long distance from the hurricane center. It happens in almost all quadrants (except the right-rear quadrant) with larger than six times Rmax away from hurricane center, where the swell and local wind-sea have distinct features that can be identified easily. Waves in the right-rear quadrant are locally generated without such a bi-modal feature. Type II (Figure 5, O3) is bi-modal in frequency with a large directional deviation. It occurs in the two left quadrants where the wave direction deviates considerably from the wind direction. Similar to Type I, two separate peaks are induced by swell and local wind-sea, respectively. The difference is that the two peaks in this situation always have large deviations in direction and can occur very close to the hurricane center. Type III (Figure 5, O1) is bi-modal in frequency with almost the same or similar directions. The two peaks normally are very close in frequency. It happens in all four quadrants, especially when the hurricane is moving towards or away from the measurement location. The nonlinear wave-wave interactions only partially transfer the energy between high frequencies and low frequencies. Type IV is the bi-modal due to coastal effects. As shown in Figure 5 (O4), Buoy 42007 is very close to the coast. In this situation, the wind and fetch reduction by land and shallow waters would affect the wave spectra. Among all bi-modal spectra, the percentages of these four types are 28%, 28%, 15% and 19%, respectively.

image

Figure 5. Observed 1D and 2D bi-modal spectra at four locations (O1-O4). The 2D directional spectra are shown in polar contours. The red and green radial lines represent the wind and wave directions, respectively.

Download figure to PowerPoint

[14] In order to test whether these bi-modal features can be reproduced by a numerical model, an experiment (Case 2) by the wave model SWAN is set up using the hurricane conditions (953 mb, 95 knot, 45 nm and 5.5 m/s for Pc, Vmax, Rmax and Vt, respectively) at the time when the spectra at O2 were generated during Hurricane Ike. The results are shown in Figure 6. Figures 6a and 6b represent the bi-modal spectra in Types I and II, respectively. By qualitative comparison with the observed spectra (see O2 and O3 in Figure 5), it can be seen that the model results reflect the similar bi-modal features in observations. However, the modeled spectra are much narrower. Type III has not been successfully simulated by SWAN because it needs high resolution in frequency, which is limited by the requirement for modeling the nonlinear wave-wave interaction by the method of DIA.

image

Figure 6. Modeled 1D and 2D bi-modal spectra at locations (a) M5 and (b) M6. The 2D directional spectra are shown in polar contours. The red and green radial lines represent the wind and wave directions, respectively.

Download figure to PowerPoint

6. Summary and Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Study Area and Data Description
  5. 3. Dominant Wave Directions
  6. 4. Mono-modal Frequency Spectra
  7. 5. Bi-modal Spectra
  8. 6. Summary and Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[15] The directional spectra of hurricane-generated waves in the Gulf of Mexico have been studied using in-situ observations at 12 buoys from 7 recent hurricanes. A series of numerical experiments using a spectral wave model have been carried out to gain insight into the cross-spectral energy transfer and origin of waves. The main scientific questions addressed are whether there are bi-modal wave spectra under complex hurricane winds and what controls the shape of the spectra.

[16] It has been found from both observations and modeling that the dominant wave directions in two front quadrants appear to be radiate out from a region to the right of the hurricane center. The wave direction in the right-rear quadrant has small deviations from the wind direction as most of them are locally generated in this quadrant. In the left-rear quadrant, some of the waves are radiated out from the right of the hurricane center, and some are also locally generated.

[17] The comprehensive datasets and numerical experiments have confirmed that most of the hurricane-induced spectra are mono-modal. The relationship between the non-dimensional wave energy and non-dimensional peak frequency is similar to the simple fetch-limited, unidirectional wave generation cases. The relationship based on the observations agrees very well with the modeled results, but has differences from the form of Donelan et al. [1985], especially for the swell part. Considerable amounts of swell energy are found in all quadrants except the right-rear quadrant where waves are mainly locally generated. The numerical experiments have suggested that, in deep waters under hurricane winds, all three source terms, namely wind energy input, whitecapping and non-linear wave-wave interactions play a crucial role in shaping the wave spectra.

[18] Bi-modal features have been found in observed wave spectra. Except for the cases significantly affected by coasts, three types of bi-modal spectra are identified. The first type is found at locations greater than 6 times Rmax from the hurricane center, where the swell and local wind-sea have distinct features. The second type is bi-modal in frequency with significant differences in direction. It happens in left quadrants when the direction of hurricane winds deviates considerably from the swell direction. The third type is bi-modal in frequency with two close peaks in almost the same mean wave direction. The first two types have been successfully simulated by the spectral wave model, although the modeled spectra are much narrower for the swell part than the observed spectra. The combined field observations and numerical experiments using hypothetical, separated wind fields have provided a comprehensive picture of hurricane-induced directional wave spectra and insight into the dynamics of the wave field. The findings presented in this paper could benefit not only the scientific community of wave modeling and air-sea interaction, but also the coastal and ocean engineering community for the design of offshore and coastal structures in hurricane prone areas.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Study Area and Data Description
  5. 3. Dominant Wave Directions
  6. 4. Mono-modal Frequency Spectra
  7. 5. Bi-modal Spectra
  8. 6. Summary and Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[19] The study has been supported by the U.S. National Science Foundation (NSF) (grant 0652859), the NSF Northern Gulf Coastal Hazards Collaboratory (grant 1010640) and the U.S. National Oceanic and Atmospheric Administration (NOAA) through the Northern Gulf Institute (grant 09-NGI-08). Computational resources were provided by the Louisiana Optical Network Initiative (LONI) and Louisiana State University. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the NSF or NOAA.

[20] The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Study Area and Data Description
  5. 3. Dominant Wave Directions
  6. 4. Mono-modal Frequency Spectra
  7. 5. Bi-modal Spectra
  8. 6. Summary and Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Study Area and Data Description
  5. 3. Dominant Wave Directions
  6. 4. Mono-modal Frequency Spectra
  7. 5. Bi-modal Spectra
  8. 6. Summary and Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

Auxiliary material for this article contains a table and a figure.

Auxiliary material files may require downloading to a local drive depending on platform, browser, configuration, and size. To open auxiliary materials in a browser, click on the label. To download, Right-click and select “Save Target As…” (PC) or CTRL-click and select “Download Link to Disk” (Mac).

Additional file information is provided in the readme.txt.

FilenameFormatSizeDescription
grl28541-sup-0001-readme.txtplain text document2Kreadme.txt
grl28541-sup-0002-ts01.txtplain text document1KTable S1. Information about 7 hurricanes and 2 test cases.
grl28541-sup-0003-fs01.epsPS document30KFigure S1. Relationships between non-dimensional wave energy and non-dimensional peak frequency given by observed data, modeled results, JONSWAP form and Donelan et al.'s [1985] form.

Please note: Wiley Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.