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  • View in gallery

    Directional spectra within tropical cyclones from the in situ data of Tamizi and Young (2020). Data presented for locations shown by the dots in each octant of the storm, which is propagating to the north (up the page). Solid arrows (and vertical lines) show mean wind direction and dashed arrows (and vertical lines) the peak wave direction. At each point the directional spreading function D(f, θ) and the one-dimensional spectrum E(f) are shown. Both D(f, θ) and E(f) have been normalized, such that they have maximum values of one. All angles are measured anticlockwise from the x axis.

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    Contour plots of significant wave height Hs within tropical cyclones. Also shown are vectors of the peak wave direction. The spatial scale is normalized by the radius to maximum winds Rm. Tropical cyclone velocity of forward movement, (a) Vfm = 2.5 m s−1 and (b) Vfm = 5.0 m s−1.

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    Nondimensional energy ε as a function of nondimensional frequency ν for waves within tropical cyclones. (a) In situ buoy data of Tamizi and Young (2020). (b) WW3 model for the case of a tropical cyclone with Δp = 50 hPa and Vfm = 2.5 m s−1.

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    Directional spectra within a tropical cyclone from the WW3 model. Data presented for locations shown by the dots in each octant of the storm which is propagating to the north (up the page). Solid arrows (and vertical lines) show mean wind direction and dashed arrows (and vertical lines) the peak wave direction. At each point the directional spreading function D(f, θ) and the one-dimensional spectrum E(f) are shown. Both D(f, θ) and E(f) have been normalized, such that they have maximum values of one. Case shown has Δp = 50 hPa and Vfm = 2.5 m s−1. All angles are measured anticlockwise from the x axis.

  • View in gallery

    Directional spectra within a tropical cyclone from the WW3 model. Data presented for locations shown by the dots in each octant of the storm which is propagating to the north (up the page). Solid arrows (and vertical lines) show mean wind direction and dashed arrows (and vertical lines) the peak wave direction. At each point the directional spreading function D(f, θ) and the one-dimensional spectrum E(f) are shown. Both D(f, θ) and E(f) have been normalized, such that they have maximum values of one. Case shown has Δp = 50 hPa and Vfm = 5.0 m s−1. All angles are measured anticlockwise from the x axis.

  • View in gallery

    Source terms for the directional spectra shown in Fig. 4. At each location, the atmospheric input Sin, nonlinear interaction Snl, and dissipation Sds are shown. Each source term has been normalized to have an absolute maximum value of one. Normalized energy levels are shaded with red for positive and blue for negative. Case shown has Δp = 50 hPa and Vfm = 2.5 m s−1. Solid vertical lines show mean wind direction and dashed vertical lines the peak wave direction. All angles are measured anticlockwise from the x axis.

  • View in gallery

    Source terms for the directional spectra shown in Fig. 5. At each location, the atmospheric input Sin, nonlinear interaction Snl, and dissipation, Sds are shown. Each source term has been normalized to have an absolute maximum value of one. Normalized energy levels are shaded with red for positive and blue for negative. Case shown has Δp = 50 hPa and Vfm = 5.0 m s−1. Solid vertical lines show mean wind direction and dashed vertical lines the peak wave direction. All angles are measured anticlockwise from the x axis.

  • View in gallery

    WW3 model directional spectra and source terms from a location in the north-northwest octant of a tropical cyclone with Δp = 50 hPa and Vfm = 5.0 m s−1. (a) One dimensional spectrum E(f) (m2 s). (b) Directional spectrum E(f, θ) (m2 s), contours drawn at [0.1, 0.5, 1, 5, 10, 50, 100, 200]. (c) Wind input source term Sin(f,θ) × 103 (m2). (d) Nonlinear source term Snl(f, θ) × 103 (m2). (e) Dissipation source term Sds(f, θ) × 103 (m2). (f) Total source term Stot(f, θ) × 103 (m2). For each panel, the vertical solid line shows the wind direction and the dashed line the peak wave direction. All angles are measured anticlockwise from the x axis.

  • View in gallery

    An example of the role played by the nonlinear term Snl in tropical cyclone wave evolution. The nonlinear term transfers energy from the wind-sea to remotely generated waves at the spectral peak. This occurs through an energy cascade along the ridge connecting these wave systems. As a result, the waves at the spectral peak act in a parasitic manner to take energy for the local wind-sea to enhance growth at the spectral peak. The process ensures that the two wave systems remain connected through the directionally skewed spectrum. The case shown is for the spectrum in the west-southwest octant of Fig. 4.

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The Physics of Ocean Wave Evolution within Tropical Cyclones

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  • 1 aDepartment of Infrastructure Engineering, University of Melbourne, Melbourne, Victoria, Australia
  • | 2 bNOAA Weather Program Office, Silver Spring, Maryland
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Abstract

A series of numerical experiments with the WAVEWATCH III spectral wave model are used to investigate the physics of wave evolution in tropical cyclones. Buoy observations show that tropical cyclone wave spectra are directionally skewed with a continuum of energy between locally generated wind-sea and remotely generated waves. These systems are often separated by more than 90°. The model spectra are consistent with the observed buoy data and are shown to be governed by nonlinear wave–wave interactions that result in a cascade of energy from the wind-sea to the remotely generated spectral peak. The peak waves act in a “parasitic” manner taking energy from the wind-sea to maintain their growth. The critical role of nonlinear processes explains why one-dimensional tropical cyclone spectra have characteristics very similar to fetch-limited waves, even though the generation system is far more complex. The results also provide strong validation of the critical role nonlinear interactions play in wind-wave evolution.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ian R. Young, ian.young@unimelb.edu.au

Abstract

A series of numerical experiments with the WAVEWATCH III spectral wave model are used to investigate the physics of wave evolution in tropical cyclones. Buoy observations show that tropical cyclone wave spectra are directionally skewed with a continuum of energy between locally generated wind-sea and remotely generated waves. These systems are often separated by more than 90°. The model spectra are consistent with the observed buoy data and are shown to be governed by nonlinear wave–wave interactions that result in a cascade of energy from the wind-sea to the remotely generated spectral peak. The peak waves act in a “parasitic” manner taking energy from the wind-sea to maintain their growth. The critical role of nonlinear processes explains why one-dimensional tropical cyclone spectra have characteristics very similar to fetch-limited waves, even though the generation system is far more complex. The results also provide strong validation of the critical role nonlinear interactions play in wind-wave evolution.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ian R. Young, ian.young@unimelb.edu.au

1. Introduction

In tropical and subtropical regions tropical cyclones represent the major extreme meteorological events, generating winds in excess of 40 m s−1 and significant wave heights above 10 m. Such systems are characterized by a well-formed translating vortex wind field with a calm eye and winds that spiral in toward the center of the storm. The strength of the winds, the rapidly changing direction and the translation of the system all pose particular challenges for our ability to understand the physics of wind-wave evolution in such systems and to develop appropriate tropical cyclone wave models. Despite the apparent complexity of the forcing wind field, observations of waves in tropical cyclones show that there are many similarities to waves generated in relatively simple cases with approximately constant unidirectional winds (Young 1998, 2006; Hu and Chen 2011; Collins et al. 2018; Tamizi and Young 2020). As a result, a range of simple parametric models have shown surprising ability to be able to predict tropical cyclone generated waves. It has often been speculated (Young 2006, 2017, Collins et al. 2018; Tamizi and Young 2020) that this is because nonlinear wave–wave interactions play a dominate role in defining the tropical cyclone directional wave spectrum and hence the significant wave height, as they do for more simple wave generation cases. Despite this speculation and the many observational studies of tropical cyclone wave spectra, no study has yet been able to demonstrate the role of nonlinear wave–wave interactions in tropical cyclones.

The present paper uses a state-of-the-art spectral wave model to investigate the source-term balance within the tropical cyclone wave field. The model is capable of modeling the tropical cyclone wind and wave fields at high resolution (2 km) and exploring the relative importance of the processes of atmospheric input, white-cap dissipation, swell decay, and nonlinear wave–wave interaction.

The arrangement of the paper is outlined below. Following this introduction, section 2 provides an overview of observations of the tropical cyclone wave field. Section 3 describes models which have been used to represent both the wind and wave fields within tropical cyclones. The WAVEWATCH III model (henceforth WW3) is used throughout the paper and the model physics and its setup for the present application are described in section 4. The spatial distribution of the wave field, as predicted by the WW3 model is described in section 5 and compared with observations. The energy balance within the directional wave spectrum is investigated for all regions of the tropical cyclone spatial wave field in section 6 and the role of nonlinear wave–wave interactions is highlighted. Finally, a discussion and conclusions are included in section 7.

2. Observations of tropical cyclone wave fields

Waves within tropical cyclones have previously been observed using in situ buoys, aircraft-borne synthetic aperture radars (SAR) and scanning radar altimeters (SRA) and satellite altimeters (ALT). In situ buoy observations have included studies by Patterson (1974), Whalen and Ochi (1978), Black (1979), Ochi and Chiu (1982), Ochi (1993), Young (1998, 2006), Hu and Chen (2011), Collins et al. (2018), and Tamizi and Young (2020). SAR observations were reported by Elachi et al. (1977), King and Shemdin (1978), Gonzalez et al. (1978), McLeish and Ross (1983), Holt and Gonzalez (1986), Beal et al. (1986), Wright et al. (2001), and Black et al. (2007). Scanning radar altimeter (SRA) observations have been reported by Wright et al. (2001), Walsh et al. (2002), Black et al. (2007), Hwang (2016), Hwang and Fan (2017), Hwang et al. (2017), Hwang and Walsh (2016, 2018a,b), and Walsh et al. (2021). In situ buoys, SAR, and SRA measure the directional wave spectrum. In contrast, satellite altimeters measure only the significant wave height but have the advantage of a much more extensive spatial distribution of observations. Such ALT measurements have been reported by Young and Burchell (1986), Young and Vinoth (2013), and Tamizi and Young (2020).

The collective dataset from these studies reveals a generally consistent picture of the tropical cyclone wave field and the directional spectrum. Whereas the wind field within tropical cyclones (TCs) can be described by a relatively simple vortex model with added asymmetry (Tamizi et al. 2020), the wave field is more complex. Ahead of the TC center, the wave field is characterized by a combination of remotely generated waves radiating out from the intense wind region to the right of the center of the TC and locally generated wind-sea. The remotely generated waves were generated in the intense wind regions near the center of the storm, but their energy has propagated (group velocity) faster than the forward speed of the storm and outrun the TC. The locally generated wind-sea is aligned with the local wind and often propagates at angles up to and exceeding 90° relative to the remotely generated waves. The remotely generated waves also have peak frequencies much lower than the locally generated wind-sea. The phase speed of these waves at the spectral peak often exceeds the local wind speed.

The maximum values of significant wave height Hs occur to the right of the center of the TC (note, throughout the paper it is assumed that Northern Hemisphere storms are being considered). In comparison to the wind field, the waves are more strongly right–left asymmetric. In the intense wind/wave region to the right of the TC center, the wind and wave directions are more closely aligned. In the left-rear quadrant of the storm, the waves become quite confused with the low-frequency remotely generated waves often at angles between 120° and 180° compared to the wind direction (and wind-sea).

Tamizi and Young (2020) compiled an extensive dataset from National Data Buoy Centre (NDBC) (Evans et al. 2003) buoys, which consists of a total of 2902 buoy records recorded during 353 individual TCs (hurricanes). This very extensive in situ dataset is consistent with previous buoy observations and summarizes the present observational understanding from this source. Tamizi and Young (2020) combined the data from multiple TCs by adopting a frame of reference moving with the TC, as previously used by Young (1998, 2006), Hu and Chen (2011), and Collins et al. (2018). Figure 1 shows the spatial distribution of the directional wave spectrum from the Tamizi and Young (2020) data. The figure shows the mean wind direction, the peak wave direction, together with the one-dimensional spectrum E(f) and the directional spreading function D(f, θ). The direction spectrum, E(f, θ) = E(f)D(f, θ) where D(f,θ)df=1. For presentation purposes, both the one-dimensional spectra and the values of D(f, θ) shown in Fig. 1 have been normalized, such that the maximum values are one [for D(f, θ), the maximum value at each frequency is one].

Fig. 1.
Fig. 1.

Directional spectra within tropical cyclones from the in situ data of Tamizi and Young (2020). Data presented for locations shown by the dots in each octant of the storm, which is propagating to the north (up the page). Solid arrows (and vertical lines) show mean wind direction and dashed arrows (and vertical lines) the peak wave direction. At each point the directional spreading function D(f, θ) and the one-dimensional spectrum E(f) are shown. Both D(f, θ) and E(f) have been normalized, such that they have maximum values of one. All angles are measured anticlockwise from the x axis.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-21-0005.1

The mean wind direction and peak wave direction vectors in Fig. 1 show the spatial distributions of wave direction propagation described above. As previously reported by Young (1998, 2006), Hu and Chen (2011), and Collins et al. (2018), despite the fact that the spectrum consists of a combination of remotely generated waves and local wind-sea, the one-dimensional spectrum is generally unimodal with a high-frequency face proportional to approximately f−4. In the left-rear quadrant of the TC, where the wind and wave directions differ by more than 90°, there is some suggestion of bimodal behavior, with a small high-frequency peak in the one-dimensional spectrum.

As also shown by Young (2006), despite the fact that the high-frequency wind-sea and remotely generated low-frequency waves can be separated by more than 90°, the directional spectrum is also generally not bimodal. Rather, the directional spectrum is directionally skewed but there is a continuum of energy from high frequency to low frequency. As noted by Tamizi and Young (2020), the generation sources for both the low-frequency remotely generated waves (intense winds to the right of the TC center at an earlier time) and the high-frequency wind-sea (local wind) are clear. However, as the low-frequency remotely generated waves are often propagating faster than the local wind, it is reasonable to assume there is no wind input to the spectrum for these and slightly higher frequencies. The question then arises as to how the wave energy in the transition region between the wind-sea and the low-frequency peak was generated and maintained?

As noted above, Wright et al. (2001), Walsh et al. (2002), Black et al. (2007), Hwang (2016), Hwang and Fan (2017), Hwang et al. (2017), Hwang and Walsh (2016, 2018a,b), and Walsh et al. (2021) reported SRA data taken from aircraft flights through a number of hurricanes. The vast majority of the results are associated with observations from two hurricanes, Bella (1998) and Ivan (2004). Walsh et al. (2021) report data from the more recent Hurricane Lorenzo (2019). SRA spectra reported for the regions ahead of and left of the storm center are very similar to the buoy data shown in Fig. 1. They show a dominant remotely generated peak radiating out from regions near the intense wind crescent of the translating tropical cyclone wind vortex with a skewed high-frequency wind-sea. Generally, these spectra are unimodal, as for the buoy data.

To the right of the storm center the SRA spectra are very broad and possibly show trimodal forms in the right-rear quadrant, becoming bimodal in the right-front quadrant (see Black et al. 2007, their Fig. 10). The trimodal spectra in the right-rear quadrant show one swell peak propagating toward the center of the tropical cyclone, apparently generated somewhere to the southeast of the storm center (assuming the TC is propagating toward the north). Such a propagation direction is difficult to explain, as a typical vortex wind field cannot explain the existence of such a wave generation source, so far from the storm center. It is possible that this swell system is generated by some meteorological system separate from the tropical cyclone, although this seems unlikely, as the reported SRA data from both Bonnie and Ivan show this feature. Another possibility is that the directional ambiguity in the SRA was not resolved correctly, in which case, this peak would be “folded back” 180°, appearing closer to one of the other swell peaks, making the spectra bimodal.

The existence of bimodal spectra is easily explained, as the combination of local wind-sea and swell generated at an earlier time in the intense wind regions of the translating tropical cyclone. However, trimodal systems, with peaks at similar wavenumbers (frequencies), do not seem consistent with a wind field of this nature. For a continuous wind field represented by a vortex and a storm propagating at a constant velocity of forward movement, swell should be continuously generated as the storm translates. How such a system would generate multiple, separated swell peaks is not obvious.

Although the buoy results shown in Fig. 1 present a consistent description of the spatial distribution of the TC directional spectrum, wave buoys have only limited directional resolving ability (they measure only three components) (Young 1994; Tamizi and Young 2020). These limitations mean that the resulting spectra tend to be smoother and directionally broader than recorded by instruments with more active sensors (e.g., spatial wave gauge arrays) (Young 1994). Hence, it is possible that some of the smooth transition in direction as a function of frequency may be an artifact of the limited directional resolving power of the buoys. If this is the case, then it may explain some of the differences with the SRA data. Unfortunately, the SRA publications mentioned above do not present comparisons with buoy overflights. However, as noted above, for most regions of the tropical cyclone wave field, the SRA spectra are generally consistent with buoy data. Resolving whether the differences are a limitation of the buoys, the SRA, the selected hurricanes, or all of these is beyond the scope of this paper and a subject for future research.

3. Models of tropical cyclone wind and wave fields

a. Wind field models

As noted above, it is common to approximate the tropical cyclone wind field using a simple vortex model (Holland 1980, Willoughby et al. 2006; Holland et al. 2010). Holland (1980) represented the radial pressure profile within a TC as
p=p0+Δpexp[(Rm/r)b],
where Δp is the central pressure drop, Rm is the radius to maximum winds, r is the radial distance from the center of the TC, and p0 is the central pressure in the TC. Using this pressure profile, Holland et al. (2010) represented the surface (10-m elevation) wind speed as
U10={100bsΔp(Rm/r)bsρexp[(Rm/r)bs]}x,
where ρ is the density of air and surface values are defined by the subscript s. Holland et al. (2010), approximated the exponent bs by
bs=4.4×105Δp2+0.01Δp+0.03pt0.014ϕ+0.15Vfmxa+1.0.
In (3), Vfm is the velocity of forward movement of the TC (m s−1), ϕ is the absolute value of the latitude (°), t is time, ∂p/∂t has units of hectopascals per hour (hPa h−1) and Δp has units of hectopascals (hPa). The exponent x in (2) can be expressed as
x={0.5,forrRm0.5+(rRm)xn0.5rnRm,forr>Rm.
The exponent xa in (3) is given by xa = 0.6 (1 − Δp/215). Following Holland et al. (2010), x = x(r) and xn = x(rn). The value xn can be determined from (2) if measurements of the surface wind are available at a radius rn from the TC center.

The wind field defined by (2)(4) is symmetric. Based on extensive scatterometer measurements in TCs, Tamizi et al. (2020) found that the asymmetry of the wind field can be approximated to first order by the vector addition of the velocity of forward movement to the wind field vectors. This approximation agrees with the findings of Holland (2008), Klotz and Jiang (2016, 2017), and Olfateh et al. (2017). Consistent with the results of Powell (1982), Zhang and Uhlhorn (2012), and Tamizi et al. (2020) found that the observed inflow angle is a function of both p0 and Vfm with the maximum values occurring in the right-rear quadrant (~35°) and the minimum values in the left-front quadrant (~10°).

b. Wave field models

The fact that one-dimensional wave spectra within TCs are unimodal leads to the obvious comparison with fetch-limited spectra. Young (1998, 2006), Hu and Chen (2011), Collins et al. (2018), and Tamizi and Young (2020) have all considered the detailed spectral shape of the one-dimensional spectrum under TC conditions. Despite the complex wind conditions described above [(1)(4)], these studies show that TC wave spectra are remarkably similar to fetch-limited spectra measured during approximately constant unidirectional winds. In particular, the fetch-limited scaling between nondimensional energy and nondimensional peak frequency, proposed by Hasselmann et al. (1973) (JONSWAP) and Donelan et al. (1985) holds
ε=aνm,
where ε=g2Etot/U104 is the nondimensional energy and ν = fpU10/g is the nondimensional peak frequency. The total energy is represented by Etot=E(f)df, fp is the spectral peak frequency, and g is gravitational acceleration. The coefficients a and m are typically determined from recorded data.

This same scaling has also been confirmed from airborne SRA data (Hwang 2016; Hwang and Fan 2017; Hwang et al. 2017; Hwang and Walsh 2016, 2018a,b).

Noting this “JONSWAP-type” scaling, several authors have developed relationships to determine the significant wave height in TCs based on JONSWAP fetch-limited relationships. Young and Burchell (1986), Young (1988a), and Young and Vinoth (2013) use the concept of an “extended fetch” where waves move forward with the TC to define an “equivalent fetch,” which is a function of the velocity of forward movement Vfm and the maximum wind velocity in the storm Vmax. The equivalent fetch is used to define the maximum significant wave height in the TC. Values at other locations in the storm are then related to this maximum value. Hwang (2016) and Hwang and Walsh (2016) use a “circular racetrack” model in which a fetch is defined as a function of the distance from the center of the TC. The significant wave height is then determined at that point from a JONSWAP-type relationship (Hasselmann et al. 1973). These models are described in detail and reviewed in Young (2017).

Our understanding of the physics of fetch-limited growth is that there is an active balance between three physical processes: atmospheric input Sin, white-cap dissipation Sds, and nonlinear wave–wave interactions Snl (Hasselmann et al. 1973). The balance between these processes results in the observed spectral form and the scaling represented by (5). In the case of TC wave generation, Fig. 1 shows that the spectra are a combination of remotely generated waves (at the spectral peak) and locally generated wind-sea. As the spectral peak waves are generally propagating faster than the local wind, they receive no positive input from the wind. Hence, it is reasonable to assume that the spectral balance near the peak of the spectrum is very different to fetch-limited cases (i.e., Sin ≈ 0 or negative). Despite this, the JONSWAP-type scaling represented by (5) holds, even when the peak frequencies in this relationship are apparently disconnected from the local wind. Further, even though the directional spectra in TCs are directionally skewed, the one-dimensional spectra are very similar to fetch-limited cases. Note, however, that the peak frequencies in TC cases are at much lower values of fp and the wave ages of these peak waves, U10/Cp < 1 (Tamizi and Young 2020), where Cp is the phase speed of waves at the spectral peak.

The aim of this paper is to investigate the spectral balance under the complex forcing of a TC and understand the physical processes responsible for the observed spectra. To date, there has been speculation on the possible energy balance (i.e., Snl dominates; Tamizi and Young 2020) but no detailed evaluation.

4. Translating grid WW3 model

The WW3 model (Tolman 1991, 2002; WAVEWATCH III Development Group 2019) is widely used as a state-of-the-art operational and research wave model. The model has been validated in TC conditions by Moon et al. (2003), Tolman and Alves (2005), and Liu et al. (2017). One of the challenges in modeling TC waves is to have a grid of sufficient spatial extent to model the translation of the storm, while having a grid resolution which is sufficiently small to define the intense wind vortex near the eye of the storm, where much of the important wave generation occurs. These requirements (fine grid of large spatial extent) pose computation limitations. To overcome these challenges, Tolman and Alves (2005) developed a moving grid version of WW3, where the computational grid can move forward with the TC. Following Tolman and Alves (2005) the deep water governing radiative energy balance equation relative to this moving grid system becomes
E(f,θ)t+(CgVfm)xE(f,θ)=S(f,θ),
where Cg is the group velocity of the spectral component, Vfm is the velocity of forward movement of the TC (the velocity of translation of the computational grid), and ∇x is the spatial gradient differential operator. As noted above, the source term S represents the physical processes active in wind-wave evolution. A number of source term (ST) packages have been proposed for WW3, including ST3 (Janssen 1991, 2004; Bidlot et al. 2007; Bidlot 2012), ST4 (Ardhuin et al. 2010; Leckler et al. 2013), and ST6 (Donelan et al. 2006; Babanin et al. 2007; Babanin 2011; Rogers et al. 2012; Zieger et al. 2015). Liu et al. (2017) have compared these source term packages in the context of TC wave prediction and determined that they produce similar results. Hence, we have opted for ST4, as it is the default package in WW3 and has an extensive user history.

A potential limitation in this (or any other) source term package is the representation of the nonlinear term Snl. All of these source term packages use the Discrete Interaction Approximation (DIA) (Hasselmann et al. 1985) to represent Snl. This simplification is used because a full solution to the nonlinear term is computationally too expensive for two-dimensional spatial domains. The limitations of the DIA are well documented (Cavaleri et al. 2007; Resio and Perrie 2008; Perrie et al. 2013; Tolman 2013; Rogers and van Vledder 2013). As noted by Liu et al. (2017) a simple substitution of the full solution for Snl, although computationally expensive, may be possible for a limited range of computations. However, as each of the ST packages has been calibrated using the DIA, such a substitution would generally not produce acceptable results. A full recalibration of the WW3 model would be required. Therefore, the only practical option is to use the DIA, but to note the limitations that this brings.

Our simulations used the moving grid feature in WW3 and a computational grid of spatial extent 2000 km × 1900 km with a spatial resolution of Δx = 2 km. The directional wave spectrum E(f, θ) was defined with a directional resolution of Δθ = 5° (i.e., 72 direction band) and 50 frequency bands, defined by fn = 1.1fn−1 with the first band f1 = 0.04 Hz and the last band f50 = 4.27 Hz.

The wind field used to drive the translating WW3 model grid consisted of the Holland vortex defined by (2)(4) with first-order asymmetry provided by the vector addition of the velocity of forward movement Vfm to the wind field vectors and an assumed constant inflow angle of 20° (Tamizi et al. 2020). The wind field model is then fully specified for given values of Δp, Vfm, and two spatial-scale parameters Rm (radius to maximum winds) and R34 (the radius to gales), where the wind speed is equal to 34 kt (17.5 m s−1). For the results presented here, we consider a moderately intense TC with Δp = 50 hPa, Rm = 30 km, R34 = 300 km and two cases of velocity of forward movement, Vfm = 2.5 and 5.0 m s−1. These values are typical for mature TCs (Tamizi et al. 2020).

5. Spatial distribution of wave spectra

Figure 2 shows contour plots of the spatial distribution of Hs for the two cases described above, as predicted by the WW3 model. The peak wave directions (arrows) are also shown. Note that for all plots, the TC is assumed to be propagating to the north (up the page) and is located at coordinates (0, 0). The wave field shows the characteristic crescent shaped distribution with the largest waves to the right of the TC center (Bretschneider 1972; Patterson 1974; Ross 1976; U.S. Army Corp of Engineers 1977; Whalen and Ochi 1978; Black 1979; Young and Burchell 1986; Young 1988a,b, 2006; Young and Vinoth 2013, Liu et al. 2017, Tamizi and Young 2020). As shown in Fig. 1, the wind direction (spiraling in toward the TC center with an inflow angle of 20°) and the peak wave direction are quite different. As is clear in Fig. 2 and also shown from the buoy data of Fig. 1, the peak waves radiate out from the intense wind regions near the TC center. The largest waves to the right of the TC center increase with increasing Vfm, consistent with the concept of an extended fetch to the right of the TC center (Young 1988a; Young and Burchell 1986; Young and Vinoth 2013). However, ahead of the TC, values of Hs decay more rapidly for the faster moving storm. This is consistent with the wave field being dominated by remotely generated waves. For the slower storm, these waves can more easily outrun the storm and hence appear at large distances ahead of the TC center.

Fig. 2.
Fig. 2.

Contour plots of significant wave height Hs within tropical cyclones. Also shown are vectors of the peak wave direction. The spatial scale is normalized by the radius to maximum winds Rm. Tropical cyclone velocity of forward movement, (a) Vfm = 2.5 m s−1 and (b) Vfm = 5.0 m s−1.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-21-0005.1

Figure 3 shows the nondimensional energy, ε as a function of nondimensional frequency ν. Figure 3a shows the buoy data of Tamizi and Young (2020) and Fig. 3b shows the data for all grid points for the WW3 model case with Vfm = 2.5 m s−1. Note that points in the eye of the TC, where U10 ≈ 0 m s−1 have been excluded from the plot. Also shown on the figures are the fetch-limited result of Donelan et al. (1985), ε = 6.36 × 10−6ν−3.3 (5) and the commonly applied demarcation between swell and wind-sea of ν = 0.13. Noting that the model data are drawn from every grid point in the spatial domain (excluding the TC eye) and hence every quadrant, it is in reasonably good agreement with the buoy data. Both the buoy and model are consistent with the fetch-limited result of Donelan et al. (1985) for ν > 0.13. Forν < 0.13 both the buoy data and model fall below the Donelan et al. (1985) result, with the model apparently containing less energy at smaller values of nondimensional frequency ν (see section 7). There are, however, limited buoy data for very small values of ν and the model values still lie within the data scatter of the buoys. For both the model and buoy data, a significant proportion of the data are for values of ν less than the swell wind-sea limit. This is consistent with the situation shown in Fig. 1, where much of the wave field is dominated by remotely generated waves which propagate at phase speeds greater than the local wind speed.

Fig. 3.
Fig. 3.

Nondimensional energy ε as a function of nondimensional frequency ν for waves within tropical cyclones. (a) In situ buoy data of Tamizi and Young (2020). (b) WW3 model for the case of a tropical cyclone with Δp = 50 hPa and Vfm = 2.5 m s−1.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-21-0005.1

Figures 4 and 5 show the one-dimensional spectrum E(f) and the directional spreading function D(f, θ) for each quadrant of the TC for the cases of Vfm = 2.5 and 5.0 m s−1, respectively. These results can be compared to the buoy data in Fig. 1. As for the buoy observations, the model results show unimodal one-dimensional spectra throughout the spatial domain. There is some suggestion of bimodality in the left-rear quadrant for the case of Vfm = 5.0 m s−1 (Fig. 5). This same feature is seen for the buoy data in Fig. 1. The vectors showing the peak wave directions are similar for the two model cases, with both in good agreement with the buoy data.

Fig. 4.
Fig. 4.

Directional spectra within a tropical cyclone from the WW3 model. Data presented for locations shown by the dots in each octant of the storm which is propagating to the north (up the page). Solid arrows (and vertical lines) show mean wind direction and dashed arrows (and vertical lines) the peak wave direction. At each point the directional spreading function D(f, θ) and the one-dimensional spectrum E(f) are shown. Both D(f, θ) and E(f) have been normalized, such that they have maximum values of one. Case shown has Δp = 50 hPa and Vfm = 2.5 m s−1. All angles are measured anticlockwise from the x axis.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-21-0005.1

Fig. 5.
Fig. 5.

Directional spectra within a tropical cyclone from the WW3 model. Data presented for locations shown by the dots in each octant of the storm which is propagating to the north (up the page). Solid arrows (and vertical lines) show mean wind direction and dashed arrows (and vertical lines) the peak wave direction. At each point the directional spreading function D(f, θ) and the one-dimensional spectrum E(f) are shown. Both D(f, θ) and E(f) have been normalized, such that they have maximum values of one. Case shown has Δp = 50 hPa and Vfm = 5.0 m s−1. All angles are measured anticlockwise from the x axis.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-21-0005.1

As for the buoy data, the model spectra show a continuum of energy between the locally generated wind-sea and the remotely generated waves at the peak of the spectrum. This holds across all quadrants of the TC wave field and the model and buoy results are in reasonably good agreement. In addition to the model reproducing this continuum of energy, the rate at which the spectrum rotates from the wind direction to the direction of the remotely generated peak is also in good agreement with the buoy data. In the right-forward quadrant, the rotation occurs (i.e., spectrum fully aligned with the wind) by 2fp (buoy and model), where fp is the frequency of the spectra peak. In the left-forward quadrant, the rotation occurs by 2.5fp (buoy and model). In the left-rear quadrant, where the most confused wave conditions occur, there is some divergence between model (1.5fp) and buoy (2fp). In the right-rear quadrant, buoy and model are again in good agreement (1.5fp).

Interestingly, the model spectra appear narrower than the buoy spectra. The reason for this is not clear. It is perhaps that the analysis technique for the buoy data (Fourier expansion; Longuet-Higgins et al. 1963; Young 1994) yields excessively broad spectra or that the nonlinear coupling between wind-sea and remotely generated waves in the DIA representation of the source term Snl in the model yields this result. It is unusual, as the DIA typically results in excessively broad spectra (Komen et al. 1984).

The results in Figs. 25 add confidence that the WW3 model is reproducing the broad trends reported from measured buoy data for the TC wave field (noting the differences to the SRA data reported earlier). In addition, the model also appears to be able to capture the measured properties of the spectrum throughout the spatial wave field of the TC. This includes both the shape of the one-dimension spectrum, together with the directionally skewed spectra. As a result, the physics of the model, as represent by the propagation of energy and the energy balance of the source terms S (6) within the model, produce results consistent with recorded data.

6. Energy balance in a translating tropical cyclone

Figures 6 and 7 show the source terms in the radiative transfer Eq. (6), S(f, θ) = Sin(f, θ) + Snl(f, θ) + Sds(f, θ) for the two cases of Vfm = 2.5 and 5.0 m s−1, respectively. Because the magnitudes of the spectra and wind speeds vary for different regions of the tropical cyclones, the magnitudes of the source terms also vary. To display in a single figure, each of the source terms in Figs. 6 and 7 have been normalized to have a maximum value of one.

Fig. 6.
Fig. 6.

Source terms for the directional spectra shown in Fig. 4. At each location, the atmospheric input Sin, nonlinear interaction Snl, and dissipation Sds are shown. Each source term has been normalized to have an absolute maximum value of one. Normalized energy levels are shaded with red for positive and blue for negative. Case shown has Δp = 50 hPa and Vfm = 2.5 m s−1. Solid vertical lines show mean wind direction and dashed vertical lines the peak wave direction. All angles are measured anticlockwise from the x axis.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-21-0005.1

Fig. 7.
Fig. 7.

Source terms for the directional spectra shown in Fig. 5. At each location, the atmospheric input Sin, nonlinear interaction Snl, and dissipation, Sds are shown. Each source term has been normalized to have an absolute maximum value of one. Normalized energy levels are shaded with red for positive and blue for negative. Case shown has Δp = 50 hPa and Vfm = 5.0 m s−1. Solid vertical lines show mean wind direction and dashed vertical lines the peak wave direction. All angles are measured anticlockwise from the x axis.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-21-0005.1

Directly to the right of the TC center, the wind and waves are more aligned than other regions of the TC and the source terms are similar to our understanding of the energy balance in fetch-limited growth (Komen et al. 1984). The atmospheric input Sin is positive throughout the spectrum, including at the spectral peak. The dissipation Sds is approximately the mirror image of Sin, being negative across the spectrum. The nonlinear term Snl shows the characteristic plus–minus signature of this term (Komen et al. 1984), transferring energy from frequencies above the spectral peak to lower frequencies, thus supporting growth of energy near the peak and its migration to lower frequencies.

As the mean wind direction and peak wave direction become separated in direction (e.g., left of TC center), the energy balance becomes more complex. The atmospheric input, Sin is positive for the wind-sea (e.g., f > 2fp). As the spectral peak is at a significant angle to the wind direction and often has a phase speed, C > U10, Sin becomes negative at the spectral peak (opposing wind). The dissipation, Sds is negative for all frequencies. As a result, both Sin and Sds would result in decay of the remotely generated waves at the spectral peak. The nonlinear term, Snl shows a plus–minus structure along the “ridge” of energy which joints the wind-sea spectrum at higher frequencies to the remotely generated spectral peak at lower frequencies. This results in a cascade of energy from the wind-sea to the remotely generated spectral peak components. Thus, although the spectral peak receives no direct positive input from the wind, it indirectly remains coupled to the wind by the energy cascade from the wind-sea. This continual flux of energy from the wind-sea to the remotely generated spectral peak results in the continuum of energy between the two systems seen in the spectra in Figs. 1, 4, and 5. Thus, the spectral peak acts in a “parasitic” manner, continually taking energy from the wind-sea and hence sustaining both the continuum of energy between the two systems and the sustained growth of the remotely generated spectra peak.

Even when the waves are separated by more than 90°, this continual feed of energy from the wind-sea ensures the two systems remain connected. In none of the cases shown in Figs. 6 or 7 do the wind-sea and remotely generated peak become decoupled systems and appear as separate spectral peaks.

As the spectra in Figs. 4 and 5 and the source terms in Figs. 6 and 7 are normalized, they do not provide information on the relative magnitudes of the source terms. To address this, Fig. 8 shows the spectra and source terms for the case of Vfm = 5 m s−1 and the octant to the north-northwest of the storm center (Fig. 5 for spectra and Fig. 7 for source terms). In Fig. 8, the terms are not normalized. Figure 8a shows the 1D spectrum E(f) and Fig. 8b the directional spectrum E(f, θ) (both have units of m2 s). The unimodal structure of the 1D spectrum, as described above is clear, with the high-frequency face being approximately proportional to f−4. The directional spectrum E(f, θ) clearly shows the low-frequency peak at a frequency of approximately 0.07 Hz and a direction of 118° (note only the directional spreading function D(f, θ) was shown in Fig. 5). This clearly seems to represent remotely generated waves propagating out from the intense wind regions of the tropical cyclone vortex. These waves are propagating at an angle of more than 90° to the local wind direction (216°). The directional spectrum becomes directionally skewed, with high-frequency components aligning with the local wind direction. However, both the 1D and directional spectra remain unimodal.

Fig. 8.
Fig. 8.

WW3 model directional spectra and source terms from a location in the north-northwest octant of a tropical cyclone with Δp = 50 hPa and Vfm = 5.0 m s−1. (a) One dimensional spectrum E(f) (m2 s). (b) Directional spectrum E(f, θ) (m2 s), contours drawn at [0.1, 0.5, 1, 5, 10, 50, 100, 200]. (c) Wind input source term Sin(f,θ) × 103 (m2). (d) Nonlinear source term Snl(f, θ) × 103 (m2). (e) Dissipation source term Sds(f, θ) × 103 (m2). (f) Total source term Stot(f, θ) × 103 (m2). For each panel, the vertical solid line shows the wind direction and the dashed line the peak wave direction. All angles are measured anticlockwise from the x axis.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-21-0005.1

The source terms Sin, Snl, and Sds are shown in Figs. 8c, 8, and 8e, respectively [units m2], and the total source term Stot(f, θ) = Sin(f, θ) + Snl(f, θ) + Sds(f, θ), in Fig. 8f. The results confirm the energy balance seen in the normalized results (i.e., the source terms are of similar magnitude). The wind input Sin (Fig. 8c) is positive for the wind-sea and negative for the remotely generated waves at the spectral peak. That is, the peak is decaying as it is propagating at greater than 90° to the wind direction. The wind input in the region between the wind-sea and remotely generated waves (continuum of energy referred to above as the ridge) is approximately zero, demonstrating that the local wind does not generate these waves. The dissipation Sds (Fig. 8e) is, not surprisingly, negative for all components, being largest for the high-frequency (steeper) components. Again, in the region between these two systems the dissipation is small.

The nonlinear term Snl (Fig. 8d) is conservative, redistributing energy within the spectrum. It is clear that this term transfers energy from the wind-sea to the region of the ridge, which joins the wind-sea and the remotely generated waves (Fig. 8d). The nonlinear term is also negative at the spectral peak and appears to be transferring energy to frequencies lower than the spectral peak and at even increasing angles to the local wind direction. As seen in the spectrum in Fig. 8b, this transfer from the peak results in a continued downshift in frequency and rotation in direction of the remotely generated peak. The total source term Stot (Fig. 8f) in the ridge region between the wind-sea and the remotely generated peak is largely the same as Snl (as the other source terms are approximately zero), confirming that the energy balance in this region is dominated by the nonlinear term. Hence, as noted above, it is the nonlinear term which is largely responsible for the ridge of energy between the wind-sea and remotely generated peak. Hence, it is Snl that sustains the unimodal but skewed directional spectrum seen in both model and buoy data.

7. Discussion and conclusions

The results above show that the WW3 model is capable of reproducing the general features of the spectral shape throughout the spatial wave field of a TC. As a result, it is reasonable to assume that, to some level of accuracy, the source terms of the model capture the main physical processes active in this complex wave field. Importantly, the results confirm the speculation of Young (2006) and Tamizi and Young (2020) that the directionally skewed spectra which result in such cases, are dominated by nonlinear energy transfer from the high-frequency wind-sea to the low-frequency remotely generated spectral peak (Fig. 8f). This coupling between the wind-sea and spectral peak is sustained even when the two systems are directionally separated by more than 90°. Figure 9 shows a diagrammatic representation of the cascade of energy from the wind-sea, along the continuum of energy (ridge) joining the wave systems to the spectral peak. This form of coupling across direction and frequency has been demonstrated for simpler systems by Young and van Vledder (1993). The present, results are, however, the first demonstration of such coupling in a tropical cyclone situation.

Fig. 9.
Fig. 9.

An example of the role played by the nonlinear term Snl in tropical cyclone wave evolution. The nonlinear term transfers energy from the wind-sea to remotely generated waves at the spectral peak. This occurs through an energy cascade along the ridge connecting these wave systems. As a result, the waves at the spectral peak act in a parasitic manner to take energy for the local wind-sea to enhance growth at the spectral peak. The process ensures that the two wave systems remain connected through the directionally skewed spectrum. The case shown is for the spectrum in the west-southwest octant of Fig. 4.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-21-0005.1

It is perhaps surprising, that even with the limitations of the DIA form of Snl, the source term is still able to adequately model the interaction between the wind-sea and remotely generated waves that are essential to describing wave spectra within tropical cyclones. The fact that nonlinear interactions play such a critical role in shaping the TC spectrum, explains a number of observed features of such wave systems. Such features include: the skewed directional shape of the directional spectrum, and the fact that one-dimensional TC spectra are similar in shape to fetch-limited spectra, as well as the relationship between nondimensional energy and nondimensional frequency and the observation that JONSWAP-type power laws can be used to predict significant wave height within TCs. All are results which rely on the self-similar properties of nonlinear interactions in shaping the wave spectrum (Young and van Vledder 1993).

As noted above, the DIA form for Snl is a limitation of the present modeling approach. That limitation, together with differences in driving wind fields, probably account for the observed discrepancies between the model and observed spectra. In particular, Fig. 3 shows that model spectra seem to have less energy at low frequencies than observed spectra. This suggests that the model underestimates the energy in the parasitic remotely generated spectral peak. It is possible that the energy input in the intense wind regions of the TC is underestimated. After all, Sin has never been measured under TC conditions and hence this term is extrapolated to these wind speeds. This may result in an underestimation of the remotely generated energy. Noting the important role of Snl in defining the spectral shape, it is more likely that the magnitude of the energy cascade to the spectral peak is underestimated by the model. As the DIA considers only a very small subset of interacting wave components (Komen et al. 1984), it is plausible that the magnitude of the energy transfer is larger than shown in these calculations.

Despite these limitations, the present results have clearly shown that nonlinear interactions are critical in wind-wave evolution in tropical cyclones. The results are also a compelling validation of the central role that Hasselmann’s quadruplet interactions play in the dynamics of wind waves in a wide range of forcing scenarios (Hasselmann 1962).

Acknowledgments

The authors would like to acknowledge the support provide to AT by the University of Melbourne through a PhD scholarship. We also thank the U.S. National Oceanic and Atmospheric Administration (NOAA) for providing access to HPC resources.

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  • Zieger, S., A. V. Babanin, W. E. Rogers, and I. R. Young, 2015: Observation-based source terms in the third-generation wave model WAVEWATCH. Ocean Modell., 96, 225, https://doi.org/10.1016/j.ocemod.2015.07.014.

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