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  • View in gallery

    Roll cases from Hurricanes Jeanne (2004) and Isabel (2003): (a) Crosswind (y axis) component from the observations, (b) NWPS E(n, sj), and (c) time-averaged NWPS. The dark solid thick line, dark solid thin line, and light solid thin line represent the time-averaged NWPS of the crosswind component, along-wind component, and vertical velocity, respectively; the dashed thick line represents the averaged NWPS of crosswind component over the observations with mean wind speed 20–22 m s−1 of all hurricanes without cleanly defined rolls. (d) FT power density spectra: the dark solid thick line, dark solid thin line, and light solid thin line represent the spectra of the crosswind component, along-wind component, and vertical velocity, respectively.

  • View in gallery

    (a) Time series of 15-min-averaged wind speeds of Hurricanes Jeanne (2004) and Isabel (2003) recorded by the towers. The dashed vertical line indicates the time when the rolls shown in Fig. 1 were observed. (b) Doppler radar reflectivity images closest to the time when the rolls were observed. The long white arrow indicates the location of the tower. White lines and stars indicate the hurricane best track.

  • View in gallery

    Hurricane Ivan (2004): (a) 15-min-averaged NWPS of surface wind stress and TKE derived from WT analyses, where the red thick line indicates 15-min-averaged wind speed; (b) 15-min-averaged NWPS for four scale bands, <48, 48–236, 236–1014, and >1014 m; (c) 15-min-averaged NWPS for scales smaller than 2 and 3 km; (d) peak NWPS in the spectra; and (e) scale of the peak NWPS.

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    Satellite IR images before, at, and after the time when the peak wind speed was observed by the tower. The black star indicates the tower location. The thick black line and red circles indicate the hurricane best track.

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    (a) NWPS for scales smaller than 236 m from different hurricanes. Each point represents a 15-min average. (b) NWPS for scales larger than 236 m. (c) Peak NWPS. (d) Scale of the peak NWPS.

  • View in gallery

    Schematic of energy cascade process: (a) local interaction or staircaselike cascade and (b) long-range interaction or elevator-like cascade (adopted from Hunt and Carlotti 2001).

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Wavelet Analyses of Turbulence in the Hurricane Surface Layer during Landfalls

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  • 1 Department of Earth and Environment, Florida International University, Miami, Florida
  • | 2 National Oceanographic and Atmospheric Administration/Atlantic Oceanographic and Meteorological Laboratory/Hurricane Research Division, Miami, Florida
  • | 3 Department of Civil and Coastal Engineering, University of Florida, Gainesville, Florida
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Abstract

Using wavelet transform (WT), this study analyzes the surface wind data collected by the portable wind towers during the landfalls of six hurricanes and one tropical storm in the 2002–04 seasons. The WT, which decomposes a time series onto the scale-time domain, provides a means to investigate the role of turbulent eddies in the vertical transport in the unsteady, inhomogeneous hurricane surface layer. The normalized WT power spectra (NWPS) show that the hurricane boundary layer roll vortices tend to suppress the eddy circulations immediately adjacent to rolls, but they do not appear to have a substantial effect on eddies smaller than 100 m. For low-wind conditions with surface wind speeds less than 10 m s−1, the contributions of small eddies (<236 m) to the surface wind stress and turbulent kinetic energy (TKE) decrease with the increase of wind speed. The opposite variation trend is found for eddies greater than 236 m. However, for wind speeds greater than 10 m s−1, contributions of both small and large eddies tend to level off as wind speeds keep increasing. It is also found that the scale of the peak NWPS of the surface wind stress is nearly constant with a mean value of approximately 86 m, whereas the scale of the peak NWPS of TKE generally increases with the increase of wind speed, suggesting the different roles of eddies in generating fluxes and TKE. This study illustrates the unique characteristics of the surface layer turbulent structures during hurricane landfalls. It is hoped that the findings of this study could enlighten the development and improvement of turbulent mixing schemes so that the vertical transport processes in the hurricane surface layer can be appropriately parameterized in forecasting models.

Corresponding author address: Ping Zhu, Department of Earth and Environment, Florida International University, MARC 360, 11200 SW 8th St., Miami, FL 33199. Email: zhup@fiu.edu

Abstract

Using wavelet transform (WT), this study analyzes the surface wind data collected by the portable wind towers during the landfalls of six hurricanes and one tropical storm in the 2002–04 seasons. The WT, which decomposes a time series onto the scale-time domain, provides a means to investigate the role of turbulent eddies in the vertical transport in the unsteady, inhomogeneous hurricane surface layer. The normalized WT power spectra (NWPS) show that the hurricane boundary layer roll vortices tend to suppress the eddy circulations immediately adjacent to rolls, but they do not appear to have a substantial effect on eddies smaller than 100 m. For low-wind conditions with surface wind speeds less than 10 m s−1, the contributions of small eddies (<236 m) to the surface wind stress and turbulent kinetic energy (TKE) decrease with the increase of wind speed. The opposite variation trend is found for eddies greater than 236 m. However, for wind speeds greater than 10 m s−1, contributions of both small and large eddies tend to level off as wind speeds keep increasing. It is also found that the scale of the peak NWPS of the surface wind stress is nearly constant with a mean value of approximately 86 m, whereas the scale of the peak NWPS of TKE generally increases with the increase of wind speed, suggesting the different roles of eddies in generating fluxes and TKE. This study illustrates the unique characteristics of the surface layer turbulent structures during hurricane landfalls. It is hoped that the findings of this study could enlighten the development and improvement of turbulent mixing schemes so that the vertical transport processes in the hurricane surface layer can be appropriately parameterized in forecasting models.

Corresponding author address: Ping Zhu, Department of Earth and Environment, Florida International University, MARC 360, 11200 SW 8th St., Miami, FL 33199. Email: zhup@fiu.edu

1. Introduction

Turbulent mixing is the primary mechanism for the vertical transport of energy, moisture, and momentum in the boundary layer; yet it is a process that must be parameterized in numerical models. Over the years, many turbulence schemes have been developed and are widely used in numerical weather forecast and climate simulations. These same schemes, often without any modification, are also used in the simulation and prediction of hurricanes to account for the turbulent transport in the hurricane boundary layer (HBL) under the assumption that the turbulent mixing processes in the HBL shares the characteristics with that in nonhurricane conditions in which the parameterizations were formulated and evaluated.

Recent observations showed that the HBL turbulent flow, despite energetic, possesses highly organized vortical features. For example, the boundary layer horizontal roll vortices, which are similar to those visualized by the cloud streets often seen during cold air outbreaks over the oceans (Mourad 1996a,b), have been detected outside the deep convective rainbands and eyewall of hurricanes by Doppler radar (e.g., Wurman and Winslow 1998; Morrison et al. 2005), synthetic aperture radar (SAR) (e.g., Katsaros et al. 2000), and in situ aircraft measurements (e.g., Zhang et al. 2008). All these observations suggest that roll vortices can effectively create mixing and generate large fluxes in the boundary layer due to their highly organized updrafts and downdrafts (Zhu 2008). Morrison et al. (2005) showed that the momentum fluxes induced by rolls could be 2–3 times larger than those that would have been estimated with the standard turbulent schemes applied to the observed mean wind profiles. If rolls would be the common features in the HBL, the Morrison et al. (2005) estimation suggests that a serious bias would occur if the standard turbulent schemes developed in the nonhurricane conditions were used to parameterize the HBL turbulent fluxes. By examining a large number of aircraft data, French et al. (2007) and Zhang et al. (2008) found that only a very small proportion of the data contains cleanly defined rolls, which is somewhat a surprise considering that the HBL flow tends to produce streamwise rolls according to Foster’s (2005) local nonlinear instability theory. Although the specific orientation of aircraft measurements, sampling location, and roll definition criteria are the possible reasons that may have eliminated some of the rolls, a possible physical explanation is that the roll instability mechanism may be demolished by the storm convective activity. As reviewed by Etling and Brown (1993), the growth rates of convective instability are larger than those of inflection-point instability mode that is generally believed to be the mechanism for roll formation. This may explain why rolls are rarely observed inside rainbands and eyewall where convection is strong. The less frequently observed rolls than expected confirm previous results that the roll instability mode favors certain dynamic and thermodynamic conditions.

From the parameterization perspective, a successful scheme should account for the vertical transport induced by all types of eddies, ranging from chaotic turbulence to highly organized roll vortices and other types of circulation features including convective cells. Thus, it is essential to understand the complex interplay between eddies of various scales, how they are affected by the generation of shear or mixed shear and buoyancy instability modes, and how this interplay determines the total transport in the HBL. Classically, this can be done through the Fourier transform (FT). For example, Zhang (2010) presents a FT analysis of aircraft observations in a near hurricane force wind condition. While a FT obtains a spectrum in terms of frequency (or spatial scale), it simply loses the information of time at which a particular frequency or scale occurs. Thus, ideally a FT can only be applied to steady and homogeneous conditions. Since the swirling hurricane vortex generates a continuously evolving HBL that changes its wind speed and direction, the HBL rarely reaches a steady state. The homogeneous condition is also hardly satisfied owing to the highly asymmetric hurricane vortex structures. For this reason, the wavelet transform (WT) is chosen in this study to analyze the wind data collected during the landfall of hurricanes. The WT, which can be considered as the generalized windowed FT (Morlet et al. 1982), provides a means to separate and sort different structures of a time series on different frequencies at different times. Thus, a WT is a tool well suited to analyze turbulence in the unsteady and inhomogeneous HBL.

Although it is important to understand turbulent eddy structures and vertical transport processes in the entire HBL, this study focuses only on the surface layer because the data used in this study was collected in this layer. Besides the data limitation, there are additional motivations to focus on the turbulence in the hurricane surface layer. Previous studies show that turbulent motion in the surface layer consists of an “active” part (often called bursts and sweeps in literature) that produce the shear stress (e.g., Shaw and Businger 1985; Mahrt and Gibson 1992; Högström and Bergström 1996) and an “inactive” part (also called inactive eddies) that carries little momentum fluxes in the surface layer (Townsend 1961; Bradshaw 1967; Mahrt and Gibson 1992). Based on the experimental, theoretical, and numerical results, Hunt et al. (2001) illustrated how the structure of turbulence in the surface layer, especially in shear flows, continues to change as the Reynolds number increases well above 104 in ways that are not well understood and cannot be reproduced by numerical simulations. The analyses of the data collected in hurricane conditions may shed light on our understanding of the eddy interactions in the surface layer in terms of active and inactive motions at the extremely high Reynolds number.

In this study, using the WT method we analyzed wind data collected in the surface layer during the landfall of hurricanes. The objectives of this study are to 1) provide a quantitative analysis of the surface wind stress (i.e., momentum fluxes) and turbulent kinetic energy (TKE) induced by eddies with different scales including rolls, 2) determine the change in structure of eddies and the associated fluxes and TKE during the course of hurricane landfalls, and 3) explore how organized vortical structures such as rolls and convective cells modulate the smaller-scale eddies and affect the vertical transport in the hurricane surface layer.

2. Observations and wavelet transform

The data used in this study were collected by the portable wind towers (PWTs) deployed along the paths of landfalling hurricanes as a part of the Florida Coastal Monitoring Program (FCMP) (Yu et al. 2008). All towers were deployed on open flat terrain to avoid the influence from high level obstacles or buildings. A 3D Gill propeller anemometer (model 200–27005) and a R.M. Young vane wind monitor (model 05103V) were installed at 10 m to collect wind data at a 10-Hz temporal resolution. The extensive comparisons between these two systems show that the measured horizontal winds were consistent with each other under high hurricane wind conditions. A second Gill propeller anemometer was installed at 5 m. The sensors contain filters to remove aliasing and were calibrated to follow the cosine law within 3% over the ±30° range. The measuring system mechanically filters the amplitudes of short wavelength fluctuations due to the response characteristics of the wind anemometer (Schroeder and Smith 2003), which may cause inaccuracy of measurements at the high frequency end. An error estimation has been provided by Yu and Chowdhury (2008). For this reason, signals with frequencies higher than 2.0 Hz will not be included in this paper. For frequencies lower than 2.0 Hz, the FCMP wind measurements produced reasonable spectra (Yu et al. 2008; Yu and Chowdhury 2008). Examples of FT spectra are also provided in Fig. 1. The WT analyses show that 5-m wind data basically produces results similar to those of 10-m data. Thus, in this paper we only present the WT analyses on the 10-m 3D wind data collected during the 2002–04 hurricane seasons. These include tower observations from Hurricanes Lili (2002), Isabel (2003), Charley (2004), Frances (2004), Jeanne (2004), and Ivan (2004) as well as Tropical Storm Isidore (2002).

Unlike the FT, which breaks up a signal xk (k = 0, 1, … , N − 1) into sine/cosine waves of various frequencies, the WT decomposes the signal into scaled and translated versions of a “mother wavelet” Ψ0, called “child wavelets” or simply wavelets represented by
i1520-0469-67-12-3793-e1
where sj and n denote scale and translation (in the time axis), respectively, δt is the sampling period of the signal, and (δt/sj)1/2 is the energy normalization factor, which keeps the energy of child wavelets the same as that of mother wavelet. A WT converts the signal onto the scale-translation domain by scaling and translating the mother wavelet to match the high and low frequencies in the series to provide an improved fitting to nonlinear, irregular data. Mathematically, this process is represented by a convolution of xk with scaled and translated wavelets:
i1520-0469-67-12-3793-e2
where the asterisk indicates the complex conjugate and W(n, sj) is called a wavelet coefficient. The choice of mother wavelet is somewhat arbitrary. Since this study deals with wavelike signals, the continuous Morlet wavelet is chosen. We have tested other discrete wavelets (e.g., Daubechies, biorthogonal, Coiflets, etc.). They do not appear to affect the results to be presented in the next section.
A WT analysis at every possible scale involves a fair amount of computation and generates large volumes of data because it converts a 1D series into a 2D dataset. It turns out that the analysis will be much more efficient and retains sufficient accuracy if scales are chosen discretely based on powers of 2. The WT scales can be mapped to the equivalent Fourier frequencies in terms of the central frequency that captures the main oscillation of a given wavelet. The wavelet power spectrum (WPS) is, then, defined as |W(n, sj)|2. It can be shown that the variance of the data series xk can be written in terms of WPS as
i1520-0469-67-12-3793-e3
where Cδ = 6 is a scale-independent constant for the Morlet wavelet; sj = s02jδj (j = 0, 1, … , J) represents the discrete scales; s0 is the smallest scale that can be resolved by the wavelet, which is set to 2δt in this study; and δj determines the scale resolution of wavelet analysis. The smaller δj is, the higher the scale resolution of wavelet analysis. In this study δj is set to 0.1, which, according to Torrence and Compo (1998), gives sufficiently dense sampling in scale. Also, J = δj−1 log2(Nδt/s0) determines the largest scale that can be identified by the WT. Combined with the definition of σ2, Eq. (3) may be rewritten as
i1520-0469-67-12-3793-e4
where the overbar denotes the average of a certain time period. In this study, the time series of tower observations have been divided into segments of different lengths: 5, 10, 15, 20, and 25 min. The WT analysis results do not show much sensitivity to the selected averaging length. Thus, all results shown in this paper are from the analyses using a 15-min averaging length. With Eq. (4), one may further define the normalized WPS (NWPS) E(n, sj) as
i1520-0469-67-12-3793-e5
Equation (5) indicates that NWPS E(n, sj) may be used to describe the contribution of a certain scale sj to the total energy, in percent, at the time translation n. Finally, the wavelet coefficient W(n, sj) can also be used to reconstruct the original data series. Comparison between the original and reconstructed data series provides a way to examine the accuracy of the WT. For detailed information of the WT, please refer to Torrence and Compo (1998).

Since wind speeds change continuously during the landfall of hurricanes, the same frequency will represent different scales of eddies as wind speed changes. Thus, all WT analyses in this study were made in terms of spatial scales by applying Taylor’s frozen eddy hypothesis.

3. Results

The existence of large vortical features in the HBL can be clearly seen from the FCMP observations. As an example, Fig. 1 shows two roll cases selected from Hurricanes Jeanne (2004) and Isabel (2003). For a better illustration, we have rotated the coordinate with the x axis aligned with the mean wind direction. Figure 1a shows the y-direction wind (or crosswind) component. The alternate positive and negative transverse winds are the signature of roll circulations with the roll alignment roughly along the mean wind direction. This sinelike low frequency variation associated with the possible roll vortices is a dominant feature (roughly 1 km in scale) in the NWPS computed by Eq. (5) (Fig. 1b.1 and b.2). The reconstructed time series from the WT coefficients (light dotted lines in Fig. 1a.1 and a.2) match perfectly with the original signal (solid lines in Fig. 1a.1 and a.2), suggesting the great accuracy of the WT. The crosswind scale of rolls (approximately 1 km) is estimated based on the mean wind speed at the moment when the rolls were observed, about 21 m s−1 in both cases according to Fig. 2a. The crosswind scale of rolls identified here using the WT analysis is consistent with the dominant roll scale of the crosswind component from SAR image analyses (Mourad 1996a,b; Mourad et al. 2000) in nonhurricane conditions, as well as those detected in hurricane conditions (Morrison et al. 2005; Zhang et al. 2008).

The locations of the towers relative to the storm center at the moment of roll occurrence are shown in Fig. 2b, which are the Doppler radar images closest to the time when the rolls were observed. In both cases, the low elevation angle (0.5°) radar reflectivities at the towers are less than 30 dBZ. According to Zipser and Lutz (1994), such a value of low angle radar reflectivity suggests that the identified rolls formed in weak convective conditions. This is consistent with the previous observations that HBL rolls tend to form outside the eyewall and rainbands.

Since the oscillations associated with the rolls are quite steady within the time window shown in Fig. 1a, we computed the time-averaged NWPS, which allows us to examine the contribution of rolls to the power spectra of the crosswind component υ. To explore how rolls interact and affect other eddies with different scales, we also calculated the averaged NWPS of the crosswind component over all observations with mean wind speed 20–22 m s−1 without cleanly identified rolls. The results are shown in Fig. 1c.1 and c.2. It clearly indicates that the rolls with 1-km crosswind scale contribute significantly to the total NWPS. A rough estimation suggests that more than one-third of the variance of the transverse winds is caused by the roll vortices. The comparison between spectra with and without roll effects (thick solid and dashed lines, respectively) indicates that rolls tend to suppress the eddies with scales immediately adjacent to rolls. However, rolls have little effect on the eddies with scales smaller than 100 m. This result is similar to Mourad et al.’s (2000) SAR image analyses and is consistent with LeMone’s (1976) turbulent energy budget analyses, which shows that there is little energy exchange between the rolls and small-scale high-frequency turbulence, although rolls do play a role in redistributing turbulence-producing elements in the boundary layer.

The second peak (with scales 80–200 m) on the NWPS of the crosswind component (Fig. 1c.1 and c.2) corresponds to the “gust microfront” indicated on Mourad et al.’s (2000) power spectra, although the gust microfront is not as cleanly defined in the Isabel case as in the Jeanne case. The gust microfronts, also known as burst/sweep features in literature, are the vertical momentum flux carrier (Shaw and Businger 1985; Mahrt and Gibson 1992; Högström and Bergström 1996). The crosswind scale of gust microfront in our cases is a little larger than that identified from the Mourad et al. SAR image analyses (approximately 50–70 m). This scale difference is probably due to the strong hurricane force winds in our cases.

To obtain the basic idea of the 3D structure of rolls and gust microfronts, the time-averaged NWPSs of the along-wind component u and vertical velocity w are also plotted in Fig. 1c.1 and c.2. The NWPSs have different characteristics for all three components. The dominant scale of NWPS of the along-wind component (dark thin lines in Fig. 1c.1 and c.2) is longer than that of the crosswind component, approximately 4–5 km for the Jeanne case and slightly greater than 1 km for the Isabel case. Since the motion with the dominant scale is interpreted as a roll circulation, this result suggests that, compared with its width, the length of roll is less constrained and may vary from case to case. For the gust microfront, there is only a slight difference in the dominant scale between the along-wind and crosswind components. However, the gust microfront mode is the dominant scale on the time-averaged NWPS of vertical velocity (light thin lines in Fig. 1c.1 and c.2). It is much stronger than the roll mode. This is understandable because gust microfronts or bursts/sweeps, which may be scaled to the depth of the surface layer, are the active eddies in the surface layer. The large NWPS of vertical velocity associated with these features, to some extent, explains why they are the efficient carriers of momentum flux (to be shown later). On the other hand, as noted by Mahrt and Gibson (1992), rolls, which may be scaled to the depth of boundary layer, are likely one of the sources of inactive eddies in the surface layer. Indeed, numerical simulations by Zhu (2008) showed that the strongest vertical motion is associated with rolls is in the middle of the boundary layer, well above the surface layer. This may explain why the roll mode is weaker than that of gust microfront in the wavelet power spectra of vertical velocity.

As a comparison, we also calculated the eight-point-averaged FT power density spectra of all three components (Fig. 1d.1 and d.2). The peak NWPS of the crosswind component associated with the rolls derived from the WT analyses is consistent with that of the FT analyses. In the inertial subrange, the FT spectra nearly follow Kolmogorov’s 5/3 law, suggesting good quality of the wind data. A t test was conducted to test the hypothesis if the regressed slope (in the scale range from 12 to 330 m) is the same as “5/3.” The results show that the hypothesis is accepted at the 97.5% confidence level. At the low -frequency (large scale) end, again, we see that the FT power density spectra of vertical velocity is noticeably lower than those of horizontal wind components, consistent with the results from the WT analyses. This feature is similar to what has been found in nonhurricane or low wind conditions (e.g., Kaimal et al. 1972; Miyake et al. 1970), as well as in near-hurricane-force wind conditions (Zhang 2010).

The above analyses only focus on a short period of time so as to illustrate the HBL rolls identified from the tower observations. The available long-period tower observations collected during the hurricane landfalls allow us to depict the characteristics of the rolls and other types of vortical features and their change as the hurricanes made landfall. As an example, Fig. 3 shows the WT analyses of the entire dataset collected during the landfall of Hurricane Ivan (2004). The NWPS of the surface wind stress τ and TKE is shown in Fig. 3a. It shows that eddies tend to shift toward larger scales as wind speed increases and the contributions of different scale eddies to the surface wind stress and TKE change substantially during the course of landfall. To clearly depict this change, we further plotted the time variation of NWPS of particular scales or scale bands in Figs. 3b–e. The contribution of small eddies (e.g., <48 m in Fig. 3b) to the surface wind stress and TKE decreases as wind speed increases, whereas the contribution of larger eddies (e.g., >236 m in Fig. 3b) increases with wind speed, as the storm center approaches the tower. However, despite the apparent shifting toward larger scales, more than 85% and 90% of wind stress is generated by eddies with scales smaller than 2 and 3 km, respectively (Fig. 3c). Larger scale eddies appear to be more important to TKE than wind stress. Near the storm center, at approximately 0600 UTC 16 September (see Fig. 4), eddies larger than 2 and 3 km can make up about 30% and 20% of TKE, respectively.

Figure 3d shows the time variation of the peak NWPS (i.e., the maximum NWPS in the spectra). There is a noticeable decrease of the peak NWPS as wind speed increases. This is expected since the total wind stress and TKE are spread over a wider range of scales at higher wind speeds. What is a surprise is the scale at which the peak NWPS occurs (Fig. 3e). There is no apparent change in the spectrum peak scale of surface wind stress, which fluctuates around its mean value, ∼86 m. As shown in Fig. 3b, eddies in the narrow range 48–238 m make up about 40% of wind stress, which is fairly steady during the entire course of landfall, suggesting that the eddies in vicinity of the peak scale contribute a significant proportion of wind stress. In contrast, the peak scale of TKE behaves remarkably differently than that of wind stress. It shifts toward large scale substantially as the storm moves closer to the tower, consistent with Fig. 3c that large-scale eddies contribute more to TKE than wind stress. This result suggests that a critical scale may exist in the hurricane surface layer beyond which the correlation between horizontal and vertical velocity perturbations tends to decrease. Thus, for eddies well beyond the critical scale, although they can contribute significantly to the turbulent intensity or TKE owing to their large variance, they may not be efficient in generating fluxes due to the weak correlation between horizontal and vertical perturbations associated with eddies. We shall discuss this characteristic phenomenon further within the scope of the concept of active and inactive motions in the surface layer in a later section.

To better understand the eddy spectrum characteristics revealed by the WT analyses, we carefully examined the available satellite images during the entire tower observation period. Figure 4 shows four satellite IR images before, at, and after the time when the peak wind speed was observed by the tower. As illustrated by the figure, owing to the asymmetric hurricane structure and spinning of the storm vortex, the tower basically recorded the surface wind field in the relatively weaker quadrant of the storm before the peak wind speed was observed. After the peak, the more violent part of the storm has approached. Thus, the irregular WT spectrum features shown in Figs. 3a–e, to some extent, reflect the way the storm approaches and the asymmetrically distributed hurricane rainbands and eyewall. The large peak scale of the NWPS of TKE on 16 September shown in Fig. 3e is most likely associated with the strong deep convection of the storm.

As illustrated by Fig. 3, although eddies generally shift toward larger scales with the increase of wind speed as the storm center approaches the tower, eddies do not appear to symmetrically (in terms of the change of wind speed) shift back to smaller scales as the wind begins to decrease as the storm center moves away. A possible explanation is that large eddies have not only well-defined structures but also a certain lifetime. Large eddies are generated by instability, which has three modes in the boundary layer—convective instability, inflection-point instability, and parallel instability—although the growth rate of the parallel instability is much slower than the other two (Etling and Brown 1993). Once eddies form due to certain instability, they may continue to survive for some time, even if the ambient condition that determines the instability is changed, since it takes time to dissipate the energy of large eddies. This possible phase-delay effect in the unsteady boundary layer adds complexity to the turbulent eddy parameterization because the eddy characteristics are not uniquely determined by the mean state of the boundary layer flow as illustrated by Fig. 3. Currently, the eddy characteristics in unsteady conditions are not well understood, and further investigation is needed.

The apparent shifting of eddies toward larger scales with the increase of wind speed raises an interesting question: are the large eddies constrained by the HBL height? The answer to this question requires knowledge of the vertical structure of the HBL. Unfortunately, we do not have sounding data to examine the evolution of the entire HBL during the landfall process. Presumably, large eddies should be subjected to the depth of the HBL because the inversion at the top of the boundary layer tends to suppress the development of large eddies, which is supported by the numerical simulation of large eddy circulations in the HBL (Zhu 2008). However, we note that the spatial scales of eddies used in this study are deduced solely from Taylor’s frozen eddy hypothesis without considering other physical constraints. Whether this hypothesis is still valid for large-scale eddies, particularly for those larger than the HBL depth, is an unanswered question. All of these issues will be investigated in our future study when more data, particularly sounding data, are gathered.

To further evaluate the above results, we performed the WT analyses on the tower wind data of the hurricane cases that have sufficiently long records and computed the NWPS of the surface wind stress and TKE. Since it involves multiple hurricane cases, how to group the data is important for interpreting the analysis results. As we showed previously, the surface layer turbulence statistics seem to depend on the quadrants of a hurricane where convective activity varies. However, classification based on hurricane quadrants is difficult as we do not have an objective method to define the quadrants. We can only subjectively define quadrants based on satellite and Doppler radar images, which, in many cases, cannot provide a clean boundary. Since the change of wind speeds, to some extent, reflects the change of quadrants (although it may not be always true), in this study we bin the wind data and WT analyses as a function of wind speed (Fig. 5). However, we note that the relationship in terms of wind speed can be complicated by the asymmetry of a hurricane vortex as the case of Ivan shown previously.

As a first-order approximation, the WT analysis (Fig. 5a) shows that in all hurricane cases the contribution of smaller eddies (with scales less than 236 m) to the surface wind stress and TKE decreases with wind speed for surface wind speeds less than 10 m s−1. The opposite variation trend is seen for eddies with the scale greater than 236 m (Fig. 5b). However, the contributions from both small and large eddies start to level off as wind speed exceeds 10 m s−1. If we define a split ratio as the ratio of the contribution from the small eddies (<236 m) to that of the large eddies (<236 m), then at the leveling off the split ratio is roughly between 6:4 and 7:3 and between 4:6 and 5:5 for the surface wind stress and for TKE, respectively.

Figure 5c shows the peak NWPS as a function of wind speed. In all hurricane cases, the peak NWPS of both wind stress and TKE decreases with wind speed at low wind speeds less than 10 m s−1 and then gradually levels off as wind speed keeps increasing. A possible explanation for the leveling off is that, as more turbulent fluxes and energy are generated by large eddies with the increase of wind speed, the downscaling transport process has to speed up so as to prevent fluxes and energy from accumulating at the large-scale eddies. Thus, we speculate that the leveling off of the peak NWPS at high wind speeds might simply reflect internal change of the energy cascade process of turbulence in response to the change of ambient conditions. This picture of the energy cascade process at the leveling off described here is fundamentally the same as the model of energy cascade for the high Reynolds number turbulent flow proposed by Hunt and Carlotti (2001). They argued that the traditional eddy-to-eddy energy cascade in the surface layer at high Reynolds number flow is shortcut by the creation of an internal boundary layer (IBL) within each individual large eddy and by the shedding of vorticity from the highly deformed and relatively fast moving eddies as they descend into the slow moving air near the ground. Thus, in each large eddy above a critical scale, there is a direct, or elevator-like, transfer toward small eddies with scale less than the critical scale rather than the classical eddy-to-eddy staircaselike cascade. This picture is illustrated by Fig. 6, which is adopted from Hunt and Carlotti (2001). The leveling-off phenomenon at the high wind speeds shown in Fig. 5 may be considered as evidence to support Hunt and Carlotti’s elevator-like model for the energy cascade in the surface layer of the high Reynolds number turbulent flow.

The most distinguishing feature between the surface wind stress and TKE is the scale of the peak NWPS. The constant peak scale of wind stress near 86 m shown in Hurricane Ivan is robustly supported by all hurricane cases. Although the peak NWPS scale of TKE generally increases with the increase of wind speed, as in the case of Hurricane Ivan, there is a large spread among different hurricanes. Since large-scale eddies can contribute significantly to TKE, the large spread of peak scales of TKE implies that a wide spectrum of large eddies are generated among different hurricanes or even in a given storm due to the highly asymmetric hurricane structures, complicated land surface conditions, and a variety of ways of hurricane approaching the land. A key issue here is to understand the relationship between the different behavior of NWPS peak scales for wind stress and TKE and the dynamics of the turbulent eddies in the surface layer. As reported by Mahrt and Gibson (1992), most of the momentum transport in the surface layer is associated with coherent “bursting and sweeping” features. This was supported by Högström and Bergström (1996), who showed that 86%–95% of momentum transport is carried out within the identified bursts and sweeps. The scale of peak NWPS of surface wind stress found in this study falls inside the scale range of the intermittent bursts and sweeps (10–100 m) reported by Shaw and Businger (1985). On the other hand, although inactive large eddy features, such as rolls, are very efficient in transporting momentum in the boundary layer as indicated by Morrison et al.’s (2005) observation and Zhu’s (2008) simulation, they carry little momentum fluxes in the surface layer. For high Reynolds number turbulent flow (which is the case of hurricane surface layer), Hunt and Morrison (2000) proposed a top–down mechanism to relate inactive motion to the eddy structures dynamically significant in the surface layer. They argued that large eddies impinge onto the ground where they generate an internal layer in which smaller eddies develop. In this way, the smaller-scale bursts and sweeps, which are important to momentum transport, are generated through the interaction between larger inactive eddies and the surface. The nearly constant scale of peak NWPS of the surface wind stress shown in Fig. 5d may be considered as evidence to support the Hunt and Morrison top–down mechanism.

4. Conclusions and discussion

Turbulent transport in the HBL, which plays a key role in the development of a hurricane, is not well characterized owing to the difficulties in observations and lack of appropriate analysis methods that can be applied to the unsteady, inhomogeneous conditions. To depict a relatively complete picture of the change in turbulent eddy structures in the surface layer with the evolution of hurricanes, the WT is used to decompose the FCMP wind data onto the scale–time domain, which allows us to scrutinize closely the change of eddy power spectra and contribution of eddies with specific scales to the total fluxes and TKE during the entire course of hurricane landfalls. The analyses show that the rolls recorded in the tower observations formed at the places of landfalling hurricanes where convective activity is relatively weak and have a crosswind scale of approximately 1 km, consistent with typical crosswind scale of rolls observed in other conditions. Rolls play an important role in modulating turbulent eddies; particularly, they tend to suppress the eddies immediately adjacent to rolls. However, they do not appear to have an effect on eddies smaller than 102 m. This result is consistent with the LeMone (1976) turbulent energy budget analyses, which shows that there is little energy exchange between the rolls and small-scale (high frequency) turbulence, although rolls do play a role in redistributing turbulence-producing elements in the boundary layer.

It is found that the contribution of small and large eddies to the surface wind stress and TKE decreases and increases, respectively, with the increase of wind speed for low wind speeds approximately less than 10 m s−1 but tends to level off as wind speeds keep increasing. The split ratio of small eddies (>236 m) to large eddies (>236 m) at the leveling off is about 6:4/7:3 for wind stress and 4:6/5:5 for TKE, respectively. We argue that the leveling off is likely the outcome of the internal change of the energy cascade process of turbulence. The results that we showed here appear to support the Hunt and Carlotti (2001) “elevator like” energy cascade model.

The WT analyses also illustrate the different role of eddies in generating fluxes and TKE. The scale of the peak NWPS of wind stress is nearly a constant with the mean value of 86 m. This result suggests that a critical scale may exist. Far beyond the critical scale, the correlation between horizontal and vertical velocity perturbations associated with eddies are too weak to generate significant momentum fluxes despite the fact that these eddies can contribute substantially to turbulent intensity, which is confirmed by the general increase of the peak scale of NWPS of TKE with wind speed. The different characteristics of the NWPS peak scales for surface wind stress and TKE may be well explained by Hunt and Morrison’s top–down mechanism for eddy motion at high Reynolds number. The impingment of “inactive” large eddies onto the surface tends to generate “active” smaller-scale “bursts/sweeps,” which are dynamically important in the surface layer and efficient carriers of momentum flux.

However, our analyses show that the relationship between the peak NWPS scale for surface wind stress and TKE and wind speed appears to be strongly affected by the storm convective activity. The asymmetrically distributed rainbands and eyewall and the specific way of a storm approaching land are the possible reasons to cause the wide spread of peak NWPS of TKE. Moreover, although eddies generally shift toward larger scales with the increase of wind speed, eddies do not appear to symmetrically shift back to smaller scales as the wind begins to decrease after reaching its peak. This phenomenon suggests that the eddy characteristics may not be uniquely determined by the mean state of the boundary layer in the unsteady conditions.

Since all the data used in this study are collected over land during the landfall of hurricanes, the results shown here may not be extended to open ocean conditions in that the interaction between turbulent flow and the underlying surface may be completely different. Thus, more analyses are needed to obtain a complete picture of turbulent eddy structures in the HBL for different conditions. Finally, without temperature and moisture measurements, it is unclear if the finding on momentum flux and TKE is also valid for the vertical transport of heat and moisture. These issues will be investigated in our future studies. Despite these limitations, the results presented here nevertheless suggest that the uniqueness of the hurricane surface layer is not only reflected by its unsteadiness and inhomogeneity due to the continuously changed external environment but also because such external changes lead to an internal change in structure of turbulent eddies. Because the mixing mechanisms for large and small eddies are different, the findings of this study suggest that the existing turbulent scheme developed from steady nonhurricane conditions may not work properly under the hurricane environment. For example, in classical TKE closure schemes, all fluxes are parameterized based on the predicted TKE. However, as we demonstrated in this study for high wind conditions, eddies beyond the critical scale can still contribute significantly to TKE but are inefficient in generating momentum fluxes in the surface layer. The problem for the TKE type of schemes arises from the fact that the parameterization closure is derived based on a local downgradient diffusion assumption, whereas the large eddies fundamentally create nonlocal mixing. It is hoped that the analyses of the structure of turbulent eddies and their contribution to fluxes and energy may enlighten and provide guidance for the improvement of boundary layer schemes so that they can appropriately represent the unique transport processes in the HBL. In our future studies, the WT method will be extended to analyze the data collected for open ocean conditions.

Acknowledgments

Ping Zhu wishes to acknowledge the support for this work from the National Science Foundation under Grant AGS-0847332 and the NOAA/Florida Hurricane Alliance. We are very grateful to the three anonymous reviewers for their constructive comments. Their helpful suggestions led to improvements of this paper.

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Fig. 1.
Fig. 1.

Roll cases from Hurricanes Jeanne (2004) and Isabel (2003): (a) Crosswind (y axis) component from the observations, (b) NWPS E(n, sj), and (c) time-averaged NWPS. The dark solid thick line, dark solid thin line, and light solid thin line represent the time-averaged NWPS of the crosswind component, along-wind component, and vertical velocity, respectively; the dashed thick line represents the averaged NWPS of crosswind component over the observations with mean wind speed 20–22 m s−1 of all hurricanes without cleanly defined rolls. (d) FT power density spectra: the dark solid thick line, dark solid thin line, and light solid thin line represent the spectra of the crosswind component, along-wind component, and vertical velocity, respectively.

Citation: Journal of the Atmospheric Sciences 67, 12; 10.1175/2010JAS3437.1

Fig. 2.
Fig. 2.

(a) Time series of 15-min-averaged wind speeds of Hurricanes Jeanne (2004) and Isabel (2003) recorded by the towers. The dashed vertical line indicates the time when the rolls shown in Fig. 1 were observed. (b) Doppler radar reflectivity images closest to the time when the rolls were observed. The long white arrow indicates the location of the tower. White lines and stars indicate the hurricane best track.

Citation: Journal of the Atmospheric Sciences 67, 12; 10.1175/2010JAS3437.1

Fig. 3.
Fig. 3.

Hurricane Ivan (2004): (a) 15-min-averaged NWPS of surface wind stress and TKE derived from WT analyses, where the red thick line indicates 15-min-averaged wind speed; (b) 15-min-averaged NWPS for four scale bands, <48, 48–236, 236–1014, and >1014 m; (c) 15-min-averaged NWPS for scales smaller than 2 and 3 km; (d) peak NWPS in the spectra; and (e) scale of the peak NWPS.

Citation: Journal of the Atmospheric Sciences 67, 12; 10.1175/2010JAS3437.1

Fig. 4.
Fig. 4.

Satellite IR images before, at, and after the time when the peak wind speed was observed by the tower. The black star indicates the tower location. The thick black line and red circles indicate the hurricane best track.

Citation: Journal of the Atmospheric Sciences 67, 12; 10.1175/2010JAS3437.1

Fig. 5.
Fig. 5.

(a) NWPS for scales smaller than 236 m from different hurricanes. Each point represents a 15-min average. (b) NWPS for scales larger than 236 m. (c) Peak NWPS. (d) Scale of the peak NWPS.

Citation: Journal of the Atmospheric Sciences 67, 12; 10.1175/2010JAS3437.1

Fig. 6.
Fig. 6.

Schematic of energy cascade process: (a) local interaction or staircaselike cascade and (b) long-range interaction or elevator-like cascade (adopted from Hunt and Carlotti 2001).

Citation: Journal of the Atmospheric Sciences 67, 12; 10.1175/2010JAS3437.1

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