## 1. Introduction

Because of their catastrophic potential, wind and wave properties of tropical cyclones (TCs) are subjects of great interest. Over the years, simultaneous measurements of winds and waves have been acquired inside TCs by National Oceanic and Atmospheric Administration (NOAA) hurricane reconnaissance and research aircraft (e.g., Wright et al. 2001; Walsh et al. 2002; Moon et al. 2003; Black et al. 2007; Fan et al. 2009a,b; Fan and Rogers 2016; Liu et al. 2017). For each mission, the aircraft makes several passes through the TC center. Two-dimensional (2D) directional wavenumber spectra are obtained along several transects radiating from the TC center. A sequence of analyses of these data has shown that the waves inside TCs possess similarity properties. Several of the notable similarity properties are: the fetch- and duration-limited nature of wave growth, conformation to the general wave spectrum functions, and sinusoidal variation of dominant wave properties such as significant wave height, dominant wave period, and propagation direction (Hwang 2016; Hwang and Walsh 2016; Hwang and Fan 2017; Hwang et al. 2017; Hwang and Walsh 2018a). These similarity properties are useful for formulating parametric models to provide handy estimations of wave properties of interest, such as the maximum wave height inside TCs (Hwang and Walsh 2018b) and the L-band tilting surface slope (Hwang and Fan 2018) important to TC monitoring using microwave reflectometry techniques (Ruf et al. 2016).

In this paper, the transports and net fluxes of wave energy and momentum are investigated using the 2D spectra obtained in hurricane hunter missions. First, the energy and momentum transport vectors are computed from the spectrum. A spatial distribution model of the transports is constructed making use of their sinusoidal variations. The coefficients of the sinusoidal variations are obtained through harmonic analysis. By the definition that the gradient of transport is net flux for a quasi-steady system, the model of net fluxes can be readily derived from analytical differentiation of the transport model functions. Understanding of energy and momentum budget is of great interest for air–sea coupling research, in which the surface waves at the air–sea interface play an important role. Because of the complexity in the air, wave, and current subsystems, establishing a fully integrated air–wave–current coupled model remains a major challenge. Being able to produce a parameterization formula of the residual wave energy and momentum fluxes can be very useful for the atmospheric and ocean current subsystems to account for the wave effects in their lower (atmospheric models) or upper (ocean current models) boundary conditions. Section 2 describes the energy and momentum transport computation and model development of transports and net fluxes, and the main features of transports and net flues revealed by the parametric models. Section 3 presents results on the spatial distributions of transports and net fluxes applied to the hurricane hunter datasets, model and measurement comparison, discussion on the limitation of the parametric models, and suggestion for their improvement. Section 4 is a summary.

## 2. Transport and net flux computation and modeling

### a. Transport

The directional spectrum can be used to compute the wave energy transport vector **T**_{E} = (*T*_{Ex}, *T*_{Ey}) = (*Ec*_{gx}, *Ec*_{gy}), where *x* and *y* are Cartesian coordinates, **c**_{g} = (*c*_{gx}, *c*_{gy}) is energy transport velocity vector, and *E* is wave energy per unit sea surface area. The wave energy *E* is equal to the variance of surface wave elevation *ρ*_{w} and gravitational acceleration *g*, i.e., *x* are to the right/left of TC, and positive/negative *y* are to the front/back of TC. The Cartesian coordinates (*x*, *y*) and polar coordinates (*r*, *ψ*) are used interchangeably, i.e., *r* = (*x*^{2} + *y*^{2})^{0.5}, *ψ* = tan^{−1}(*y*/*x*). When the azimuthal angle *ϕ* is referenced to TC heading as zero, the two azimuthal angles are related by *ϕ* =*ψ* − 90°.

*S*(

*k*

_{x},

*k*

_{y}), the energy transport components are given by

**k**is the wavenumber vector:

**k**= (

*k*,

*ψ*) = (

*k*

_{x},

*k*

_{y}) are used interchangeably in this paper. The archived 2D wavenumber spectra are stored as 65 × 65 matrices with spectral resolution

*dk*= 0.0035 rad m

^{−1}, the maximum value of

*k*

_{x}and

*k*

_{y}is 0.11 rad m

^{−1}. The wave momentum

*M*is the wave energy divided by the phase speed, i.e.,

*M*=

*E*/

*c*, the momentum transport components are calculated by

*c*

_{g}=

*c*/2 has been employed. Figure 1 shows the transport components computed from one of the hurricane hunter missions (I14) reported in Fan et al. (2009a); I14 has the largest number of radial transects (11). A brief summary of all the hurricane hunter datasets used in this paper is given in Table 1. Descriptions of these datasets have been provided in previous publications (Wright et al. 2001; Walsh et al. 2002; Moon et al. 2003; Black et al. 2007; Fan et al. 2009a,b; Fan and Rogers 2016; Liu et al. 2017; Hwang and Walsh 2018a,b), a brief summary of the hurricane hunter wind and wave measurements is given in the appendix. As illustrated in Fig. 1, the sinusoidal variation is the primary feature of the azimuthal distribution of transport components, with the radial distance

*r*from the TC center as a secondary parameter. This is expected because sinusoidal variation is the main feature of the dominant wave properties such as significant wave height

*H*

_{s}and dominant wave period

*T*

_{p}(e.g., Hwang and Walsh 2016; Hwang and Fan 2017; appendix).

Some basic information of the datasets used for the analysis in this paper. The terms *U*_{10m}, *r*_{m}, *ϕ*_{m}, *H*_{sm}, *T*_{pm}, No. of *S*, No. of *T*, *V*_{h}, thN, and *Dt* in the header represent the maximum surface level wind speed, radius of maximum wind, azimuth angle of maximum wind location relative to hurricane heading (positive CCW), maximum significant wave height, maximum dominant wave period, number of spectra, number of radial transects, hurricane translation speed, hurricane heading referenced to north (positive CCW), and duration (h) in each data file.

*Q*=

*q*/

*q*

_{m},

*q*can be

*T*

_{Ex},

*T*

_{Ey},

*T*

_{Mx}, or

*T*

_{My}, and

*q*

_{m}is its maximum of absolute value used for normalization, which will be further discussed in section 3a. These dimensionless transport components are represented by adding a superscript * in the variables, i.e.,

*Q*=

*a*

_{0,Q}+ 2

*a*

_{1,Q}cos

*ψ*+ 2

*b*

_{1,Q}sin

*ψ*. The coefficients

*a*

_{0,Q},

*a*

_{1,Q}, and

*b*

_{1,Q}are functions of normalized radial distance from the TC center

*R*=

*r*/

*r*

_{m}, and

*r*

_{m}is the radius of maximum wind speed.

*a*

_{0,Q},

*a*

_{1,Q}, and

*b*

_{1,Q}for

*Q*=

*Z*represents

*a*

_{0,Q},

*a*

_{1,Q}, and

*b*

_{1,Q}. The fitting coefficients

*p*

_{1,Z},

*p*

_{2,Z}, and

*p*

_{3,Z}are listed in Table 2. The results from energy and momentum transports are very similar so the following discussion addresses

*a*

_{0}term is the mean of azimuthal variation. It is predominately negative for

*r*= 2

*r*

_{m}. The

*a*

_{1}and

*b*

_{1}terms represent the amplitude and phase of azimuthal variation. The results show that the peaks of leftward and forward transports are at

*r*/

*r*

_{m}between 2 and 3. The phase of peak variation can be obtained by tan

^{−1}(

*b*

_{1}/

*a*

_{1}).

The first three harmonic coefficients to describe the azimuthal variations of (a)–(c) *R* = *r*/*r*_{m}. Experimental results are from the first four datasets listed in Table 1.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

The first three harmonic coefficients to describe the azimuthal variations of (a)–(c) *R* = *r*/*r*_{m}. Experimental results are from the first four datasets listed in Table 1.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

The first three harmonic coefficients to describe the azimuthal variations of (a)–(c) *R* = *r*/*r*_{m}. Experimental results are from the first four datasets listed in Table 1.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

As in Fig. 2, but for

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

As in Fig. 2, but for

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

As in Fig. 2, but for

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Parameters describing the second-order polynomials fitting to the first three harmonic coefficients *a*_{0}, *a*_{1}, and *b*_{1} in (5): *Z* = *p*_{1,Z}*R*^{2} + *p*_{2,Z}*R* + *p*_{3,Z}.

These features are shown graphically with contour maps of

Spatial patterns of dimensionless transport components: (a)

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Spatial patterns of dimensionless transport components: (a)

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Spatial patterns of dimensionless transport components: (a)

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

### b. Net flux

*S*

_{Ein},

*S*

_{Edis}, and

*S*

_{Enl}are the input, dissipation, and nonlinear interaction energy source terms, and

*δS*

_{E}is the residual. The corresponding momentum source terms are given with similar variables with subscript

*M*replacing

*E*. The average aircraft speed for acquiring surface wave spectrum is about 120 m s

^{−1}and each spectrum is computed with 3D wave topography along about 9-km-long ground track (Wright et al. 2001), which represents 75 s of temporal duration between two consecutive spectra. Each mission to complete wave mapping lasts between 2.7 and 6.7 h (Table 1, last column), it is reasonable to assume that the wave field during the mission duration is quasi steady, i.e., ∂/∂

*t*terms in (6) and (7) can be dropped and the gradient terms represent the net fluxes, i.e.,

*Z*=

*a*

_{n,Q}or

*b*

_{n,Q}is given as second-order polynomial functions of

*R*(5), its radial derivative is

Figure 5 shows the results of the normalized net flux components *r*_{m} wide on both sides of the TC center and bounded by the green lines in Fig. 5a. The frontward net fluxes (positive *q*_{F} = *F*_{Ex}, *F*_{Ey}, *F*_{Mx}, and *F*_{My}, the scaling factors *q*_{Fm} are *q*_{Tm}/*r*_{m}, which will be further discussed in section 3a.

As in Fig. 4, but for the dimensionless net flux components: (a)

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

As in Fig. 4, but for the dimensionless net flux components: (a)

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

As in Fig. 4, but for the dimensionless net flux components: (a)

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

The azimuthal variation is mainly simple harmonic for transports (Fig. 4) and biharmonic for net fluxes (Fig. 5). These features are readily seen in the line traces shown in Fig. 6, which illustrates the azimuthal variations at *R* = 0.5, 1, 2, and 5 with solid, dashed, dashed–dotted, and dotted curves, respectively. The results for the momentum transport and net flux components are similar. Black and magenta colors are for the *y* and *x* components, respectively.

Examples of dimensionless energy (a) transport and (b) net flux components at selected *R* values (0.5, 1.0, 2.0, and 5.0). Black curves are the *y* component (positive for forward transport or net flux, negative for backward transport or net flux). Magenta curves are the *x* component (positive for rightward transport or net flux, negative for leftward transport or net flux).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Examples of dimensionless energy (a) transport and (b) net flux components at selected *R* values (0.5, 1.0, 2.0, and 5.0). Black curves are the *y* component (positive for forward transport or net flux, negative for backward transport or net flux). Magenta curves are the *x* component (positive for rightward transport or net flux, negative for leftward transport or net flux).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Examples of dimensionless energy (a) transport and (b) net flux components at selected *R* values (0.5, 1.0, 2.0, and 5.0). Black curves are the *y* component (positive for forward transport or net flux, negative for backward transport or net flux). Magenta curves are the *x* component (positive for rightward transport or net flux, negative for leftward transport or net flux).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

## 3. Result and discussion

### a. Scaling factors

The model functions (3)–(5) and (9)–(12) for transports and net fluxes, respectively, described in section 2 are given in dimensionless variables. The scaling factors for the transport components *q*_{T} = *T*_{Ex}, *T*_{Ey}, *T*_{Mx}, and *T*_{My} are *q*_{Tm} = *T*_{Exm}, *T*_{Eym}, *T*_{Mxm}, and *T*_{Mym}, respectively, where the *m* appended in the subscripts represents the maximum of the absolute transport values. For the net flux components *q*_{F} = *F*_{Ex}, *F*_{Ey}, *F*_{Mx}, and *F*_{My}, the scaling factors are *q*_{Fm} = *q*_{Tm}/*r*_{m}.

*q*

_{Tm}and

*q*

_{Fm}as functions of maximum wind speed estimated from the 11 hurricane hunter datasets listed in Table 1. The results are scattered, mainly caused by the fact that the hurricane hunter aircraft may or may not visit the region of maximum wind speed, thus leading to uncertainty of the critical TC parameters for scaling computation, i.e.,

*U*

_{10m},

*r*

_{m}, and the associated wave data at the maximum wind speed location. Regardless, the results show a general power function dependence on the maximum wind speed

*q*

_{m}can be

*q*

_{Tm}or

*q*

_{Fm}. The first two numbers printed at the lower edge of each panel in Fig. 7 are the fitting coefficients

*A*

_{q}and

*a*

_{q}, and the third number is the correlation coefficient between data and fitted function. The quantities

*q*

_{Tm}and

*q*

_{Fm}are the expected maximum magnitude of the transport and net flux components. For the left–right transports, the wind speed exponents are 1.7 and 1.6 for energy and momentum, respectively, the correlation coefficient of the fitting is 0.6. For the front-back transports, the exponents are 2.4 and 2.1, with correlation coefficient better than 0.82. Interestingly, the exponents of the net fluxes (lower row of Fig. 7) are between 3.2 and 4.0, which are about 1.6 steeper than their corresponding transport exponents (upper row of Fig. 7). This seems to reflect the general inverse relationship between

*U*

_{10m}and

*r*

_{m}; the factor

*r*

_{m}appears in

*q*

_{Fm}but not in

*q*

_{Tm}.

The maxima of the absolute values of measured transport and net flux components used as the scaling factors. Empirical fitting functions shown with straight lines are derived from 11 hurricane hunter missions listed in Table 1. They show power function dependence: *A*_{q} and *a*_{q} are listed as the first two numbers printed at the bottom of each panel, the third number is the correlation coefficient of the fitted results and measurements: (a) *T*_{Exm} and *T*_{Eym}, (b) *T*_{Mxm} and *T*_{Mym}, (c) *F*_{Exm} and *F*_{Eym}, (d) *F*_{Mxm} and *F*_{Mym}. The scaling factors of net flux components are computed from the transport components by *q*_{Fm} = *q*_{Tm}/*r*_{m}.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

The maxima of the absolute values of measured transport and net flux components used as the scaling factors. Empirical fitting functions shown with straight lines are derived from 11 hurricane hunter missions listed in Table 1. They show power function dependence: *A*_{q} and *a*_{q} are listed as the first two numbers printed at the bottom of each panel, the third number is the correlation coefficient of the fitted results and measurements: (a) *T*_{Exm} and *T*_{Eym}, (b) *T*_{Mxm} and *T*_{Mym}, (c) *F*_{Exm} and *F*_{Eym}, (d) *F*_{Mxm} and *F*_{Mym}. The scaling factors of net flux components are computed from the transport components by *q*_{Fm} = *q*_{Tm}/*r*_{m}.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

The maxima of the absolute values of measured transport and net flux components used as the scaling factors. Empirical fitting functions shown with straight lines are derived from 11 hurricane hunter missions listed in Table 1. They show power function dependence: *A*_{q} and *a*_{q} are listed as the first two numbers printed at the bottom of each panel, the third number is the correlation coefficient of the fitted results and measurements: (a) *T*_{Exm} and *T*_{Eym}, (b) *T*_{Mxm} and *T*_{Mym}, (c) *F*_{Exm} and *F*_{Eym}, (d) *F*_{Mxm} and *F*_{Mym}. The scaling factors of net flux components are computed from the transport components by *q*_{Fm} = *q*_{Tm}/*r*_{m}.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

### b. Comparison of model results and observations

#### 1) Transport

Figure 8 shows the comparison of transport components computed from measured spectra and those based on the model described in section 2a. B24 dataset is used as an example. It is from a mission conducted on 24 August 1998 inside Hurricane Bonnie at its category 2 stage with maximum surface wind speed (*U*_{10m}) of 44.4 m s^{−1} according to the HWIND analysis (e.g., Powell et al. 1996; Powell and Houston 1998) around the data acquisition time (Wright et al. 2001). The maximum wind at flight level is 45.7 m s^{−1} in the dataset that may or may not have passed through the location of maximum wind speed. The wind speed at flight level is about 10% higher than *U*_{10} (Franklin et al. 2003; Uhlhorn and Black 2003; Uhlhorn et al. 2007). In this paper, only those data within *R* = *r*/*r*_{m} ≤ 5 are used for discussion because the polynomial functions used for the modeling start to deviate from observations (Figs. 2 and 3). For B24 (*r*_{m} = 74 km, maximum *r* is 182 km), this condition is satisfied for all measurements in the dataset. Potentially the analysis can be extended to the regions with *R* > ~5 by adapting a two-segment fitting for the *a*_{0,Q}, *a*_{1,Q}, and *b*_{1,Q} parameters in (3). This is not done in this paper because those coefficients approach constant beyond *R* greater than about 5 (Figs. 2 and 3) thus the radial gradient terms approach 0. As shown in section 2, most actions happen in the neighborhood of *R* < ~5: for the transports the peaks/valleys are around *R* = 2.5 (Fig. 4), and for the net fluxes there is a monotonic trend of increasing net flux magnitudes toward the TC center (Fig. 5).

Comparison of measured and modeled transport components: (a) *T*_{Ex}, (b) *T*_{Ey}, (c) *T*_{Mx}, and (d) *T*_{My}. The first dataset (B24) in Table 1 is used for illustration. The statistics of bias, slope of linear regression, RMS difference, and correlation coefficient (*b*_{0}, *b*_{1}, *b*_{2}, *b*_{3}) are printed at the top edge of each panel.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Comparison of measured and modeled transport components: (a) *T*_{Ex}, (b) *T*_{Ey}, (c) *T*_{Mx}, and (d) *T*_{My}. The first dataset (B24) in Table 1 is used for illustration. The statistics of bias, slope of linear regression, RMS difference, and correlation coefficient (*b*_{0}, *b*_{1}, *b*_{2}, *b*_{3}) are printed at the top edge of each panel.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Comparison of measured and modeled transport components: (a) *T*_{Ex}, (b) *T*_{Ey}, (c) *T*_{Mx}, and (d) *T*_{My}. The first dataset (B24) in Table 1 is used for illustration. The statistics of bias, slope of linear regression, RMS difference, and correlation coefficient (*b*_{0}, *b*_{1}, *b*_{2}, *b*_{3}) are printed at the top edge of each panel.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

The mostly negative *T*_{Ex} (*T*_{Mx}) and positive *T*_{Ey} (*T*_{My}) indicate the dominance of observed leftward and forward energy (momentum) transports. The models reproduce these features reasonably well. Printed above each panel are the regression statistics of bias, slope of linear regression, root-mean-square (RMS) difference, and correlation coefficient (*b*_{0}, *b*_{1}, *b*_{2}, and *b*_{3}, respectively). These comparison statistics, together with the number of data points *N*, are also tabulated in Table 3. For this case, the correlation coefficients are all better than 0.92, and the linear regression slopes are between 0.80 and 0.84. Underestimation by the model is especially severe for the extreme condition. Table 4 lists the observed minimum, maximum, mean, and standard deviation of measured and modeled transport components. For example, the (minimum, maximum, mean, standard deviation) of left–right energy transport are (−4.32, 1.31, −1.17, 1.39) ×10^{5} J m s^{−1} m^{−2} (or W m^{−1}) in the observation, and (−2.62, 1.25, −0.99, 1.18) ×10^{5} J m s^{−1} m^{−2} in the model. The corresponding values of front–back transport component are (−1.28, 5.66, 1.43, 1.76) ×10^{5} J m s^{−1} m^{−2} in the observation, and (−1.28, 3.48, 1.19, 1.39) ×10^{5} J m s^{−1} m^{−2} in the model. Missing the extreme values is an indication of increasing nonlinearity toward the extreme wind condition and that higher harmonic terms are needed to capture the nonlinearity (Fig. 1).

Statistics (number, bias, slope of linear regression, RMS difference, correlation coefficient) of the observed and modeled energy and momentum transport components.

Statistics [minimum, maximum, mean, standard deviation (S.D.)] of the observed and modeled energy and momentum transport components.

Figure 9 shows the comparison of transport components computed from measured spectra combining the four datasets used in the model formulation, i.e., B24, I09, I12, and I14 (Table 1). The maximum flight level wind speeds in the datasets are 45.7, 74.0, 59.5, and 69.6 m s^{−1}. Applying the correction factor of 0.9 (Franklin et al. 2003; Uhlhorn and Black 2003; Uhlhorn et al. 2007), the maximum surface level wind speed *U*_{10m} are 41.1, 66.6, 53.6, and 62.6 m s^{−1}. The dominance of leftward and forward energy (momentum) transports is consistently observed and modeled. The statistics of (*b*_{0,} *b*_{1}, *b*_{2}, *b*_{3}) are printed at the top edge of each panel, the correlation coefficients range between 0.86 and 0.90. These comparison statistics, together with the number of data points *N*, are also tabulated in Table 3. The statistics of (minimum, maximum, mean, standard deviation) of transport components are listed in Table 4. These four datasets have been examined in detail (Wright et al. 2001; Moon et al. 2003; Black et al. 2007; Fan et al. 2009a,b; Hwang 2016; Hwang and Walsh 2016; Hwang and Fan 2017; Hwang and Walsh 2018a,b). The data size is generally larger and with better spatial coverage (233–600 spectra in 5–11 radial transects). However, the hurricane hunter may or may not have approached the maximum wind region, especially in the extreme category 4 and 5 hurricanes. The heavy rainbands may also render ineffective the wave sensing by the Ka-band (36 GHz) scanning radar altimeter system carried on the aircraft.

As in Fig. 8, but combining the first four datasets in Table 1; these are datasets used for formulating the model functions described in section 2.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

As in Fig. 8, but combining the first four datasets in Table 1; these are datasets used for formulating the model functions described in section 2.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

As in Fig. 8, but combining the first four datasets in Table 1; these are datasets used for formulating the model functions described in section 2.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Figure 10 shows the comparison of transport components computed from measured spectra combining the seven datasets not used in the model formulation, i.e., B26, F13, H23, H24, L30, F31, and F01 (Table 1). The maximum flight level wind speeds in the datasets are 38.8, 61.4, 49.0, 36.9, 36.7, 70.2, and 62.5 m s^{−1}. Applying the correction factor of 0.9 (Franklin et al. 2003; Uhlhorn and Black 2003; Uhlhorn et al. 2007), the maximum surface level wind speed *U*_{10m} are 34.9, 55.3, 44.1, 33.2, 33.0, 63.2, and 56.3 m s^{−1}. This group gives a good coverage of different TC intensities. The dominance of leftward and forward energy (momentum) transports is consistently observed and modeled. The statistics of (*b*_{0}, *b*_{1}, *b*_{2}, *b*_{3}) are printed at the top edge of each panel, the correlation coefficients range between 0.78 and 0.82. These comparison statistics, together with the number of data points *N*, are also tabulated in Table 3. The statistics of (minimum, maximum, mean, standard deviation) of transport components are listed in Table 4. These seven datasets are generally much smaller, and the spatial coverage is relatively sparse (78–212 spectra in 3 to 6 radial transects). It is more likely to miss the maximum wind location that provides *q*_{m} and *r*_{m} used as the normalization factors in the model development (section 2a). However, the comparison statistics remain very good with correlation coefficient between 0.78 and 0.82, and slope of linear regression between 0.76 and 0.86.

As in Fig. 8, but combining the last seven datasets in Table 1; these are datasets not used for formulating the model functions described in section 2.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

As in Fig. 8, but combining the last seven datasets in Table 1; these are datasets not used for formulating the model functions described in section 2.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

As in Fig. 8, but combining the last seven datasets in Table 1; these are datasets not used for formulating the model functions described in section 2.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

#### 2) Net flux

*E*

_{t}(Hwang and Sletten 2008) and surface wind stress

*τ*

_{a}:

*ρ*

_{a}is density of air,

*ω*

_{#}=

*ω*

_{p}

*U*

_{10}

*g*

^{−1}, and

*C*

_{10}is the drag coefficient, which uses the formula derived from microwave radiometer sensing in TCs (Hwang 2018; Hwang et al. 2019).

The left column of Fig. 11 shows the magnitude of net fluxes, |*F*_{E}| and |*F*_{M}| respectively, as functions of radial distance from the TC center. Dataset B24 is used as an illustration. The radius of maximum wind is shown with vertical dashed lines in the figure. The inputs to the models are the measurement locations (*r*, *ψ*) and the estimated *r*_{m} for generating the normalized net flux components. The scaling factors *q*_{Fm} = *q*_{Tm}/*r*_{m} are given by the observed transport maximum and radius of maximum wind speed. The most significant feature is the monotonic decreasing trend of |*F*_{E}| and |*F*_{M}| with increasing radial distance. The right column of Fig. 11 shows the normalized net fluxes |*F*_{E}|/*E*_{t} and |*F*_{M}|/*τ*_{a} as functions of radial distance, the vertical dashed line represents the radius of maximum wind speed. The ratios are generally small except near the TC center. For this case, |*F*_{E}|/*E*_{t} and |*F*_{M}|/*τ*_{a} exceed 0.5 at *r* less than about 40 and 25 km, respectively.

The magnitudes of energy and momentum net fluxes and their fractions of the expected wind input counterparts: (a) |*F*_{E}|, (b) |*F*_{E}|/*E*_{t}, (c) |*F*_{M}|, and (d) |*F*_{M}|/*τ*_{a}. There is a generally decreasing trend with the radial distance from the TC center. For this dataset (B24), the |*F*_{E}|/*E*_{t} and |*F*_{M}|/*τ*_{a} ratios can exceed 0.5 for *r* less than about 25 to 30 km.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

The magnitudes of energy and momentum net fluxes and their fractions of the expected wind input counterparts: (a) |*F*_{E}|, (b) |*F*_{E}|/*E*_{t}, (c) |*F*_{M}|, and (d) |*F*_{M}|/*τ*_{a}. There is a generally decreasing trend with the radial distance from the TC center. For this dataset (B24), the |*F*_{E}|/*E*_{t} and |*F*_{M}|/*τ*_{a} ratios can exceed 0.5 for *r* less than about 25 to 30 km.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

The magnitudes of energy and momentum net fluxes and their fractions of the expected wind input counterparts: (a) |*F*_{E}|, (b) |*F*_{E}|/*E*_{t}, (c) |*F*_{M}|, and (d) |*F*_{M}|/*τ*_{a}. There is a generally decreasing trend with the radial distance from the TC center. For this dataset (B24), the |*F*_{E}|/*E*_{t} and |*F*_{M}|/*τ*_{a} ratios can exceed 0.5 for *r* less than about 25 to 30 km.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Figure 12 shows |*F*_{E}|/*E*_{t} and |*F*_{M}|/*τ*_{a} based on B24 (upper row) and I14 (lower row) datasets with (*U*_{10m}, *r*_{m}) = (41.4 m s^{−1}, 74 km) and (62.6 m s^{−1}, 42 km), respectively. The ratios are plotted against the local wind speed. The energy flux ratio is given in the left column and the net momentum flux ratio in the right column. Depending on *U*_{10m} and *r*_{m}, the wind speed at which the ratios of fluxes exceeding 0.5 is well over about 30 m s^{−1} for |*F*_{E}|/*E*_{t} and 15–20 m s^{−1} for |*F*_{M}|/*τ*_{a}. Further inward to the TC center, the ratios can even exceed unity, signifying that substantial energy and momentum from outer region are carried into the central region where wind speed is low.

(a),(c) |*F*_{E}|/*E*_{t} and (b),(d) |*F*_{M}|/*τ*_{a} ratios plotted as functions of wind speed with color showing the inverse wave age *ω*_{#}. The ratios can exceed 0.5 for *U*_{10} as high as 30 m s^{−1}. Datasets (top) B24 and (bottom) I14 (bottom row) are used for illustration.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

(a),(c) |*F*_{E}|/*E*_{t} and (b),(d) |*F*_{M}|/*τ*_{a} ratios plotted as functions of wind speed with color showing the inverse wave age *ω*_{#}. The ratios can exceed 0.5 for *U*_{10} as high as 30 m s^{−1}. Datasets (top) B24 and (bottom) I14 (bottom row) are used for illustration.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

(a),(c) |*F*_{E}|/*E*_{t} and (b),(d) |*F*_{M}|/*τ*_{a} ratios plotted as functions of wind speed with color showing the inverse wave age *ω*_{#}. The ratios can exceed 0.5 for *U*_{10} as high as 30 m s^{−1}. Datasets (top) B24 and (bottom) I14 (bottom row) are used for illustration.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

The local inverse wave age *ω*_{#} in the central region is frequently less than 0.8 and has been classified as swell in the conventional definition of the term. However, in the central region of a TC, there are substantial energy and momentum exchange between the remotely generated long waves and local winds as revealed by the large |*F*_{E}|/*E*_{t} and |*F*_{M}|/*τ*_{a} ratios. Figure 13 shows |*F*_{E}|/*E*_{t} and |*F*_{M}|/*τ*_{a} as functions of *ω*_{#} (left column) and *r*/*r*_{m} (right column). The results are from combining the first four datasets in Table 1, with *U*_{10m} between 41.4 and 66.6 m s^{−1} and *r*_{m} between 13 and 74 km. Many of the cases that are classified as swell (*ω*_{#} less than about 0.8, left column) have large values of |*F*_{E}|/*E*_{t} and |*F*_{M}|/*τ*_{a}. In the region with *r*/*r*_{m} less than about 0.75 (right column, top panel), there are many cases with |*F*_{E}|/*E*_{t} > 0.5. Similarly, in the region with *r*/*r*_{m} less than about 0.5 (right column, bottom panel), there are many cases with |*F*_{M}|/*τ*_{a} > 0.5. This observation has an important implication, that is, the waves in the TC central region are dynamically very different from the conventional swell that almost has no interaction with the local wind and current. These waves propagated from an upstream region into the TC central area are actively interacting with the local wind and current systems with large energy and momentum exchanges.

(a),(b) |*F*_{E}|/*E*_{t} and (c),(d) |*F*_{M}|/*τ*_{a} ratios plotted as functions of (left) *ω*_{#} and (right) *r*/*r*_{m}. The ratios can exceed 0.5 for wave conditions traditionally classifies as swell (*ω*_{#} < ~0.8) and within *r*/*r*_{m} less than about 0.5. The first four datasets in Table 1 are used for illustration.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

(a),(b) |*F*_{E}|/*E*_{t} and (c),(d) |*F*_{M}|/*τ*_{a} ratios plotted as functions of (left) *ω*_{#} and (right) *r*/*r*_{m}. The ratios can exceed 0.5 for wave conditions traditionally classifies as swell (*ω*_{#} < ~0.8) and within *r*/*r*_{m} less than about 0.5. The first four datasets in Table 1 are used for illustration.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

(a),(b) |*F*_{E}|/*E*_{t} and (c),(d) |*F*_{M}|/*τ*_{a} ratios plotted as functions of (left) *ω*_{#} and (right) *r*/*r*_{m}. The ratios can exceed 0.5 for wave conditions traditionally classifies as swell (*ω*_{#} < ~0.8) and within *r*/*r*_{m} less than about 0.5. The first four datasets in Table 1 are used for illustration.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

There are many formulas of drag coefficient *C*_{10} that can be used in (14) to compute wind stress with *U*_{10} input. Results on using different *C*_{10} have been discussed briefly in Hwang and Fan (2017, their Fig. 8). The various *C*_{10} equations differ mainly in high winds with *U*_{10} greater than about 30 m s^{−1}. In the present analysis, the interesting result is the large fraction of residual energy and momentum fluxes compared to the expected energy dissipation rate and wind stress near the central region within about 30 km radius from the TC center. In this region for TCs with relatively large eyes, the wind speed is less than about 30 m s^{−1} and most of the drag coefficient formulas are similar, so the conclusion is not changed by using different *C*_{10} formulas.

### c. Tropical cyclone as a source of global wave generation

*Q*=

*f*

_{Q}(

*R*), where

*Q*can be

*x*components are left–right (LR) transports and fluxes, and

*y*components are front–back (FB) transports and fluxes. The total transports over a given area inside TC can be evaluated by integration over the area. Figure 14 shows the area-integrated dimensionless positive and negative elements of the transport and net flux components

*Q*

_{A}=

*R*=

*n*= 1, 2, 3, 4, and 5, i.e.,

*y*: FB positive) and leftward (

*x*: LR negative). Interestingly, the magnitudes of leftward components are almost as large as the forward components.

Area integrated transport and net flux components: (a) *R* = 1, 2, 3, 4, and 5, for all positive and negative *Q*_{A} elements in the integration region. Positive and negative of *x* component indicate rightward and leftward transport or net flux, positive and negative of *y* component indicate frontward and backward transport or net flux.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Area integrated transport and net flux components: (a) *R* = 1, 2, 3, 4, and 5, for all positive and negative *Q*_{A} elements in the integration region. Positive and negative of *x* component indicate rightward and leftward transport or net flux, positive and negative of *y* component indicate frontward and backward transport or net flux.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Area integrated transport and net flux components: (a) *R* = 1, 2, 3, 4, and 5, for all positive and negative *Q*_{A} elements in the integration region. Positive and negative of *x* component indicate rightward and leftward transport or net flux, positive and negative of *y* component indicate frontward and backward transport or net flux.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

*R*= 1. Outward from

*R*= 1, there is a local amplitude maximum near

*R*between 2 and 3, as shown in Fig. 6a. The net flux (Fig. 15b) shows biharmonic pattern, as in the alternative representation of Fig. 6b. These mono and biharmonic patterns become more conspicuous in the bottom row of Fig. 15, which shows the area integrated energy transport and net energy flux

*R*= 1–2 into leftward and forward two branches of about equal strength. The TC center, with rather large local transport and net flux (top row), seems to act as a quasi-singular barrier or facilitator and directs the influx from the right back quarter toward left and front. The bifurcation flux pattern may play a role in stabilizing the TC propagation.

Azimuthal variations of energy transport and net flux vectors: (a) *R* = 1, 2, 3, 4, and 5.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Azimuthal variations of energy transport and net flux vectors: (a) *R* = 1, 2, 3, 4, and 5.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Azimuthal variations of energy transport and net flux vectors: (a) *R* = 1, 2, 3, 4, and 5.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

The dimensionless results presented in Fig. 14 are expressed as the quantity *Q* = *q*/*q*_{m} per *q*_{m} as illustrated in Fig. 7, the dimensional quantities can be obtained. Using *r*_{m} = 40 km = 4 × 10^{4} m as an example, Fig. 16 shows the leftward, rightward, frontward, and backward transports and net fluxes integrated over 1, 2, 3, 4, and 5 *r*_{m} for a TC with *U*_{10m} = 50 m s^{−1}_{.} The resulting magnitudes are rather large (Fig. 16). For example, integrated to 5 *r*_{m} (i.e., 200 km), the forward energy transport is more than 10^{16} W m (or J m s^{−1}), the forward net energy flux is about 10^{11} W; the forward momentum transport and net flux are about 6 × 10^{14} N m and 6 × 10^{9} N.

Numerical values of (a) *T*_{EAx} and *T*_{EAy}, (b) *T*_{MAx} and *T*_{MAy}, (c) *F*_{EAx} and *F*_{EAy}, (d) *F*_{MAx} and *F*_{MAy}. The TC intensity is *U*_{10m} = 50 m s^{−1}, and *r*_{m} = 40 km.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Numerical values of (a) *T*_{EAx} and *T*_{EAy}, (b) *T*_{MAx} and *T*_{MAy}, (c) *F*_{EAx} and *F*_{EAy}, (d) *F*_{MAx} and *F*_{MAy}. The TC intensity is *U*_{10m} = 50 m s^{−1}, and *r*_{m} = 40 km.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Numerical values of (a) *T*_{EAx} and *T*_{EAy}, (b) *T*_{MAx} and *T*_{MAy}, (c) *F*_{EAx} and *F*_{EAy}, (d) *F*_{MAx} and *F*_{MAy}. The TC intensity is *U*_{10m} = 50 m s^{−1}, and *r*_{m} = 40 km.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

For TCs with the same *r*_{m}, the effects of TC intensity are determined by the scaling factor (Fig. 7). Figure 17 shows examples of energy transport and net flux for two TCs with *U*_{10m} = 35 and 70 m s^{−1}; and *r*_{m} = 40 km. Integrated to 5*r*_{m} (200 km), the forward energy transport is about 5 × 10^{15} and 2.5 × 10^{16} W m, respectively. The net forward flues are about 2.5 × 10^{10} and 4 × 10^{11} W, respectively. Waves leaving the generation area of a big storm are known to travel long distances of tens of thousands of kilometers. It is a manifestation of the tremendous energy and momentum produced by the large storm.

TC intensity and numerical values of (a),(b) *T*_{EAx} and *T*_{EAy} and (c),(d) *F*_{EAx} and *F*_{EAy}. The left and right columns show results for *U*_{10m} = 35 and 70 m s^{−1}, respectively; *r*_{m} = 40 km.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

TC intensity and numerical values of (a),(b) *T*_{EAx} and *T*_{EAy} and (c),(d) *F*_{EAx} and *F*_{EAy}. The left and right columns show results for *U*_{10m} = 35 and 70 m s^{−1}, respectively; *r*_{m} = 40 km.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

TC intensity and numerical values of (a),(b) *T*_{EAx} and *T*_{EAy} and (c),(d) *F*_{EAx} and *F*_{EAy}. The left and right columns show results for *U*_{10m} = 35 and 70 m s^{−1}, respectively; *r*_{m} = 40 km.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

### d. Limitation of the parametric model and suggestion of improvement

The scanning radar altimeter (SRA) 2D spectra are archived in 65 × 65 matrices with *k*_{x} and *k*_{y} ranging from 0.0035 to 0.112 rad m^{−1}. The maximum wavenumber for the corner elements is 0.158 rad m^{−1}. For a conservative estimation, the shortest resolved wavelength corresponding to *k* = 0.112 rad m^{−1} is 56 m. It is clearly a concern whether the measured spectrum captures sufficient spectral energy of the wave field.

The observed peak wavelength is typically in the range of 125 to 400 m. Figures 18a and 18b show the histogram and cumulative distribution function (CDF) of the dominant wavelengths in L30, B24, and I14 datasets, with *U*_{10m} = 33, 41, and 63 m s^{−1}, respectively (Table 1). The number of cases with relatively short peak wavelength increases as the wind speed inside TC decreases. For example, the percentage of cases with wavelength shorter than 200 m is about 20% or less for B24 and I14, but goes up to about 80% for L30.

(a) Histograms of the observed dominant wavelengths in datasets L30, B24, and I14, with maximum wind speeds 33, 41, and 63 m s^{−1}, respectively. (b) Cumulative distribution functions of the observed dominant wavelengths in datasets L30, B24, and I14.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

(a) Histograms of the observed dominant wavelengths in datasets L30, B24, and I14, with maximum wind speeds 33, 41, and 63 m s^{−1}, respectively. (b) Cumulative distribution functions of the observed dominant wavelengths in datasets L30, B24, and I14.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

(a) Histograms of the observed dominant wavelengths in datasets L30, B24, and I14, with maximum wind speeds 33, 41, and 63 m s^{−1}, respectively. (b) Cumulative distribution functions of the observed dominant wavelengths in datasets L30, B24, and I14.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

*G*function) can be expressed as

*ω*/

*ω*

_{p})]

^{2}/2

*σ*

^{2}} is the peak enhancement factor. The spectral coefficients

*α*,

*γ*, and

*σ*vary with spectral slope −

*s*

_{f}and wave development stage, which is frequently represented by the dimensionless spectral peak frequency or inverse wave age:

*ω*

_{#}=

*ω*

_{p}

*U*

_{10}/

*g*. Equation (17) can be degenerated to the spectral functions of Pierson and Moskowitz (1964), Joint North Sea Wave Project (JONSWAP) or Hasselmann et al. (1973, 1976), Donelan et al. (1985), and Young (1998). The analysis and full expressions of

*α*,

*γ*, and

*σ*parameters are given in Hwang et al. (2017). Additional analysis result from incorporating the low-pass mean square slope data obtained in TCs through microwave reflectometry is given in Hwang and Fan (2018); the study leads to the recommendation of the following wind speed function for the average

*s*

_{f}:

*G*spectrum (17) with

*s*

_{f}defined by (18) is denoted the G18 spectrum function.

The wave spectral energy is concentrated near the peak region as indicated by the exponential term and power function dependence on frequency in (17). Based on (18), the average *s*_{f} increases gradually from 4.70 for *U*_{10} ≤ 18 m s^{−1} to 5.66 at 80 m s^{−1}. Figure 19a shows examples of wave energy (black curves) and transport (red curves) spectra at 15, 30, 45, and 60 m s^{−1} based on the G18 spectrum function, with *ω*_{#} values typically observed in TCs. The frequency spectrum is converted to the wavenumber spectrum by *S*(*k*)*dk* = *S*(*ω*)*dω*. Applying the deep-water gravity wave dispersion relation: *ω*^{2} = *gk*, then *S*(*k*) = *S*(*ω*)*g*/2*ω*. The spectral density is further normalized by its maximum value for this discussion. The two dashed–dotted green vertical lines mark the resolved spectral wavenumber range of the SRA spectra (0.035 and 0.112 rad m^{−1}). Figure 19b shows the cumulative variances of the energy and transport spectra normalized by their maximum values given by integration to *k* approaching ∞ (in the example given the maximum value of the wavenumber *k*_{max} in the computed spectra is 2363 rad m^{−1}). It is clear that the spectra measured by the SRA, which is designed for high-wind applications, only miss a small portion of the ocean surface wave energy and transport. Spectral energy and transport computations using the SRA spectra are judged to be reliable especially in high winds.

(a) Examples of the wind wave energy and energy transport spectra normalized by their maximum spectral densities. The wind speed, inverse wave age, and dominant wavelength (*U*_{10}, *ω*_{#}, *L*_{p}) are listed in the legend. (b) The normalized cumulative variances for the cases shown in (a).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

(a) Examples of the wind wave energy and energy transport spectra normalized by their maximum spectral densities. The wind speed, inverse wave age, and dominant wavelength (*U*_{10}, *ω*_{#}, *L*_{p}) are listed in the legend. (b) The normalized cumulative variances for the cases shown in (a).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

(a) Examples of the wind wave energy and energy transport spectra normalized by their maximum spectral densities. The wind speed, inverse wave age, and dominant wavelength (*U*_{10}, *ω*_{#}, *L*_{p}) are listed in the legend. (b) The normalized cumulative variances for the cases shown in (a).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Another set of limiting factors influencing the accuracy of the parametric models discussed in section 2 are the scaling variables *U*_{10m} and *r*_{m} [Fig. 7 and Eq. (13)]. *U*_{10m} and *r*_{m} are among the most important TC properties but they are also among the most difficult to get accurate measurements. The best track analysis provides resolution to about 5 kt and 5 n mi (about 2.5 m s^{−1} and 9 km), respectively. For TCs with low maximum wind or small eye, the error introduced by the uncertainty can be quite large. For example, in the 11 cases examined in this study, two have eyes smaller than 20 km, and another three have eyes less than 35 km. The 9-km *r*_{m} resolution is a source of large uncertainty. Empirically, the scaling factors for energy and momentum transports and net fluxes are approximately power functions of *U*_{10m} with the exponents of power functions ranging between 1.6 and 4.0 (Fig. 7). To improve the accuracy of the parametric functions for obtaining estimates of the energy and momentum transports and net fluxes, the most important step is to improve the accuracy of *U*_{10m} and *r*_{m} determination.

The small number of SRA datasets contains limited variety of TC parameters, especially on the range of propagation speed, stage of development, and steadiness of TC system. The parametric models of energy and momentum transports and net fluxes as developed in this paper probably work better for slow-moving and steady stages of TC development, which characterize most of the cases used in this study (Table 1).

Finally, the number of radial transects in the SRA wind wave datasets is limited by the hurricane hunter operation time of typically less than 6 h per mission excluding transit time. The 10 and 11 radial transects of B24 and I14 are the largest numbers conducted by hurricane hunter wave sensing so far. Unless multiple aircraft are used in a coordinated fashion in the same TC, the number of radial transects is not likely to increase due to the limited operation duration. At the early stage of this study, several attempts to use datasets with less number of radial transects produce relatively poor results; *O*(10+) transects with similar range coverage is crucial for the parametric model development. Increasing the number of radial transects will allow expanding to higher harmonics the Fourier series that describes the energy and momentum transports (3). The analysis of the net fluxes of energy and momentum can also be carried out to higher harmonics.

## 4. Summary

The surface wave energy and momentum transports and net fluxes inside tropical cyclones are investigated using the 2D wavenumber spectra measured in 11 hurricane reconnaissance and research missions in six hurricanes (Table 1). The prominent characteristic of transports is their sinusoidal azimuthal variation. Harmonic analysis is used to derive the coefficients of the Fourier components. The coefficients vary with radial distance systematically. The results are used to formulate a parametric model for describing the spatial patterns of transports inside TCs (Figs. 4 and 6). The modeled results are in good agreement with direct transport computation using the measured 2D spectra (Figs. 8–10).

The net fluxes are obtained from the spatial gradients of transports. Biharmonic azimuthal variation is the dominant feature of the net flux components (Figs. 5 and 6). Compared to the expected energy and momentum inputs from local wind, the net fluxes are small except in the region near the TC center. Depending on the TC maximum wind speed and radius of maximum wind, the net fluxes can exceed more than one-half of the expected local wind input values in winds as high as 30 m s^{−1} (Fig. 12). In the central region, the wave conditions are frequently classified as swell based on the criterion of *ω*_{#} less than about 0.8. However, the net fluxes from these waves propagated from an upstream region are a significant portion of the expected local wind input (Fig. 13); so, unlike the conventional definition of swell, these upstream waves actively interact with local wind and current systems with large energy and momentum exchanges (Figs. 11–13).

The transport and net flux models described in this paper can be used to perform quantitative estimation of various transport and net flux properties of interest. For example, Fig. 14 shows the normalized results of area integrated forward, backward, leftward, and rightward transports and net fluxes. The bottom row of Fig. 15 shows the azimuthal variations of the normalized area integrated transports and net fluxes. It is feasible that the bifurcation pattern of net fluxes may play a role in stabilizing the TC propagation (Figs. 15b,d). Applying the scaling factors of the transport and net flux components (Fig. 7), and the normalization area factor

## Acknowledgments

This work is sponsored by the Office of Naval Research (Funding Doc. N0001416WX00044). Comments and suggestions from two anonymous reviewers have helped to improve the presentation and analysis. Datasets used in this analysis are given in the references cited. The processing codes and data segments can also be obtained by contacting the corresponding author. U.S. Naval Research Laboratory Publication Number JA/7260—20-0654.

## APPENDIX

### Brief Summary of Hurricane Hunter Wind and Wave Measurements

In the 1990s NOAA research aircraft installed the NASA 36-GHz SRA and flew eight seasons starting in 1998. The postprocessing is complex and publication of results is usually delayed. For example, the first published measurements of Bonnie (1998) (B24) is in 2001 (Wright et al. 2001) and results of B26 are documented in 2002 (Walsh et al. 2002). Another well-documented SRA measurements are from Ivan 2004 (I09, I12, and I14) and published in 2009 (Fan et al. 2009b). The other six datasets listed in Table 1, acquired between 1999 and 2004, are introduced much later (Hwang and Walsh 2018a,b). Of all the available SRA datasets B24 and I14 are the only ones with double-digits radial transects, 10 and 11, respectively. Here are some examples of the SRA wind and wave measurements. They are excerpts from Hwang and Fan (2017) and Hwang et al. (2017), and placed in this appendix so that readers do not need to go through previous papers.

Figure A1, reproducing part of Fig. 1 in Hwang and Fan (2017), shows examples of the spatial distributions of *U*_{10}, *H*_{s}, and *T*_{p} from two hurricane reconnaissance and research missions during Hurricane Ivan 2004. On the top row are the results for I14, which has the best spatial coverage (with 11 radial transects and 600 spectra) in all the hurricane hunter wind and wave datasets ever published. On the bottom row are results for I09, which has six radial transects of considerably different range coverage; the relatively sparse and uneven spatial coverage poses severe challenge for the task of formulating a parametric model to describe the spatial distribution pattern.

Spatial distributions of (a),(d) *U*_{10}; (b),(e) *H*_{s}; and (c),(f) *T*_{p} for cases (top) I14 and (bottom) I09. [Reproducing part of Fig. 1 in Hwang and Fan (2017)].

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Spatial distributions of (a),(d) *U*_{10}; (b),(e) *H*_{s}; and (c),(f) *T*_{p} for cases (top) I14 and (bottom) I09. [Reproducing part of Fig. 1 in Hwang and Fan (2017)].

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Spatial distributions of (a),(d) *U*_{10}; (b),(e) *H*_{s}; and (c),(f) *T*_{p} for cases (top) I14 and (bottom) I09. [Reproducing part of Fig. 1 in Hwang and Fan (2017)].

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Figure A2, reproducing part of Fig. 13 in Hwang and Fan (2017), shows the azimuthal distributions of *U*_{10}, *H*_{s}, and *T*_{p} at several radial distances from the TC center, the radial distance is given in dimensionless range *R* = *r*/*r*_{m} in the figure legend. The data are from I14. Sinusoidal variation is the dominant feature of the *U*_{10}, *H*_{s}, and *T*_{p} azimuthal distributions.

Azimuthal variation of (a) *U*_{10}, (b) *H*_{s}, and (c) *T*_{p} at several radial distances shown as *R* = *r*/*r*_{m} in the legend for case I14. [Reproducing part of Fig. 13 in Hwang and Fan (2017).]

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Azimuthal variation of (a) *U*_{10}, (b) *H*_{s}, and (c) *T*_{p} at several radial distances shown as *R* = *r*/*r*_{m} in the legend for case I14. [Reproducing part of Fig. 13 in Hwang and Fan (2017).]

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Azimuthal variation of (a) *U*_{10}, (b) *H*_{s}, and (c) *T*_{p} at several radial distances shown as *R* = *r*/*r*_{m} in the legend for case I14. [Reproducing part of Fig. 13 in Hwang and Fan (2017).]

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Figure A3, reproducing Fig. 12 in Hwang et al. (2017), shows examples of the directional spectrum obtained by the hurricane hunter. The data are from B24. The four spectra illustrate the complicated directional spectral properties in different locations inside a TC. The four quadrants of the four spectra in the figure correspond approximately to the physical quadrants of the measurement locations relative to the TC center. The position vector proportional to its distance from the hurricane center, its normal, wind and dominant wave vectors, and hurricane heading are superimposed with arrows of different colors as labeled in the legend. The location of the local spectral peaks (up to five) are shown in descending order with symbols circle, plus, triangle, square, and penta-star. The Earth coordinates (E, N, W, and S) are used in the plotting, the position vector (*x*_{h}, *y*_{h}) and (*r*, *ϕ*) rotated with reference to the hurricane heading are shown in the upper text string, wind speed, significant wave height, dominant wave period, and dimensionless spectral peak frequency are shown in the lower text string.

Examples of the normalized SRA 2D wavenumber spectra. The position vector proportional to its distance from the hurricane center, its normal, wind and dominant wave vectors, and hurricane heading are superimposed with arrows of different colors as labeled in the legend. The location of the local spectral peaks (up to five) are shown in the descending order with symbols circle, plus, triangle, square, and penta-star. The Earth coordinates (E, N, W, and S) are used in the figure, the position vector (*x*_{h}, *y*_{h}) and (*r*, *ϕ*) rotated with reference to the hurricane heading are shown in the upper text string, wind speed, significant wave height, dominant wave period, and dimensionless spectral peak frequency are shown in the lower text string. The four quadrants of the four spectra correspond approximately to the physical quadrants of the measurement locations relative to the TC center. [Reproducing Fig. 12 in Hwang et al. (2017).]

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Examples of the normalized SRA 2D wavenumber spectra. The position vector proportional to its distance from the hurricane center, its normal, wind and dominant wave vectors, and hurricane heading are superimposed with arrows of different colors as labeled in the legend. The location of the local spectral peaks (up to five) are shown in the descending order with symbols circle, plus, triangle, square, and penta-star. The Earth coordinates (E, N, W, and S) are used in the figure, the position vector (*x*_{h}, *y*_{h}) and (*r*, *ϕ*) rotated with reference to the hurricane heading are shown in the upper text string, wind speed, significant wave height, dominant wave period, and dimensionless spectral peak frequency are shown in the lower text string. The four quadrants of the four spectra correspond approximately to the physical quadrants of the measurement locations relative to the TC center. [Reproducing Fig. 12 in Hwang et al. (2017).]

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

Examples of the normalized SRA 2D wavenumber spectra. The position vector proportional to its distance from the hurricane center, its normal, wind and dominant wave vectors, and hurricane heading are superimposed with arrows of different colors as labeled in the legend. The location of the local spectral peaks (up to five) are shown in the descending order with symbols circle, plus, triangle, square, and penta-star. The Earth coordinates (E, N, W, and S) are used in the figure, the position vector (*x*_{h}, *y*_{h}) and (*r*, *ϕ*) rotated with reference to the hurricane heading are shown in the upper text string, wind speed, significant wave height, dominant wave period, and dimensionless spectral peak frequency are shown in the lower text string. The four quadrants of the four spectra correspond approximately to the physical quadrants of the measurement locations relative to the TC center. [Reproducing Fig. 12 in Hwang et al. (2017).]

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0104.1

## REFERENCES

Black, P. G., and Coauthors, 2007: Air–sea exchange in hurricanes: Synthesis of observations from the coupled boundary layer air–sea transfer experiment.

,*Bull. Amer. Meteor. Soc.***88**, 357–374, https://doi.org/10.1175/BAMS-88-3-357.Donelan, M. A., J. Hamilton, and W. H. Hui, 1985: Directional spectra of wind-generated ocean waves.

,*Philos. Trans. Roy. Soc. London***315A**, 509–562, https://doi.org/10.1098/rsta.1985.0054.Fan, Y., and W. E. Rogers, 2016: Drag coefficient comparisons between observed and model simulated directional wave spectra under hurricane conditions.

,*Ocean Modell.***102**, 1–13, https://doi.org/10.1016/j.ocemod.2016.04.004.Fan, Y., I. Ginis, and T. Hara, 2009a: The effect of wind-wave-current interaction on air–sea momentum fluxes and ocean response in tropical cyclones.

,*J. Phys. Oceanogr.***39**, 1019–1034, https://doi.org/10.1175/2008JPO4066.1.Fan, Y., I. Ginis, T. Hara, C. W. Wright, and E. J. Walsh, 2009b: Numerical simulations and observations of surface wave fields under an extreme tropical cyclone.

,*J. Phys. Oceanogr.***39**, 2097–2116, https://doi.org/10.1175/2009JPO4224.1.Franklin, J. L., M. L. Black, and K. Valde, 2003: GPS dropwindsonde wind profiles in hurricanes and their operational implications.

,*Wea. Forecasting***18**, 32–44, https://doi.org/10.1175/1520-0434(2003)018<0032:GDWPIH>2.0.CO;2.Hasselmann, K., and Coauthors, 1973: Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Hydraulic Engineering Rep., Deutches Hydrographisches Institut, 95 pp.

Hasselmann, K., D. B. Ross, P. Müller, and W. Sell, 1976: A parametric wave prediction model.

,*J. Phys. Oceanogr.***6**, 200–228, https://doi.org/10.1175/1520-0485(1976)006<0200:APWPM>2.0.CO;2.Hwang, P. A., 2016: Fetch- and duration-limited nature of surface wave growth inside tropical cyclones: With applications to air–sea exchange and remote sensing.

,*J. Phys. Oceanogr.***46**, 41–56, https://doi.org/10.1175/JPO-D-15-0173.1.Hwang, P. A., 2018: High wind drag coefficient and whitecap coverage derived from microwave radiometer observations in tropical cyclones.

,*J. Phys. Oceanogr.***48**, 2221–2232, https://doi.org/10.1175/JPO-D-18-0107.1.Hwang, P. A., and M. A. Sletten, 2008: Energy dissipation of wind-generated waves and whitecap coverage.

,*J. Geophys. Res.***113**, C02012, https://doi.org/10.1029/2007JC004277.Hwang, P. A., and E. J. Walsh, 2016: Azimuthal and radial variation of wind-generated surface waves inside tropical cyclones.

,*J. Phys. Oceanogr.***46**, 2605–2621, https://doi.org/10.1175/JPO-D-16-0051.1.Hwang, P. A., and Y. Fan, 2017: Effective fetch and duration of tropical cyclone wind fields estimated from simultaneous wind and wave measurements: Surface wave and air–sea exchange computation.

,*J. Phys. Oceanogr.***47**, 447–470, https://doi.org/10.1175/JPO-D-16-0180.1.Hwang, P. A., and Y. Fan, 2018: Low-frequency mean square slopes and dominant wave spectral properties: Toward tropical cyclone remote sensing.

,*IEEE Trans. Geosci. Remote Sens.***56**, 7359–7368, https://doi.org/10.1109/TGRS.2018.2850969.Hwang, P. A., and E. J. Walsh, 2018a: Propagation directions of ocean surface waves inside tropical cyclones.

,*J. Phys. Oceanogr.***48**, 1495–1511, https://doi.org/10.1175/JPO-D-18-0015.1.Hwang, P. A., and E. J. Walsh, 2018b: Estimating maximum significant wave height and dominant wave period inside tropical cyclones.

,*Wea. Forecasting***33**, 955–966, https://doi.org/10.1175/WAF-D-17-0186.1.Hwang, P. A., Y. Fan, F. J. Ocampo-Torres, and H. García-Nava, 2017: Ocean surface wave spectra inside tropical cyclones.

,*J. Phys. Oceanogr.***47**, 2393–2417, https://doi.org/10.1175/JPO-D-17-0066.1.Hwang, P. A., N. Reul, T. Meissner, and S. H. Yueh, 2019: Whitecap and wind stress observations by microwave radiometers: Global coverage and extreme conditions.

,*J. Phys. Oceanogr.***49**, 2291–2307, https://doi.org/10.1175/JPO-D-19-0061.1.Liu, Q., A. Babanin, Y. Fan, S. Zieger, C. Guan, and I.-I. Moon, 2017: Numerical simulations of ocean surface waves under hurricane conditions: Assessment of existing model performance.

,*Ocean Modell.***118**, 73–93, https://doi.org/10.1016/j.ocemod.2017.08.005.Moon, I.-J., I. Ginis, T. Hara, H. L. Tolman, C. W. Wright, and E. J. Walsh, 2003: Numerical simulation of sea surface directional wave spectra under hurricane wind forcing.

,*J. Phys. Oceanogr.***33**, 1680–1706, https://doi.org/10.1175/2410.1.Pierson, W. J., and L. Moskowitz, 1964: A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii.

,*J. Geophys. Res.***69**, 5181–5190, https://doi.org/10.1029/JZ069i024p05181.Powell, M. D., and S. H. Houston, 1998: Surface wind fields of 1995 Hurricanes Erin, Opal, Luis, Marilyn, and Roxanne at landfall.

,*Mon. Wea. Rev.***126**, 1259–1273, https://doi.org/10.1175/1520-0493(1998)126<1259:SWFOHE>2.0.CO;2.Powell, M. D., S. H. Houston, and T. A. Reinhold, 1996: Hurricane Andrew’s landfall in south Florida. Part I: Standardizing measurements for documentation of surface wind fields.

,*Wea. Forecasting***11**, 304–328, https://doi.org/10.1175/1520-0434(1996)011<0304:HALISF>2.0.CO;2.Ruf, C. S., and Coauthors, 2016: New ocean winds satellite mission to probe hurricanes and tropical convection.

,*Bull. Amer. Meteor. Soc.***97**, 385–395, https://doi.org/10.1175/BAMS-D-14-00218.1.Uhlhorn, E. W., and P. G. Black, 2003: Verification of remotely sensed sea surface winds in hurricanes.

,*J. Atmos. Oceanic Technol.***20**, 99–116, https://doi.org/10.1175/1520-0426(2003)020<0099:VORSSS>2.0.CO;2.Uhlhorn, E. W., P. G. Black, J. L. Franklin, M. Goodberlet, J. Carswell, and A. S. Goldstein, 2007: Hurricane surface wind measurements from an operational stepped frequency microwave radiometer.

,*Mon. Wea. Rev.***135**, 3070–3085, https://doi.org/10.1175/MWR3454.1.Walsh, E. J., and Coauthors, 2002: Hurricane directional wave spectrum spatial variation at landfall.

,*J. Phys. Oceanogr.***32**, 1667–1684, https://doi.org/10.1175/1520-0485(2002)032<1667:HDWSSV>2.0.CO;2.Wright, C. W., and Coauthors, 2001: Hurricane directional wave spectrum spatial variation in the open ocean.

,*J. Phys. Oceanogr.***31**, 2472–2488, https://doi.org/10.1175/1520-0485(2001)031<2472:HDWSSV>2.0.CO;2.Young, I. R., 1998: Observations of the spectra of hurricane generated waves.

,*Ocean Eng.***25**, 261–276, https://doi.org/10.1016/S0029-8018(97)00011-5.