Transports and Net Fluxes of Surface Wave Energy and Momentum inside Tropical Cyclones: Spectrum Computation and Modeling

Paul A. Hwang

doi:10.1175/JPO-D-20-0104.1

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Journal of Physical Oceanography

Abstract
1. Introduction
2. Transport and net flux computation and modeling
- a. Transport
- b. Net flux
3. Result and discussion
4. Summary
Brief Summary of Hurricane Hunter Wind and Wave Measurements

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Fig. 1.

Azimuthal variations of energy and momentum transport components calculated from 2D spectra acquired by hurricane hunter in Hurricane Ivan 2004 (I14 dataset): (a) T_Ex, (a) T_Ey, (a) T_Mx, and (a) T_My.

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Fig. 2.

The first three harmonic coefficients to describe the azimuthal variations of (a)–(c) $T_{E x}^{*}$ and (d)–(f) $T_{E y}^{*}$ . The coefficients are functions of R = r/r_m. Experimental results are from the first four datasets listed in Table 1.

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Fig. 3.

As in Fig. 2, but for $T_{M x}^{*}$ and $T_{M y}^{*}$ .

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Fig. 4.

Spatial patterns of dimensionless transport components: (a) $T_{E x}^{*}$ , (b) $T_{E y}^{*}$ , (c) $T_{M x}^{*}$ , and (d) $T_{M y}^{*}$ using the parametric models describe in section 2.

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Fig. 5.

As in Fig. 4, but for the dimensionless net flux components: (a) $F_{E x}^{*}$ , (b) $F_{E y}^{*}$ , (c) $F_{M x}^{*}$ , and (d) $F_{M y}^{*}$ .

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Fig. 6.

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).

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Fig. 7.

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: $q_{m} = A_{q} U_{10 m}^{a_{q}}$ . The fitted 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.

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Fig. 8.

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 (b₀, b₁, b₂, b₃) are printed at the top edge of each panel.

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Fig. 9.

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.

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Fig. 10.

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.

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Fig. 11.

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.

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Fig. 12.

(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 U₁₀ as high as 30 m s⁻¹. Datasets (top) B24 and (bottom) I14 (bottom row) are used for illustration.

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Fig. 13.

(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.

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Fig. 14.

Area integrated transport and net flux components: (a) $T_{E A x}^{*}$ and $T_{E A y}^{*}$ , (b) $T_{M A x}^{*}$ and $T_{M A y}^{*}$ , (c) $F_{E A x}^{*}$ and $F_{E A y}^{*}$ , and (d) $F_{M A x}^{*}$ and $F_{M A y}^{*}$ . The area integration is carried out from the TC center to 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.

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Fig. 15.

Azimuthal variations of energy transport and net flux vectors: (a) $T_{E}^{*}$ and (b) $F_{E}^{*}$ ; the area integrated transport and net flux vectors: (c) $T_{E A}^{*}$ , and (d) $F_{E A}^{*}$ . The area integration is carried out from the TC center to R = 1, 2, 3, 4, and 5.

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Fig. 16.

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.

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Fig. 17.

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.

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Fig. 18.

(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.

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Fig. 19.

(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 (U₁₀, ω_#, L_p) are listed in the legend. (b) The normalized cumulative variances for the cases shown in (a).

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Fig. A1.

Spatial distributions of (a),(d) U₁₀; (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)].

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Fig. A2.

Azimuthal variation of (a) U₁₀, (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).]

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Fig. A3.

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).]

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Editorial Type:: Article

Article Type:: Research Article

Transports and Net Fluxes of Surface Wave Energy and Momentum inside Tropical Cyclones: Spectrum Computation and Modeling

Paul A. Hwang

Paul A. Hwang Remote Sensing Division, U.S. Naval Research Laboratory, Washington, D.C.

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https://orcid.org/0000-0001-5683-9348

Online Publication:: 10 Nov 2020

Print Publication:: 01 Nov 2020

DOI:: https://doi.org/10.1175/JPO-D-20-0104.1

Page(s):: 3309–3329

Received:: 14 May 2020

Accepted:: 09 Sep 2020

Published Online:: 10 Nov 2020

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

Transports and net fluxes of surface wave energy and momentum inside tropical cyclones (TCs) are analyzed with wave spectra acquired by hurricane hunters. Previous analyses of dominant wave properties show a primary feature of sinusoidal azimuthal variation. Transports calculated from directional wave spectra are also primarily sinusoidal, which is modeled as a harmonic series. The result reveals that forward transport peaks are in the right-front quarter relative to the TC heading, and somewhat weaker valleys of backward transports are in the left-back quarter. Rightward transport peaks are in the right-back quarter and stronger leftward transport valleys are in the left-front quarter. Net fluxes are derived analytically from the gradients of transports. Their azimuthal variations are primarily biharmonic with forward trend confined in a slightly left-tilted parallel channel about a width two to three radius of maximum wind (RMW) on each side of the TC center. Leftward net fluxes are in a parallel channel of similar size and normal to that of the forward net fluxes. In vectors, the right-back quarter is a region of net influxes of energy and momentum. The TC central region has strong local fluxes that lead to bifurcation of the flux lines into leftward and forward paths. This may play a role in stabilizing the TC propagation. The net fluxes are a small fraction of the expected energy and momentum inputs from local wind except near the eye region. Within about 30 km from the TC center the local wind speed may exceed 30 m s⁻¹ and the net fluxes can exceed 50% of the expected local wind input.

Corresponding author: Paul A. Hwang, paul.hwang@nrl.navy.mil

Abstract

Corresponding author: Paul A. Hwang, paul.hwang@nrl.navy.mil

Keywords: Energy transport; Fluxes; Momentum; Wind stress; Wind waves; Tropical cyclones

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 $η_{rms}^{2}$ times the product of water density ρ_w and gravitational acceleration g, i.e., $E = ρ_{w} g η_{rms}^{2}$ . Throughout this paper, the coordinates are rotated such that the TC heading is toward the top of the page, so positive/negative 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 = (x² + y²)^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°.

With a 2D wavenumber spectrum S(k_x, k_y), the energy transport components are given by

\begin{matrix} T_{E x} = \iint S (k_{x}, k_{y}) c_{g} \cos ψ d k_{x} d k_{y}, \\ T_{E y} = \iint S (k_{x}, k_{y}) c_{g} \sin ψ d k_{x} d k_{y}, \end{matrix}

(1)

where 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

\begin{matrix} T_{M x} = \iint [S (k_{x}, k_{y}) \cos ψ / 2] d k_{x} d k_{y}, \\ T_{M y} = \iint [S (k_{x}, k_{y}) \sin ψ / 2] d k_{x} d k_{y}, \end{matrix}

(2)

where the deep-water gravity wave dispersion relation 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).

Table 1.

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.

The sinusoidal variation can be represented as a sum of harmonic components:

Q (r, ψ) = a_{0} (R, ψ) + 2 \sum_{n = 1}^{N} [a_{n, Q} (R, ψ) \cos n ψ + b_{n, Q} (R, ψ) \sin n ψ],

(3)

where 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.,

T_{E x}^{*} = T_{E x} / T_{E x m}

T_{E y}^{*} = T_{E y} / T_{E y m}

T_{M x}^{*} = T_{M x} / T_{M x m}

, and

T_{M y}^{*} = T_{M y} / T_{M y m}

. Making use of the orthogonal property of trigonometric functions, then

\begin{matrix} a_{n, Q} = \int Q \cos^{n} ψ d ψ, \\ b_{n, Q} = \int Q \sin^{n} ψ d ψ . \end{matrix}

(4)

With relatively small numbers of radial transects in the hurricane hunter wave datasets (Table 1), the harmonic analysis is carried out to the first sinusoidal components, that is, Q = a_0,Q + 2a_1,Q cosψ + 2b_1,Q sinψ. 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.

Figures 2 and 3 show a_0,Q, a_1,Q, and b_1,Q for Q =

T_{E x}^{*}

T_{E y}^{*}

T_{M x}^{*}

, and

T_{M y}^{*}

calculated from four hurricane hunter datasets: B24, I09, I12, and I14 (Table 1). The fitted curve superimposed on each panel is a second-order polynomial function:

Z = p_{1, Z} R^{2} + p_{2, Z} R + p_{3, Z},

(5)

where 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

T_{E x}^{*}

and

T_{E y}^{*}

(Fig. 2). The a₀ term is the mean of azimuthal variation. It is predominately negative for

T_{E x}^{*}

and positive for

T_{E y}^{*}

thus indicating a net transport in the leftward and forward directions. Interestingly, the leftward transport magnitude shows a monotonically increasing trend toward the TC center. The forward transport, on the other hand displays a local peak at about r = 2r_m. The a₁ and b₁ 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(b₁/a₁).

Table 2.

Parameters describing the second-order polynomials fitting to the first three harmonic coefficients a₀, a₁, and b₁ in (5): Z = p_1,ZR² + p_2,ZR + p_3,Z.

These features are shown graphically with contour maps of $T_{E x}^{*}$ , $T_{E y}^{*}$ , $T_{M x}^{*}$ , and $T_{M y}^{*}$ in Fig. 4. The upper and lower rows show energy and momentum transport components, respectively. The left column (Figs. 4a,c) gives the left–right transports and the right column (Figs. 4b,d) shows the front-back transports. The contours are 0.1 apart with red solid lines for positive and black dashed lines for negative. The 0-transport line is given with a thick green line. The transport components of energy and momentum are again very similar, and the discussion focuses on the top row (energy transport). With respect to the TC heading, the peak of rightward transport (positive $T_{E x}^{*}$ ) is at the right-back quarter and extends to a region slightly less than a half plane as marked by the green 0-transport line in Fig. 4a. A steeper valley of leftward transport (negative $T_{E x}^{*}$ ) is axially opposite to the positive peak, and the region of leftward transport is slightly larger than a half plane. The peak of forward transport (positive $T_{E y}^{*}$ ) is at the front-right quarter (Fig. 4b) and the region of forward transport occupies slightly larger than a half plane. A weaker valley of backward transport (negative $T_{E y}^{*}$ ) is axially opposite to the forward peak, and the backward transport occupies a region slightly smaller than a half plane.

b. Net flux

The energy and momentum balance equations can be expressed as

\frac{\partial E}{\partial t} + \frac{\partial T_{E x}}{\partial x} + \frac{\partial T_{E y}}{\partial y} = S_{E in} + S_{E dis} + S_{E nl} = δ S_{E},

(6)

\frac{\partial M}{\partial t} + \frac{\partial T_{M x}}{\partial x} + \frac{\partial T_{M y}}{\partial y} = S_{M in} + S_{M dis} + S_{M nl} = δ S_{M} .

(7)

where 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.,

\begin{matrix} F_{E} = \nabla \cdot T_{E} = δ S_{E}, \\ F_{M} = \nabla \cdot T_{M} = δ S_{M} . \end{matrix}

(8)

Explicitly, the net flux components can be obtained from the transport components

F_{E x} = \frac{\partial T_{E x}}{\partial x}; F_{E y} = \frac{\partial T_{E y}}{\partial y}; F_{M x} = \frac{\partial T_{M x}}{\partial x}; F_{M y} = \frac{\partial T_{M y}}{\partial y} .

(9)

With the transport models given in polar functions (3), the fluxes can be converted to the Cartesian coordinated through the transformation functions

\begin{matrix} \frac{\partial}{\partial x} = \cos ψ \frac{\partial}{\partial r} - \frac{1}{r} \sin ψ \frac{\partial}{\partial ψ}, \\ \frac{\partial}{\partial y} = \sin ψ \frac{\partial}{\partial r} + \frac{1}{r} \cos ψ \frac{\partial}{\partial ψ} . \end{matrix}

(10)

The polar coordinate derivatives of the harmonic function (3) are

\begin{matrix} \frac{\partial Q}{\partial R} = \frac{\partial a_{0, Q}}{\partial R} + 2 \frac{\partial a_{1, Q}}{\partial R} \cos ψ + 2 \frac{\partial b_{1, Q}}{\partial R} \sin ψ, \\ \frac{\partial Q}{\partial ψ} = - 2 a_{1, Q} \sin ψ + 2 b_{1, Q} \cos ψ . \end{matrix}

(11)

Because Z = a_n,Q or b_n,Q is given as second-order polynomial functions of R (5), its radial derivative is

\partial Z / \partial R = 2 p_{1, Z} R + p_{2, Z} .

(12)

Figure 5 shows the results of the normalized net flux components $F_{E x}^{*} = F_{E x} / F_{E x m}$ , $F_{E y}^{*} = F_{E y} / F_{E y m}$ , $F_{M x}^{*} = F_{M x} / F_{M x m}$ , and $F_{M y}^{*} = F_{M y} / F_{M y m}$ . The energy and momentum net flux components are on the upper and lower rows, respectively. The left column (Figs. 5a,c) gives the left–right net fluxes and the right column (Figs. 5b,d) shows the front–back net fluxes. The contours are 0.1 apart with red solid lines for positive and black dashed lines for negative. The 0-net-flux line is given with a thick green line. The leftward net fluxes (negative $F_{E x}^{*}$ and $F_{M x}^{*}$ ) are confined in a channel about 2 to 3 r_m wide on both sides of the TC center and bounded by the green lines in Fig. 5a. The frontward net fluxes (positive $F_{E y}^{*}$ and $F_{M y}^{*}$ ) are in a channel of similar size and oriented about normal to that of the leftward net fluxes (Fig. 5b). Rightward net fluxes are in the front and back outer regions (Fig. 5a), whereas backward net fluxes are in the left and right outer regions (Fig. 5b). For the net flux components 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.

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.

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.

Figure 7 shows the 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} = A_{q} U_{10 m}^{a_{q}} .

(13)

where 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.

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 U₁₀ (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).

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 (b₀, b₁, b₂, and b₃, 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) ×10⁵ J m s⁻¹ m⁻² (or W m⁻¹) in the observation, and (−2.62, 1.25, −0.99, 1.18) ×10⁵ 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) ×10⁵ J m s⁻¹ m⁻² in the observation, and (−1.28, 3.48, 1.19, 1.39) ×10⁵ 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).

Table 3.

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

Table 4.

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, b₁, b₂, b₃) 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.

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 (b₀, b₁, b₂, b₃) 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.

2) Net flux

The spatial density of hurricane hunter wave measurements is not sufficient for obtaining the net flux components from directly computing the gradient of transport components. Here the model results are compared with the parameterized wind input energy exchange rate E_t (Hwang and Sletten 2008) and surface wind stress τ_a:

\begin{matrix} E_{t} = 0.20 ω_{#}^{3.3} η_{#} ρ_{a} U_{10}^{3}, \\ τ_{a} = ρ_{a} C_{10} U_{10}^{2}, \end{matrix}

(14)

where ρ_a is density of air,

η_{#} = η_{rms}^{2} g^{2} U_{10}^{- 4}

, ω_# = ω_pU₁₀g⁻¹, and C₁₀ 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.

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.

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.

There are many formulas of drag coefficient C₁₀ that can be used in (14) to compute wind stress with U₁₀ input. Results on using different C₁₀ have been discussed briefly in Hwang and Fan (2017, their Fig. 8). The various C₁₀ equations differ mainly in high winds with U₁₀ 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 C₁₀ formulas.

c. Tropical cyclone as a source of global wave generation

The models in section 2 describe the transports and net fluxes per unit area inside TCs in dimensionless representations: Q = f_Q(R), where Q can be

T_{E x}^{*}

T_{E y}^{*}

T_{M x}^{*}

T_{M y}^{*}

F_{E x}^{*}

F_{E y}^{*}

F_{M x}^{*}

, and

F_{M y}^{*}

. Referenced to the TC heading, the 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 =

T_{E A x}^{*}

T_{E A y}^{*}

T_{M A x}^{*}

T_{M A y}^{*}

F_{E A x}^{*}

F_{E A y}^{*}

F_{M A x}^{*}

, and

F_{M A y}^{*}

. The integration is carried out from the TC center to R = n = 1, 2, 3, 4, and 5, i.e.,

\begin{matrix} Q_{A n}^{+} = \int_{0}^{2 π} \int_{0}^{n} Q R d R d ψ, Q \geq 0, \\ Q_{A n}^{-} = \int_{0}^{2 π} \int_{0}^{n} Q R d R d ψ, Q \leq 0. \end{matrix}

(15)

The primary transports and net fluxes are forward (y: FB positive) and leftward (x: LR negative). Interestingly, the magnitudes of leftward components are almost as large as the forward components.

The azimuthal variations of the energy and momentum transport and net flux components are displayed in Fig. 15. The energy and momentum patterns are similar so only the energy transport and net flux are illustrated here. Given in vectors the top row of Fig. 15 is an alternative representation of Fig. 6, which shows

T_{E}^{*} (ϕ)

and

F_{E}^{*} (ϕ)

. The sinusoidal azimuthal variation of the cyclonic transport pattern is clearly illustrated (Fig. 15a). The amplitude of sinusoidal variation increases monotonically inward from 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

Q_{A n} (ψ) = \int_{ψ - Δ ψ / 2}^{ψ + Δ ψ / 2} \int_{0}^{n} Q R d R d ψ .

(16)

The amplitude of

T_{E A}^{*} (ψ)

sinusoidal variation increases monotonically outward from the TC center (Fig. 15c) due to the weighting of the integration area. The biharmonic pattern of the area integrated net flux shows inflow in the right back quarter toward the TC center and bifurcation starting near 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.

The dimensionless results presented in Fig. 14 are expressed as the quantity Q = q/q_m per $r_{m}^{2}$ . Applying the empirical wind speed functions for the scaling factors q_m as illustrated in Fig. 7, the dimensional quantities can be obtained. Using r_m = 40 km = 4 × 10⁴ 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 10¹⁶ W m (or J m s⁻¹), the forward net energy flux is about 10¹¹ W; the forward momentum transport and net flux are about 6 × 10¹⁴ N m and 6 × 10⁹ N.

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 5r_m (200 km), the forward energy transport is about 5 × 10¹⁵ and 2.5 × 10¹⁶ W m, respectively. The net forward flues are about 2.5 × 10¹⁰ and 4 × 10¹¹ 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.

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.

The general form of the wind wave spectrum (the G function) can be expressed as

S (ω) = α g^{2} ω_{p}^{- 5} {(\frac{ω}{ω_{p}})}^{- s_{f}} \exp [- \frac{s_{f}}{4} {(\frac{ω}{ω_{p}})}^{- 4}] γ^{Γ}

(17)

where Γ = exp{−[1 − (ω/ω_p)]²/2σ²} 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_pU₁₀/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:

s_{f} = {\begin{matrix} 4.7, & U_{10} \leq 18 {m s}^{- 1}, \\ 4.7 {(U_{10} / 18)}^{1 / 8}, & U_{10} > 18 {m s}^{- 1} . \end{matrix}

(18)

The 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 U₁₀ ≤ 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: ω² = gk, then S(k) = S(ω)g/2ω. 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.

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 $r_{m}^{2}$ , Figs. 16 and 17 show examples of applying the model to get area integrated transports and net fluxes. These large numbers are reflected in the well-observed phenomenon that waves leaving the generation area of a big storm can travel long distances over tens of thousands of kilometers.

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 U₁₀, 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.

Figure A2, reproducing part of Fig. 13 in Hwang and Fan (2017), shows the azimuthal distributions of U₁₀, 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 U₁₀, H_s, and T_p azimuthal distributions.

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.

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Journal of Physical Oceanography

Abstract
1. Introduction
2. Transport and net flux computation and modeling
- a. Transport
- b. Net flux
3. Result and discussion
4. Summary
Brief Summary of Hurricane Hunter Wind and Wave Measurements

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Fig. 1.

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Fig. 2.

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Fig. 3.

As in Fig. 2, but for $T_{M x}^{*}$ and $T_{M y}^{*}$ .

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Fig. 4.

Spatial patterns of dimensionless transport components: (a) $T_{E x}^{*}$ , (b) $T_{E y}^{*}$ , (c) $T_{M x}^{*}$ , and (d) $T_{M y}^{*}$ using the parametric models describe in section 2.

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Fig. 5.

As in Fig. 4, but for the dimensionless net flux components: (a) $F_{E x}^{*}$ , (b) $F_{E y}^{*}$ , (c) $F_{M x}^{*}$ , and (d) $F_{M y}^{*}$ .

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Fig. 6.

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Fig. 7.

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Fig. 8.

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Transports and Net Fluxes of Surface Wave Energy and Momentum inside Tropical Cyclones: Spectrum Computation and Modeling

Abstract

Abstract

1. Introduction

2. Transport and net flux computation and modeling

a. Transport

b. Net flux

3. Result and discussion

a. Scaling factors

b. Comparison of model results and observations

1) Transport

2) Net flux

c. Tropical cyclone as a source of global wave generation

d. Limitation of the parametric model and suggestion of improvement

4. Summary

Acknowledgments

APPENDIX

Brief Summary of Hurricane Hunter Wind and Wave Measurements

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