Effect of Wave Directions on Orientation and Magnitude of Surface Wind Stress under Typhoon Megi (2010)

Je-Yuan Hsu aInstitute of Oceanography, National Taiwan University, Taipei, Taiwan

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Abstract

The relationship between the wind-wave spectrum and surface wind stress τ in tropical cyclones, especially for the misalignment |ϕ| between the wind and τ, was investigated using data from three Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats deployed in Super Typhoon Megi (2010). The floats measured τ by integrating downward momentum flux in the ocean and, in a recent development, directional spectra of surface waves. The wind was captured by aircraft surveys. At wind speeds from 25 to 40 m s−1, the |ϕ| increased with the increasing angle between the wind and dominant waves. The |ϕ| was small near the eyewall, where wave energy concentrated in a narrow frequency band. At the location far away from the eyewall, where most spectra were bimodal in directions with similar frequencies, the stress direction might be similar to the high-frequency waves. The misalignment between the wind and propagating swell might affect the growth and directional spreading of wind waves under tropical cyclones. The resulting wave breaking might then release wave momentum into the ocean as most stress clockwise from the wind direction. C, the downwind drag coefficient, increased with increasing inverse wave age of dominant waves. |C|, the magnitude of the crosswind drag coefficient, was significant when low-frequency waves deviated from the wind by more than 90°. The wave directions are used in the inverse wave age for scaling drag coefficients. The new parameterization based on wave dynamics can be useful for improving the prediction of wind stress curl under storms.

Significance Statement

Tropical cyclones can impact the maximum sea surface temperature cooling through the curl of surface wind stress τ, which causes divergence under the storm eye. To investigate the effect of surface waves on τ, measurements of downward momentum flux and surface waves from three EM-APEX floats deployed under Typhoon Megi in 2010 are used. The inverse wave age involving wind forcing on the wave directions of dominant waves and swells can significantly influence the momentum transfer efficiency in the downwind and crosswind directions, respectively. That is, the propagating swell under storms should play a crucial role in the downward momentum flux during its interaction with high-frequency waves and wind. Incorporating wave directions to parameterize the magnitude and orientation of τ will improve future models’ ability to predict wind stress curl and thereby the heat fluxes to storm intensification.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Je-Yuan Hsu, jyahsu@ntu.edu.tw

Abstract

The relationship between the wind-wave spectrum and surface wind stress τ in tropical cyclones, especially for the misalignment |ϕ| between the wind and τ, was investigated using data from three Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats deployed in Super Typhoon Megi (2010). The floats measured τ by integrating downward momentum flux in the ocean and, in a recent development, directional spectra of surface waves. The wind was captured by aircraft surveys. At wind speeds from 25 to 40 m s−1, the |ϕ| increased with the increasing angle between the wind and dominant waves. The |ϕ| was small near the eyewall, where wave energy concentrated in a narrow frequency band. At the location far away from the eyewall, where most spectra were bimodal in directions with similar frequencies, the stress direction might be similar to the high-frequency waves. The misalignment between the wind and propagating swell might affect the growth and directional spreading of wind waves under tropical cyclones. The resulting wave breaking might then release wave momentum into the ocean as most stress clockwise from the wind direction. C, the downwind drag coefficient, increased with increasing inverse wave age of dominant waves. |C|, the magnitude of the crosswind drag coefficient, was significant when low-frequency waves deviated from the wind by more than 90°. The wave directions are used in the inverse wave age for scaling drag coefficients. The new parameterization based on wave dynamics can be useful for improving the prediction of wind stress curl under storms.

Significance Statement

Tropical cyclones can impact the maximum sea surface temperature cooling through the curl of surface wind stress τ, which causes divergence under the storm eye. To investigate the effect of surface waves on τ, measurements of downward momentum flux and surface waves from three EM-APEX floats deployed under Typhoon Megi in 2010 are used. The inverse wave age involving wind forcing on the wave directions of dominant waves and swells can significantly influence the momentum transfer efficiency in the downwind and crosswind directions, respectively. That is, the propagating swell under storms should play a crucial role in the downward momentum flux during its interaction with high-frequency waves and wind. Incorporating wave directions to parameterize the magnitude and orientation of τ will improve future models’ ability to predict wind stress curl and thereby the heat fluxes to storm intensification.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Je-Yuan Hsu, jyahsu@ntu.edu.tw

1. Introduction

During tropical cyclone–ocean interaction, storm wind can affect the thermal and momentum structure in the upper ocean in various ways. The surface wind stress τ as downward momentum flux can induce strong ocean currents, thereby sea surface temperature (SST) cooling via shear instability. Besides the significant vertical shear of wind-induced current, a moving cyclone can generate a complex surface wave field at the air–sea interface. The breaking of short waves and the change of ocean surface roughness can affect the downward momentum flux (Melville and Rapp 1985; Jones and Toba 2012). Although recent model studies have discussed the important roles of surface waves in the tropical cyclone–wave–ocean coupling dynamics (e.g., Kudryavtsev et al. 2014), such as the effect of Langmuir circulation (e.g., Reichl et al. 2016), the influence of surface waves on the τ in the extreme sea state is still not fully understood due to limited observations. Measurements including both τ and surface waves under tropical cyclones will thus be useful for guiding the parameterizations of air–sea momentum fluxes in the extreme sea state.

Under the storms, the propagation direction of the swell on the right of the track is more aligned with the local wind than in the front-left quadrant (Young 2003). The region on the right of the storm track is thus defined as the along-swell sector, and that in the front-left quadrant as the cross-swell sector (Black et al. 2007; Holthuijsen et al. 2012). The wind duration and fetch in the along-swell sector are longer than those in the cross-swell sector, termed the “extended fetch” effect (Young 2003; Young and Vinoth 2013), which is favorable to the growth of surface waves. It will yield higher significant wave height and younger wave age in the along-swell sector of tropical cyclones than those in the cross-swell sector. In contrast to the bulk properties of surface waves, the spatial variation of surface wave directional spectra under storms can be even more complicated. Most spectra of surface waves concentrate the energy within a narrow band of frequency (∼0.08–0.12 Hz) in the along-swell sector, but some wave spectra at the location far away from the eye are bimodal in direction due to the influence of propagating swell (Hu and Chen 2011).

When the wind blows over a calm sea state, the capillary waves are generated via the resonance between a fluctuating pressure field and surface waves (i.e., wind waves; Phillips 1957). Only part of the wind momentum downward transports via Reynolds flux. The breaking of these ephemeral capillary waves will then dissipate the wave momentum into the ocean as stress, which is the main pathway for the downward momentum flux felt by the ocean (the oceanic stress in this study; Jones and Toba 2012). Because the energy of fetch-limited waves is proportional to the wind speed |U10| (the wind at 10-m height above the sea surface), |U10| is often used for parameterizing the drag coefficient Cd in the bulk formula of surface wind stress |τ| = ρairCd|U10|2 (Table 1; Large and Pond 1981; Powell et al. 2003; Black et al. 2007; Bryant and Akbar 2016), where ρair is the air density. At low wind speeds |U10| < 20 m s−1, Cd increases with increasing |U10| (e.g., Edson et al. 2013; Richter et al. 2021). At extreme wind speeds |U10| > 40 m s−1, Cd can be either a constant or decrease with increasing |U10|. Despite the consistent trend of Cd = f(|U10|) between different studies (Fig. 1 in Hsu et al. 2017), the values of Cd can vary from 1.5 × 10−3 to 4.5 × 10−3 at |U10| = 25–40 m s−1. Although the uncertainty in tropical cyclones’ wind fields may result in the scattering of Cd (Zhou et al. 2022), complex surface wave fields can be an alternative cause for changing the equilibrium range of wind waves (D’Asaro 2015; Hwang et al. 2017).

Table 1

List of abbreviations in this study.

Table 1

Several studies then discuss how surface waves may influence the dependence of Cd on |U10| in different sectors of storms. Holthuijsen et al. (2012) report that Cd in the cross-swell sector is larger than that in the along-swell sector at |U10| = 25–40 m s−1, assuming the wind speed increases with height logarithmically. However, other model studies argue that Cd in the along-swell sector should be larger than that in the cross-swell sector because the variation of surface wave breaking may affect the magnitude of form drag (e.g., Moon et al. 2004; Chen et al. 2013). According to the observed τ by Hsu et al. (2019), the drag coefficient in the front-right quadrant of five storms (i.e., within the along-swell sector) varies from 2.0 × 10−3 to 3.2 × 10−3 at |U10| = 25–40 m s−1, higher than that in the front-left quadrant (i.e., within the cross-swell sector). The effective wind duration ζ, which considers the wave effect, is thus proposed for reducing the scattering of Cd from different tropical cyclones. Zhou et al. (2022) simulate the surface wave field under these five storms using an atmosphere–wave–ocean coupled model. The Cd is inversely proportional to the angle between the wind and dominant wave direction ψ when 45° < ψ < 90°. The wave age (the ratio of phase speed of dominant waves to wind speed) may have negligible influence on the Cd under tropical cyclones, different from the dependence of Cd on wave age at low wind speeds (e.g., Geernaert et al. 1986). Measuring wave propagation direction is critical for improving the parameterization of Cd.

Note that surface wind stress τ is the forcing at the air–sea interface, but the anemometers capture the wind several meters above the sea surface (e.g., mounted on a buoy). The stress τ may not be in the same direction as the measured wind, though the traditional Monin–Obukhov similarity theory assumes wind direction in the atmospheric boundary layer does not change with height (Geernaert 1988; Drennan et al. 1999; Grachev et al. 2003; Zhang et al. 2009; Chen et al. 2019). On the left of Typhoon Chaba’s track in 2010, the τ can be clockwise from the wind direction at a buoy with an angle ϕ > 30° (Potter et al. 2015). Other studies also report the misalignment |ϕ| between the wind and τ under tropical cyclones (Hsu et al. 2017, 2019). Assuming the alignment between τ and U10 in the bulk formula of τ or Monin–Obukhov similarity theory near the eyewall of storms may not be reliable (Richter et al. 2021). Although some model studies are devoted to simulating ϕ under the storms, the results of |ϕ| are mostly <10° outside the eyewall (e.g., Chen et al. 2013; Reichl et al. 2014). Collins et al. (2018) explore the propagation direction of waves at different frequency bands. Similar to the orientation of τ (Potter et al. 2015), high-frequency wind waves propagate in directions between the wind and dominant waves when the peak waves deviate from the wind direction significantly. The nonlinear wave–wave interaction may affect the directional spreading of high-frequency waves (Young 2006; Tamizi and Young 2020), thereby the orientation of downward momentum flux.

In 2010, the Impact of Typhoons on the Ocean in the Pacific (ITOP) programs (D’Asaro et al. 2014) conducted a series of experiments under several typhoons (Potter et al. 2015; Hsu et al. 2017, 2018). Seven Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats were deployed from the aircraft on 16 October 2010. After one day of the float deployment, Megi’s eye passed the southern edge of the float array (Hsu et al. 2017). The τ was estimated by integrating the downward momentum flux captured by the floats (Hsu et al. 2017, 2019). Hsu (2021) later developed a new method for estimating the directional spectra of surface waves at the EM-APEX floats under Megi. The observed spatial variation of surface waves agreed with the previous observations well (e.g., Wright et al. 2001).

Therefore, this study will further discuss the roles of wave propagation direction in the variations of τ under Megi by using the measurements of τ and waves on three EM-APEX floats. Because the curl of surface wind stress τ can affect storm-induced SST cooling via divergence under the eye, the results will favor future model forecasts on tropical cyclone–wave–ocean interaction. Section 2 will describe the methods for measuring wind, surface wind stress, and surface wave spectra under Megi. The results at each float will be presented in section 3. Section 4 will explore the possible factors causing the misalignment between the wind and τ in the view of surface waves. Sections 5 and 6 will report and discuss the results of drag coefficients in different bins regarding the wave directions in the inverse wave age.

2. Methods for measuring wind, stress, and surface waves under Megi

Megi was formed around 12 October 2010, and its maximum wind speed could be more than 85 m s−1 at a radius of 15 km during its translation in the western Pacific (Fig. 1). The ITOP program deployed EM-APEX floats and more than two hundred GPS dropsondes under Megi using the C130 aircraft (D’Asaro et al. 2014; Hsu et al. 2017). Along the flight track, the stepped frequency microwave radiometer (SFMR) mounted on the bottom of the aircraft measured the brightness temperature, and thereby the wind speed at 10-m height above the sea surface |U10|. As autonomous vehicles, EM-APEX floats could continuously profile the temperature, salinity, and horizontal current velocity (Sanford et al. 2005). The floats stayed below 30-m depth during the passage of Megi to protect floats against strong turbulence. The data were then transmitted via Iridium satellites. We also collected the raw voltage measurements via ship around 19 October 2010. The section will describe the methods for deriving the wind, surface wind stress, and surface wave spectra under Megi and show part of the results of wind and surface wind stress.

Fig. 1.
Fig. 1.

(a) Trajectories of three EM-APEX floats (colored lines), Megi’s track (black line with arrow), and interpolated wind map (contour lines: wind speed at 10-m height above the sea surface |U10|; blue arrows: wind vector) at the time when Megi’s eye passes the floats. The results of (b) wind speed, (c) wind direction, (d) downwind surface wind stress τ, and (e) the angle ϕ between the wind and surface wind stress at the float positions are from Hsu et al. (2017). The black dashed line in (b) marks the wind speed |U10| at 25 m s−1, and the gray area in (d) and (e) indicates unreliable estimates of surface wind stress after the passage of Megi’s eye. Dashed lines with error bars in (c) are the wind direction computed using the results from Zhang and Uhlhorn (2012).

Citation: Journal of Physical Oceanography 53, 7; 10.1175/JPO-D-22-0193.1

a. Wind map processing

Hsu et al. (2017) reconstruct three wind maps of Megi using the wind measurements taken by the SFMR and dropsondes, assuming the symmetric eyewall. Three floats (em3763c, em4913a, and em3766c) that capture ocean response at |U10| > 25 m s−1 are used for the analysis in this study. The maximum |U10| at em3763c can be up to 65 m s−1. The results are similar to the model-simulated wind field in Zhou et al. (2022). Note that the radial component of U10 (i.e., with an inflow angle) under a tropical cyclone is not negligible. Because the inflow angle can vary spatially under a storm (Zhang and Uhlhorn 2012), we derive the wind direction and inflow angle using the dropsondes at the front-right quadrant of Megi. The results of the inflow angle are then used for reconstructing the wind direction (Hsu et al. 2017). The estimated wind direction is similar to those from an empirical model of inflow angle (Zhang and Uhlhorn 2012; Fig. 1c).

b. Surface wind stress and drag coefficient in linear momentum budget

Several studies have used the linear momentum budget to estimate surface wind stress τ before the passage of tropical cyclones’ eyes (Jarosz et al. 2007; Sanford et al. 2011; Hsu et al. 2017, 2019), assuming the τ balanced with the integration of temporal change of horizontal ocean current v and Coriolis force in the upper ocean. The results of τ as the downward momentum flux in the ocean are the surface wind stress “felt” by the ocean current. Because τ may deviate from the wind direction (e.g., Chen et al. 2019), we define the τ as
τ=τU10̂+τU10̂ρ0H0(vt+f0k^×v)dz,
with
ϕ=tan1(τ/τ),
where τ is the crosswind surface wind stress in the direction perpendicular to the wind U10̂, τ is the downwind surface wind stress in the direction aligned with the wind U10̂, ρ0 is mean seawater density in the upper 100 m, f0 is the Coriolis frequency, and H is the integration depth set as 100 m. The ϕ is the orientation offset between the τ and wind, positive for the clockwise rotation of τ from the wind direction. The drag coefficients (crosswind drag coefficient C and downwind drag coefficient C) can be computed as
C=τρair|U10|2andC=τρair|U10|2,
where ρair is the air density.

In Hsu et al. (2017), the profiles of horizontal current velocity are gridded in 30-min intervals. The missing measurements in the upper 30 m are extrapolated using the mean of the available current data from 30- to 35-m depth. Because the surface mixed layer depth is about 50 m one day before the passage of Megi’s eye, it is reliable to assume that the strong wind has mixed the velocity field in the upper 30 m homogeneously. We also validate the assumption using floats under other tropical cyclones for which upper-30-m measurements are available, confirming the negligible influence of the extrapolation (Hsu et al. 2019). Because the floats measure not only the wind-driven current but also tides and background mean current, signals not due to the wind of Megi are excluded from the current velocity measurements first. The τ is derived by integrating the measurements of ∂u/∂t and fk × u [Eq. (1)] (Hsu et al. 2019). The uncertainties of estimated τ are mostly due to the tides. Details for correcting the effect of nonwind driven current are described in Hsu et al. (2017). In addition to the correction for deriving the wind-driven current, the roles of other momentum terms absent in the linear momentum budget, such as advection and pressure gradient terms, need to be quantified and excluded. According to the simulated ocean response to storm wind in the PWP3D model (Price et al. 1994), the change of horizontal pressure gradient due to the upwelling is negligible to the τ at the front-right quadrant of storms except under the eye. The results of τ at the floats are therefore corrected by using the simulated advection and pressure gradient terms in the PWP3D model, as that did in Hsu et al. (2017).

Under Megi, the downwind surface wind stress τ can be up to 8 N m−2 at the front-right quadrant of Megi (Fig. 1). The angle ϕ between the τ and wind [Eq. (2)] is mostly greater than 15° at |U10| < 40 m s−1. The corresponding C reaches the peak of ∼3.2 × 10−3 at |U10| = 25–40 m s−1 [Eq. (3)], similar to that under other storms (Zhou et al. 2022). In section 5, we will scale the results of drag coefficients as a new parameterization. 1000 simulations of drag coefficients are generated using the mean and standard deviation of drag coefficients at each float position, assuming a normal distribution based on the Monte Carlo method (Hsu et al. 2017). The 1000 realizations of the drag coefficient averages within each interval are derived by averaging the generated simulations. We can then compute the mean and standard deviation of the drag coefficient averages using 1000 realizations. The empirical results of drag coefficients can be used for simulating the storm-induced momentum flux in the ocean response models (e.g., Powell et al. 2003).

c. Estimation of surface wave spectra at EM-APEX floats

We use the 1-Hz electrode and magnetometer measurements for deriving horizontal velocity and vertical acceleration of surface waves under Megi (Hsu 2021). Only ascending profiles are used for wave analysis because the direction of wind waves may rotate rapidly during the period between two successive profiles. The spectrograms and normalized directional moments at each layer are derived via the cross spectrum (Kuik et al. 1988; D’Asaro 2015; Thomson et al. 2018) in each 120-s window. The biases of spectrograms due to the depth variation of wave measurements are corrected following D’Asaro (2015). The influence of overlapped measurements on bulk wave properties is negligible (Hsu 2021). We randomly average 80% of spectrograms 100 times to derive 100 realizations of one-dimensional wave energy spectra E(f), which are then used for computing the mean and standard deviation of E(f). The cutoff frequency of mean E(f) is determined if less than 25% realizations of E(f) containing values at that frequency band. The directional spectra of surface waves E(f, θ) are reconstructed using the mean of E(f) and normalized directional moments via the maximum entropy method (Thomson et al. 2018). The properties of surface waves can then be estimated using the results of E(f, θ), where f and θ are the frequency and direction bands, respectively (appendix).

3. Spatial variation of surface wind stress and surface waves

a. Surface wind stress and drag coefficients

According to the results of τ at three floats under Megi (Fig. 2), the misalignment ϕ between the τ and wind at all floats may decrease with the decreasing zonal distance to Megi’s eye |X|. As compared with the similar ϕ captured by three floats near the eyewall, the ϕ is not spatially homogeneous when |X| > 40 km. At two floats near Megi’s track (em4913a and em3763c), the direction of τ is clockwise (ϕ > 0) from the wind, unlike the negative ϕ found at em3766c (>70 km to the right of Megi’s track). Because of the limited results of negative ϕ, it is hard to conclude whether τ will change its orientation direction from the wind vector with the increasing meridional distance to the tropical cyclones’ track.

Fig. 2.
Fig. 2.

Results of surface wind stress τ (blue arrows), wind U10 (red arrows), (a) downwind drag coefficient C, and (b) crosswind drag coefficient C at three floats (dots connected with color lines) under Megi. The black contour lines are the wind speed |U10|. The coordinates of the maps are slightly rotated assuming the westward translation of Megi, although the motion of Megi is not completely westward during this period (∼181° clockwise from the east).

Citation: Journal of Physical Oceanography 53, 7; 10.1175/JPO-D-22-0193.1

The results of τ are used for computing the downwind C and crosswind drag coefficients C (Fig. 2). At |U10| = 25–40 m s−1, C at em4913a is less than 2.0 × 10−3, slightly smaller than that at em3766c. At em3763c, which captures the ocean response under the eye of Megi, C first increases with the increasing |U10| (or decreasing meridional distance to the track), and then saturated at |U10| > 40 m s−1 (or near the eyewall of Megi). Similar to the change of ϕ before the passage of Megi’s eye, C decreases with the decreasing |X|. C can be more than 2.0 × 10−3 at |U10| < 40 m s−1, but generally less than 1.0 × 10−3 at |U10| > 40 m s−1. At |U10| < 40 m s−1, although C at em3763c is slightly larger than that at em4913a, C at these two floats is similar. The spatial variations of C and C should result from different factors.

b. Spectra and bulk properties of surface waves

The directional spectra of surface waves E(f, θ) are estimated using the upper-70-m measurements in the ascending profiles (Fig. 3). The results of E(f, θ) are in good agreement with previous studies (Hsu 2021). At X < −100 km, the surface wave spectra are mainly constituted by two groups of dominant waves with similar frequencies but different propagation directions. One of them propagates westward, and the other travels northward. The swell that propagates northward deviates from the wind direction significantly. The wave direction at the two floats on the right of Megi’s track (em4913a and em3766c) is more westward than on the track (em3763c). When Megi moves closer to the floats (X > −80 km), E(f, θ) becomes more monomodal in both direction and frequency bands, but the wave energy may spread in the wider range of direction bands. Interestingly, dominant waves at X > −80 km propagate in the direction between the wind and northward propagating swell. After strong wind induces wind waves locally, the nonlinear wave–wave interaction will transport energy from the wind direction to the swell direction (Young 2006), as the E(f, θ) at X ∼ −40 km. That is, the evolution of the surface wave field captured by the floats may involve complicated interactions between the wind and swell.

Fig. 3.
Fig. 3.

Directional spectra of surface waves E(f, θ) using measurements at each ascending profile (small circular plots) at three floats under Megi. The direction bands of E(f, θ) indicate the direction for waves propagating ahead. The red arrows are the wind direction, and the color shading is the wind speed |U10|. The coordinates of the maps are slightly rotated assuming the westward translation of Megi (black arrow), although the motion of Megi is not completely westward during this period (∼181° clockwise from the east).

Citation: Journal of Physical Oceanography 53, 7; 10.1175/JPO-D-22-0193.1

The one-dimensional wave energy spectra [E(f) and E(θ) in the appendix] are computed by using the results of E(f, θ) (Fig. 4). At X < −160 km, most wave energy of E(f) concentrates at the peak frequency fp ∼ 0.10 Hz. The waves at the location far away from the track (em3766c) are more monomodal in frequency than those at the other two floats. When the eye of Megi moves closer to the floats, different fp is found at different floats. At em3766c, the fp decreases to 0.08 Hz near the eyewall. At em3763c, part of the dominant waves’ frequency is at 0.10 Hz consistently, but the frequency band that contains the other group of high-energy waves is downshifted to 0.08 Hz. The decrease of fp is mainly due to the growth of duration-limited waves (Fontaine 2013). At X < −160 km, part of the dominant waves propagates westward, and the other part propagates northward, clockwise from the westward-propagating waves (Figs. 4d–f). Because the wind direction on the track of Megi (em3763c) should not turn northward until the passage of the eye, these northwestward-propagating waves may come from the rear-right quadrant of the storm (Hwang et al. 2017). Waves near the track of Megi remain the bimodal features in frequency and direction until X > −60 km, presumably due to the increasing wind speed and alignment between the wind and swell.

Fig. 4.
Fig. 4.

One-dimensional wave energy spectra (a)–(c) E(f) and (d)–(f) E(θ) at floats (top) em3766c, (middle) em4913a, and (bottom) em3763c with regard to their zonal distance to Megi’s eye, computed using the results of directional spectra of surface waves E(f, θ). The black dashed lines are the time when Megi’s eye passes the floats. The blue lines with dots are the estimated peak frequency fp (appendix).

Citation: Journal of Physical Oceanography 53, 7; 10.1175/JPO-D-22-0193.1

At X < −160 km, the significant wave height Hs at em4913a is slightly lower than that at the other two floats (appendix; Fig. 5) due to the energy difference in low-frequency bands (Fig. 4). The observed Hs then increases up to 12 m at em3763c on the track of Megi, consistent with the increase of wind speed. Because the period of the peak waves Tp is similar at all floats, the change of wind speed decreases the wave age cp/|U10|. The ratio r of the downwind mean square slope of surface waves to that in the crosswind direction is about one at X < −160 km, consistent with the previous results at the wind speed of 20 m s−1 (appendix; Hwang and Fan 2018). The waves with strong bimodal directional distribution may yield the crosswind component of the mean square slope equivalent to the downwind component (Hwang and Wang 2001). At X > −100 km, r at two floats on the right of Megi’s track increases to more than 1.25. That is, the waves may become more monomodal due to the increasing wind forcing (Hwang and Wang 2001). The timing for the increase of r is consistent with that for decreasing wave age. The difference between the mean wave direction θm and the peak direction θp is mostly within 45°.

Fig. 5.
Fig. 5.

(a) Significant wave height Hs, (b) mean wave period Tm, (c) wave age cp/|U10|, (d) ratio r of mean square slope in the downwind direction to that in the crosswind directions, and (e) mean wave direction θm (solid lines labeled on the left axis) and wave direction at the peak frequency θp (dashed lines labeled on the right axis) at three floats (colored lines with dots). See the appendix for the computation of wave properties.

Citation: Journal of Physical Oceanography 53, 7; 10.1175/JPO-D-22-0193.1

4. Misalignment between wind, stress, and waves

Hsu et al. (2019) estimate the surface wind stress τ using float measurements under five tropical cyclones. Although the influence of dominant wave direction θp on the variation of drag coefficients from five tropical cyclones at |U10| = 25–40 m s−1 has been demonstrated by Zhou et al. (2022), it will be interesting to study how waves at different frequencies result in the change of momentum transfer efficiency under a single storm, especially for the misalignment |ϕ| between the wind and τ (section 1). Below, we will discuss the effect of wave direction and roughness on τ at |U10| = 25–40 m s−1, the crucial wind speed regime for the wind stress curl to induce maximum SST cooling (Pun et al. 2021).

a. Variations of waves at different frequency bands

We compute the mean propagation direction θ of surface waves at the high-frequency (subscript h as θh) and low-frequency bands (subscript l as θl), respectively (Fig. 6), by using the results of one-dimensional energy spectra E(f) and dominant wave direction at each frequency band θ1(f) (appendix). The frequency bands of high-frequency waves span from the peak frequency fp to 0.16 Hz and low-frequency waves from 0.02 Hz to fp. As compared with the definitions of wind waves at the frequency bands >1.5 × fp (Collins et al. 2018) or 2.5 × fp (Young 2006) in some previous studies, one may regard the θh as the mean wave directions during the energy cascade from wind waves to dominant waves (Tamizi and Young 2020). Because the fp is ∼0.09 Hz before the passage of Megi’s eye, the mean wave properties are derived from the results in at least five frequency bands. The θh, which is directly affected by the energy cascade from wind waves, rotates more rapidly than the θl of the swell during the storm translation. Near the eyewall, most wave energy concentrates in narrow bands of frequency, so the angle Δθhl between θh and θl decreases with the decreasing |X|.

Fig. 6.
Fig. 6.

Mean wave direction at low-frequency (green arrows) and high-frequency (magenta arrows) bands, surface wind stress τ (blue arrows), and wind U10 under Megi. The color dots mark the angle between the wind and stress, positive for τ clockwise from the wind direction. The black contour lines are the wind speed |U10|. The coordinates of the maps are converted assuming the westward translation of Megi, although Megi moves almost westward during this period.

Citation: Journal of Physical Oceanography 53, 7; 10.1175/JPO-D-22-0193.1

Under Megi, θl is >90° clockwise from the wind direction until X < −20 km (Fig. 7). The temporal change of Δθhw (the difference between the wind and θh) is more significant than Δθlw (the difference between the wind and θl). At em3763c, which is near the cross-swell sector of Megi, the observed Δθlw and Δθpw are smaller than those at the other two floats (within the along-swell sector). The misalignment between the wind and waves can be complicated even within the along-swell sector of tropical cyclones. Because Δθhw < 30° at em3763c, Δθhl can be more than 90°, larger than that at the other two floats. The Δθhl also increases with the decreasing distance to the right of Megi’s track. Although the breaking of high-frequency waves is crucial for downward momentum flux in the ocean (Melville and Rapp 1985), it is hard to directly correlate the change of ϕ with either Δθhw or Δθhl. There may be multiple factors for altering the orientation of τ.

Fig. 7.
Fig. 7.

Times series of the (a) positive angle ϕ clockwise from the wind to surface wind stress, (b) positive angle Δθpw clockwise from the wind to dominant waves, (c) positive angle Δθlw clockwise from the wind to low-frequency waves, (d) positive angle Δθhw clockwise from the wind to high-frequency waves, (e) positive angle Δθhl clockwise from the high-frequency waves to low-frequency waves, and (f) the ratio of mean square slope in the downwind direction to that in the crosswind direction at the high-frequency bands rh (solid lines labeled on the left axis) and low-frequency bands rl (dashed lines on the right axis) at three floats (color lines). The black-outlined circles indicate the results at wind speed |U10| > 25 m s−1.

Citation: Journal of Physical Oceanography 53, 7; 10.1175/JPO-D-22-0193.1

Besides the wave direction, we compute the ratio of the downwind mean square slope to that in the crosswind directions at the high- (rh) and low-frequency (rl) bands (appendix), respectively. The rh may increase with the decreasing distance to the track of Megi. The rl remains almost constant at 1, consistent with previous results at low wind speed regimes (Hwang and Wang 2001). When the crosswind component of the mean square slope becomes more significant, the wave spectra will be more bimodal directional distribution (Hwang and Wang 2001). Increasing wind speed with the decreasing distance to the eyewall of Megi may enhance the spectral energy above fp as a monomodal feature. Since rl does not vary with the wind speed, the low-frequency waves should remain in the feature of bimodal spreading regardless of the change of wind speed. Wind mainly forces the growth of short waves in the generation of τ (Jones and Toba 2012). Note that ϕ > 30° occurs when rh > 1.25. The misalignment between the τ and wind should be associated with the evolution of wave energy at high-frequency bands.

b. Dependence of the angle between the wind and τ

Wave directions and ocean surface roughness are used to explore the variation of absolute angle |ϕ| at |U10| = 25–40 m s−1 (Fig. 8). The correlation coefficient between |ϕ| and Δθpw is 0.79 within the 95% confidence interval, consistent with the results in Zhou et al. (2022). The low-frequency swell may alter the propagation direction of dominant waves at the location far away from the eyewall. For the monomodal spectra near the eyewall (|Δθhl| < 30°), the |ϕ| is less than 20° (Fig. 8b). At 30° < Δθhl < 90° due to the bimodal spectra located far from the eyewall, the direction of τ is similar to the high-frequency waves (the mean and standard deviation of the difference between positive ϕ and Δθhw is 2.6 ± 9.8°). The correlation coefficient between |ϕ| and Δθhw is 0.61 within the 95% confidence interval. However, even if the wind is in a similar direction to the high-frequency waves (i.e., |Δθhw| is small), a significant orientation offset Δθhl > 90° between the high and low-frequency waves may still alter the stress from the wind direction. Because the sea state under tropical cyclones is not a pure wind sea, the presence of low-frequency swell under tropical cyclones may affect the high-frequency wind waves, thereby stress direction.

Fig. 8.
Fig. 8.

Dependence of the misalignment |ϕ| between the wind and stress on the (a) positive angle Δθpw clockwise from the wind to dominant waves, (b) positive angle Δθhw clockwise from the wind to high-frequency waves, (c) positive angle Δθlw clockwise from the wind to low-frequency waves, (d) the ratio of mean square slope in the downwind direction to that in the crosswind direction at the high-frequency bands rh, and (e) the ratio of significant wave height to the wavelength at peak frequency. The results at each float position are marked by the color based on the angle from low-frequency waves to high-frequency waves Δθhl. The symbols of upward-pointing triangles mark the results of ϕ < 0.

Citation: Journal of Physical Oceanography 53, 7; 10.1175/JPO-D-22-0193.1

The ratio of mean square slope at the high-frequency bands rh, and the ratio of significant wave height Hs to wavelength λp at the peak frequency fp are used for studying the effects of ocean surface roughness on the |ϕ| (appendix; Figs. 8d,e). The correlation coefficient between rh and |ϕ| is 0.42 with the p value ∼0.07, slightly smaller than the dependence of |ϕ| on Δθhw at 30° < Δθhl < 90°. The wave directions may affect |ϕ| more significantly than the change of ocean surface roughness. On the other hand, rh can be greater than 1.25 at the two floats near the track of Megi, where the |U10| increases rapidly to more than 35 m s−1 within a few hours. Significant |ϕ| and the change of the monomodal feature at high-frequency bands may occur simultaneously. Note that Hs generally increases with the increasing wind speed, so Hs/λp can be regarded as the inverse wave age. As compared with the dependence of |ϕ| on rh, the change of |ϕ| is irrelevant to the variation of Hs/λp, similar to the negligible effect of Hs/λp on the downwind drag coefficient (Zhou et al. 2022). The misalignment between the wind and τ may be relevant to the change of fine-scale ocean surface roughness instead of the slope of dominant waves, in agreement with the previous results on the importance of the breaking of short waves (Jones and Toba 2012).

5. Scaling of drag coefficients using wave age in propagation directions

Previous studies have emphasized the important role of wind waves in the generation of downward momentum flux in the ocean (Jones and Toba 2012; Takagaki et al. 2012). The wave age (cp/|U10|, where cp is the phase speed of dominant waves) is used as the ocean surface roughness for exploring the variation of drag coefficients (e.g., Geernaert et al. 1986; Oost et al. 2002; Grare et al. 2013). Younger waves tend to generate rougher ocean surface and higher drag coefficients. However, because the storm translation causes the rapid rotation of wind vectors, the wind under tropical cyclones is not aligned with the propagating swell. Some studies have noticed that the swell can complicate the dependence of the drag coefficient on wave age (Donelan et al. 1997; Drennan et al. 2003). Its fast group velocity may affect the wind forcing on the surface wave field (Donelan et al. 2012), similar to the effect of surface current on the drag coefficient (Edson et al. 2013). Below, the inverse wave age based on the wind components in the directions of different waves |U10| cos(Δθaw)/ca (Miles 1957; Grachev and Fairall 2001) will be used for scaling drag coefficients (subscript a = l for waves at low-frequency bands; a = h for waves at high-frequency bands; a = p for waves at peak frequency), where ca = g/ωa is the phase speed for waves at deep water and g the gravity (appendix).

As compared with the negligible correlation coefficient R between C and |U10| (Fig. 9), the inverse wave age involving alignment between wind and waves may significantly affect the spatial variation of C. For the dependence of C on high-frequency waves at |U10| cos(Δθhw)/ch = 0.5–2.5, the R can be 0.60 within the 95% confidence interval, consistent with the previous finding on the important roles of wind waves in τ (Jones and Toba 2012). On the other hand, the R between C and the inverse wave age of dominant waves is 0.66 within the 95% confidence interval, even higher than either high- or low-frequency waves. Part of the downwind momentum flux should be affected by low-frequency swell because the evolution of dominant waves depends on the direction of low-frequency waves significantly (Young 2006).

Fig. 9.
Fig. 9.

Dependence of the downwind drag coefficient C on the (a) wind speed |U10|, (b) wind forcing on the slope of dominant waves |U10| cos(Δθpw), (c) wind forcing on the slope of high-frequency waves |U10| cos(Δθhw), and (d) wind forcing on the slope of low-frequency waves |U10| cos(Δθlw). The vertical red bars in (b) mark the standard deviation of bin-averaged results of C in each bin (horizontal red bars). The correlation coefficient and the corresponding p value are labeled in a lower corner of each panel.

Citation: Journal of Physical Oceanography 53, 7; 10.1175/JPO-D-22-0193.1

We then explore how the inverse wave age may affect the absolute values of |C| (Fig. 10). Most |C| decreases with the increasing |U10|, presumably due to the decreasing distance to the storm eye. The correlation coefficient R between |C| and |U10| cos(Δθlw)/cl of −0.57 within 95% confidence level is higher than between |C| and |U10|. Using the propagation direction of the low-frequency swell may improve the prediction of |C|. Note that |C| is mostly greater than 1.0 × 10−3 at the negative |U10| cos(Δθlw)/cl but may be less than 0.5 × 10−3 at positive |U10| cos(Δθlw)/cl. The presence of crosswind stress τ at Δθlw > 90°, i.e., part of the wind is opposite to the swell direction, demonstrates the importance of the wind-swell misalignment on stress direction (Donelan et al. 1997; Grachev et al. 2003). Because the variation of |C| is nearly irrelevant to the wave age of high-frequency waves, the dynamics for generating the crosswind stress may be negligibly affected by the local wind waves (Grachev et al. 2003).

Fig. 10.
Fig. 10.

Dependence of the crosswind drag coefficient C on the (a) wind speed |U10|, (b) wind forcing on the slope of dominant waves |U10| cos(Δθpw), (c) wind forcing on the slope of high-frequency waves |U10| cos(Δθhw), and (d) wind forcing on the slope of low-frequency waves |U10| cos(Δθlw). The vertical red bars in (b) mark the standard deviation of bin-averaged results of C in each bin (horizontal red bars). The correlation coefficient and the corresponding p value are labeled in the lower-right corner of each panel. The symbols of upward-pointing triangles mark the results of C < 0.

Citation: Journal of Physical Oceanography 53, 7; 10.1175/JPO-D-22-0193.1

Because of the significant effects of wave directions on the wind forcing in the inverse wave age, we compute the mean and standard deviation of bin-averaged C and |C| when the correlation coefficient R is >0.5 and within the 95% confidence interval (section 2; Table 2). At the negative inverse wave age for dominant waves [|U10| cos(Δθpw)/cp = −0.9–0], C is (1.9 ± 0.6) × 10−3, occurring when large Δθlw yields Δθpw > 90°. At the old wave age of dominant waves [|U10| cos(Δθpw)/cp = 0–0.5], C is (3.3 ± 0.6) × 10−3. If the wave age becomes younger at |U10| cos(Δθpw)/cp > 0.5 due to either increasing wind speed or alignment between wind and dominant waves, C is (3.9 ± 0.7) × 10−3. The dependence of downwind C on wave age is consistent with the previous studies. At the negative inverse wave age for low-frequency waves [|U10| cos(Δθlw)/cl < 0], |C| is (1.5 ± 0.3) × 10−3. At the positive and old wave age of long waves [|U10| cos(Δθlw)/cl = 0–0.5], |C| is (0.5 ± 0.5) × 10−3. Although bin-averaged |C| at |U10| cos(Δθlw)/cl > 0.5 is not computed due to insufficient results, the |C| should decrease with decreasing wave age of low-frequency waves, according to the decreasing ϕ at |U10| > 40 m s−1.

Table 2

Bin-averaged drag coefficient using intervals of inverse wave age involving wave directions at the wind speed |U10| = 25–40 m s−1.

Table 2

In short, a new parameterization of drag coefficients is proposed using the data at the front-right quadrant of Megi. We believe the results can be applied to the other storm quadrants because the inverse wave age at different frequencies involves the dynamics in the wave growth and breaking. For example, C (or |C|) at the front-left quadrant of the storm, where the wind is significantly misaligned with the dominant wave direction, should be lower (higher for |C|) than that at the front-right quadrant, where the wind is more aligned with the swell (Hsu et al. 2019). Even for those tropical cyclone–wave–ocean coupled models, this drag coefficient parameterization is useful for guiding the model results of waves and stress. Note that the correlation coefficient of C to the dominant or low-frequency waves differs from the |C|. There should be multiple factors for changing the momentum transfer efficiency in the downwind and crosswind directions, which will be discussed in the next section.

6. Discussion on the directions of waves and stress under tropical cyclones

This study reports the strong correlation between the drag coefficients and inverse wave age under Megi. The heterogeneity of wave directions should affect the misalignment ϕ between the wind and stress τ significantly. Under Megi, most high-frequency waves propagate in the direction between the wind and low-frequency waves (or dominant waves). When waves are bimodal in frequency and direction at a location far from the eyewall (Δθhl > 30°), the orientation of τ changes with the increasing angle between the wind and dominant waves. When waves concentrate the spectral energy in a narrow band of frequency and direction near the eyewall (Δθhl < 30°), the τ is nearly aligned with the wind. Below, we will discuss why the direction of high-frequency waves may deviate from the wind direction. It will help explain the dependence of downwind C on the inverse wave age of dominant waves (or high-frequency waves). Then, the interaction between the wind and low-frequency waves (swell) will be studied to understand the variations of crosswind |C|.

The translation of tropical cyclones causes the propagating swell clockwise from the wind at all quadrants under storms (Young 2003; Hwang et al. 2017). When the wind blows over the oscillating air–sea interface due to the swell, the water surface at the leeward side will experience less wind than that at the windward side (termed airflow separation zone; Ebuchi et al. 1993; Fig. 11a). Although a portion of the airflow separates from the water surface, another part can still generate wind waves on the leeward side when the swell is not too high (at a location far from the eyewall). The nonlinear wave–wave interaction then leads to the energy cascade from wind waves toward the frequency and direction of the swell (Young 2006). It results in the skewness of the directional spectra toward low-frequency bands, so the direction of high-frequency waves (or dominant waves) should be clockwise from the wind direction. The wind components aligned with these high-frequency waves, i.e., the inverse wave age of high-frequency waves |U10| cos(Δθhw)/ch, will enhance wave growth, thereby wave breaking for releasing momentum. Because high-frequency waves propagate clockwise from the wind direction, most τ caused by the breaking of high-frequency waves is also clockwise from the wind (em3763c and em4913a), in good agreement with the clockwise-orientated τ in previous studies (e.g., Potter et al. 2015; Hsu et al. 2019). Because the low-frequency swell affects the directional spreading of high-frequency waves, the components of τ (and drag coefficients) in the downwind/crosswind direction may also decrease/increase with the increasing Δθlw (Fig. 8), even if a constant |U10| cos(Δθhw)/ch should result in a similar magnitude of drag coefficient.

Fig. 11.
Fig. 11.

Schematic illustrations of (a) the interaction between the wind and low-frequency swell and (b) the phenomenon of airflow separation regarding the steepness of dominant waves. The photographs in (b) are used to illustrate how the sea state may affect the roughness felt by the wind.

Citation: Journal of Physical Oceanography 53, 7; 10.1175/JPO-D-22-0193.1

On the other hand, although the nonlinear wave–wave interaction is useful for explaining the dependence of crosswind |C| on the orientation of high-frequency waves, it cannot answer why the magnitude of |C| at negative |U10| cos(Δθlw)/clθlw > 90°) is at least twice higher than that at the positive |U10| cos(Δθlw)/clθlw < 90°). The observed angle Δθlw between the wind and low-frequency waves is from 80° to 120° (Fig. 7a). When the low-frequency swell propagates in the direction opposite to a portion of the wind (Δθlw > 90°), the wind forcing may result in more efficient growth and breaking of low-frequency swell (Donelan et al. 2012). The breaking of low-frequency waves then releases momentum that induces an additional stress in the direction of low-frequency waves, causing an increase of |C|. The swell direction relative to the wind, i.e., following swell versus counter swell, should play an important role in the crosswind momentum flux (Donelan et al. 1997; Grachev and Fairall 2001; Grachev et al. 2003). Although the breaking of low-frequency swells in the counter-swell sea state (Δθlw > 90°) may also lead to a stress in the downwind direction, its magnitude should be relatively small in comparison with the momentum flux from the breaking of short waves and Reynolds stress under strong wind conditions (Donelan et al. 1997; Jones and Toba 2012). The C in the counter-swell sea state does not differ from the following-swell state significantly (Fig. 9d).

Note that some estimates of τ at em3766c (>70 km to the right of the track when Δθlw > 90°) are counterclockwise (ϕ < 0) from the wind at |X| = 60–100 km (Fig. 6), unlike the positive ϕ found at the two floats near the track of Megi. Because of the limited results of negative ϕ, it is challenging to further explore why those estimates of τ at em3766c orientated differently than the other two floats. Grachev and Fairall (2001) have observed the upward transport of momentum from the strong swell to the low-speed winds. This means that crosswind swell may induce a “negative” crosswind stress (i.e., counterclockwise from the wind direction) from the perspective of the ocean if an upward momentum transport occurs. As those counterclockwise-oriented τ found in the environment with strong propagating swell far away from the eyewall of Megi, it is reasonable to suspect that the energy of low-frequency swell relative to the wind in the counter-swell sea state may also affect the orientation direction of τ.

Near the eyewall of Megi, strong wind is more aligned with the observed direction of dominant waves than other locations far from the eye. The alignment between the wind and dominant waves (Young 2003) will increase the wave steepness and then wave breaking. Due to these breaking waves near the eyewall, the wind may reattach to the water surface at the crest or windward side of the preceding wave instead of blowing over the trough—the separation zone at the leeward side increases (Fig. 11b). Donelan et al. (2004) argue that this phenomenon of separation between the airflow and water surface may lead to the saturation of momentum transfer efficiency at the wind speed >33 m s−1. Therefore, when the waves become more monomodal in direction and frequency near the eyewall (Δθhl < 30°), the wind may feel the aerodynamic roughness of the sea surface rather than the geometric roughness of high-frequency waves—only the Reynolds flux of wind can induce stress. It may cause the saturation of the drag coefficient and the alignment between the τ and wind direction (regardless of the Δθhw).

A flowchart is used to summarize how wave directions in the inverse wave age may affect the pathways of downward momentum flux (Fig. 12; more details can be found in Jones and Toba 2012), assuming the constant inverse wave age. When the wind can feel the geometric roughness of waves, the wave breaking due to the increase of inverse wave age of high-frequency waves may enhance the momentum transfer efficiency, i.e., drag coefficients. However, the large angle of Δθlw may cause an increase of C/|C| less/more than that at the small Δθlw. On the other hand, θlw > 90° as part of the wind in the counterswell sea state may enhance |C|, as compared with the following-swell sea state at θlw < 90°. When the ocean surface is ubiquitous with the breaking of dominant waves near the eyewall of the storm, only the Reynolds flux of wind can induce the downward momentum flux, yielding the saturation of C and negligible |C|.

Fig. 12.
Fig. 12.

Flowchart for summarizing the hypothesis on the effect of inverse wave age on downwind drag coefficient C and magnitude of crosswind drag coefficient |C|. The physics in the red boxes occurs at all wind speeds, and that in the blue boxes occurs only when the wind can feel the geometric roughness of waves.

Citation: Journal of Physical Oceanography 53, 7; 10.1175/JPO-D-22-0193.1

Some atmosphere–wave–ocean coupled models have simulated form drag via the dissipation due to wave breaking (e.g., Tolman and Chalikov 1996; Donelan et al. 2012). The model results of τ constituted by the shear stress and form drag have an angle ϕ clockwise from the wind due to the propagating swell. Unfortunately, according to the previous analysis (Tamizi et al. 2021), wave models may underestimate the energy of low-frequency waves under tropical cyclones. The Discrete Interaction Approximation form of nonlinear wave–wave interaction terms based on the data at the low wind speeds may fail to predict the propagating swell’s energy. The unreliable prediction of high-frequency wave direction may then affect the stress, yielding simulated ϕ (Reichl et al. 2014) smaller than those observed by the field experiments (Potter et al. 2015). Improving the nonlinear wave–wave interaction parameterizations is thus critical to future forecasts on tropical cyclone–wave–ocean interaction.

7. Summary and conclusions

Seven EM-APEX floats were deployed under Typhoon Megi 2010, a devastating storm with a fast translation speed. Measurements taken by three floats in the along-swell sector of Megi are used for deriving surface wind stress τ and drag coefficients via linear momentum budget (Hsu et al. 2017). The bin-averaged results of drag coefficients and corresponding uncertainties have been studied carefully (Hsu et al. 2017; Zhou et al. 2022). Because of the recent progress in deriving the measurements of surface waves at EM-APEX floats (Hsu 2021), this study further discusses the effect of wave direction on the spatial variation of τ under Megi. Since C is saturated and ϕ is negligible at |U10| > 40 m s−1, the main objective is to find critical conditions for altering the orientation and magnitude of τ at |U10| = 25–40 m s−1 of a tropical cyclone, the wind speed regime that can significantly affect the maximum SST cooling via wind stress curl.

We explore the spatial variations of τ and directional spectra of surface waves at the front-right quadrant of Megi. When the distance of Megi’s eye to the floats |X| is >50 km, the results of τ are mostly >30° clockwise from the wind direction. The |ϕ| decreases with the decreasing distance of the eye to the floats. At |X| > 50 km, the spectra of surface waves are bimodal in directions with similar frequencies. The wave energy then concentrates in a narrow frequency band near the eyewall of Megi. At each float, the temporal change of wave direction at low-frequency bands is slower than at high-frequency bands. The directions of τ and high-frequency wind waves are mostly between the wind and low-frequency swell. The angle between the wind and waves decreases with the decreasing |X|.

The effect of wave directions on the misalignment |ϕ| between the wind and τ is then studied. The |ϕ| increases with the increasing angle between the wind and dominant waves. When high-frequency waves propagate in a similar direction to low-frequency swell near the eyewall of Megi, τ is nearly aligned with the wind (σ |ϕ| < 20°). At the location far from the eyewall, the τ is in a similar direction to the high-frequency waves unless the low-frequency swell significantly deviates from the wind direction. The low-frequency swell may affect the equilibrium of wind waves and thereby stress direction. We also study the effect of the ratio of mean square slope on the change of |ϕ|. The wave slope Hs/λp as the conventional inverse wave age should not affect the orientation of τ, similar to the negligible effect of Hs/λp on the drag coefficient (Zhou et al. 2022). The significant |ϕ| may occur when the wind enhances the downwind mean square slope at the high-frequency bands of surface waves.

The inverse wave age at different frequencies is used for exploring factors for the variations of downwind C and absolute values of crosswind drag coefficients |C|, respectively. Because the correlation coefficient between C and dominant waves is 0.66, the wind forcing on the slope of dominant waves may affect C under tropical cyclones more significantly than only the wind speed |U10|. The correlation coefficient between |C| and wind forcing on low-frequency waves is −0.57, different from the dependence of C on the low-frequency waves. Although wind is the most critical source for the growth of wind waves, the bimodal directional distribution under the storms may alter the wind forcing, thereby downward momentum released by the wave breaking. Therefore, the results of drag coefficients are scaled using the inverse wave age that wave directions have modulated. This new databased parameterization of drag coefficients will be useful for guiding the simulation of wind stress curl in tropical cyclone–wave–ocean coupled models.

As compared with previous studies using the float measurements under Megi (Hsu et al. 2017, 2019; Zhou et al. 2022), this study presents the importance of wave directions on the spatial variation of τ in the along-swell sector of the storm. Ocean response models often compute τ as a vector aligned with the storm wind (e.g., Price et al. 1994), which implicitly assumes the negligible effect of surface waves in the Monin–Obukhov theory. Our results, which highlight the surface wave effect on wind stress curl, should directly favor the prediction of maximum SST cooling and thereby storm intensification (Price et al. 1994; Emanuel 1995). Although deploying free-drifting floats from aircraft near the track of a storm is more challenging than other autonomous vehicles that can sail by themselves (Lenain and Melville 2014; Foltz et al. 2022), measurements at EM-APEX floats are valuable for studying the effect of surface waves on ocean response to tropical cyclones’ wind. Microstructure sensors have recently been mounted on some floats for profiling eddy diffusivity in the upper ocean (Kunze et al. 2021). Future studies may use floats to study the effect of wave-induced Langmuir circulation on turbulent mixing under storms.

Acknowledgments.

The author expresses his appreciation to the Ministry of Science and Technology in Taiwan for funding this work (110-2636-M-002-012-), the Office of Naval Research Physical Oceanography Program for their support of the ITOP experiment in 2010, and the 53rd Weather Reconnaissance Squadron for their deployment of the EM-APEX floats. In addition, the author extends his thanks to Dr. Sanford, the inventor of the EM-APEX floats; this work could not be finished without his visionary design of a float capable of profiling seawater motion.

Data availability statement.

Results of surface wind stress are available from the Mendeley data repository (https://doi.org/10.17632/gnc6hz2nwm.1). Wave data are available upon request.

APPENDIX

Estimation of Surface Waves Properties Using Spectra

This study estimates the directional spectra of surface waves E(f, θ) using the wave measurements taken by EM-APEX floats (Hsu 2021). The spectral estimates are available from 0.02 to 0.16 Hz. By integrating the E(f, θ) over the frequency and direction bands, respectively, we can derive the one-dimensional wave energy spectra as
E(f)=E(f,θ)dθandE(θ)=E(f,θ)df
The E(f) and E(θ) are used for discussing the temporal change of spectral energy at the frequency and direction bands, respectively.
Several bulk properties are derived using the results of E(f), including the significant wave height Hs as the mean wave height of the highest one-third of surface waves and the mean wave period (Thomson et al. 2018); that is,
Hs4E(f)dfandTm=E(f)dffE(f)df.
To minimize the unknown perturbation for biasing the estimates of peak frequency fp, which is the frequency of the spectral peak, we use the weighted method for computing fp (Young 1995; Collins et al. 2018); that is,
fp=fE4(f)dfE4(f)df.
The wavelength λp at fp can be computed using the dispersion relation of surface waves in the deep water.
During the estimation of directional spectra E(f, θ), we derive the dominant wave direction at each frequency band θ(f) using the normalized directional moments (Hsu 2021). The wave properties at different frequency bands can then be computed using the weighted-average method, similar to that for the mean wave period Tm, i.e.,
Xm=X(f)E(f)dfE(f)df,Xh=fp0.16X(f)E(f)dffp0.16E(f)df,andXl=0.02fpX(f)E(f)df0.02fpE(f)df
For example, the θl and ωl are the mean wave propagation direction and mean wave angular frequency at the low-frequency bands. Because the mean direction is computed based on the E(f) as the weighting factor, a narrow spectral peak of E(f) may yield similar estimates between the Xh and Xl.
The mean square slope relevant to the steepness of surface waves has been used for parameterizing the wave breaking in the model simulations (Kleiss and Melville 2010). Since the downward momentum flux depends on the equilibrium range of wind waves via wave breaking, the mean square slope in the downwind (MSSdw) and crosswind directions (MSScw) are computed (Hwang and Wang 2001) as
MSSdw=(kcosδθ)2E(f,δθ)dδθdk andMSScw=(ksinδθ)2E(f,δθ)dδθdk,
where k is the wavenumber, computed using the dispersion relation of surface waves in deep water, and δθ the angle between the wind and each direction band. The ratio r between the MSSdw and MSScw is used for exploring the change of ocean surface roughness due to the wind (Hwang and Fan 2018); that is,
r=MSSdw/MSScw.
The range for computing the ratio at the low-frequency (rl) and high-frequency (rh) bands is the same as the computation of the mean wave direction at different frequency bands.

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    • Export Citation
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  • Hsu, J.-Y., 2021: Observing surface wave directional spectra under Typhoon Megi (2010) using subsurface EM-APEX Floats. J. Atmos. Oceanic Technol., 38, 19491966, https://doi.org/10.1175/JTECH-D-20-0210.1.

    • Search Google Scholar
    • Export Citation
  • Hsu, J.-Y., R.-C. Lien, E. A. D’Asaro, and T. B. Sanford, 2017: Estimates of surface wind stress and drag coefficients in Typhoon Megi. J. Phys. Oceanogr., 47, 545565, https://doi.org/10.1175/JPO-D-16-0069.1.

    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Hwang, P. A., and D. W. Wang, 2001: Directional distributions and mean square slopes in the equilibrium and saturation ranges of the wave spectrum. J. Phys. Oceanogr., 31, 13461360, https://doi.org/10.1175/1520-0485(2001)031<1346:DDAMSS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • 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, 73597368, https://doi.org/10.1109/TGRS.2018.2850969.

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