## 1. Introduction

### a. Surface wind stress and drag coefficient

Tropical cyclones with strong winds, heavy rain, and storm surges produce severe damage to coastal and inland regions annually. To improve the prediction of tropical cyclones’ intensity, and the associated storm surges and precipitation, extensive studies have been devoted to typhoon–ocean interactions. The surface wind stress ** τ** generated by tropical cyclone winds extracts energy and momentum from the storm, limiting its intensity, but also forces ocean currents (Emanuel 1995). It is often parameterized by a drag coefficient

*C*

_{d}, expressed as |

**| =**

*τ**ρ*

_{air}

*C*

_{d}|

**U**

_{10}|

^{2}, where

*ρ*

_{air}is the air density, and |

**U**

_{10}| is the wind velocity at 10 m above the sea surface. Previous studies suggest various empirical forms of

*C*

_{d}as a function of |

**U**

_{10}|, atmospheric stability, surface roughness, surface wave height, and wave age (e.g., Charnock 1955; Dyer 1974; Johnson et al. 1998; Drennan et al. 2003). Better understanding of the surface wind stress and its parameterization

*C*

_{d}is thus crucial for forecasting tropical cyclones and improving the prediction of the oceanic response to them.

### b. Methods to estimate surface wind stress in tropical cyclones

Most previous studies have computed *C*_{d} under tropical cyclones from atmospheric measurements. The wind speed taken by the anemometers on buoys can be used to compute the momentum flux from tropical cyclones to the ocean (e.g., Potter et al. 2015). Powell et al. (2003) and Holthuijsen et al. (2012) estimated the surface roughness length and *C*_{d} using the profiles of tropical cyclones’ wind speed taken by GPS dropsondes, assuming the wind speed increases logarithmically with the height above the sea surface. Bell et al. (2012) studied the *C*_{d} using the atmospheric angular momentum budget and measurements of tropical cyclones’ wind speed taken by a stepped frequency microwave radiometer (SFMR) mounted on the aircraft.

Alternatively, *C*_{d} can be estimated from the momentum flux to ocean currents by measuring velocity under tropical cyclones. Jarosz et al. (2007) and Sanford et al. (2011) analyzed velocity measurements taken by the ADCP on moorings and three electromagnetic autonomous profiling explorer (EM-APEX) floats under tropical cyclones, respectively. Using the depth-integrated linear momentum equation (hereinafter, the linear momentum budget method), they estimated the magnitude of surface wind stress and parameterized *C*_{d} as a function of wind speed. The present study uses a similar approach to compute *C*_{d} under Typhoon Megi. Note that the *C*_{d} estimated using the oceanic velocity measurements as a bottom-up approach may be inconsistent with that using the atmospheric wind speed measurements.

### c. Previous drag coefficient estimation methods

The Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) has made observations of the air–sea flux from low to moderate wind speeds (|**U**_{10}| < 20 m s^{−1}) using ships and buoys since the 1990s (Edson et al. 2013). Employing the TOGA COARE bulk algorithm 3.5 (Edson et al. 2013), the drag coefficient depends more strongly on wind speed than atmospheric stability. It increases from 1 × 10^{−3} to 2.4 × 10^{−3}, >100%, as the wind speed |**U**_{10}| increases from 5 to 20 m s^{−1}, whereas the drag coefficient changes only about 5% when the atmospheric stability (*z*/*L*, where *z* is the height above the sea surface and *L* is the Monin–Obukhov length) varies from −1 to −0.2. Large and Pond (1981) report that *C*_{d} is a constant for |**U**_{10}| = 4–11 m s^{−1} and increases linearly with wind speed for |**U**_{10}| = 11–25 m s^{−1} (Fig. 1a). The latest TOGA COARE bulk algorithm 3.5 also proposes that *C*_{d} increases linearly with wind speed for |**U**_{10}| = 6–25 m s^{−1} (Edson et al. 2013), with a value slightly greater than that reported by Large and Pond (1981).

The parameterization of *C*_{d} in tropical cyclone wind conditions has been studied extensively using atmospheric and oceanic measurements, laboratory experiments, and model simulations (e.g., Powell et al. 2003; Donelan et al. 2004; Jarosz et al. 2007; Black et al. 2007; Sanford et al. 2011; Holthuijsen et al. 2012; Chen et al. 2013). These studies generally support the linear increase of *C*_{d} with wind speed for |**U**_{10}| < 25 m s^{−1}. For stronger winds, |**U**_{10}| > 30 m s^{−1}, *C*_{d} is “saturated,” either remaining at a constant value or decreasing with wind speed (Fig. 1a). Bell et al. (2012) quantified *C*_{d} at wind speeds greater than 52 m s^{−1} in two hurricanes and report that *C*_{d} scattered for extremely high wind speeds |**U**_{10}| = 52–72 m s^{−1}, with a mean of 2.4 × 10^{−3} and a standard deviation of 1.1 × 10^{−3}.

Recent studies suggest that the parameterization of *C*_{d} by |**U**_{10}| varies in different sectors of the tropical cyclone. Holthuijsen et al. (2012) report that a maximum *C*_{d} (~4.6 × 10^{−3}) is located at the front-left quadrant of the tropical cyclone, and a minimum (~1.7 × 10^{−3}) is located on the right side for |**U**_{10}| = 30–40 m s^{−1}. Chen et al. (2013) used an atmosphere–wave–ocean coupled model to investigate *C*_{d} under Hurricane Frances. In contrast to results reported by Holthuijsen et al. (2012), they conclude that *C*_{d} is generally greater at the front-right quadrant of tropical cyclones than at the left side. They suggest that the variation in different quadrants is due to the spatial variability of surface waves forced by the rapid change of tropical cyclones’ wind.

Most previous studies assume that the crosswind stress is insignificant compared to the downwind stress (e.g., Large and Pond 1981). Recent field experiments investigating the effect of surface waves on surface wind stress report significant crosswind stress (Geernaert 1988; Drennan et al. 1999; Grachev et al. 2003; Zhang et al. 2009; Potter et al. 2015). Zhang et al. (2009) report that the direction difference between the surface wind and stress vectors varies from −40° to 60° for wind speeds 5–20 m s^{−1} (Fig. 1b). Under tropical cyclones, extremely complex surface waves can be generated. The effect of surface waves on the crosswind stress has been studied using numerical models coupled with the surface wave field in tropical cyclones (Moon et al. 2004; Chen et al. 2013; Reichl et al. 2014). Chen et al. (2013) report that the direction difference between the surface wind and the stress vectors is more than 20° within the eyewall of a tropical cyclone, again suggesting a significant crosswind stress.

Studies report a large variability in *C*_{d} at wind speeds greater than 25 m s^{−1} (Fig. 1a) and contradictory results for *C*_{d} distribution in different sectors of tropical cyclones (Holthuijsen et al. 2012; Chen et al. 2013). Numerical model studies show significant crosswind stress under tropical cyclones, but field observations are meager and vary greatly.

### d. Drag coefficient study of Typhoon Megi 2010

In 2010 an international joint experiment, the Impact of Typhoons on the Ocean in the Pacific (ITOP), was conducted in the western Pacific to study the oceanic response to and recovery from tropical cyclones (D’Asaro et al. 2014). One of the primary scientific goals was to investigate surface wind stress under extreme wind conditions. During ITOP, seven EM-APEX floats were deployed to the right of Typhoon Megi’s track, a category 5 typhoon in October 2010. Following Sanford et al. (2011), we estimate downwind and crosswind stress using velocity measurements taken by these floats via the linear momentum budget method.

Typhoon Megi and EM-APEX float measurements are described in section 2, and the linear momentum budget method is discussed in section 3. The downwind and crosswind drag coefficients are defined. The tidal and low-frequency current velocities may introduce uncertainties to wind stress estimates and are discussed in section 4. The apparent drag coefficients are estimated in section 5. The Price–Pinkel–Weller model (PWP3D) has been used to study the ocean momentum response to tropical cyclones (Price et al. 1994; Sanford et al. 2011). Here, the PWP3D is used to assess the assumed linear momentum budget to estimate surface wind stress (section 6). A correction to the derived wind stress estimates is made to yield the adjusted wind stress (section 7), which is investigated using the PWP3D model. Our drag coefficient estimates and the direction difference between the surface wind and stress vectors are discussed and compared with previous studies (section 8).

## 2. Experiment and measurements

ITOP targeted Typhoons Fanapi and Megi using measurements taken from various atmospheric and oceanic platforms (D’Asaro et al. 2014). In this analysis, we focus on the drag coefficient estimated using measurements taken during Super Typhoon Megi only (Fig. 2a). Megi formed in the western Pacific on 12 October 2010 and intensified rapidly becoming a category 5 typhoon on 17 October (Wang and Wang 2014). Typhoon Megi moved primarily westward in the western Pacific, passed the northern Philippines on 18 October, turned northwestward into the South China Sea, and dissipated on 23 October after making landfall in China (D’Asaro et al. 2014; Wang and Wang 2014). This study focuses on measurements of Megi in the western Pacific only.

Between 12 and 18 October, 221 GPS dropsondes were deployed from the C130 aircraft to measure vertical profiles of wind speed and direction (Hock and Franklin 1999), and SFMR mounted on the bottom of the C130s measured the microwave brightness temperature. The measurements of microwave brightness temperature were processed to estimate the wind speed at 10 m above the sea surface (|**U**_{10}|) as described in Uhlhorn and Black (2003). These were cross calibrated and combined to construct a map of the surface winds (Fig. 2b), as described in appendix A. During the measurement period, the radius of maximum wind speed was 15 km, smaller than the average size of tropical cyclones in the western Pacific (~40 km), and the westward translation speed was ~7 m s^{−1}, faster than typical tropical cyclones at the same latitude (4–5 m s^{−1}).

Seven EM-APEX floats were deployed by a C130 aircraft, at a horizontal separation of ~25 km, along 128.3°E between 18.7° and 21°N on 16 October 2010, 1 day before the arrival of the eye of Typhoon Megi (Fig. 2a; Table 1). Floats were recovered by the R/V *Roger Revelle* on 19 October, 3 days after the deployment. Three EM-APEX floats measured the oceanic response to winds greater than 25 m s^{−1}. One float (em3763c) passed directly under the eye of Megi; the other two floats (em4913a and em3766c) passed at ~42 km and ~73 km north of Megi’s eye on the right side of the storm track. Data obtained from these three floats are used to compute the surface wind stress in this study.

EM-APEX float deployment locations at the time they began profiling during the ITOP experiment.

EM-APEX floats measure the electric and magnetic fields in the ocean (Sanford et al. 2005). The oceanic current velocity, relative to a conductivity-weighted average current ^{−1}, estimated using the white spectral level of the observed velocity spectra. Temperature and salinity measurements were taken by a SeaBird Electronics SBE-41 CTD sensor mounted on the top end of the floats. The vertical resolution of velocity, temperature, and salinity was 3–4 m. GPS positions and data were transmitted by Iridium satellites when floats surfaced.

Before the arrival of Megi, floats profiled vertically from near the surface to 230-m depth at a profiling speed of 0.1–0.12 m s^{−1}. Between 1000 UTC 16 October and 2100 UTC 18 October when Megi passed the float array, EM-APEX floats profiled between 30- and 230-m depth to prevent damage by storm-induced surface waves. In the following analysis, the current velocity in the upper 30 m is assumed constant and extrapolated to the surface using the shallowest velocity measurement below 30-m depth. The floats’ positions during this period are estimated using the time integration of current velocity measured by the floats.

Strong near-inertial waves were generated on the right side of Megi due to the inertial resonance of the wind pattern. At 42 km to the right of the storm track (em4913a; Fig. 2a), the near-inertial current was greater than 1 m s^{−1} (Figs. 3d,e). The surface mixed layer, defined as the shallowest depth where the density gradient is greater than 0.03 kg m^{−4} and the density is greater than the surface values by more than 0.3 kg m^{−3}, deepened by more than 20 m, from ~40- to 70-m depth, within one-half day after Megi’s eye arrived at the float array, ~2030 UTC 16 October. The base of the surface mixed layer oscillated by ~10 m near the inertial period due to the convergence and divergence of near-inertial waves, in agreement with observations reported by Gill (1984). The surface mixed layer cooled from 29.3° to 28.2°C in 1 day, presumably due to vertical mixing (Sanford et al. 2011).

The background oceanic current measured by the floats from the north to Megi’s track varied from 0.4 (em3766c) to 0.1 m s^{−1} (em3763c) at 12 h before Megi’s arrival (Fig. 3), consistent with the surface geostrophic current estimated from AVISO (Fig. 2b). Tidal currents were also present, though at velocities less than inertial waves, especially in the surface mixed layer. Detailed analysis of the tides is given in appendix B.

Several moorings were deployed during ITOP on the prevailing path of tropical cyclones (D’Asaro et al. 2014). One of the moorings, SA1, was located about 200 km north of Typhoon Megi’s track. The mooring was equipped with a 75-kHz upward-looking ADCP to measure current velocity between 50- and 550-m depth. In the following analysis, mooring velocity measurements will be used to quantify the tidal current during the observational period.

## 3. Estimates of surface wind stress and drag coefficients

### a. Linear momentum budget method

*g*is the gravity,

*f*is the local Coriolis frequency of ~5 × 10

^{−5}rad s

^{−1}at 20°N,

*p*is the pressure,

*ρ*

_{0}is the Boussinesq density,

*ρ*is the in situ density, and

**is the stress vector.**

*τ**−H*becomes

*z*= −

*H*, respectively. We assume that the vertical velocity vanishes at the sea surface. Following Sanford et al. (2011), we choose

*H*= 150 m and assume that

**v**and surface wind stress

**for convenience.**

*τ***is the uncertainty of**

*τ***represents the error in estimates of surface wind stress because of either the neglect of nonlinear terms and pressure gradient, that is, the third–fifth terms on the left side of Eq. (3), or the existence of non-wind-driven ocean current measured by the floats. The latter is discussed in section 4. To distinguish the estimates of surface wind stress using the linear momentum budget from the true wind stress**

*τ***, the**

*τ***= 0, so**

*τ*### b. Downwind and crosswind drag coefficients

**on the ocean surface can be projected to the respective directions along and perpendicular to the wind vector at 10-m height above the sea surface**

*τ***U**

_{10}as

**U**

_{10}, and

*ϕ*between

**U**

_{10}and the surface wind stress

**is defined as**

*τ**ϕ*are greater than zero when the orientation of

**is clockwise from the |**

*τ***U**

_{10}|. Following previous studies (Smith 1980; Powell et al. 2003; Donelan et al. 2004), we parameterize

*C*

_{d}is the magnitude of the drag coefficient. The estimates of downwind and crosswind drag coefficients computed using the linear momentum budget expressed in Eq. (4) are termed apparent downwind and crosswind drag coefficients

## 4. Storm-induced current velocity under Typhoon Megi

**v**taken before the arrival of Typhoon Megi’s eye are used to estimate the surface wind stress and drag coefficient using the depth-integrated linear momentum budget [Eq. (4)]. The primary constituents of current velocity can be assumed as

**v**

_{wind}is the velocity of wind-driven current,

**v**

_{tide}is the velocity of tides,

**v**

_{low}is the velocity of low-frequency currents, constant in amplitude and direction for at least a half day, such as the surface geostrophic current, and

**is the instrumental noise in the velocity measurements.**

*ϵ*Only the wind-driven ocean current velocity **v**_{wind} should be used to estimate surface wind stress in Eq. (4). The background currents, such as tides **v**_{tide} and low-frequency currents **v**_{low}, which are not directly forced by Megi, will cause the linear momentum budget to unbalance. For example, the depth-integrated linear momentum of a diurnal barotropic tide K_{1} with amplitude 0.1 m s^{−1} from the ocean surface to 150-m depth is ~0.7 N m^{−2}, leading to an error of ~20% in surface wind stress estimates if *C*_{d} *=* 4 × 10^{−3} at |**U**_{10}| = 30 m s^{−1}. The magnitude of low-frequency currents can also influence the Coriolis rotation term in the linear momentum budget. The integrated momentum of randomly distributed ** ϵ** in the linear momentum budget was investigated but is negligible compared to the contributions of

**v**

_{wind},

**v**

_{tide}, and

**v**

_{low}.

To estimate the apparent surface wind stress **v**_{wind}, the tides **v**_{tide} and low-frequency currents **v**_{low} are both estimated and removed from EM-APEX velocity measurements. The amplitude and phase of tides at the float locations are estimated by harmonic fitting of the velocity measurements between 200- and 220-m depth (appendix B, section a) and then used to extrapolate tides to the ocean surface by assuming the first-mode baroclinic tide. The uncertainty of estimating **v**_{tide} is primarily due to vertical phase propagation and amplitude difference in tidal extrapolation, implemented by the analysis of tides on the mooring SA1 (appendix B, section b). The low-frequency current is estimated by averaging the profiles of (**v** − **v**_{tide}) at different selected periods, assuming the **v**_{wind} had not been forced by Megi (appendix B, section c). The uncertainty of estimating **v**_{low} is affected by the estimates of **v**_{tide} and the selected averaging period.

Simulations of 40 000 normally distributed (**v**_{tide} + **v**_{low}) are generated in the stochastic simulation (appendix B, section d) and removed from each of the observed EM-APEX float velocity profiles so as to generate 40 000 realizations of apparent wind-forced velocity profiles

## 5. Estimates of apparent drag coefficients

The |**U**_{10}| for the computation of drag coefficients is simulated in the normal distribution using the measured wind speed and root-mean-square (RMS) error; RMS is ~4 m s^{−1} outside Megi’s eyewall, and ~5 m s^{−1} within the eyewall (appendix A, section a). Using the

Estimates of apparent drag coefficient are presented as the function of wind speed at 10 m above the sea surface |**U**_{10}| (small dots and thin vertical lines in Fig. 4). The standard deviations of estimates of apparent drag coefficients ^{−3} for |**U**_{10}| = 25–30 m s^{−1}. The large uncertainty in estimates of drag coefficient for |**U**_{10}| < 25 m s^{−1} is due primarily to contamination by the tides and mean currents on estimates of wind-driven currents. We discuss drag coefficients for |**U**_{10}| > 25 m s^{−1} exclusively in the following.

To summarize the effect of wind speed on apparent drag coefficients, we further average drag coefficients in different bins of wind speed (±2 m s^{−1} at |**U**_{10}| = 27 m s^{−1}, ±3 m s^{−1} at |**U**_{10}| = 31 m s^{−1}, ±3 m s^{−1} at |**U**_{10}| = 37 m s^{−1}, and ±9 m s^{−1} at |**U**_{10}| = 56 m s^{−1}). We use the mean and standard deviation of apparent drag coefficients from each profile of EM-APEX floats to generate 1000 simulations assuming a normal distribution. Within each bin of wind speed, the apparent drag coefficient averages are computed using the generated simulations from at least three different profiles, and then the mean and the standard deviation of the apparent drag coefficient averages are computed. The estimates of ^{−3} for |**U**_{10}| = 30–40 m s^{−1}. The *ϕ* are significantly different from zero, indicating the crosswind component of surface wind stress is not negligible.

## 6. PWP3D model

The apparent drag coefficient under Typhoon Megi is estimated assuming balance of the linear momentum budget. Simulations of oceanic response under Typhoon Megi in the PWP3D model are used to validate the assumption of linear momentum balance here and to correct the apparent drag coefficient due to the neglect of nonlinear and pressure gradient terms (section 7).

### a. Model description

Price et al. (1994) developed the PWP3D numerical model using momentum, continuity, temperature, and salinity equations to study oceanic responses to moving tropical cyclones. The initial temperature and salinity fields are assumed horizontally homogeneous. In the following model simulations, horizontally and temporally averaged vertical profiles of EM-APEX float measurements of temperature and salinity taken within 18–19 h before the arrival of Megi’s eye are used as the initial conditions. The horizontal spatial resolution is 3 km, and the temporal resolution is 180 s. The spatial domain of the ocean is ±375 km in the zonal direction and ±300 km in the meridional direction. The vertical resolution is 5 m from the ocean surface to 300-m depth.

In the model, vertical turbulent mixing is parameterized using the bulk Richardson number *υ* and Δ*ρ* are the difference of velocity and density, respectively, across the base of the surface mixed layer *h* (Price et al. 1986). In the PWP3D model, the turbulent mixing is enforced when Ri_{g} < 0.25, or Ri_{b} < 0.65, and the momentum and mass are mixed until Ri_{g} and Ri_{b} are beyond their stability criteria. For the model simulation, the spatial differencing terms are computed in the second-order scheme, and the leapfrog-trapezoidal method is used for the temporal integration (Price et al. 1994). Solar radiation is assumed as a sinusoidal function with a peak of 500 W m^{−2} at noon and zero at midnight. Longwave radiation is computed using the Boltzmann constant and sea surface temperature assuming the blackbody (Price et al. 1986). Constant values of dry (26°C) and wet (25°C) bulb air temperature in tropical cyclones are assumed (Sanford et al. 2011). The sensible and latent heat flux are computed using the wind speed and sea surface temperature in the model.

For Typhoon Megi’s wind forcing in the model, the storm’s wind speed and direction are interpolated linearly in time, moving in the real translation track and passing the grid point at the center of the spatial domain at 2030 UTC 18 October. The wind speed at the floats’ positions in the model simulations is the same as the observations (Figs. 7g,h,i). The surface wind stress for |**U**_{10}| < 25 m s^{−1} in all simulations presented in this study is computed using the drag coefficient parameterization proposed by Large and Pond (1981) and mapped |**U**_{10}|, assuming no crosswind stress.

### b. Model simulations and momentum budget

In the first model simulation presented in this study (hereafter referred to as simulation A), the surface wind stress ** τ** in the model for |

**U**

_{10}| > 25 m s

^{−1}is computed using the mapped |

**U**

_{10}| advected over the ocean and our estimates of apparent downwind drag coefficient but assuming no crosswind stress. Results of PWP3D model simulations of oceanic currents at three float positions as Megi approached the float array show the surface wind stress increasing and strong currents generated in the surface mixed layer (Fig. 5). At the arrival time of Megi, the simulated currents of ~1.5 m s

^{−1}are similar to those observed by floats (Fig. 3), and the modeled mixed layer deepening, ~30 m, is consistent with the observed deepening at the float on the track of Megi (em3763c).

The estimates of surface wind stress computed from Eqs. (3) and (4) are compared with the input surface wind stress (Fig. 5). The estimates of surface wind stress from Eq. (3) agree very well with the input surface wind stress, indicating that the momentum equation is implemented correctly and supporting the choice to neglect turbulent fluxes at 150-m depth in the model. Furthermore, the estimates of surface wind stress from Eq. (4) (the depth-integrated linear momentum budget) agree with the input surface wind stress with an uncertainty of <10% only before the arrival of the storm’s eye. The agreement within Megi’s eyewall and after the passage of Megi’s eye is poor because the pressure gradient and nonlinear advection terms induced by Typhoon Megi are important, as suggested by Sanford et al. (2011). Based on PWP3D model results, drag coefficient estimates using Eq. (4) are reliable only before the arrival of Typhoon Megi, and some corrections near the eyewall of Megi are required. Note that variations in air temperature or solar radiation have little effect on the momentum budget balance. The estimate of surface wind stress using the linear momentum equation varies less than 1%, even with a change of ±3°C in air temperature or an insolation peak increase from 500 to 1000 W m^{−2}.

Because the nonlinear and pressure gradient terms at two floats near Megi’s track (em4913a and em3763c) led to an uncertainty of <15% before the passage of Megi’s eye, another PWP3D model run at the float positions of em3763c (9 km from Megi’s eye) was performed to investigate the corrections to the assumed linear momentum balance (section 7).

## 7. Estimates of adjusted drag coefficients

In simulation A (section 6), the downwind component of surface wind stress is computed using the mapped |**U**_{10}| and apparent downwind drag coefficient for |**U**_{10}| > 25 m s^{−1} but assuming no crosswind stress. The nonlinear and pressure gradient terms, which are excluded from the estimates of apparent drag coefficients in the linear momentum budget [Eq. (4)], ^{−3} (~10% at the peak) and ^{−3} at wind speeds greater than 25 m s^{−1}. We subtract the effects of nonlinear and pressure gradient terms on drag coefficients (the difference between green dots and black curve in Fig. 6b and between purple dots and black line in Fig. 6c) from our estimates of apparent downwind and crosswind drag coefficients (section 5) and call these the adjusted downwind and crosswind drag coefficients.

Another simulation (simulation B) computes the surface wind stress using the mapped |**U**_{10}| and adjusted downwind and crosswind drag coefficients for |**U**_{10}| > 25 m s^{−1} (black curves in Figs. 6e,f). Drag coefficients computed using Eq. (4) (green dots and purple dots in Figs. 6e,f) agree well with the apparent drag coefficients discussed in section 5 (red dots with vertical bars in Figs. 6e,f), suggesting that the adjusted drag coefficients (black curves in Figs. 6e,f) are the better estimates of the true drag coefficients.

Table 2 summarizes the estimated adjusted drag coefficients in Typhoon Megi. The ^{−3} at |**U**_{10}| = 27 m s^{−1}, reaches its peak of 3.5 × 10^{−3} at |**U**_{10}| = 31 m s^{−1}, and decreases to 2.0 × 10^{−3} for |**U**_{10}| > 55 m s^{−1}. The saturation of **U**_{10}| < 27 m s^{−1}, reaches its peak of 1.6 × 10^{−3} at |**U**_{10}| = 37 m s^{−1}, and decreases to 0.7 × 10^{−3} for |**U**_{10}| > 45 m s^{−1}. The **U**_{10}| > 37 m s^{−1}, but the orientation offset between stress and |**U**_{10}| changes only slightly, from 30° to 20° for |**U**_{10}| = 37–56 m s^{−1}.

The mean and standard errors of adjusted downwind and crosswind drag coefficients at different wind speed intervals during Typhoon Megi. Same as the values shown in Fig. 1.

To further validate our estimates of adjusted **U**_{10}| and the *C*_{d} reported by Holthuijsen et al. (2012) for |**U**_{10}| > 25 m s^{−1} but assuming no crosswind stress. The correlation coefficient of depth-integrated linear momentum components between simulation B results and float observations for |**U**_{10}| > 25 m s^{−1} is 0.83. This is slightly better than that between simulation C results and observations (0.68; Fig. 8). That is, simulation results using our adjusted drag coefficients agree with observations better than those using the *C*_{d} reported by Holthuijsen et al. (2012). The discrepancy between simulated ocean momentum response in simulation B at 42 km to the right of Megi’s track and observations (Fig. 7e) suggests that the drag coefficient may be affected by factors other than the wind speed.

## 8. Discussion

Many prior studies use wind speed profiles observed in the atmospheric boundary layer to investigate the neutral drag coefficient under tropical cyclones (Powell et al. 2003; Holthuijsen et al. 2012). In this study we instead use oceanic momentum response to study tropical cyclone surface wind stress (Jarosz et al. 2007; Sanford et al. 2011). The estimates of surface wind stress are parameterized by wind speed as drag coefficients.

For |**U**_{10}| < 30 m s^{−1}, the drag coefficient increases with wind speed, in agreement with previous studies (Fig. 1a). For |**U**_{10}| = 30–40 m s^{−1}, the magnitude of ^{−3} agrees with that reported in an atmosphere–wave–ocean coupled model at the front-right quadrant of tropical cyclones (Chen et al. 2013) but is much greater than the neutral drag coefficient reported by others (Powell et al. 2003; Donelan et al. 2004). The unstable planetary boundary layer and complex surface wave field under Typhoon Megi may have caused the high drag coefficient (Dyer 1974; Chen et al. 2013). For |**U**_{10}| > 40 m s^{−1}, the drag coefficient decreases with wind speed, in agreement with other studies (Powell et al. 2003; Holthuijsen et al. 2012) that propose drag coefficient saturation by sea foam and spray (Powell et al. 2003; Donelan et al. 2004).

For the crosswind component of surface wind stress for |**U**_{10}| < 30 m s^{−1}, the angle between wind and stress is nearly zero, indicating that the assumed alignment between wind and stress in Monin–Obukhov similarity theory is valid under Typhoon Megi at low wind speeds. For |**U**_{10}| > 30 m s^{−1}, the angle between wind and stress vectors is >15° (Fig. 1b), slightly greater than a numerical model result from the front-right quadrant of tropical cyclones (Chen et al. 2013). Previous studies suggest swell traveling under the tropical cyclone may yield significant crosswind stress (Chen et al. 2013; Reichl et al. 2014; Potter et al. 2015). The **U**_{10}| > 40 m s^{−1}, similar to the dependence of

## 9. Summary

Velocity, temperature, and salinity measurements were taken by seven EM-APEX floats air deployed on the right side of Typhoon Megi, a small and fast-moving category 5 typhoon, during the ITOP experiment in 2010. Downwind and crosswind drag coefficients were computed from three floats closest to the eye using the depth-integrated linear momentum equation.

Extensive efforts are devoted to estimate the uncertainty of the derived drag coefficients due to the uncertainty in velocity measurements and imperfections in removing non-wind-driven currents. Estimates of the “apparent” drag coefficients are made assuming a linear momentum equation. PWP3D model simulations show that the momentum balance is approximately linear before the arrival of the eye but with significant components of nonlinear and pressure gradient force after the passage of Megi. The effects of nonlinear and pressure gradient terms before the arrival of the typhoon are corrected using the model to obtain the adjusted downwind and crosswind drag coefficients (Fig. 1; Table 2).

At |**U**_{10}| = 27 m s^{−1}, our estimates of the downwind drag coefficient ^{−3} are greater than the *C*_{d} reported in previous studies (Powell et al. 2003; Donelan et al. 2004; Black et al. 2007; Jarosz et al. 2007). At |**U**_{10}| = 31 m s^{−1}, our estimates of the downwind drag coefficient ^{−3} are much greater than previously reported and a factor of 2 greater than the drag coefficient on the right side of tropical cyclones reported by Holthuijsen et al. (2012). Our results are in agreement with the numerical model study by Chen et al. (2013), which reports a stronger drag coefficient, >3.0 × 10^{−3}, in the front-right quadrant of tropical cyclones. At higher wind speeds our estimates of ^{−3} at |**U**_{10}| = 50 m s^{−1} and remain nearly constant to our observed maximum wind speed of 62 m s^{−1}, consistent with the *C*_{d} reported by Bell et al. (2012).

We present, for the first time, measurements of the crosswind drag coefficient **U**_{10}| in tropical cyclones. For |**U**_{10}| ≤ 25 m s^{−1}, estimates of ^{−3}. At |**U**_{10}| = 27 m s^{−1}, the wind stress vector is about 10° clockwise from the wind vector, consistent with the report by Zhang et al. (2009) for lower wind speeds. For |**U**_{10}| > 30 m s^{−1}, ^{−3}, and the wind stress vector is mostly >20° clockwise from the wind vector. Chen et al. (2013), in their atmosphere–wave–ocean model simulations, report a similar result near the center of a tropical cyclone. As the wind speed increases from 40 to 60 m s^{−1}, ^{−3}. The decrease of ^{−1}, is similar to the decrease in

This paper focuses on the drag coefficients under Typhoon Megi and on the details of the analysis method. Similar data are available for four other tropical cyclones (Hurricane Frances 2004, Hurricane Gustav 2008, Hurricane Ike 2008, and Typhoon Fanapi 2010). Drag coefficient analyses using the same method will be reported in a subsequent publication.

## Acknowledgments

The authors appreciate the Office of Naval Research Physical Oceanography Program (N00014-08-1-0560, N00014-08-1-0577, N00014-10-1-0313, N00014-11-1-0375, and N00014-14-1-0360) for their support, the 53rd Weather Reconnaissance Squadron for their deployments of the EM-APEX floats, and J. F. Price for use of the PWP3D model. J. Carlson and J. Dunlap designed and built the EM sensor systems on the EM-APEX float. Constructive comments from J. F. Price and one anonymous reviewer are appreciated and greatly improved the presentation of this paper.

## APPENDIX A

### Typhoon Megi Wind Field

#### a. Wind speed processing

Continuous wind estimates at 10-m height above the sea surface **U**_{10} at each float were computed using data from aircraft penetrations of Typhoon Megi. Data from Vaisala RS92 dropsondes deployed by two WC-130J and one Taiwanese DOTSTAR aircraft (Wu et al. 2005) and from a nadir-looking SFMR (Uhlhorn and Black 2003) on the WC-130Js were used. These data can be obtained online (at http://catalog.eol.ucar.edu/itop_2010/). The positions of all measurements are determined by GPS data with an accuracy of better than 100 m.

The dropsondes measure horizontal wind velocity, temperature, and humidity from flight level to within a few meters above the sea surface. Wind speed accuracy is ±0.5 m s^{−1} (UCAR/NCAR 1993). The **U**_{10} was computed directly from the RS92 dropsonde profiles.

The SFMR measures the brightness temperature, which is used to estimate surface wind speed continuously along the aircraft track with a nominal error of 4.0 m s^{−1} (Uhlhorn and Black 2003) for winds greater than 20 m s^{−1}. The SFMR data were calibrated to the dropsonde 10-m winds using all SFMR and dropsonde data collected during ITOP. The SFMR measurement with only the highest quality data (Q2 = 4) was used to calibrate each nearby dropsonde measurement of **U**_{10}. Measurement pairs separated by more than 18 s were rejected. A total of 343 pairs remained. Significant bias errors were found after comparing dropsonde and SFMR measurements at the same wind speed and were corrected using a cubic polynomial in log_{10}(|**U**_{10}|). Corrections are small, 2 m s^{−1} or less for |**U**_{10}| = 20–50 m s^{−1}, but rise rapidly below this. The SFMR winds below 20 m s^{−1} were discarded in this study. The corrected SFMR data matched the dropsonde data with an RMS error of 3.5 m s^{−1}, less than the nominal SFMR error of 4.0 m s^{−1} (Uhlhorn and Black 2003) and much more than the dropsonde error of 0.5 m s^{−1}. The differences showed no trends in time, with aircraft number, with the distance between the plane and the dropsonde, or with the logged quality of the data.

The corrected SFMR winds, regardless of logged quality, and the 10-m dropsonde winds were used to create maps of the |**U**_{10}| of Megi. Data from three aircraft surveys were used. Flight 530W deployed 16 dropsondes during two passes through the storm from ~0000 to 0400 UTC 16 October, immediately after deploying the EM-APEX floats. Flight 630W deployed 44 dropsondes during three passes through the storm from ~2100 UTC 16 October to ~0300 UTC 17 October. This was coordinated with a DOTSTAR flight that deployed 10 additional dropsondes around the periphery of the storm. The 744 data points (13 dropsondes) during this survey were within 100 km of the storm center at six different azimuths. This survey coincided with the passage of the storm’s eye through the float array and is the primary source of data for the drag coefficient calculations. Flight 830W deployed 27 dropsondes during four passes through the storm from ~1100 to 1600 UTC 17 October.

For each survey, a continuous wind field was created in two steps. First, a radially symmetric wind field was fit to the observations. For radii *r* > *R*_{max} the form *V* = (*V*_{max}*R*_{max}/*r*)^{n} was used. For *r* < *R*_{max}, the form *V* = *V*_{max}(*r*/*R*_{max}) was used. Within the eye, *r* < *R*_{eye}, and *V* = *V*_{eye}. Figure A1 shows this function for the 17 October survey (*n* = 0.5, *R*_{max} = 12 km, *V*_{max} = 70 m s^{−1}, *R*_{eye} = 6 km, and *V*_{eye} = 10 m s^{−1}). Although this captures much of the wind variations, azimuthal deviations of up to ±10 m s^{−1} are apparent. Both true azimuthal variations in the storm structure and errors in the location of the storm center used to make the radial map can cause the deviations. These deviations were then mapped (red regions in Fig. A2 representing a higher wind speed than the radial model and blue regions in Fig. A2 representing a lower wind speed) by smoothing the observations in polar coordinates using a Gaussian smoother with scales of 0.1 in log_{10}(*r*) and *π*/4 in azimuth and added to the radially symmetric model to form a wind field for the storm at the time of the survey. The standard deviation of the maps from the data is 1.8 m s^{−1} for Typhoon Megi, well within the expected uncertainty in the SFMR data. The deviations at the peak winds are <1 m s^{−1}. Two maps of the surface wind speed between the surveys, one earlier and one later, were constructed by moving the survey fields along the storm track, for example, the wind map of 17 October in Fig. A3. The |**U**_{10}| on the floats is computed by linearly interpolating these two maps in time.

The float positions are computed by linearly interpolating between roughly hourly GPS fixes, when these are available. Between approximately yeardays 289.4 and 291.5, the floats stopped surfacing to avoid damage and no GPS fixes are available. During these times, float positions were estimated by integrating the depth-averaged velocity measured by the floats, starting from the last GPS fix and adjusting this trajectory with a linear trend to hit the next GPS fix. The error is estimated from the size of this adjustment, typically a few centimeters per second, resulting in a position error of a few kilometers. The wind speeds computed at these floats’ positions are used to calculate drag coefficient in this study (Fig. A4).

For the outer two floats, the lateral gradients are small. The uncertainty in the estimated wind speed is dominated by measurement and mapping errors. The former uncertainty is taken as the RMS SFMR error ~3.5 m s^{−1}, and the latter is conservatively taken as the RMS deviation of the data from the map ~1.8 m s^{−1}. Combining these yields 4.0 m s^{−1} RMS. This error is conservative because much of the estimated SFMR error is undoubtedly due to variability in the dropsonde velocity estimates due to boundary layer turbulence.

The central float, under the greatest wind speed, passed under the storm eyewall north of the eye but inside the maximum wind region, one of high spatial gradients. The resulting time series of wind at the float is highly sensitive to the exact float track and the details of the wind map. In particular, the double wind peak results from the float passing under the comma-shaped wind maximum (Fig. A4). Neither the float track nor the wind map are sufficiently accurate to capture the details of this feature correctly. This large uncertainty only occurs for about 1 h while the float was in Megi’s eyewall, with a plausible estimate of the RMS wind speed uncertainty at any given time ±10 m s^{−1}. The uncertainty in the peak hourly average wind speed is less because the float position is known to about 1 km from a recent GPS fix and the small advective velocities. Because the storm moved nearly westward, the float certainly went through the eyewall north of the eye in a region of 50–80 m s^{−1} winds. The average wind during this time is estimated at 71 ± 5 m s^{−1} computed by fitting a smoothing spline to the nearby data points and resampling them randomly.

#### b. Inflow angle of Typhoon Megi

*u*

_{t}and radial wind

*u*

_{r}components. The storm-relative inflow angle

*α*is defined as

*α*is positive when the air mass is transported inward to the eye of tropical cyclones.

Following Zhang and Uhlhorn (2012), we use data taken from GPS dropsondes deployed in Typhoon Megi between 0100 UTC 16 October and 1200 UTC 17 October to compute the inflow angle *α*. The mean of *α* (*α* (*σ*_{α}) are computed within the layer between 10 and 50 m above the sea surface (blue dots and vertical lines in Fig. A5). The distance of each GPS dropsonde profile to the eye of Typhoon Megi is normalized by the radius of the storm’s maximum wind (*R*_{max}) as *r**. At the front-right quadrant of the storm beyond *R*_{max}, that is, *r** > 1, the standard deviation *σ*_{α} is less than 5°.

We average *r** corresponding to the distances of three EM-APEX floats nearest Typhoon Megi’s eye. The average of *r** = 1–1.5, 32° ± 8° at *r** = 3.4–4.4, and 30° ± 13° at *r** = 5.2–6.8. Our results of *α* increases linearly from zero at Megi’s eye to a peak value of 35° at *r** = 2 and decreases to 32° at *r** = 6. The effect of the storm motion is added to yield the interpolated vector wind field.

## APPENDIX B

### Estimates of Tides and Low-Frequency Ocean Currents on EM-APEX Floats

Tides and low-frequency currents can result in the surface wind stress induced by Typhoon Megi to fall out of balance with the integrated wind-driven momentum in the linear momentum budget. The following analysis focuses on the methods to estimate tidal and low-frequency currents and their corresponding uncertainties to isolate the current velocity for surface wind stress estimates. The subsurface mooring SA1, with an upward-looking 75-kHz ADCP mounted at 550-m depth, was located about 90 km west of the float array (Fig. 2b). The estimates of tides using the velocity measurements taken by the ADCP on the subsurface mooring SA1 are used to estimate tidal amplitude and phase at the EM-APEX floats positions.

#### a. Estimates of tidal amplitude and phase

We use harmonic fitting to estimate the phase and amplitude of the tides, assuming that the observed velocity is a linear superposition of mean current, inertial waves, and K_{1} and M_{2} tides. Velocity measurements on mooring SA1 show an amplitude of K_{1} greater than O_{1} and M_{2} greater than S_{2} (not shown in this study). The short time interval of velocity measurements on EM-APEX floats, ~3 days, is also not capable of separating the similar frequency signals for K_{1} from O1 and S_{2} from M_{2}.

The harmonic analysis is applied to EM-APEX float velocity measurements between 0200 UTC 16 October and 0100 UTC 19 October, with an inertial period at 18°–22°N of 32–38 h and a K_{1} tidal period of 24 h and M_{2} of 12.42 h. Note that the periods of inertial waves and diurnal tides are not far apart and therefore estimates of diurnal tides in the linear harmonic analysis could be biased by the presence of inertial waves induced by Megi. To minimize these contamination effects on the estimates of tidal currents, we apply the harmonic analysis to the velocity data between 200- and 220-m depth, which is much deeper than the penetration depth of the storm-induced inertial waves.

Table B1 summarizes tidal current amplitudes and phases estimated from velocity measurements taken from EM-APEX floats and on mooring SA1. The K_{1} on SA1 leads the K_{1} on the float (em4390d) at about 20.2°N by about 71° ± 9°. Assuming the K_{1} tide propagates eastward from the Luzon Strait (Zhao 2014), we estimate a zonal wavelength of 310–400 km and a phase speed of 3.6–4.6 m s^{−1}. Our estimate of the phase speed agrees with the climatology K_{1} phase speed from AVISO satellite observations of 4–5 m s^{−1} (Zhao 2014). Estimates of the M_{2} tidal phase at the float array vary greatly, presumably due to multiple M_{2} tide sources arriving on the experimental site, as suggested by Zhao (2014). The M_{2} amplitude estimates from the floats are 0.04–0.14 m s^{−1} and are generally weaker than the K_{1} tide.

The estimates of amplitude and phase of K_{1} and M_{2} tides from EM-APEX float and mooring SA1 observations. The tidal function is assumed as *A* cos(*ωt* + *θ*), where *A* is the amplitude, *ω* is the tidal frequency, and *θ* is the phase at *t* = 0, which is the arrival time of Typhoon Megi at the float array, about 2030 UTC 16 Oct. The subscripts *x* and *y* are the zonal and meridional directions, respectively. The harmonic fitting is applied to the velocity measurements between 200- and 220-m depth from 0200 UTC 16 Oct to 0100 UTC 19 Oct.

#### b. Estimation of tides assuming first-mode baroclinic tide

We extrapolate tidal amplitude and phase estimated using EM-APEX float velocity measurements at 200–220-m depth to the layer above 200-m depth. The extrapolation depends on the vertical phase propagation and vertical structure of amplitude, which are not available from limited float measurements. The vertical structure of tidal amplitudes and vertical phase propagation observed on mooring SA1 before Typhoon Megi (Fig. B1) are used to guide the vertical extrapolation of tidal amplitudes and phases on EM-APEX float positions.

We divide the velocity measurements on SA1 into three layers: (A) 60–150-m depth, (B) 150–200-m depth, and (C) 200–220-m depth. The phases of diurnal and semidiurnal tides estimated on SA1 do not show significant differences between layers A and B, with a mean phase difference of ~0 and a standard deviation of ~30°, and are independent of tidal amplitudes (Figs. B2a,b), suggesting a vertical standing feature of diurnal and semidiurnal tides.

We extrapolate the amplitude of tides on SA1 from layer C to layer A based on the vertical structure of the first-mode baroclinic tide. The eigenmode structure is obtained using the density profiles taken by Argo floats between the surface and 1600-m depth near SA1 and EM-APEX floats during 14–18 October. The difference between the tidal amplitude estimated directly from velocity in layer A and that estimated from the extrapolation from layer C is computed. Their mean values and one standard deviation of difference are computed as a function of the tidal amplitude in layer C. They vary from −0.08 to 0.05 m s^{−1} with one standard deviation about 0.03 m s^{−1} (Figs. B2c,d). The standard deviation of amplitude within layer A is 0.02 m s^{−1}. The amplitude difference due to extrapolation (Figs. B2c,d) is applied to the extrapolation of tidal amplitudes on EM-APEX float measurements.

#### c. Estimation of low-frequency current

Because the surface geostrophic current and the barotropic adjustment velocity

#### d. Tides and low-frequency current in the stochastic simulation

The velocity of tides on the EM-APEX floats is extrapolated to the ocean surface assuming the vertical structure of the first-mode baroclinic tide; 10 000 simulations of normally distributed tides (K_{1} + M_{2}) are generated in a stochastic simulation, using the mean and standard deviation of vertical phase propagation and amplitude difference according to the analysis of tides on mooring SA1 (appendix B, section b). The average velocity of tides near the ocean surface at the arrival of Typhoon Megi is ~0.1–0.15 m s^{−1} (Fig. B3). The low-frequency current is estimated in four different selected averaging periods (appendix B, section c) after removing the 10 000 simulations of tides. The average low-frequency current speed in the surface mixed layer is less than 0.1 m s^{−1} at the float em3763c on Typhoon Megi’s track (Fig. B3). The 40 000 simulations of background current velocity (tides + low-frequency current) are removed from the EM-APEX velocity profiles before estimating surface wind stress.

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