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  • View in gallery

    Photograph of the bow of the R/V Saramiento de Gamboa with indications of the positions of the 3D sonic and the 2D vane anemometers. Also shown is an illustration of the overestimation of the pitch of the relative wind vector by the classic tilt-motion correction (4). The relative wind vector is tilted by the angle due to the presence of the ship and measured as with vertical component w. Applying the DR after the correction for gives , which overestimates the tilt.

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    Time series of (a) normalized wind speed and significant wave height, (b) friction velocity, (c) measured relative wind direction and ship speed, and (d) true wind direction and distance between ship and buoy position. The buoy values are COARE3.0 bulk flux results calculated from 1-min measurements on the mooring, which were averaged to 12 min and interpolated on the bow mast time series using the GPS time stamps. Bow mast , true wind direction, and are calculated using (15) and (16), respectively. The shaded areas mark periods that were selected for the estimation of the yaw and acceleration bias in the measured relative wind direction and speed on the bow mast. These periods are also used for the comparison of the direct measurements of on the bow mast with the COARE3.0 estimates from the buoy.

  • View in gallery

    Pitch and roll angles ( and , respectively) and the vertical wind speed residual c3, determined by the linear regression (21) over 15° wind direction bins, are plotted as functions of the measured relative wind direction. The shading shows the 95% confidence intervals of the regression.

  • View in gallery

    Wind vector pitch at the 3D sonic calculated using the classic DR (○) and using DRx as a function of the measured relative wind direction and the ratio of ship velocity and measured wind speed. The results for pitch and roll from the rPF method (see Fig. 3) are shown for comparison (black and blue lines).

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    Ratio of the measured relative wind speed and the expected relative wind speed, calculated from the buoy wind speed and the ship velocity using (24) as a function of the measured relative wind direction. The uplift was assumed to be zero . Individual measurements that were selected for the ship buoy comparison are shown as black dots. The red line shows averages that were taken over 15° relative wind direction bins. The blue dashed line shows the bin average values for an assumed uplift . The error bars show the standard deviation of the mean.

  • View in gallery

    Expected relative wind direction calculated from the buoy wind speed and the ship velocity using (24) as a function of the measured relative wind direction. Individual measurements that were selected for the ship buoy comparison are shown as black dots. The red line shows averages that were taken over 15° relative wind direction bins. The error bars show the standard deviation of the mean.

  • View in gallery

    Observed ratio of the wind speed measured by the 3D sonic (10 m MSL) and the 2D vane anemometer (16 m MSL), as a function of the measured relative wind direction. Also shown are the bin averages of r and the ratio of the accelerations , calculated with (14), for and .

  • View in gallery

    Ratio of the EC measurements of on board the R/V Saramiento de Gamboa and the COARE3.0 bulk flux estimates from the mooring as a function of the measured relative wind direction measured on the ship. Individual 12-min measurements are shown as scatter and average values as lines. Results from the rPF method (16) are shown as blue and —, and results using DR as black and , and DRx as red ○ and . For this comparison, the data were restricted to the selected time periods marked in Fig. 2.

  • View in gallery

    Direct measurements of as a function of and the measured relative wind direction: (a) results obtained using the standard motion correction and rotation into the stream (4) and (b) results using the rPF method (16). Individual measurements (12 min) are shown as (○) and bin averages over 30° wind direction and 2 m s−1 wind speed bins are shown as thick lines and big , respectively (error bars show the standard deviation). The COARE3.0 and COARE3.5 bulk flux predictions calculated from are shown as green and black lines, respectively.

  • View in gallery

    As in Fig. 9, but for the sign-preserving square root of the crosswind stress component . Note that positive values indicate downward crosswind momentum flux.

  • View in gallery

    Measured stress angle (using rPF results) as a function of the measured relative wind direction and . The bin averages (shown as black line) are taken for over 15° wind direction bins. Error bars indicated standard deviation from the mean.

  • View in gallery

    Ratio of the measured with EC and the COARE3.5 bulk flux prediction using as a function of the relative wind direction and the wind speed. The ratios are presented as bin-averaged values over 15° wind direction and 1 m s−1 wind speed bins (minimum six measurements per bin). Also shown are 15° wind direction bins for (minimum 12 measurements per bin). The error bars indicate the standard deviation. Results using (a) DR, (b) DRx, and (c) rPF.

  • View in gallery

    The bin average ratios from Fig. 12 for DR, DRx, and rPF, for the wind speed range . Also shown are the results for setting the roll angle to zero in the rPF correction method (rPF = 0). Each bin average is based on a minimum of 12 individual measurements.

  • View in gallery

    Ratios of computed with DR, DRx, and rPFϕ = 0 and computed with rPF as a function of the relative wind direction. The data are presented as bin averages with the standard deviation indicated as the shaded area or error bars. The same 15° wind direction bins as in Fig. 3 are used.

  • View in gallery

    As in Fig. 14, but for the ratios of the sonic sensible heat flux .

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Direct Flux Measurements from Mobile Platforms at Sea: Motion and Airflow Distortion Corrections Revisited

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  • 1 School of Physics, and Ryan Institute, National University of Ireland, Galway, Galway, Ireland
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Abstract

Ship-based measurements of wind speed and direct fluxes are affected by airflow distortion that can lead to a tilt of the wind vector as well as acceleration or deceleration of the wind speed. Direct flux measurements are additionally affected by the fluctuating velocity of the platform. The classic approach is to first correct the wind speed for angular and translational platform velocities and thereafter rotate the wind vector into the mean flow. This study finds that for ships under way, this leads to an overestimation of the vector tilt and biased flux estimates. This may explain the common observation that flux estimates from ships in transit have lower quality than measurements taken on station. Here an alternative approach is presented, where the flow-distortion-induced tilt of the wind vector is estimated from the 3D wind speed measurements and applied to the apparent wind vector. The tilt correction is carried out after correction for the fluctuating part of the platform velocity but before removing the ship’s mean translational velocity. This new method significantly improved the agreement of direct momentum flux measurements made from a ship under way with the parameterization of the COARE3.5 bulk model. The sensitivity of the eddy covariance measurements of momentum and scalar fluxes to the choice of the tilt-motion correction method is analyzed, and this study proposes that a reanalysis of previous direct flux measurements with the new method discussed here can improve researchers’ understanding of air–sea interaction.

Corresponding author address: Brian Ward, School of Physics, National University of Ireland, Galway, University Road, Galway, Ireland. E-mail: bward@nuigalway.ie

Abstract

Ship-based measurements of wind speed and direct fluxes are affected by airflow distortion that can lead to a tilt of the wind vector as well as acceleration or deceleration of the wind speed. Direct flux measurements are additionally affected by the fluctuating velocity of the platform. The classic approach is to first correct the wind speed for angular and translational platform velocities and thereafter rotate the wind vector into the mean flow. This study finds that for ships under way, this leads to an overestimation of the vector tilt and biased flux estimates. This may explain the common observation that flux estimates from ships in transit have lower quality than measurements taken on station. Here an alternative approach is presented, where the flow-distortion-induced tilt of the wind vector is estimated from the 3D wind speed measurements and applied to the apparent wind vector. The tilt correction is carried out after correction for the fluctuating part of the platform velocity but before removing the ship’s mean translational velocity. This new method significantly improved the agreement of direct momentum flux measurements made from a ship under way with the parameterization of the COARE3.5 bulk model. The sensitivity of the eddy covariance measurements of momentum and scalar fluxes to the choice of the tilt-motion correction method is analyzed, and this study proposes that a reanalysis of previous direct flux measurements with the new method discussed here can improve researchers’ understanding of air–sea interaction.

Corresponding author address: Brian Ward, School of Physics, National University of Ireland, Galway, University Road, Galway, Ireland. E-mail: bward@nuigalway.ie

1. Introduction

Air–sea exchange plays an important role in the global transport of momentum, heat, and mass. Our level of understanding of these processes feeds directly into the quality of weather forecasting and climate models. Direct flux measurements from oceanographic platforms like ships, buoys, and coastal towers have become key to studying these processes and have allowed the development and verification of bulk parameterizations, like the COARE algorithm (Fairall et al. 2000, 2003).

Wind speed plays an important role in all air–sea exchange processes. However, accurate measurement of the in situ wind speed over the ocean are complicated by the fact that the large structures of ships and the meteorological instrumentation itself obstruct the air flow and thus lead to distortions in the wind field. Even at well-exposed measurement positions, this can lead to errors in the wind speed by about 10% (e.g., Yelland et al. 2002; Popinet et al. 2004). Acceleration of the air flow was found to be mainly a function of the measurement location and the relative wind direction (Popinet et al. 2004), with some evidence for a dependence on the relative wind speed (O’Sullivan et al. 2013).

In addition to the acceleration, the relative wind vector is tilted and displaced when the streamlines circumvent the obstacle (Yelland et al. 1998). The tilt of the wind vector is especially relevant for direct flux measurements (Wilczak et al. 2001; Griessbaum and Schmidt 2009). Knowledge of the displacement is required for the correct normalization of the wind speed, measured at height z, to the standard height of 10 m MSL and neutral stability. The displacement is even more important for the estimation of the momentum flux using the inertial dissipation method (IDM), where the third power of the estimate of the friction velocity is proportional to the effective measurement height (Yelland and Taylor 1996).

Direct flux measurements on moving platforms require accurate quantification and correction for the effects of platform motion and distortion of the air flow.

The vertical flux of a quantity x can be directly measured as the covariance with the vertical component (w) of the wind speed in the mean streamline coordinate system:
e1
The eddy covariance (EC) method exploits (1) to measure the vertical flux of momentum, heat, and trace gases; for example, the friction velocity is defined as1
e2
Before (1) can be applied, the coordinate system needs to be rotated into the local mean streamline coordinate system. This is typically achieved by a double rotation (DR) that aligns the coordinate system with the mean flow by rotating it first in the horizontal plane xy to achieve and then in the new x′–z′ plane to obtain , for a given averaging interval of typically 10–30 min.

The DR method leaves one remaining degree of freedom, the orientation in the resulting y″–z″ plane. McMillen (1988) suggested that the angle about which the coordinate system needs to be rolled around the new x″ can be defined by requiring . This results in a triple rotation (TR), where the coordinate system is first yawed, then pitched, and finally rolled. The underlying assumption for the TR method is that the crosswind surface stress is zero. This might not always be valid over the open ocean when wind and swell direction are not aligned. Furthermore, uncertainties in the measurements of due to the limited averaging time can lead to erratic corrections with the TR method (Wilczak et al. 2001).

Wilczak et al. (2001) found that the limited averaging time and small measurement errors in the three wind speed components can lead to large errors in the tilt corrections using the DR or TR method, and they proposed an alternative method, “planar fit” (PF), to derive pitch and roll angles of the wind vector that force for a large ensemble average (indicated by the overbar). Each sample interval then requires only a single rotation (SR) in the new x′–y′ plane to force .

It has to be stressed here that the fluxes determined from (1) are highly sensitive to the correct choice of the coordinate system (Deacon 1969); that is, the tilt of the air flow due to large structures needs to be accounted for. For small tilt errors, the fractional bias in the turbulence fluxes is proportional to the tilt and can vary with stability and boundary layer depth that define the ratios of and , or (Wilczak et al. 2001).

Wyngaard (1981) showed that the effect of small nearby structures (e.g., the turbulence probe itself) is not sufficiently corrected by the tilt corrections and presented a set of equations to correct flow distortion in the turbulence quantities arising from objects of size . Modern sensor design and calibration tries to minimize the distortion effects of the turbulence probe itself. The corrections, which were proposed by Wyngaard (1981), have been applied by Oost et al. (1994) and Edson et al. (1991) to correct momentum flux measurements for the effect of a horizontal mounting pole near a 3D sonic anemometer. This involved measurements of the tilt of the wind vector made with the sonic anemometer in two different positions (upright and pointing downward). Oost et al. (1994) and Edson et al. (1991) showed that applying the Wyngaard (1981) correction reconciled the values from the up and down measurements.

Oost et al. (1994) expanded the approach of Wyngaard (1981) to multiple objects but also reported that the corrections would overestimate the airflow distortion by objects that are, in size, comparable to the measurement height, that is, the measurement platform. We are not aware that the corrections proposed by Wyngaard (1981) and Oost et al. (1994) have found wider application in other open ocean flux measurements. There appears to be general consensus in the meteorological community to minimize turbulent flow distortion by identifying optimal sensor locations, rather than applying empirical corrections to a complex problem. Thus, generally only mean wind speed and tilt corrections are applied to the measurements.

On moving platforms the wind speed measurements are additionally biased by the changing sensor orientation (pitch and roll of the platform) and the relative velocity at the probe location. Anctil et al. (1994) reported direct flux measurements from a discus buoy and presented an equation for the true motion-corrected wind speed using fast measurements of the tilt and acceleration of the platform. This approach was expanded to ships under way by Edson et al. (1998), who also presented methods to calculate the tilt angles using the angular rate and acceleration measurements from a strapped-down motion sensor. An addition to Edson et al. (1998) was made by Miller et al. (2008) to explicitly account for misalignments between the anemometer and the motion sensor. The true wind speed in the earth reference frame is computed as the sum:
e3
where is the measured wind speed rotated from the platform coordinate system (denoted by subscript p) into the earth reference system (denoted by e) and is the platform motion contamination [for the explicit formulas of and the coordinate transformation matrix , see Edson et al. (1998) and Miller et al. (2008)]. In order to rotate the wind vector into the local streamline coordinate system for the computation of fluxes, Anctil et al. (1994) applied a double rotation to :
e4
Note that Anctil et al. (1994) applied (4) to eddy correlation (EC) measurements from a moored buoy, hence . This equation has become the baseline for all direct flux measurements from mobile platforms, including ships under way.

Edson et al. (1998) compared momentum flux measurements on board two different research vessels, the Iselin and the Wecoma, with simultaneous measurements on (i) the Research Platform (R/P) Floating Instrument Platform (FLIP) and (ii) a small autonomous catamaran. In both cases the along-wind stress component measured on board the ships was approximately 15% higher than the measurements from either FLIP or the catamaran. The difference was attributed to airflow distortion biases in the ship’s EC measurements, with the assumption that the measurements taken on R/P FLIP and the catamaran were free from airflow distortion (Edson et al. 1998). For R/V Iselin, Edson et al. (1998) reported tilts in the true wind vector of approximately 3° to 5° for bow-on and beam-on wind directions, respectively. Pedreros et al. (2003) supported these results, reporting that measured on board the R/V L’Atalante was on average 18% higher than direct flux measurements made from an air–sea interaction spar (ASIS) buoy, for a maximum distance of 5 km. However, this number is based on a linear regression to the direct flux measurements from the ship and from the ASIS buoy, which had a relatively large intercept (see Fig. 7a in Pedreros et al. 2003). Assuming no intercept, the average overestimation becomes approximately 30%. Pedreros et al. (2003) also reported a 30% systematic overestimation of the neutral drag coefficient by the ship’s EC measurements, when compared to the EC measurements from the ASIS buoy and shipborne IDM estimates (see Fig. 11 in Pedreros et al. 2003). This overestimation includes all measurements made during the experiment. Pedreros et al. (2003) argued that the difference between the direct measurement on the ship and the buoy might be caused by different fetches encountered by the ASIS buoy and the ship during the course of the experiment. However, the IDM results for from the ship agree with the measurement of from the buoy (see Fig. 11 in Pedreros et al. 2003). We therefore assume that the overestimation of by the EC measurements on board the R/V L’Atalante was on average +30% and was due to flow distortion on the ship. Pedreros et al. (2003) also reported an overestimation of the sensible heat flux by 9% (±9%) when compared to direct measurements from the ASIS buoy.

Pedreros et al. (2003) restricted the dataset to relative wind directions less than ±30° and and corrected the measured wind speeds for mean airflow distortion. The corrections for acceleration, relative wind direction, and tilt are reported in Dupuis et al. (2003): the average vertical tilt of the wind vector was 7°, and it decreased slightly with increasing relative wind direction. The direct fluxes were calculated using (4) and no attempt was made for a flow distortion correction on the fluxes.

The apparent bias in the direct flux measurements motivated Edson et al. (2013) to exclude shipborne EC momentum flux measurements from the recent update to the COARE bulk flux algorithm (version 3.5). However, earlier versions of COARE (Fairall et al. 2000, 2003) did include shipborne EC momentum flux measurements. For the bulk values from COARE3.0 are 10% higher than for the newest version, COARE3.5, and for COARE3.0 predicts 2.5% higher values than COARE3.5. The ratio decreases approximately linear over the wind speed range . Direct EC flux measurements of trace gases from buoys are rare, mainly due to the usually higher power and maintenance requirements and therefore shipborne flux measurements are still the main input to bulk flux formulations for gas exchange (Fairall et al. 2011).

Here we analyze mean wind speed and direct flux measurement taken on board the R/V Saramiento de Gamboa during the Middle Atmosphere Dynamics and Structure (MIDAS)–Salinity Processes in the Upper Ocean Regional Study (SPURS) field campaign in March/April 2013. We compare them to the mean wind speed measurements and bulk flux estimates from a surface mooring. The wind speed and direction measurements from the surface mooring are used to estimate a mean flow distortion correction for the wind speed measurements taken on the bow mast of the research vessel. The wind vector tilts are estimated using an adaptation of the planar fit method and are used in a new way to correct the wind speed measurements for ship-motion and airflow distortion, prior to the direct flux calculations.

Our new approach to airflow distortion and motion correction is presented in section 2, followed by the description of the experiment and the methods used to correct the mean wind speeds for acceleration and tilt of the streamlines (section 3). The results obtained using the classic approach (4) and our new method are compared in section 4. Our conclusions are provided in section 5.

2. Theory

a. Parameterization of the wind vector distortion

Before the measured 3D wind speed can be used to calculate mean wind speed and direct fluxes, it has to be corrected for airflow distortion and ship motion. The effect of airflow distortion on the mean wind speed measurement can be described with a flow distortion transformation matrix:
e5
where a is the acceleration of the wind speed; α is the relative wind direction; and are the rotations of the wind vector by the Euler angles , respectively. Although the relative wind direction fluctuates over time, we assume that when the variations of α are small, the effect of flow distortion can be approximated by . This means that becomes essentially a function of the relative horizontal wind direction, as the average undisturbed vertical wind speed is close to zero over the open ocean. Increased motion (heave, pitching, rolling, or yawing) of the platform can potentially create large deviations from the expected mean flow distortion.
The relation between the measured 3D wind speed average and the average of the true wind vector can now be described with the following equation:
e6
It is important to note that it is the relative wind vector and not that is transformed by .

b. Height displacement of the air flow

There are several reports of significant height displacements of the air flow by several meters, and the wind speed measured at height z might originate from . Yelland et al. (2002) found that the magnitude of the displacement varies with the relative wind direction and that variations with the relative wind speed could be neglected. In order to make wind speed measurements comparable, they need to be normalized to a height of 10 m MSL and neutral stability. Therefore, it is necessary to know the original height of the measured wind speed. Assuming a universal logarithmic profile, we can express the wind speed at height as
e7
where is the von Kármán constant, assumed to be close to 0.4; and is an empirical function that describes the effect of the buoyancy flux on the shape of the logarithmic wind profile (Monin and Obukhov 1954). The function depends on the ratio of the measurement height and the Monin–Obukhov length:
e8
where g is the acceleration due to gravity, T is temperature (K), and q is the specific humidity (kg kg−1).
In (7) we defined the correction function as
e9
For neutral stability (i.e., ) and typical , it can be estimated that . For unstable conditions : . However, for stable conditions has the same sign as and can potentially become the order of 1.
For a ship under way, the measured wind speed at height z is given by
e10
Here we simplified the notation using and , and we defined the ratios
e11
e12
which depend on the true wind speed, the ship velocity, and the true relative wind direction , which is given as the difference between the true wind direction and the ships heading:
e13
This gives for bow-on winds and for winds from the stern. The measured relative wind direction is defined analogs to (13) with replacing . Both and are ≤1 for . For the last step in (10), we used the approximation .
We can also relate the flow-distorted wind speed measurements at two different heights ( and ) if we express and , respectively, based on the wind speed at height z:
e14
where was used, and the terms involving and higher were omitted.
Equations (10) and (14) show that for the estimation of the wind speed acceleration , the ship speed and atmospheric stability need to be taken into account. Further, an estimate of the height displacement is required to normalize the flow distortion and ship-motion-corrected true wind speed (relative to the surface current ) for height and stability:
e15

If the height displacement is unknown, then (10) and (14) can be used to estimate the uncertainty arising from a potential .

c. Motion correction and rotation into the mean flow

We find that rotating the wind vector after the motion correction [i.e., using (4)] leads to an overestimation of the wind vector tilt and can cause potentially large errors in the computed fluxes if the ship’s mean velocity is not zero.

We suggest here that the tilt of the wind vector, which is mainly caused by airflow distortion, needs to be removed before the wind vector is corrected for the mean ship velocity. The derived true wind vector should only be rotated within the horizontal plane.

We find that applying the rotations , , and before correcting the measured wind vector for pitch, roll, yaw, and heave of the ship leads to a less efficient removal of the platform motion signal from the co- and power spectra. Our interpretation is that the velocity signal caused by this fast platform motion is imprinted on the measurements of the already distorted wind vector, and therefore it needs to be removed before rotating the wind vector. Our modified version of (4) becomes
e16
where the ship motion is separated into the mean horizontal velocity and the fluctuating part (pitch, roll, yaw, and heave) with .
The transformation denotes a modification to (5): the relative wind vector is not divided by but is subtracted. Thus, (5) becomes
e17
Note that if transformation were used in (16), then the covariance moment would be scaled with the correction for the mean wind speed acceleration .2 There is, however, no evidence in literature nor in our results that the acceleration of the mean wind speed reflects equally on the turbulent quantities. Also note that after the motion and tilt correction in (16), only a single rotation over the z axis is applied to the true wind vector in order to align it with the local streamline (thereby forcing ).
If the flow distortion correction is not available—for example, if the quantity of data from one wind direction sector is not large enough to perform a planar fit—then the wind speed measurements should be corrected in the following way:
e18
Equation (18) describes (i) rotation of the measured wind speeds into earth frame and correction for ; (ii) double rotation to achieve first and then ; (iii) reversal of the rotation around the z axis, indicated by ; (iv) correction for ; and (v) a final rotation around the z axis to achieve .
For on-station measurements, (18) is equivalent to (4), but it avoids the overestimation of the wind vector tilt if the mean platform velocity is >0. Unlike (16), (18) cannot account for potential roll or yaw of the relative wind vector and can therefore lead to biased flux estimates. The flow-distortion-induced yaw of the wind vector becomes relevant only when the platform is moving and the vector addition of and leads to an incorrect estimate of the true wind direction. This bias in has no effect on the scalar fluxes or [if defined as in (2)]. It does, however, lead to incorrect estimates of the wind stress angle, defined as
e19
The roll and pitch of the wind vector are relevant for all fluxes. Further, the planar fit reduces the effect of calibration errors and the undersampling of large eddies, as discussed by Wilczak et al. (2001). Therefore, (16) is superior to (18), even if mean acceleration a and yaw bias are unknown, and should be applied when possible.

3. Methods

The measurements presented here were taken on board the R/V Saramiento de Gamboa during the SPURS–MIDAS cruise (from Las Palmas de Gran Canaria on 16 March to Ponta Delgada, Azores, on 17 April 2013). Relative wind speed and direction were measured on the bow mast of the ship at two measurement heights, and , with a R3A 3D sonic anemometer (Gill) and a 2D vane anemometer Wind Monitor-MA (R. M. Young), respectively. Figure 1 shows a photograph of the bow of the ship on which the positions of the two anemometers are indicated. Note that the 3D sonic was located approximately 1.5 m in front of a large tower with a cross section of 0.75 m. An inertial motion unit NAV440 (Moog Crossbow) recorded 3D acceleration and angular rates. The unit was installed 1.16 m behind and 0.82 m below the R3A. This position vector was accounted for in the motion correction following Edson et al. (1998). A differential GPS system, VS-100 (Hemisphere GPS), recorded the ship’s heading and course and provided a universal time stamp. Mean meteorological data (downwelling longwave and shortwave, air temperature, relative humidity, atmospheric pressure, and the relative wind speed at ) were recorded at 0.1 Hz and averaged to 12 min. The 3D wind speed , the speed of sound temperature , and ship motion were recorded at a measurement frequency of 20 Hz. They were used to calculate true wind speed and the covariance moments. The details of the motion correction procedures are described in section 3d. The direct comparison of the ship’s heading and the heading recorded by the VS-100 revealed a yaw of 5° in the orientation or the flux mast in respect to the centerline of the ship. All measurements were corrected for this tilt offset prior to further analysis.

Fig. 1.
Fig. 1.

Photograph of the bow of the R/V Saramiento de Gamboa with indications of the positions of the 3D sonic and the 2D vane anemometers. Also shown is an illustration of the overestimation of the pitch of the relative wind vector by the classic tilt-motion correction (4). The relative wind vector is tilted by the angle due to the presence of the ship and measured as with vertical component w. Applying the DR after the correction for gives , which overestimates the tilt.

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

A buoy, the WHOI station 41061,3 was moored in September 2012 at the measurement site (24.5811°N, 38°W) and provided mean meteorological measurements with 1-min resolution. EC flux measurements were also performed on the buoy, but the data are not available at the current time. The wind speed and direction measurement from the buoy are used as a freestream reference to estimate the effects of mean airflow distortion in the shipborne measurements. Figure 2 shows a time series of wind speed, direction, and friction velocity measured on board the ship and by the buoy (the buoy is computed with the COARE3.0 bulk flux algorithm). Also shown are the ships speed and the distance to the buoy.

Fig. 2.
Fig. 2.

Time series of (a) normalized wind speed and significant wave height, (b) friction velocity, (c) measured relative wind direction and ship speed, and (d) true wind direction and distance between ship and buoy position. The buoy values are COARE3.0 bulk flux results calculated from 1-min measurements on the mooring, which were averaged to 12 min and interpolated on the bow mast time series using the GPS time stamps. Bow mast , true wind direction, and are calculated using (15) and (16), respectively. The shaded areas mark periods that were selected for the estimation of the yaw and acceleration bias in the measured relative wind direction and speed on the bow mast. These periods are also used for the comparison of the direct measurements of on the bow mast with the COARE3.0 estimates from the buoy.

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

a. Quality control

The 12-min time series of was used to calculate average values and the direct fluxes , , and ; the same was done for 4-min averaging periods.

For mean variables like and , intervals were rejected when one of the following empirical criteria was fulfilled:

  • Measured relative wind direction:
    eq1
  • Variation of the three subinterval wind directions:
    eq2
  • Variation of the ship velocity:, where is the average difference between two consecutive measurements.
  • The vector average of the instantaneous course/heading vector was smaller than 0.95 (1 indicates a perfectly stable course).
  • Measured average relative wind speed: .

In order to reduce the loss of sample intervals due to maneuvering of the ship, the 12-min intervals were defined every 4 min; thus, three intervals were overlapping. If more than one of three contiguous (overlapping) intervals passed the quality control, then only the first interval was accepted. Of the 1515 independent 12-min intervals, 964 passed the first quality control. For the direct fluxes, the following additional rejection criteria were applied:

  • Measured relative wind direction: .
  • Variation of the three subinterval wind directions: .
  • The cross correlation suggested a shift of sample of the time series.

In total, 612 intervals passed these additional criteria.

We used periods with relatively constant conditions and short distances between the buoy and the ship (≤100 km) for the estimation of the wind speed acceleration and yaw bias as well as for the comparison of the direct flux measurements with the bulk flux estimates from the buoy. These periods are marked as shaded areas in Fig. 2. For these tasks, intervals were additionally rejected when criteria for the ship’s mean variables were fulfilled (see above) or based on the following:

  • Ship speed minimum: .
  • Difference between ship’s course and heading: .
  • Stability (at buoy and ship): .
The first two criteria were implemented to reduce the uncertainty in the wind direction measurement. The third criterion accounts for the uncertainty arising from large wind profile corrections for . These controls were passed by a total of 560 with 12-min intervals, of which 185 passed the criteria for direct fluxes (see above). Note that for these intervals, the significant wave height was always below 3.1 m and thus below the height of the wind speed is measured at the buoy (3.3 m MSL).

b. Estimation of the wind vector pitch and roll

The pitch and roll of the wind vector were estimated from the 3D relative wind speed measurement at the bow mast using a method similar to the anemometer tilt correction algorithm, PF, presented by Wilczak et al. (2001). The underlying hypothesis is that, over the ocean, the time average of the vertical wind speed component is close to zero. An observed nonzero vertical wind speed can be caused by (i) a tilt of the anemometer (i.e., the platform), (ii) a tilt of the wind vector due to flow distortion, or (iii) measurement errors.

In this experiment, the anemometer was mounted collinearly with the motion sensor. However, even after accounting for the platform tilt by using the apparent wind speed measurements , the mean vertical wind speed was nonzero and increased with growing relative wind speed. Therefore, we assume that the observed nonzero vertical component of the average apparent wind speed is caused by flow distortion.

The method presented here is an adaptation of the PF for the situation where the structure of the wind field cannot be described with a single set of tilt angles. In the case of shipborne measurements, the flow distortion pattern will likely vary with the relative wind direction. To account for this, the PF method is applied on wind direction sectors. A sector planar fit was also used by Yuan et al. (2010) to account for airflow patterns over forestry. In Yuan et al. (2010) the roll and pitch angles are defined in a fixed coordinate frame. Here the pitch and roll angles are defined relative to the average relative wind direction. This has the advantage that (i) a direct comparison with the pitch estimate from the DR method is possible and (ii) the effect on the EC fluxes can be estimated using small angle approximations (see Wilczak et al. 2001).

To determine the tilt angles and , the apparent wind speed measurements were averaged over 12-min intervals and sorted into bins based on the average relative wind direction . From the members of each bin, a mean relative wind direction was computed and each individual averaged wind vector was rotated into this ensemble mean flow:
e20
The wind vector pitch and roll in the streamline coordinate system was then derived from a linear regression to the measured vertical wind speed with the horizontal wind speed components:
e21
The mean pitch and roll of the relative wind vector for each sector are given by
e22
e23
The result of the regression and the residual c3 are shown in Fig. 3. An asymmetry between port and starboard winds can be seen. This could be caused by a slight misalignment of the bow mast to the ship, which was estimated to 5°. We acknowledge that the definition of a roll angle in a radial coordinate system is questionable, as will vanish for sufficiently small wind direction bins. However, we found that including in the analysis did significantly improve the results when compared to using . In order to distinguish the abovementioned variation of the PF method from others, it will be dubbed radial planer fit method (rPF).
Fig. 3.
Fig. 3.

Pitch and roll angles ( and , respectively) and the vertical wind speed residual c3, determined by the linear regression (21) over 15° wind direction bins, are plotted as functions of the measured relative wind direction. The shading shows the 95% confidence intervals of the regression.

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

In Fig. 4 the pitch angles estimated using (4) and (18) are plotted as functions of the relative wind direction. Also shown are the results from the rPF method . The pitch estimates from the classical DR method show a strong dependence on the relative wind direction and on the ratio . The highest values of are reached when the ship steams into the wind. The estimates from the DRx method show much less variability and agree well with the estimates from the rPF method, however, they give on average slightly higher pitch values than the rPF method, especially at low wind speeds. This is because the DRx method cannot account for .

Fig. 4.
Fig. 4.

Wind vector pitch at the 3D sonic calculated using the classic DR (○) and using DRx as a function of the measured relative wind direction and the ratio of ship velocity and measured wind speed. The results for pitch and roll from the rPF method (see Fig. 3) are shown for comparison (black and blue lines).

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

The apparent overestimation of the wind vector tilt by the DR correction method is illustrated in Fig. 1 for the example of the ship steaming into the wind . The relative wind speed is pitched upward by θ, which leads to a measured average vertical velocity . Deriving the pitch angle from and the corrected horizontal wind speed as done in (4) gives .

c. Estimation of yaw bias and acceleration

While pitch and roll angle can be estimated directly from the 3D wind speed measurement on board the ship, the estimation of the yaw angle and the acceleration of the mean wind speed a requires a freestream reference. Here we utilized wind speed and current measurements from the WHOI mooring.

The wind speed measurements on the mooring were made at a height of 3.3 m MSL and corrected to a standard height and neutral stability using the bulk flux parameterization COARE3.0 (Fairall et al. 2003). The values were converted to using the local stability at the location of the R/V Saramiento de Gamboa, derived from the direct flux measurements and the measurement height of the bow mast sonic z = 10 m MSL. In order to calculate , we made use of the approximation . The freestream wind vector relative to Earth at the location of the ship was calculated, assuming the same surface currents as measured by the buoy. The expected relative wind vector was calculated as
e24
and used to estimate the acceleration of the relative wind speed and the yaw bias as functions of the measured relative wind direction. Note that the speed of the surface current measured at the buoy location was on average 0.2 m s−1 with a maximum value of 0.4 m s−1.

Figure 5 shows the ratio of the relative wind speed measured at the bow mast and the expected relative wind speed calculated from the buoy. The ratio is given by the acceleration/deceleration of the wind speed when the streamlines circumvent the ship superstructure and by a potential uplift of the streamlines by the height difference . For this study we assumed , that is, and thus . For this could cause an underestimation of that will increase with the relative wind direction. To estimate the uncertainty in caused by the unknown height displacement, we used (10) to calculate for a constant (dashed blue line in Fig. 5). Even for the relatively large , the average difference of +3% acceleration is small compared to the large uncertainty in due to the variability in the wind field.

Fig. 5.
Fig. 5.

Ratio of the measured relative wind speed and the expected relative wind speed, calculated from the buoy wind speed and the ship velocity using (24) as a function of the measured relative wind direction. The uplift was assumed to be zero . Individual measurements that were selected for the ship buoy comparison are shown as black dots. The red line shows averages that were taken over 15° relative wind direction bins. The blue dashed line shows the bin average values for an assumed uplift . The error bars show the standard deviation of the mean.

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

The relative wind direction measurements and expected values are compared in Fig. 6. Positive relative wind directions are overestimated and negative relative wind directions are underestimated; that is, the magnitude of the measured relative wind direction is always higher than expected. This means that the approaching air is deflected away from the ship’s structure. The magnitude of the bias increases with increasing relative wind direction but starts to decrease beyond ±90°; these data exhibit an asymmetric behavior similar to θ and ϕ (see Fig. 3).

Fig. 6.
Fig. 6.

Expected relative wind direction calculated from the buoy wind speed and the ship velocity using (24) as a function of the measured relative wind direction. Individual measurements that were selected for the ship buoy comparison are shown as black dots. The red line shows averages that were taken over 15° relative wind direction bins. The error bars show the standard deviation of the mean.

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

d. Correction for platform motion and airflow distortion

The ship motion, platform tilt , and the velocity bias were calculated for subintervals of 12 min using the motion correction script from Miller et al. (2008). The code was modified to allow for differences in the ship’s heading and the actual course. The velocity bias was separated into a mean and fluctuating part as outlined in section 2.

The average relative wind direction was calculated from the apparent wind vector. The tilt angles , , , and the acceleration were derived by linearly interpolating the results from the two nearest wind direction bins. The time series of the true wind speed in the local streamline coordinate system were computed (i) using (4), (ii) using (16), and (iii) using (18). The covariances calculated with (4), (16), and (18) will be labeled standard DR method, rPF, and DRx, respectively.

4. Discussion of results

a. Intercomparison of the bow mast wind speed measurements

Figure 7 shows the observed ratio of the wind speeds recorded by the 3D sonic anemometer at height and the 2D vane anemometer at , which was expected to experience less airflow distortion due to the more exposed measurement location. The observed ratio does not only depend on the mean relative wind direction but also on the true wind speed, the platform velocity, and the local stability, which are combined in the factor [see (14)]. Note that the approximation is only valid for and near-neutral or unstable conditions. The color scale in Fig. 7 indicates the magnitude of the correction factor assuming and . Also shown are bin averages of calculated with (14) by (i) ignoring , (ii) assuming m, and (iii) assuming m. Note the similar shape of and in Fig. 5.

Fig. 7.
Fig. 7.

Observed ratio of the wind speed measured by the 3D sonic (10 m MSL) and the 2D vane anemometer (16 m MSL), as a function of the measured relative wind direction. Also shown are the bin averages of r and the ratio of the accelerations , calculated with (14), for and .

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

b. Intercomparison of the bow mast fluxes to the buoy measurements

In Fig. 8 the ratio of measured on the ship and the bulk estimates from the buoy using COARE3.0 is plotted as function of the relative wind direction. Ratios are shown for EC fluxes calculated with DR, DRx, and the rPF tilt correction. The direct , calculated with DR, overestimates the bulk flux estimates from the mooring by 46%; this ratio varies with the relative wind direction, with the largest values occurring at the wind sectors ±(20°–60°). The values calculated using DRx are on average 20% higher than the estimates from the buoy and 5% for rPF. From the three ratios, the one with rPF shows the least variation with the relative wind direction. Note that at , the values computed with COARE3.0 are 10% higher than for the newest version of COARE, COARE3.5, and for the difference is 2.5%. For this comparison, the data were restricted to the selected time periods marked in Fig. 2.

Fig. 8.
Fig. 8.

Ratio of the EC measurements of on board the R/V Saramiento de Gamboa and the COARE3.0 bulk flux estimates from the mooring as a function of the measured relative wind direction measured on the ship. Individual 12-min measurements are shown as scatter and average values as lines. Results from the rPF method (16) are shown as blue and —, and results using DR as black and , and DRx as red ○ and . For this comparison, the data were restricted to the selected time periods marked in Fig. 2.

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

c. Drag coefficient and stress angle

To demonstrate the effect of the choice of the tilt-motion correction on the measurement of the momentum flux from moving platforms, we show scatterplots of and the square roots of the crosswind component of the momentum flux calculated with (4) and (16) and in Figs. 9 and 10.

Fig. 9.
Fig. 9.

Direct measurements of as a function of and the measured relative wind direction: (a) results obtained using the standard motion correction and rotation into the stream (4) and (b) results using the rPF method (16). Individual measurements (12 min) are shown as (○) and bin averages over 30° wind direction and 2 m s−1 wind speed bins are shown as thick lines and big , respectively (error bars show the standard deviation). The COARE3.0 and COARE3.5 bulk flux predictions calculated from are shown as green and black lines, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

Fig. 10.
Fig. 10.

As in Fig. 9, but for the sign-preserving square root of the crosswind stress component . Note that positive values indicate downward crosswind momentum flux.

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

The direct measurements of calculated with (4) show a strong dependence on the relative wind direction and a weak correlation with (see Fig. 9a). When compared to the COARE3.5 bulk flux formulation (Edson et al. 2013), the EC values appear largely overestimated. The magnitude of this effect increases with decreasing wind speed as the ratio of and , and thus the tilt bias, increases (see Fig. 4). In contrast the use of (16) leads to a significantly better correlation between and (see Fig. 9b). The values are mostly higher than the COARE3.5 results.

The results for the square root of the crosswind stress of the momentum flux are shown in Fig. 10. Over land, the crosswind stress is expected to be zero. Wind–wave interactions have been proposed as explanation for observed nonzero stress angle (e.g., Rieder et al. 1994). Tilt- and motion-corrected open ocean crosswind stress measurements on R/P FLIP presented by Miller et al. (2008) are on average close to zero and show no correlation with the along-wind stress. Edson et al. (1998) and Pedreros et al. (2003) presented evidence that shipborne direct measurements overestimate the crosswind stress when compared to small platforms (a catamaran and an ASIS buoy), which they attributed to airflow distortion. The results using (4) in Fig. 10a show a strong dependence of the average crosswind stress on the relative wind direction, with the highest magnitudes at wind direction from ±(30°–60°). The application of (16) results in much smaller average crosswind stress estimates (Fig. 10b). The observed strong correlation of the crosswind stress with wind speed for −90° to −30° appears mostly unchanged by the choice of the tilt correction, but the absolute values of are changed. Note that the positive sign in Fig. 10 denotes a downward momentum flux. The wind stress angle for the rPF results was calculated with (19) and is plotted in Fig 11 as a function of the relative wind direction and . Large stress angles up to ±90° are found for low wind speeds. This might be caused by wind–wave interaction, or simply by the fact that the wind direction estimate is not very accurate for low wind speeds. For , a bin average was calculated for 15° wind direction sectors.

Fig. 11.
Fig. 11.

Measured stress angle (using rPF results) as a function of the measured relative wind direction and . The bin averages (shown as black line) are taken for over 15° wind direction bins. Error bars indicated standard deviation from the mean.

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

We could not identify a physical reason for the high crosswind stress values at −90° to −30° relative wind direction. Inaccuracies in the estimated wind direction (see Fig. 6), which would lead to parts of the along-wind stress being seen as crosswind stress, or the effects of airflow distortion could be a possible explanation.

In order to compare the performance of the different corrections for different relative wind directions, the wind speed dependence of is removed (on average) by calculating the ratio of the measured EC and the COARE3.5 bulk flux prediction for using the flow-distortion-corrected wind speed . These ratios are plotted in Fig. 12 as a function of the relative wind direction and the wind speed. Also shown are the bin averages of the ratios over the wind speed range 6–10 m s−1. These are replotted in Fig. 13 together with the result for using the rPF method but ignoring the roll angle (rPFϕ = 0). The ratios obtained from the DR method show large variations with the relative wind direction and also with the wind speed; note that the ship was under way with for most of the measurements. The highest values are reached at ±(20°–60°), but even for bow-on wind direction is overestimated by ~25% when compared to the COARE3.5 bulk flux prediction. The results from the DRx method show no clear dependence on the wind speed but qualitatively the same dependence on the relative wind direction as the DR results. For rPF the variation of the ratio with the wind direction is further reduced. But elevated values for indicated that rPF does not completely eliminate the effects of airflow distortion on the EC flux estimates. However, a significant improvement is obvious when compared to DR and DRx. Also shown in Fig. 13 is the effect of ignoring the roll offset in the rPF correction. The values for rPF are very similar to the DRx results. In fact, the difference between the rPF and DRx results arises from the differences in and (see Fig. 4). The rPF method accounts for the residual vertical wind speed measured by the 3D sonic anemometer (see Fig. 3), while DRx does not, which leads to higher estimates for .

Fig. 12.
Fig. 12.

Ratio of the measured with EC and the COARE3.5 bulk flux prediction using as a function of the relative wind direction and the wind speed. The ratios are presented as bin-averaged values over 15° wind direction and 1 m s−1 wind speed bins (minimum six measurements per bin). Also shown are 15° wind direction bins for (minimum 12 measurements per bin). The error bars indicate the standard deviation. Results using (a) DR, (b) DRx, and (c) rPF.

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

Fig. 13.
Fig. 13.

The bin average ratios from Fig. 12 for DR, DRx, and rPF, for the wind speed range . Also shown are the results for setting the roll angle to zero in the rPF correction method (rPF = 0). Each bin average is based on a minimum of 12 individual measurements.

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

Note that a ratio of 1 in Figs. 8, 12, and 13 does not necessarily imply that the flux measurements are free from airflow distortion. The variation of (rPF) with the relative wind direction implies, however, that the presented EC flux measurements are significantly biased by airflow distortion, which cannot be fully accounted for by the tilt correction. It is likely that obstruction of the airflow by the tower on the bow of the R/V Saramiento de Gamboa did contribute substantially to the observed flow distortion bias in the EC flux measurements.

d. Errors in the direct fluxes due to the standard motion correction

In this section the results for , and , obtained using DR and DRx, are directly compared with the results obtained by using the rPF method. This is put in context with previous reports about direct comparisons of EC fluxes from ships and buoys.

In Fig. 14 the ratios of the values computed with DR, DRx, and rPFϕ = 0 and the values computed with rPF are plotted as a function of the relative wind direction. The ratios are presented as bin averages over the same 15° wind direction bins as used in Fig. 3. This allows for studying the part of the flow distortion bias in that is caused by the biased estimation of the wind vector tilts if the DR and DRx methods are used. Note that DR and DRx give identical results for station measurements. Figure 15 shows the ratios obtained for the scalar flux .

Fig. 14.
Fig. 14.

Ratios of computed with DR, DRx, and rPFϕ = 0 and computed with rPF as a function of the relative wind direction. The data are presented as bin averages with the standard deviation indicated as the shaded area or error bars. The same 15° wind direction bins as in Fig. 3 are used.

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

Fig. 15.
Fig. 15.

As in Fig. 14, but for the ratios of the sonic sensible heat flux .

Citation: Journal of Atmospheric and Oceanic Technology 32, 6; 10.1175/JTECH-D-14-00137.1

The bias in is caused by a combination of the pitch and roll offset. This can be seen by comparing the ratio (DRx)/(rPF) with the ratio (rPF = 0)/(rPF) that depends only on the roll offset. This shows that for the DRx results, the bias caused by ignoring the roll is comparable to the bias caused by the slightly overestimated pitch (see Fig. 4). The large overestimation of the pitch for the underway measurements in the DR method (see Fig. 4) leads to much higher values for the ratio (DR)/(rPF) when compared to (DRx)/(rPF). Because of the contribution of the roll, the ratio (DR)/(rPF) reaches a maximum of 2 for negative and 1.5 for positive relative wind directions.

For the scalar example , the roll offset has almost no effect [(rPF = 0)/(rPF)] and the ratio (DR)/(rPF) has one distinct maximum for the bow-on wind direction. The DRx method leads to flux values that are 1%–3% higher than the results obtained with the rPF method.

Edson et al. (1998) observed an overestimation of by 8% for and also reported a wind vector tilt angle of 3°–5°. The tilt angles observed in this study ranged from 6° to 9°; a simple scaling of Edson et al. (1998) results would thus lead to 16% overestimation in . This falls somewhere between our observations for the DR method and the DRx method. Unfortunately, Edson et al. (1998) provide no details about the ratio of wind speed and ship velocity during their experiments. Pedreros et al. (2003) observed a 30% overestimation in (which equates to for ) and 9% in , for a relative wind direction range restricted to and with the restriction . The tilt angle in this study was 7° (Dupuis et al. 2003). Restricting our dataset in the same way, we obtain average differences between rPF and DR of 26% for and ~14% for . Note that the narrow restriction of the relative wind direction in Pedreros et al. (2003) increased the ratio of the average overestimation of the scalar flux and . This simple comparison shows that the flow distortion effects in direct fluxes, which were reported by Edson et al. (1998) and Pedreros et al. (2003), are likely explained by the overestimation of the pitch angle and the neglecting of the roll and yaw of the relative wind vector.

5. Conclusions

We have shown that the commonly used motion correction procedure (4) leads to an incorrect estimation of the wind vector tilt, which is reflected in an overestimation of the direct fluxes. This bias is caused mainly by errors in the pitch of the wind vector, when (4) is applied to measurements from a ship under way. This is because scales with the relative wind speed and not with . Additionally, (4) does not account for a roll of the wind vector. This is relevant for both underway and station measurements. Airflow distortion can also lead to a biased measurement of the wind direction. Typically, the magnitude of the measured relative wind direction is higher than the true relative wind direction. This can lead to errors in the stress angle; however, the yaw of the wind vector has no effect on scalar fluxes and on the estimation of with (2). For the typical measurement situation of a research vessel heading into the wind, covariance moments calculated with (4) will overestimate the vertical flux; the sign of the crosswind stress bias does, however, depend on the relative wind direction.

In this study, the bias caused by the motion correction procedure (4) was in the range of 0%–50% for and 0%–25% for for a wind direction range of ±90°. The effect for other air–sea gas exchange studies will depend on the magnitude of the wind vector tilt, the ratio of true and relative wind speed, and, to some extent, the atmospheric stability and the height of the boundary layer (Wilczak et al. 2001). We suggest that this is the main cause for the observed overestimation of air–sea fluxes, when measured on research vessels (e.g., Edson et al. 1998; Pedreros et al. 2003).

We have presented a different approach to correct for ship motion and wind vector tilt (16) and have shown that it can successfully reduce airflow distortion bias in the direct flux measurements. The required pitch and roll of the relative wind vector can be derived from the measured wind speeds using the rPF method (21) based on wind direction sectors. The estimation of yaw bias and acceleration of the mean wind speed does, however, require an independent freestream measurement or alternatively the use of computational fluid dynamics models (O’Sullivan et al. 2015).

With (18) we present an alternative to (4) that avoids the large overestimation of the wind vector pitch for underway measurements. It does not account for a potential roll or yaw of the wind vector due to airflow distortion and is subject to random measurement errors caused by relatively short averaging periods and statistical errors due to imperfections in the anemometer calibration. We suggest the rPF method to be used instead of the DR and DRx methods.

The technique presented here could be used to extend the number of high-quality flux measurements available for air–sea interaction studies.4 For example, Fairall et al. (1996) restricted covariance data not only by relative wind direction within 30° of the bow but also by the pitch of the wind vector (estimated from the DR method) less than 10°. This essentially removes underway measurements at relative low wind speeds. A reanalysis of previous EC datasets could be a straightforward solution to provide more data to improve the COARE algorithm.

These findings are most relevant for shipborne direct flux measurements and derived bulk parameterizations of air–sea exchange processes, but they also apply, to a lesser extent, to stationary flux measurements when airflow distortion cannot be avoided.

Acknowledgments

This research was funded by Science Foundation Ireland as part of the U.S.-Ireland R&D Partnership Programme under Grant 08/US/I1455 and the EU FP7 project CARBOCHANGE under Grant Agreement 264879. We would like to thank Tom Farrar at WHOI for making available the buoy data (ftp://ftp.whoi.edu/pub/users/jfarrar/SPURS/). We thank the captain and crew of the R/V Saramiento de Gamboa, who supported the measurements and the Chief Scientist Jordi Font.

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1

It is often assumed that the crosswind stress can be neglected . This leads to the also common definition .

2

This follows from the distributivity of the multiplication with a scalar over the addition .

3

More information about station 41061 can be found online (http://www.ndbc.noaa.gov/station_page.php?station=41061).

4

Two MATLAB scripts that derive the pitch and roll angle via the rPF method and to apply it to the wind speed measurements, are made available at http://airsea.nuigalway.ie/scripts.

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