## 1. Introduction

Monin–Obukhov similarity theory (MOST) and the Charnock relationship are the basis for a large fraction of the air–sea bulk flux models embedded in general circulation models and mesoscale models in use today. Despite the widespread usage and apparent overall success, physical interpretation of MOST can be ambiguous because of circular dependencies and self-correlation (Hicks 1978; Kenney 1982; Andreas and Hicks 2002; Klipp and Mahrt 2004; Baas et al. 2006). MOST requires an iterative process to predict the turbulent momentum and heat fluxes in terms of transfer coefficients that depend on stability functions, which depend on the Obukhov length *L*, which itself depends on the turbulent fluxes of heat, moisture, and momentum, and so on. This is particularly problematic for stable stratification where relating the drag coefficient or the nondimensional shear to *z*/*L* (where *z* is height) leads to self-correlation that is the same sign as the expected physical correlation.

Over the sea, the ambiguity is compounded using the Charnock relationship (Charnock 1955), where the roughness length is specified to be a function of the surface stress, while the surface stress is specified to be a function of the roughness length. Such a circular feedback can lead to extremely small estimates of the roughness length and to momentum fluxes that are less than smooth flow values (Mahrt et al. 2001).

Additional motivation for a simple model that does not rely on MOST is the large uncertainty associated with estimating the Obukhov length. Measurements of the friction velocity over the sea have minimum uncertainties estimated to be about 10% (Fairall et al. 1996), and the relative uncertainty increases with decreasing wind speed. Because determining the Obukhov length requires the friction velocity cubed, *L* may be uncertain by at least 30% (Andreas et al. 2012) without even considering the uncertainty in the heat and moisture fluxes. In weak winds less than about 4 m s^{−1}, additional uncertainty can arise as a result of the choice of analysis methods when evaluating the 10-m neutral equivalent drag coefficient

Here, we develop a formulation for the friction velocity that depends in a straightforward way on two fundamental characteristics of the mean flow known to influence the turbulence strength and the transfer of momentum: wind speed and bulk stability. The simple formulation does not use MOST, the Obukhov length, the Charnock relationship, or the aerodynamic roughness length and does not assume any special shape of the wind speed profile. We do not standardize the measured wind speed to a uniform height, say 10 m, because doing so would require introducing the quantity that we are estimating, friction velocity, into the independent variable, wind speed. This work represents a philosophical shift away from the drag coefficient and the roughness length because these are poorly behaved in weak winds (Mahrt et al. 2001; Andreas et al. 2012; Vickers et al. 2013).

The functional forms and coefficients for the wind speed and bulk stability dependencies in the simple model are determined from a large observational dataset including multiple experiments and aircraft covering a wide range of atmospheric conditions (Fig. 1). Based on the sign of the air–sea virtual potential temperature difference, conditions are unstable 57% of the time and stable 43% of the time. The wind speed ranges from 0.01 to 27.1 m s^{−1}.

## 2. Aircraft data

The aircraft dataset consists of over 5000 eddy-covariance measurements collected by four different aircraft in nine different experiments (Table 1). These data have recently been used in studies by Mahrt et al. (2012), Andreas et al. (2012), and Vickers et al. (2013), and additional information can be found in those studies and references therein.

The aircraft datasets used in this study, where *N* is the number of flux estimates, *U* is the mean wind speed (m s^{−1}), *U*_{max} is the maximum wind speed (m s^{−1}), and ^{−1}). Information on these datasets can be found in Vickers et al. (2013), Andreas et al. (2012, 2015), Mahrt et al. (2012), and references therein. Sampling rates are 40, 25, 20, and 50 Hz for the Twin Otter, C-130, Electra, and LongEZ, respectively, and nominal ground speeds are 65, 100, 100, and 55 m s^{−1} for the Twin Otter, C-130, Electra, and LongEZ, respectively. Here, CIRPAS is the Center for Interdisciplinary Remotely Piloted Aircraft Studies, NCAR is the National Center for Atmospheric Research, NOAA is the National Oceanic and Atmospheric Administration, CARMA is the Cloud-Aerosol Research in the Marine Atmosphere experiment, RED is the Rough Evaporation Duct experiment, POST is the Physics of Stratocumulus Top experiment, and CBLAST is the Coupled Boundary Layers and Air Sea Transfer experiment.

The Twin Otter, C-130, and Electra used a five-port 20- or 25-Hz radome (gust probe) mounted on the nose of the aircraft to obtain the fast-response pressure measurements. The LongEZ used the Best Atmospheric Turbulence Probe (BAT), a 50-Hz, nine-port radome on a boom extending 2 m ahead of the nose and five wing widths in front of the canard (Crawford and Dobosy 1992; Garman et al. 2006). The basic principles for obtaining the 3D wind vector from fast-response aircraft measurements of pressure are given in Lenschow (1986). To calculate the true ground speed, all aircraft employed the global positioning system to correct the aircraft’s inertial navigation (Khelif et al. 1999). The Twin Otter, Electra, and C-130 measured the sea surface radiative temperature with a Heitronics Infrarot Messtechnik GmbH Model KT 19.85; the LongEZ used an Everest Interscience, Inc., Model 4000.4GXL. Air temperature was measured using microbead thermistors. The LongEZ instrumentation is further described in Sun et al. (2001).

## 3. Methods

### a. Flux calculations

To ensure consistency for the data collected by the different aircraft in different experiments, we obtained the fast-response data and applied identical screening procedures, quality control testing, and flux calculations to each dataset. The eddy-covariance fluxes, mean wind speed, temperature, and humidity are calculated from data collected during low-altitude (10–50 m) flight segments where aircraft altitude, roll, pitch, and heading fluctuations remained within prescribed limits. The fast-response measurements of the wind, temperature, and humidity that pass this first level of screening are then scanned by quality control software to identify suspected instrumental errors. The quality control procedure tests for the following manifestations of instrument problems: a high frequency of spikes, data outside a specified range, very large skewness, very small or large kurtosis, a large local Haar mean transform (discontinuity in the mean), and a local standard deviation outside a specified range. Flagged data are plotted for visual inspection, and data exhibiting implausible behavior are removed from further analysis.

For computing the eddy-covariance turbulence fluxes, we use a short 4-km window to reduce the impact of surface heterogeneity (changes in sea surface temperature). Multiresolution decomposition of the flux (Howell and Mahrt 1997; Vickers and Mahrt 2006) indicates that a 4-km window is more than sufficient to capture the largest eddies that contribute to the turbulence fluxes of momentum, heat, and moisture, even for the high wind speed and strong turbulence Gulf of Tehuantepec Experiment (GOTEX) data (Fig. 2). In fact, Fig. 2, sometimes called an ogive plot, shows that there is very little additional flux at scales exceeding 1 km. Mahrt et al. (2012) recently pointed out a similar scale dependence of the heat flux. Individual 4-km flux estimates do suffer from large random flux sampling errors; however, our multiresolution flux decomposition analysis indicates that they do not suffer from systematic errors due to using a window that is too small.

### b. Monin–Obukhov similarity theory and the Charnock relation

*M*), sensible heat (

*H*), and latent heat (LE) at the air–sea interface are formulated in almost all atmospheric models using the following flux–gradient relationships:

*U*, potential temperature

*θ*, and specific humidity

*q*are evaluated at some height

*z*above the surface;

*q*

_{s}is the saturated specific humidity at the surface temperature. In our analysis, primes denote perturbations from a 4-km mean, and the overbar denotes 4-km averaging. In Eq. (1),

*α*is the Charnock coefficient and

*ν*is the kinematic viscosity of dry air (Charnock 1955; Smith 1988; Fairall et al. 1996). We use the usual value of

^{−1}than in COARE 2.6, thus slightly increasing the modeled fluxes in strong winds.

### c. Model for based on the 10-m neutral wind

*L*. The formulation must be solved iteratively because

^{−3}.

## 4. Results

*U*is the wind speed and

To obtain

*U*have units of meters per second. This third-order polynomial explains 83% of the variance of

^{−1}; however, we do not have the data to test it.

Estimates of the uncertainty of the regression coefficients are developed using a random sampling scheme, where we sampled the full set of near-neutral data points randomly and did the regression analysis for each subset individually. Each of 20 such subsets has approximately one-half of the data. The standard deviation of the regression coefficients over the 20 runs is considered an estimate of the uncertainty. The mean and 95% confidence limits (±2 standard deviations) for the four coefficients in Eq. (11) are, respectively, 0.17 ± 0.03, −0.019 ± 0.009, 0.0042 ± 0.0008, and −8.4 × 10^{−5} ± 2.0 × 10^{−5}.

The polynomial fit described by Eq. (11) and shown in Fig. 3 handles the problematic weak wind case by maintaining nonzero friction velocity as the mean wind speed approaches zero. The friction velocity may actually vanish or become undefined with very weak winds or for winds following swells (Vickers and Mahrt 2010). Such cases can be found in the observations; however, imposing zero surface stress in a numerical model may not be appropriate because numerical models represent grid-box area-averaged fluxes, while collapsed turbulence is likely a local transient phenomenon where zero surface stress leads to flow acceleration and generation of turbulence (Vickers and Mahrt 2010).

This third-order polynomial based on the combined dataset [Eq. (11)] is also a reasonable representation of the wind speed dependence of the friction velocity in near-neutral conditions for each dataset individually (Fig. 4). It is encouraging that one formulation can describe all nine datasets individually and collectively. This fact suggests that wave state effects may be secondary to wind speed effects. Figure 4 also highlights the importance of having multiple datasets when the range of wind speeds for an individual dataset is small.

Because we fitted the stability function by eye, the uncertainty analysis that we performed for

It is encouraging that the bulk Richardson number dependence is well behaved for strong stability. For strongly unstable conditions with

As found for

While the model [Eq. (9)] is “empirical,” we note that all models of turbulence and boundary layer processes are empirical to some degree because they have some coefficient or sets of coefficients that are based on observations. The physical basis for the formulation of the simple model is that we know from previous studies that stronger winds lead to stronger shear generation of turbulence and larger momentum transport in the presence of mean wind shear, and that stronger bulk stability leads to stronger buoyancy destruction of turbulence and relatively smaller momentum transport in the presence of mean wind shear.

### a. Reproducing the observations

Figure 7 compares the friction velocities from the simple model and the COARE version-2.6 scheme with the observed friction velocities. Comparing how well the two models agree with the observations may not be fair to the COARE scheme because the simple model coefficients are tuned to the observations. On the other hand, recall that the simple model coefficients based on the combined nine experiments also describe reasonably well each of the individual experiments (Figs. 4 and 6). The fact that the simple model tends to fit all nine experiments individually and collectively indicates that it is not overly tuned to any single experiment and in fact may be fairly general. The differences in the squared correlation (*R*^{2}), the root-mean square (RMS), and the bias between the simple model and the observations and the COARE scheme and the observations are likely insignificant, suggesting that differences between the simple model and the COARE scheme are small relative to the observational uncertainty (Fig. 7). Note that the COARE scheme has approximately 70 empirical parameters while the simple model has only 10.

The formulation of the friction velocity in terms of the 10-m neutral wind proposed by Andreas et al. (2012) yields estimates of the friction velocity that are similar to those from the simple model and the COARE scheme (not shown). Differences in the *R*^{2}, RMS, and bias values between the Andreas model and the COARE scheme are small.

### b. Dependence on measurement height

The COARE model

The simple model residuals are closer to zero than the COARE residuals for all measurement heights. Thus, even though we might expect a strong height dependence for the simple model residuals because there is no normalization of wind speed to a reference level, we actually find that the COARE scheme yields a stronger height dependence. The stronger height dependence implies that Monin–Obukhov similarity is not working, and may do more harm than good. Note, however, that the bin-averaged

### c. Distribution of residuals

The probability distributions of relative model residuals for the simple model and the COARE scheme for each dataset individually are shown in Figs. 9 and 10 , respectively. We generally find good results for both models and all datasets in that small residuals occur more frequently, and large residuals occur more rarely. A possible exception is the Electra TOGA COARE dataset, where the simple model systematically overpredicts

The narrowest distribution of relative residuals, indicating that the model almost always performs well, is found for the C-130 GOTEX dataset, which also has the strongest wind speeds. In terms of the relative error, both models perform better in strong winds than in weak winds. This could be due, in part, to the larger uncertainty in the calculated fluxes in weak winds.

We have demonstrated that the simple model can provide a reasonable description of the friction velocity for all nine datasets individually and collectively. That is, the functional forms and the coefficients in

### d. Independent model evaluation

In this section, we compare the simple model predictions of

The variance of the observed *R*^{2} values are due in part to the lack of weak wind data, where the observed fluxes are most uncertain. The average ^{−1} for FASTEX, GFDEX, and HEXOS, respectively. In addition, the stability correction is relatively small because the wind speeds are strong, and thus the uncertainty associated with the stability function becomes less important. The nondimensional slopes from linear regression are 1.13, 0.87, and 1.16 for FASTEX, GFDEX, and HEXOS, respectively (Fig. 11). The mean biases in ^{−1} for FASTEX, GFDEX, and HEXOS, respectively. The mean relative biases in

## 5. Conclusions

Our simple model is based on a third-order polynomial that predicts the friction velocity from the wind speed in near-neutral conditions. An additional function based on the bulk Richardson number applies a correction for stability. The simple model does not require an estimate of the Obukhov length or the aerodynamic roughness length, both of which are subject to large uncertainty; and it does not require iteration. The model coefficients are tuned using data from four different aircraft in nine different experiments comprising 5000 observations.

We do not correct the independent variable, wind speed, for height or stability but instead use the measured value of the mean wind speed in the third-order polynomial. The height of the measured wind speed is constrained to be between 10 and 50 m above the sea surface. We do not standardize the wind speed to a constant height, say 10 m, as is typically done in observational studies because doing so would require introducing the quantity that we are estimating, friction velocity, into the independent variable, wind speed.

The friction velocities from both the simple model and the COARE scheme, which is based on a full implementation of Monin–Obukhov similarity theory and the Charnock relation, compare well to the observations. The simple model with coefficients tuned to a combination dataset also reasonably reproduces the observed friction velocities for each of the nine experiments individually. Similar close agreement was found between the simple model and a recently published formulation of the friction velocity based on a linear dependence on the 10-m neutral equivalent wind. In addition, the simple model was effective in predicting the friction velocity for three independent datasets. This work shows that discarding the complexity of Monin–Obukhov similarity theory and avoiding the large uncertainty in estimating the Obukhov length and the roughness length can lead to a credible model for the friction velocity for most situations.

## Acknowledgments

We thank the three reviewers and the many dedicated scientists who collected and made available the data. The U.S. Office of Naval Research supported this work with Award N00014-11-1-0073 to NorthWest Research Associates. The Office of Naval Research also supported ELA with Award N00014-12-C-0290.

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