1. Background
The no-slip condition that results in loss of fluid momentum to surface drag is an important feature of atmospheric-surface-layer flow. Measuring momentum flux toward the surface is relatively straightforward for steady flow over a smooth, flat, horizontal surface. Sonic anemometers are widely used in micrometeorology to measure surface fluxes of momentum and heat by using the eddy covariance method. This technique is complicated by the reality that sonic anemometers are not always perfectly aligned with vertical or wall normal during field campaigns. Large errors in calculating momentum flux can arise from seemingly small deviations from a wall-normal alignment of the sonic anemometer (Lee et al. 2004; Wilczak et al. 2001; Kaimal and Haugen 1969). The planar tilt correction method is widely used to correct for instrument misalignment (Wilczak et al. 2001), but the method only applies to data taken over relatively flat terrain, either level or sloping. In a complex environment, such as steep, narrow valleys or urban canyons, the planar tilt correction method cannot be used without modification (Oldroyd et al. 2016; Stiperski and Rotach 2016). Other tilt correction methods are problematic, and none are as universally accepted as planar tilt correction is for flat terrain.
For current methods of momentum flux calculation, knowledge of the wall-normal direction is crucial, but in complex environments, the wall normal can vary considerably within the flux footprint of a meteorological tower (Oldroyd et al. 2016; Stiperski and Rotach 2016). In narrow canyons, two distinct wall normals could be significant influences on the flow. Most current methods also assume the flow follows the terrain, although flow separation is sometimes observed in sloping terrain (Hocut et al. 2015)
One ideal property of a momentum flux computation for use in complex environments is that the method should avoid using knowledge of the wall-normal direction, or other information about the alignment of the sonic anemometer relative to the surrounding terrain. Another ideal property is that the momentum flux calculation depends only on values that do not vary with changes in coordinate system. A third ideal property is that the calculation does not depend on the flow being perpendicular to the wall normal. In order to meet these requirements, assumptions about the turbulence are made based on results from thermally neutral laboratory flows.
This study will focus on evaluating an alternative momentum flux computation method by applying the new method to data from simple terrain using the 1999 Cooperative Atmosphere–Surface Exchange Study (CASES-99) main tower data (Poulos et al. 2002). This study will provide the basis for future application of the new method to data from complex terrain. Since this new method is derived from a different theoretical basis than the well-established methods, it is important to evaluate to what degree the new method reproduces the well-established method from flat terrain before applying the new method to data from complex terrain locations. The CASES-99 dataset was chosen because of the high confidence in the quality of the data and confidence in the standard momentum flux computation method for this surface. The momentum flux computed with the standard method will be used as a ground-truth point of reference to evaluate the performance of the alternative method.
2. Data and analysis
a. CASES-99 field campaign
The CASES-99 field campaign took place in October 1999 (Poulos et al. 2002). The main tower was located in a flat rural area near Leon, Kansas. The data have been used extensively and many of the problems with field data have been located and corrected or eliminated, making this dataset trustworthy and well respected in the boundary layer meteorology community. (More information and the data are available from NCAR at https://www.eol.ucar.edu/projects/cases99/.)
Data from the sonic anemometers located on the 60-m-tall main tower are used. Four were ATI-K probes (Applied Technologies, Incorporated) located at 10, 20, 40, and 55 m above ground level (AGL). Three of the four CSAT3 sonic anemometers (Campbell Scientific) were located 5, 30, and 50 m AGL. The fourth CSAT3 was moved from 1.5 to 0.5 m AGL on 19 October 1999. All sonic anemometers recorded the three components of the wind vector and sonic temperature at a rate of 20 times per second. Average wind speed data from four propeller anemometers located at 15, 25, 35, and 45 m AGL are also used to calculate wind shear in section 3b. The data as provided by NCAR have been carefully corrected for instrument tilt and are in the geographic coordinate system, where positive u is wind from west to east, positive υ is wind from south to north, and positive w is upward.
When computing turbulence statistics, a time scale must be chosen over which to calculate means, variances, and covariances. The scale is chosen depending on the needs of the research. Shorter times are often useful for eliminating mesoscale motion, which can dominate fluxes in extremely stable conditions. Longer times can be useful for convective conditions in order to improve statistics. To evaluate the performance of the alternative momentum flux calculation, turbulent statistics were prepared with 5-, 10-, 15-, 30-, and 60-min averaging times. Unless indicated otherwise, qualitative results are nearly identical between the different averaging times.
b. Surface stress/momentum flux
The terms surface stress, momentum flux, and friction velocity are often used interchangeably in micrometeorology. Over flat, horizontal terrain, they are synonymous in that all the terms are proportional to 
c. Matrix properties
For the Reynolds stress tensor this means that the three invariant eigenvalues λB, λM, and λS are the fundamental variances of the turbulence acting in the directions of the corresponding eigenvectors ΛB, ΛM, and ΛS. Since these eigenvectors are mutually orthogonal they form a coordinate system where the turbulent Reynolds tensor has only diagonal elements, no covariances. If the data were to be rotated into the eigencoordinate system, it would seem that there is no turbulent stress since the off-diagonal terms all vanish.
Note that 
Because they are invariants of the Reynolds stress tensor, the eigenvalues will be the same and eigenvectors will point in the same direction regardless of the coordinate system used to represent the data. Whereas the data must be expressed in the perfect coordinate system before Eq. (1) can be correctly applied, the eigenvalues and eigenvectors will be the same no matter how imperfect the instrument alignment. The data can be in the form they come off the sonic anemometer with u and υ oriented in a way that is advantageous for the site, or the data can be used after rotation. See appendix A for caveats in processing data. The resulting eigenvalues will be identical for the same block of data no matter in what coordinate system the data are presented.
Note that the eigenvectors might appear different in different coordinate systems since the reference axes are different for each coordinate system. Also remember that none of the individual components of the Reynolds stress tensor will necessarily remain constant after a change in coordinate system of the data, but the invariants will remain the same under coordinate system changes. The values of individual terms such as 
As a side note, another matrix invariant is the trace, which is the sum of the diagonal elements from upper left to lower right, and therefore also equal to λB + λM + λS. The turbulence kinetic energy (TKE) is an invariant of the Reynolds stress tensor since TKE is equal to half the trace. The total measured TKE will be the same regardless of the coordinate system used to represent the data even though the relative magnitudes of 
d. Quantify inclination
For neutrally stratified, shear-driven, laboratory flows the relationship between the eigenvectors and the streamwise coordinates is commonly reported to have a relative inclination of 17° (Hanjalić and Launder 1972; Liberzon et al. 2005). Hanjalić and Launder (1972) observed 2D flow, which was symmetric in the cross-stream direction. Liberzon et al. (2005) observed this in a confined 3D flow. In both cases, the value of 17° is purely empirical. Atmospheric flows are rarely 2D even when turbulence production is primarily due to shear. For a three-dimensional system, more than one angle is needed to uniquely quantify the relationship between the mutually orthogonal eigenvectors and the streamwise coordinates; however, in imitation of ideal shear-dominant conditions, only one angle will be used in this alternative process to calculate the turbulent friction velocity. This decision is motivated by the requirement to use only invariants of the flow as measured by a single instrument such as a sonic anemometer. To fully quantify the relationship between the eigenvectors and streamwise coordinates would require external information about the orientation of the instrument with respect to the surrounding terrain.
Two candidates for a simplified relationship between the eigenvectors and streamwise coordinates have been investigated and are illustrated in Figure 1. The angle between the wall normal direction and the eigenvector associated with the smallest eigenvalue ΛS is labeled α. Although the eigenvectors never coincide with streamwise coordinate directions, in most cases the smallest eigenvector is the one oriented closest to the direction of the wall normal, although there are exceptions in slower–wind speed conditions. The angle between the direction of the mean 3D wind vector and the plane defined by the eigenvectors associated with the two larger eigenvalues ΛB and ΛM is labeled β. This plane was chosen as a reference since the direction of the mean wind vector is only weakly associated with the direction of the large eigenvector and can align more closely with the intermediate eigenvector. For brevity, the eigenvector associated with the smallest eigenvalue is referred to as the small eigenvector, and similarly for the large and intermediate eigenvectors, even though the eigenvectors are by convention normalized to be of unit length.
Streamwise coordinates (black), and the eigenvectors (green), which are mutually perpendicular, and two ways to measure the relative inclination between them. The angle α is measured between the smallest eigenvector ΛS and the wall normal 
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-16-0282.1
To calculate α, not only is the direction of the small eigenvector needed, but also the direction of the wall normal. The wall normal can be assumed to be the sonic anemometer’s vertical coordinate if no tilt correction is needed. The wall normal can also be derived by planar tilt correction or careful measurement of the sonic vertical and the orientation of the slope of the terrain under the sonic anemometer. These measurements are possible, but even in simple terrain, they are subject to measurement uncertainty. To calculate β, no information about the orientation of the sonic with respect to its environment is needed since the eigenvectors and the mean wind vector are all calculated from the 3D wind data. For a review of the practical aspects of calculating α and β, see appendix B.
Since atmospheric flows are rarely exactly 2D even for simple terrain, α and β are rarely equal to each other, but are often close in value. In complex terrain where the wall normal is not well known, β will be the only reliably computable angle. To verify that knowledge of the wall normal is not critical, α was investigated to be a control condition to compare to β. This comparison is possible since the wall normal is well known in the CASES-99 data.
Examining the distribution of α values at 50 m above the surface (Fig. 2), the peak is near the commonly found laboratory value of 17°, but only when wind speeds are faster than the threshold value of 9.2 m s−1. For winds slower than the threshold value, little definitive can be said about the angle. Even for fast-wind conditions the range of α values is significantly large, so for any given data point α is not likely to be close to 17°. The distributions are similar for β. The wind speed threshold used here represents a distinct change in turbulence characteristics and is a function of height above the surface z (see appendix C).
Distribution of α values for data from the sonic anemometer at 50 m, averaged over 15-min blocks, including all stratification conditions. The angle α is measured between the smallest eigenvector ΛS and the wall normal, which is well known for the CASES-99 field campaign. For 50 m, the threshold wind value is 9.2 m s−1.
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-16-0282.1
The average inclination angle between the eigencoordinate system and the streamwise system varies as a function of distance from the surface (Fig. 3). This angle is also a function of the averaging time used to calculate the variances and covariances. Also, α and β yield different values for the inclination angle. Since the angle values cover a wide range of values with respect to the mean value, the differences in the means between α and β, between averaging times, and even between elevations are less significant than the standard deviation of α and β values. This large variation is the reason a previous attempt using a fixed angle to calculate an alternate turbulent stress did not work well (Klipp 2008). For both α and β below 10 m, the angles are significantly smaller than the values at 10 m and above. Shallow angles near the surface are consistent with the variances being more anisotropic near the surface and are also consistent with laboratory results closer to the surface (Hanjalić and Launder 1972).
Profiles of relative inclination angles for near-neutral stratification and above-threshold wind speeds for different averaging times for (a) α and (b) β with a dashed line at 17° for reference. Error bars indicate the extent of plus and minus one standard deviation.
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-16-0282.1
The mean inclination is also a function of thermal stratification (Fig. 4). Near-neutral thermal stratification is defined using the bulk temperature gradient on the main CASES-99 tower such that −0.005° < (T55 − T5)/50 < 0.02°C m−1, where T55 and T5 are the temperatures at 55 and 5 m, respectively. On average, unstable conditions have β values consistently 2°–3° shallower than for the neutral data at all levels. The β values for stable conditions are nearly the same as neutral at 10 m and are 2°–3° larger at elevations below that. Above 10 m the mean β values become progressively smaller than the near-neutral values, reaching a maximum difference of 8° at 55 m. This pattern is the same for all the averaging times.
Profiles of β for near-neutral, stable, and unstable conditions and above-threshold wind speeds for 15-min averaging times. Error bars indicate the extent of plus and minus one standard deviation.
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-16-0282.1
3. Reynolds stress–derived turbulent friction velocity
a. Derivation
Although this derivation uses a rotation, it is not a tilt correction. Neither is it an attempt to reproduce streamwise coordinates. Also, since the values of all six terms in the Reynolds stress tensor are used to calculate each of the eigenvalues, this method is inherently a three-dimensional analysis even with only one rotation. Since the eigenvalues, eigenvectors, and mean wind direction are invariants, rotating the data into another coordinate system, even the eigencoordinate system, is not needed. Especially notable is that by using β instead of α, no information is needed about how the sonic is oriented with respect to the landscape.
b. Evaluation
1) Compare 
To begin, data closely resembling laboratory flows are used to compare Reynolds stress–derived value of 
The scatterplot (Fig. 5) of 
Comparison of Reynolds stress–derived 
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-16-0282.1
Comparison of 
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-16-0282.1
Comparison of 
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-16-0282.1
Bulk temperature difference is chosen to classify thermal stratification instead of stability measures such as Richardson number or z/L. For Figs. 5–7, data need to have the same stability classification whether 
2) Monin–Obukhov similarity
Since the CASES-99 location is flat and level and the data have been carefully processed to remove sonic tilt, 
The relationship between 
For these plots, only data from the four Campbell Scientific CSAT3s are used. The fluxes were computed from 30-min blocks of data to match the analysis in Högström (1988). MOST assumes the fluxes represent surface flux values. For these plots, all the fluxes are calculated at the level of the instrument. In general this is local scaling, not MOST, except when the flux values represent the surface flux, and then the relationship evaluated is MOST. Instruments closer to the surface are more likely to represent surface flux values.
Times when the wind speed is faster at lower levels than above are not used since this condition violates the assumptions of MOST. In addition, times when measured quantities are near zero are also not used on the assumption that the measurement uncertainty for such small quantities will be large compared to the measured value. For all plotted points in Figs. 8–12 
For the above-threshold wind data, where it is often the case that 
MOST relationship using (a) 
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-16-0282.1
To evaluate whether or not there is a significant difference in the amount of scatter between Figs. 8a and 8b, two different measures of scatter are evaluated with a bootstrap technique. The first measure of scatter is the sum of the absolute value of the difference between the 
In order to evaluate whether differences between the measures are statistically valid, bootstrap analyses of 5000 iterations were done. To illustrate the process, the z/L > 0 data in Fig. 8a are used as an example. The N = 428 differences between the data and the reference line are taken as the original sample. From this sample, 428 values are chosen at random with replacement such that some of the original values will be chosen more than once and some will be omitted. This random sample is used to calculate both measures of scatter. The process is then repeated 5000 times. From these 5000 calculations, both mean and standard deviation can be calculated in order to set uncertainty values on the measures of scatter.
The results for Fig. 8 are listed in Table 1. The difference between the first measure of scatter for the z/L > 0 data between Figs. 8a and 8b are deemed statistically insignificant since the difference between 180.3 and 184.6 is smaller than the uncertainty values from the 5000 bootstrap estimates. Similarly for the z/L < 0 data. For the second measure of scatter, the difference is also not statistically significant for the z/L > 0 data. For the z/L < 0 data, the second measure of scatter does indicate a statistically significant difference. The scatter using 
Measures of scatter for Fig. 8, plots of 
For below-threshold wind conditions, when the two 
MOST relationship using (a) 
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-16-0282.1
Measures of scatter for Fig. 9, plots of 
As in Fig. 9, but with different axes ranges. Both 
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-16-0282.1
MOST also includes scaling of 
Scaling relationship 
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-16-0282.1
Scaling relationship 
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-16-0282.1
Measures of scatter for Fig. 11, plots of 
Measures of scatter for Fig. 12, plots of 
Scaling individual variances becomes problematic in complex terrain. Since the individual variances are not invariants and the perfect coordinate system is not easy to define in complex terrain, the correct magnitude of the individual terms may not be possible to determine. New scaling relations using the eigenvalues λB, λM, and λS could be found, but it is outside the scope of this paper to establish new scaling relationships.
Note that use of 
4. Summary of the process to calculate 
Although this alternative process to calculate the friction velocity seems complex, the process does not require significantly more effort than the standard method where one not only calculates fluxes, but must account for instrument or streamline tilt, and rotate into mean wind coordinates before calculating 
Calculate variances and covariances (fluxes) as well as the 3D wind vector, preferably in sonic anemometer coordinates before the data have undergone any tilt correction. Although data in any coordinate system can be used, it is important that the fluxes and wind vector be in the same coordinates. Also, use the same averaging time: 5-min mean winds with 5-min fluxes, 15-min mean winds with 15-min fluxes, etc. It is easiest to use the quality-controlled data in the original sonic anemometer coordinates without any tilt correction, but there is no need to undo tilt correction if the data are already in that format. See appendix A for more caveats about overprocessing data.
Use your preferred math package software (Matlab, NumPy, etc.) to calculate the eigenvalues and eigenvectors of the Reynolds stress tensor using the calculated variances and covariances.
Use the eigenvectors and 3D mean wind vector to calculate β using the definition of the dot product.
Use β and the eigenvalues to calculate the friction velocity using Eq. (2).
5. Conclusions
An alternative method to calculate 
Since the 
One drawback of this alternative turbulent stress calculation method is that it does not incorporate transport of scalars in any rigorous way, whereas the more traditional approaches can be extended to scalar flux computation. Although it is feasible to declare the β-rotated eigencoordinate system used in the derivation of Eq. (2) to be a Reynolds stress–derived frame of reference, in complex terrain there is no reason to assume that a coordinate system that was ideal for calculating turbulent friction velocity is the same one that would be ideal for computing scalar fluxes.
Acknowledgments
I would like thank Chris Hocut for his internal review and comments. I would also like to thank Jielun Sun and two anonymous reviewers for their thoughtful, constructive, and persistent comments. Their efforts have truly made this a better paper. And thank you to the CASES-99 researchers and NCAR for providing quality data for all to use.
APPENDIX A
Avoid Overprocessing Data
Measuring devices of all types are imperfect to some degree. All field data need to go through a quality-assurance process (Foken 2008; Vickers and Mahrt 1997). Better data is a commendable goal, but being overzealous can lead to altering the data rather than assuring quality.
Data can be inadvertently altered by blindly applying tilt correction. For example, the Wilczak et al. (2001) method includes an offset correction. This was included to correct for any possible electronic bias that was smaller than the resolution of the instrument. In many real-world cases, the data contain a significant upward or downward bias as a result of persistent updrafts or downdrafts as a consequence of subtle inhomogeneities at the site location, thus violating the assumptions for the offset in Wilczak et al. (2001).
Ideally, after a rotation from one frame of reference (the sonic) into another (true vertical), the location of the origin will remain the same between the two coordinate systems. This preserves the length of vectors; in other words, the wind speed will remain the same before and after the rotation. Adding an offset, with or without a rotation, means a different location for the origin after the offset, thus changing the measured length of vectors. The wind speed will have a different magnitude after the offset is applied compared to before (Arfken 1985; Boas 1983; Sun 2007).
If the magnitude of the offset derived from the planar tilt is on the order of magnitude of the instrument resolution, the offset is probably correcting for an otherwise unmeasurable instrument bias. Applying this offset is likely to improve the data, in that the data will better represent the truth we are trying to measure. If the magnitude of the planar tilt–derived offset is large compared to the resolution of the sonic anemometer, including the offset in the tilt correction will alter the data in such a way that the data are less like the truth we wish to measure. In these cases, one should just apply the rotation portion of the planar fit, not the offset. By setting the offset term to zero, the length of the vectors is preserved.
Although in theory the eigenvalues and eigenvectors will be identical no matter how the instrument is oriented, in reality, sonic anemometers are designed and optimized for the mean streamlines to flow roughly perpendicular to the sonic anemometer’s vertical. Flow from directions far from this ideal is subject to uncompensated effects of transducer shadowing (Horst et al. 2015). In order to obtain optimal data in steeply sloping terrain, in most cases the sonic anemometer will need to be aligned with the predominant streamlines in mind instead of being aligned with the gravity vector. Although no information about the sonic anemometer’s orientation with respect to the terrain is required to compute 
Another stage where data might be inadvertently altered is in testing for noisy data. Testing data for values varying from the mean by greater than 3.0 or 3.5 times the standard deviation σ can be appropriate to locate possible instances of electronic spikes (Foken 2008; Vickers and Mahrt 1997), but not all such points can be assumed to be electronic noise. Even for Gaussian-distributed data, one can expect 0.04% of data to be >3.5σ. This means that for 1 h of 20-Hz data, on average, one can expect 28 points to fall outside the 3.5σ range. But turbulence is not Gaussian (Frisch 1995), so having more data outside the ±3.5σ range does not automatically indicate noisy data. Similar caveats apply to using kurtosis as a quality check. If we keep only data that fit our assumptions about turbulence, we will only have results that affirm our assumptions.
Triple rotation of data in order to force 
APPENDIX B
Calculating Angles—A Review
The angle between two vectors is calculated by using the definition of the dot product 
The angle between a vector and a plane is the complement of the angle between the vector and the normal to the plane. Since the three eigenvectors of the Reynolds stress tensor are mutually orthogonal, ΛS is the normal to the plane defined by ΛB and ΛM. Thus β = 90° − γ where γ is the angle between ΛS and the mean wind vector.
Note that it is the orientations of the three eigenvectors that are invariant, not the sign convention indicating the direction of each eigenvector. Since the math package is unaware of the physical system being modeled, the positive direction the math package chooses for the eigenvectors does not always match what a human would pick. If β > 90°, it is assumed that the math package chose the direction that is the opposite of our requirements, so when the initial calculation yields β > 90°, β is set to 180° − β. By convention, the inverse cosine function should return a principal value between 0° and 180°. Some mathematics packages adopt a different convention.
APPENDIX C
Threshold Wind Speeds
Others have noted a change in turbulent statistics in the CASES-99 data depending on a wind speed threshold. Sun et al. (2012) found that turbulence strength (TKE1/2) increased linearly with increasing wind speeds until a threshold wind speed was reached above which the turbulence strength increased linearly but with a much steeper slope. Klipp (2014), in looking at the behavior of the transition scale between isotropic and anisotropic turbulence, found a double peak in the histogram of transition scales where one peak contained data from mostly below-threshold wind speed conditions and the other mostly above threshold. The threshold wind speeds were found empirically for only 5 and 50 m above ground, but are similar to those published in Sun et al. (2012). For this paper, the thresholds in Table C1 were determined from plots of β as a function of wind speed for 60-min averaging times. For wind speeds below threshold, β values vary from 0° to greater than 50°, while for wind speeds above threshold, β values are greater than 5° with an approximately Gaussian distribution around the mean β value (Fig. 2). Slightly different values result, within a few tenths of a meter per second, using 60-min-averaged α values and also from using the ratio β/α. Also, slightly larger threshold wind values are determined as shorter averaging times are used, again within a few tenths of a meter per second of the numbers in Table C1.
Threshold wind speeds.
Determining a threshold in turbulence data is somewhat subjective, which is why it is notable that similar thresholds are found from seemingly different approaches and by more than one research group. Is this a hint at underlying physics that is not currently part of turbulence theory? Will similar wind speed threshold effects be observed in data from sites other than the flat and level location of the CASES-99 main tower?
Unlike the mean wind profile, there is no theory dictating that the threshold wind 
Note that this relationship does not go to zero at the surface, so just above the surface all winds are below threshold. Also, since the mean wind profile is logarithmic or log linear, for any given moment there can be levels with winds above threshold at some elevations while higher up the tower the winds are below threshold. The nonlogarithmic profile of threshold wind speeds could be a hint at currently poorly understood physics of turbulence. The implications of this are beyond the scope of this paper.
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