## 1. Introduction

Reducing and quantifying uncertainties in ensemble-average estimates are essential for turbulent flow studies. From a numerical perspective, the Reynolds-averaged Navier–Stokes (RANS) equations solved in operational weather and climate forecasting models require closure of ensemble-averaged turbulence statistics. Large-eddy simulation (LES) results also must be evaluated against ensemble-averaged statistics (Pope 2000, 612–614). From an observational perspective, analyzing eddy correlation data relies on ensemble-averaged velocity components to interrelate data from multiple independently and imperfectly deployed sensors (e.g., Tanner and Thurtell 1969; McMillen 1988; Wilczak et al. 2001).

For environmental flows, time averaging is used to approximate ensemble averaging under an ergodicity assumption for statistically stationary processes (Wyngaard 2010, 34–35). Pan and Patton (2017) recently constructed a multisensor stationarity analysis technique (MSAT) that eliminates the need for visual inspection (e.g., Dias et al. 2004) or specifying a fixed duration analysis period within extended time series (e.g., Mahrt 1998; Andreas et al. 2003; Večenaj and de Wekker 2015). MSAT uses the reverse arrangement test (Kendall et al. 1979) as the basic statistical operation to determine both the occurrence and duration of stationary periods across multiple sensors. With an overarching goal to refine the understanding of turbulent canopy flows, Pan and Patton (2017) applied the first version of MSAT (i.e., MSATv1) to eddy correlation data from the Canopy Horizontal Array Turbulence Study (CHATS). The technique is capable of detecting sudden changes in mean wind directions, showing that the mean shear stress direction lags behind changes in mean wind direction, which contributes to nonnegligible crosswind component of the mean canopy drag force. Stationary intervals showing spanwise velocity integral time scales an order of magnitude larger than expected values are associated with the occurrence of microscale internal gravity waves in very stable above-canopy flows, a phenomenon supported by Raman-Shifted Eye-Safe Aerosol Lidar (REAL) measurements (see details in Mayor 2010, 2017). MSATv1 results suggest physically meaningful strategies to reduce uncertainties in estimating the bulk drag–wind relationship (e.g., removing periods associated with nonequilibrated mean wind and mean shear stress or the presence of microscale internal gravity waves).

MSATv1 applied an initial interrogation of 5-min-averaged data to identify periods with wind directions between 150° and 210°. At the time, Pan and Patton (2017) did not realize that this initial interrogation unnecessarily broke the dataset into periods of 245 min or shorter. An initial interrogation of time-averaged wind directions not only is redundant for a technique that is capable of detecting changes in mean wind directions, but also potentially limits the range of stability by confining the variations in 5-min-averaged wind directions. This paper targets the understanding of turbulent canopy flows [the same overarching goal as that in Pan and Patton (2017)] by expanding the ability to study canopy turbulence response to atmospheric stability. Broadening the stationary mean-flow periods to encompass a range of atmospheric stability requires generalizing MSAT’s applicability, which we accomplish by eliminating the need to initially interrogate time-averaged data and correspondingly by allowing time periods with extended duration. Data collected during the defoliated phase of CHATS [the same dataset interrogated by Pan and Patton (2017); see brief description in section 2a] are used to develop and evaluate the technique. Section 2b explains potential issues that limit MSATv1’s applicability to periods of extended duration. Section 2c proposes an improved approach, MSATv2 (i.e., the second version of MSAT), to determine stationary periods across multiple sensors. Section 3 shows successful application of MSATv2 to two CHATS case studies, both of which are an order of magnitude longer than those in Pan and Patton (2017). The resulting time-averaged statistics during stationary periods are used as ensemble-average estimates in section 4 to refine the understanding of canopy turbulence response to atmospheric stability. Section 5 outlines the major findings.

## 2. Experimental data and stationarity analysis procedures

### a. CHATS data and analysis procedures

CHATS took place in a walnut orchard in Dixon, California, where the trees were uniformly planted on a flat topography with roughly 7-m spacing in both north–south and east–west directions, and the canopy height *h* was 10 m (Earth Observing Laboratory 2011; Patton et al. 2011). The local time was 7 h behind the coordinated universal time (UTC) during the experiment period (March–June 2007). The experiment provided a unique dataset of canopy turbulence that sampled a wide range of atmospheric stability. Previous studies used fixed-time (e.g., 5 min, 30 min, or 3 h)-averaged CHATS data to investigate the canopy turbulence response to atmospheric stability and vegetation density (e.g., Dupont and Patton 2012a,b), to estimate parameters for RANS closure models (e.g., Shapkalijevski et al. 2016), and to evaluate LES results (e.g., Kiefer et al. 2013; Patton et al. 2016).

The orchard had been pruned every third row in the north–south direction each year. The 30-m CHATS tower was located near the northernmost border of the orchard along a north–south row of trees. The row-middle (between two rows of trees) to the west of the tower pruned 2 years prior was slightly denser than the row-middle to the east of the tower pruned 1 year prior. A total of 13 Campbell Scientific CSAT3 sonic anemometers were mounted on the tower with booms pointing to the west, measuring all three velocity components (in a local east–north–up coordinate system) and virtual temperature at a frequency of 60 Hz from the ground to approximately three canopy heights.

Table 1 provides a summary of terminology and definitions used in this paper. A *stationary interval* is defined as a period during which all three velocity components at all 13 heights are statistically stationary. For convenience, zonal (positive eastward), meridional (positive northward), and vertical (positive upward) directions are used to describe velocity components in the instrument coordinate system. All CSAT3 data have undergone transducer shadowing correction following Horst et al. (2015) prior to any analysis presented here. The current analysis focuses on data collected during the defoliated phase of CHATS (March–April 2007). Data analysis begins with interrogating missing values in raw time series of individual velocity components sampled by individual sonic anemometers at 60 Hz. When the number of consecutive missing values does not exceed five, the missing values are filled using linear interpolation of adjacent values in the time series. Otherwise the dataset is broken at any location where the number of consecutive missing values exceeds five. A criterion of five consecutive missing values is selected to guarantee the accuracy of integral time-scale estimates (see explanation in the following paragraph). Although the missing data break up the dataset into shorter periods, two resulting periods are longer than 50 h, which are an order of magnitude longer than the longest period interrogated using MSATv1 (Pan and Patton 2017). In other words, these two periods are sufficiently long to test MSATv2.

A summary of terminology and definitions.

A statistical technique requires mutually independent input samples. In turbulence, the integral time scale provides a measure of time over which velocity fluctuations remain correlated (Batchelor and Bondi 1953, 47–48). Therefore, raw data should be averaged over time blocks *δt* of at least twice the integral time scale to yield approximately independent input samples for MSATv2. In this work, integral time scales are computed using two practical methods (see appendix 7.2 in Kaimal and Finnigan 1994): 1) the *e*-folding time of the autocorrelation function, and 2) integrating the autocorrelation function to the first zero crossing. Because autocorrelation function is highly sensitive to the strategy employed to fix missing values, we only fix no more than five consecutive missing values. A resulting stationary interval is reliable only if integral time scales computed using both practical methods for all velocity components at all heights are less than *δt*/2.

MSAT hinges on the reverse arrangement test (Kendall et al. 1979, 505–513) as the basic statistical operation, which requires at least 10 input samples to yield reliable measure of mean trend (Bendat and Piersol 2011, 97–99). Therefore, any resulting stationary interval is 10*δt* or longer. In this work, the stationarity analyses are conducted using two choices of *δt*: 3 and 6 min. The reason to choose *δt* = 3 min is that 30 min or longer is required for adequate sampling of atmospheric boundary layer (ABL) turbulent motions for typical daytime conditions (Kaimal and Finnigan 1994, 254–256). The ABL-scale convective eddies require approximately 15 min to overturn (Wyngaard 1988), and therefore, 30 min corresponds to the passage of two or three of ABL-scale structures (Kaimal and Finnigan 1994, p. 255). Pan and Patton (2017) chose *δt* = 3 min for the same reason, but their velocity integral time-scale estimates for stationary intervals were up to 3 min. For this reason, analysis with *δt* = 6 min is conducted to understand MSAT’s sensitivity to the choice of *δt*.

For each of the choices of *δt*, MSATv2 analyses are conducted using significance levels of 0.05 and above (*α* ≥ 0.05) specified for the reverse arrangement test. A significance level of *α* means that *α* × 100% possible permutations are identified as nonstationary, and the other (1 − *α*) × 100% possible permutations are identified as stationary. Increasing *α* increases the technique’s stringency on identifying stationary intervals.

### b. Potential issues with MSATv1

MSATv1 uses the probability distributions of starting and ending points of *stationary time series* (see definition in Table 1) to determine 1) the nonstationary event at which a period should be split into two subperiods, and 2) the nonstationary input samples that should be removed from further analysis (Pan and Patton 2017). The basic premise is that if a time series is stationary, but including one additional input sample on either end of that time series yields a nonstationary time series, then the additional input sample must contribute to nonstationarity.

MSATv1 measures the nonstationarity of an input sample using the number of sensors showing zero probability distribution of the starting or ending points of stationary intervals. It first uses the reverse arrangement test to obtain probability distributions of starting and ending points of stationary intervals for each sensor individually, and then combines probability distributions across all sensors. MSATv1 inadequately considers the overlap between probability distributions of the starting and ending points of stationary intervals, which results in identifying nonstationary events that do not necessarily contribute the most to nonstationarity. Two issues reveal this weakness of MSATv1: 1) a nonstationary event may belong to one or more stationary intervals identified when applying a given stringency *α*, and 2) a nonstationary event identified when applying an arbitrary *α* does not always continue to be identified as nonstationary with increasing *α*. Taking the case study in section 4b in Pan and Patton (2017) for example (i.e., within-canopy velocities during 0255–0455 UTC 4 April 2007), we notice that the nonstationary event at 0413 UTC identified by MSATv1 with *α* = 0.05 belongs to multiple stationary intervals identified with the same *α* (see Fig. 7a in Pan and Patton 2017). In addition, MSATv1 with *α* ≥ 0.10 identifies nonstationary events no longer at 0413 UTC, but at slightly different locations (e.g., 0404 and 0409 UTC).

The known weakness of MSATv1 does not cause severe problem as long as the probability distributions of stationary interval starting and ending points are sufficiently simple, a condition typically satisfied when time series of interest are short. Pan and Patton (2017) obtained reasonable results using MSATv1 because they employed an initial interrogation of 5-min-averaged data to identify periods with wind directions between 150° and 210°, a process breaking the dataset into periods mostly shorter than 2 h. However, to generalize the applicability of MSAT, the initial interrogation of time-averaged wind directions should be eliminated, which could enable corresponding periods of interest to extend well beyond a couple of hours. Periods of extended duration can be associated with excessively complicated probability distributions of stationary interval starting and ending points, for which MSATv1 often yields unreasonable results.

### c. An improved approach: MSATv2

Figure 1 outlines the steps that MSATv2 uses to determine nonstationary events. Unlike MSATv1, which measures the nonstationarity of an input sample, MSATv2 measures the nonstationarity at the border between two consecutive input samples. MSATv2 first identifies periods during which all velocity components across all sensors are stationary (i.e., “stationary intervals” defined in Table 1), and then investigates whether two consecutive input samples can ever belong to the same stationary interval. If two consecutive input samples can never belong to the same stationary interval, then extending any stationary interval containing one of those input samples to include the other makes the interval no longer stationary. The border between these two input samples must involve an event contributing to nonstationarity. MSATv2 satisfies the two basic expectations that are not guaranteed in MSATv1: 1) nonstationary events identified when applying a given *α* do not belong to any stationary intervals identified when applying the same *α*, and 2) nonstationary events identified when applying an arbitrary *α* should continue to be identified as nonstationary events when applying an increased *α*. MSATv2 accurately determines events that contribute the most to nonstationarity, eliminating any potential duration upper limits to the periods of interest. Consequently, MSATv2 no longer requires the redundant and potentially biasing initial interrogation of time-averaged wind directions.

## 3. Applying MSATv2 to south-wind episodes during CHATS

MSATv2 is now challenged against the two longest periods obtained after interrogating and fixing missing values in raw time series of velocity components during the defoliated phase of CHATS. The first period starts from 2212 UTC 31 March and ends at 0121 UTC 3 April (a duration of 51.15 h). The second period starts from 0124 UTC 3 April and ends at 1651 UTC 5 April (a duration of 63.45 h). Before the first period, a nonnegligible portion of data are missing during 2209–2212 UTC 31 March from the sonic anemometer at 1.5 m. Between the two periods, a nonnegligible portion of data are missing during 0121–0124 UTC 3 April from sonic anemometers at 1.5 and 10 m. After the second period, data are missing from 1651 UTC 5 April to 0136 UTC 6 April due to tower climbing.

The discussion in sections 3a and 3b focuses on the connections between MSATv2 results of *nonstationary intervals* (see definition in Table 1) and nonstationary transitions in dynamic processes. Visually inspectable nonstationary transitions in mean wind speeds and directions are used to validate MSATv2. For nonstationary intervals associated with visually nontrivial transitions in mean wind speeds and directions, turbulence statistics that are highly sensitive to nonstationarity (e.g., integral time scales) are computed to demonstrate the value of MSATv2. In section 3c, strategies are proposed to obtain ensemble-average estimates for further analysis (in section 4) of canopy turbulence response to atmospheric stability.

### a. Case 1: A 51.15-h period from 2212 UTC 31 March to 0121 UTC 3 April 2007

Applying MSATv2 with *α* = 0.05 to 3- and 6-min-averaged velocity components at all heights (i.e., *δt* = 3 min and *δt* = 6 min) consistently suggests three nonstationary transitions (2306–2309 UTC 1 April, 1106–1118 UTC 2 April, and 0106–0121 UTC 3 April) coinciding with south-to-north or north-to-south changes in wind directions (see yellow shaded areas in Fig. 2b). Central Valley in California (where Dixon is located) is south–north oriented, and therefore, a switch between south and north winds is expected to be associated with strongly nonstationary events. MSATv2 using *δt* = 3 min with *α* = 0.05 determines two additional nonstationary intervals: 1) 0639–0642 UTC 1 April, coinciding with a shift of above-canopy mean shear directions from near zero to approximately 10° toward west, and 2) 1433–1436 UTC 1 April, coinciding with a dramatic drop in the magnitude of above-canopy mean shear. Here “above-canopy mean shear” is defined as the difference between tower-top and canopy-top mean wind vectors (i.e., the difference between red and black symbols in Fig. 2). Increasing *δt* from 3 to 6 min increases the length of a nonstationary interval associated with multiple transitions within 1 h (e.g., comparing gray and yellow shaded areas to cyan shaded area near 2300 UTC 1 April), but decreases the length of a nonstationary interval associated with an isolated transition (e.g., comparing yellow shaded area to gray shaded area near 1100 UTC 2 April).

MSATv2 results obtained with increasing significance level *α* confirm the expectation that nonstationary events identified with an arbitrary *α* continue to be identified as nonstationary with increasing *α* values (gray shaded areas remain gray shaded with increasing *α* in Figs. 3 and 4). The number of nonstationary intervals first increases with increasing *α* as an increasing number of nonstationary events are identified, and then decreases with increasing *α* as nonstationary intervals merge with each other. Results obtained using *δt* = 3 and 6 min consistently suggest that the most stationary intervals occur during afternoon periods (e.g., 2212–2400 UTC 31 March, 1900–2300 UTC 1 April, and 2100–2400 UTC 2 April). This finding agrees with the knowledge that nighttime and morning periods involve various sources of nonstationarity like inertial oscillation, internal gravity waves, and rapid growth of ABL depth. Results obtained using *δt* = 3 and 6 min are most different when applying intermediate *α* values to periods associated with frequent occurrence of nonstationary transitions (e.g., 0700–1900 UTC 1 April at *α* = 0.25 and 2300 UTC 1 April–0800 UTC 2 April at *α* = 0.30 in Figs. 3 and 4).

The sensitivity of MSATv2 results to *α* uncovers details of the time evolution of dynamic processes. For example, using *δt* = 3 min with *α* = 0.20 determines a stationary period (see definition in Table 1), 2351 UTC 1 April–0230 UTC 2 April (marked by a black circle at *α* = 0.20 in Fig. 3), bookended by two adjacent nonstationary intervals, 2254–2315 UTC 1 April and 0230 UTC 2 April. All stationary intervals between 2315 UTC 1 April and 0230 UTC 2 April show velocity integral time scales less than 1.5 min across all heights, ensuring the reliability of MSATv2 results. Increasing *α* from 0.20 to 0.25 determines two additional nonstationary events, one at 0003 UTC 2 April coinciding with a change in wind directions from northwest to southwest (see Fig. 2b), and the other at 0227 UTC 2 April consecutive with the nonstationary event at 0230 UTC 2 April. Increasing *α* from 0.25 to 0.30 yields an additional nonstationary interval, 0057–0139 UTC 2 April (i.e., 14 consecutive nonstationary events), where the dynamic transition is nontrivial based on visual inspection of mean wind speeds and directions. Specifically, the mean wind profile remains approximately constant during 0003–0227 UTC 2 April (not shown, but can be inferred from Fig. 2).

Velocity integral time scales are computed to investigate the visually nontrivial nonstationary transition during 0057–0139 UTC 2 April. Using *δt* = 3 min with *α* = 0.25, MSATv2 identifies a total of 11 stationary intervals between 0003 and 0227 UTC 2 April (Fig. 5a). Two of those stationary intervals (0021–0224 and 0039–0203 UTC, represented by gray dashed lines) contain 0057–0139 UTC, which is a nonstationary interval identified when applying a more stringent criterion, *α* = 0.30. Seven stationary intervals cover 0003–0100 UTC (0003–0057, 0003–0054, 0006–0051, 0006–0054, 0006–0057, 0009–0100, and 0012–0057 UTC, represented by black solid lines), which primarily occur before 0057–0139 UTC. Two stationary intervals cover 0133–0227 UTC (0133–0218 and 0139–0227 UTC, represented by red dash–dotted lines), which primarily occur after 0057–0139 UTC. The velocity integral time scales computed using the *e*-folding time of the autocorrelation function decrease with time (transition from black to gray and then to red lines in Figs. 5b–d), because the ABL becomes less unstable as the time approaches sunset at 0231 UTC. The horizontal velocity integral time scales computed by integrating the autocorrelation function to the first zero crossing, however, can easily double due to nonstationarity identified with *α* = 0.30 (comparing gray lines to black and red lines in Figs. 5e,f). Note that nonstationarity identified with *α* = 0.30 is much weaker than that identified with *α* = 0.05, an arbitrary criterion employed by previous reverse arrangement test analyses of ABL stationarity (e.g., Dias et al. 2004; Večenaj and de Wekker 2015). In addition, the nonstationarity during 0057–0139 UTC is revealed by variability in velocity integral time scales across stationary intervals involving similar shear and buoyancy conditions, rather than an arbitrary upper limit in velocity integral time scales [e.g., 2 min employed by Dias et al. (2004)].

### b. Case 2: A 63.45-h period from 0124 UTC 3 April to 1651 UTC 5 April 2007

Visual inspection of 3-min-averaged wind directions at canopy top (*z*/*h* = 1) and tower top (*z*/*h* = 2.9) from 0124 UTC 3 April to 1651 UTC 5 April 2007 (Fig. 6b) suggests a total of six south-to-north or north-to-south transitions in wind directions (around 0330 UTC 3 April, 2000 UTC 3 April, 1400 UTC 4 April, 0030 UTC 5 April, 0600 UTC 5 April, and 1500 UTC 5 April, respectively). The first and the last switches between south and north winds are determined by MSATv2 with *α* = 0.05 using both *δt* = 3 min and *δt* = 6 min (see the first three and the last two yellow shaded areas in Fig. 6). MSATv2 using *δt* = 3 min with *α* = 0.05 also determines the south-to-north-wind transition around 0600 UTC 5 April (see the corresponding gray shaded area in Fig. 6b). However, the other three switches between south and north winds are not identified by MSATv2 with *α* = 0.05 using either *δt* = 3 min or *δt* = 6 min. As shown in Figs. 7 and 8, determining the nonstationary transitions around 2000 UTC 3 April, 1400 UTC 4 April, and 0030 UTC 5 April requires *α* ≥ 0.20, 0.10, and 0.10, respectively, for MSATv2 using *δt* = 3 min or *α* ≥ 0.10, 0.15, and 0.15, respectively, for MSATv2 using *δt* = 6 min. These results suggest that not all nonstationary transitions are as clean cut as those interrogated in the first case study. MSATv2 must become sufficiently stringent on stationary intervals to capture all visually inspectable transitions between south-wind and north-wind episodes.

The sensitivity of MSATv2 results to the choice of *α* yields findings consistent to those obtained in the first case study: 1) nonstationary events identified with an arbitrary *α* continue to be identified as nonstationary with increasing *α* values, 2) the number of nonstationary intervals peaks at some intermediate *α* value, 3) the most stationary intervals occur during afternoon periods (e.g., 2100–2400 UTC 3 April and 2100–2300 UTC 4 April), and 4) results obtained using *δt* = 3 and 6 min are most different when applying intermediate *α* values to nighttime and morning periods (e.g., 0300–1900 UTC 4 April at 0.25 ≤ *α* ≤ 0.30 and 0500–1651 UTC 5 April at 0.20 ≤ *α* ≤ 0.25 in Figs. 7 and 8). Note that visually inspecting the most stationary period during 2100–2400 UTC 3 April is impractical due to vigorous fluctuations of 3-min-averaged wind speeds and directions (Fig. 6). Without statistical assistance, one tends to infer that 2100–2400 UTC 3 April involves a positive mean trend in mean wind speeds similar to that identified in the period from 1900 UTC 3 April to 0100 UTC 4 April. These results demonstrate the capability of MSATv2 in separating out stationary intervals bookended by nonstationary intervals.

### c. Obtaining ensemble-averaged statistics for investigating canopy turbulence response to atmospheric stability

In this section, the two case studies from sections 3a and 3b are now combined to enable further analysis of canopy turbulence in section 4. The target data are ensemble-averaged statistics representative of statistically stationary and horizontally homogeneous turbulent flows within and above the canopy. Ensemble-average estimates encounter two major sources of uncertainties: nonstationarity and finite averaging time (Wyngaard 2010, p. 204 and 35–37, respectively). Increasing *α* specified for MSATv2 eliminates an increased number of nonstationary events, reducing the uncertainty associated with nonstationarity. However, a desire to investigate canopy turbulence response to atmospheric stability requires interrogation across a wide range of possible variations in atmospheric stability and large-scale forcing, preventing us from choosing high *α* values. Thus, we must seek a balance between sufficient data to investigate the target scientific question and tolerable uncertainties associated with nonstationarity. In other words, the resulting optimal choice of *α* is practical and specific for the scientific question of interest.

A desire to interrogate across a wide range of possible variations calls for breaking the two case studies into a maximum possible number of periods bookended by nonstationary intervals (i.e., “stationary period” defined in Table 1). Combining the results in sections 3a and 3b suggests that the number of stationary periods peaks at *α* = 0.15 for MSATv2 using both *δt* = 3 min and *δt* = 6 min (Fig. 9a). Interestingly, *α* ≥ 0.15 is required for the maximum duration of stationary periods to become shorter than 12 h (Fig. 9b). In other words, *α* ≤ 0.10 is insufficiently stringent to identify nonstationary transitions associated with the diurnal cycle, and *α* ≥ 0.15 should be recommended for analyses of ABL stationarity using the reverse arrangement test.

Figures 3, 4, 7, and 8 suggest that *α* = 0.60 is the maximum possible stringency. For any stationary period identified with *α* < 0.60, one must avoid averaging data across stationary intervals associated with distinct dynamic processes. Analyzing individual stationary intervals within a stationary period is meaningful only if MSATv2 results are reliable, that is, velocity integral time scales are less than *δt*/2 across all sensors and all stationary intervals within the given stationary period (explained in section 2a). The number of reliable stationary periods peaks at *α* = 0.25 for MSATv2 using *δt* = 3 min (the black line in Fig. 9c showing the number of black circles in Figs. 3 and 7) and at *α* = 0.40 for MSATv2 using *δt* = 6 min (the red line in Fig. 9c showing the summation of black and blue circles in Figs. 4 and 8). In addition, the need of sufficient fetch for the turbulence to fully reflect canopy influences requests mean wind from south, that is, between 90° and 270°. MSATv2 using *δt* = 3 min yields a maximum of five south-wind reliable stationary periods with *α* = 0.25 (the black line in Fig. 9d): 1700–1742 UTC 1 April, 1757–1836 UTC 1 April, 0003–0227 UTC 2 April, 0012–0112 UTC 4 April, and 0642–0721 UTC 4 April. MSATv2 using *δt* = 6 min yields a maximum of three south-wind reliable stationary periods with *α* = 0.40 (the red line in Fig. 9d): 2212 UTC 31 March–0024 UTC 1 April, 2048–2248 UTC 1 April, and 2106–2342 UTC 3 April. The south-wind reliable stationary periods identified using *δt* = 3 min and *δt* = 6 min do not overlap because 1) those identified using *δt* = 3 min are too short (≤60 min) to be identified using *δt* = 6 min, and 2) those identified using *δt* = 6 min involve turbulent motions associated with too large integral time scales (>1.5 min) for an averaging block of *δt* = 3 min to provide approximately independent input samples.

For a given stationary period, turbulence statistics should be computed using only data during stationary intervals associated with similar dynamic processes. Stationary intervals starting from a set of consecutive input samples and ending at another set of consecutive input samples are likely associated with similar dynamic processes, because nonstationary transitions are likely associated with zero probability distributions of starting or ending points of stationary intervals (Pan and Patton 2017). Here we use 0003–0227 UTC 2 April, a stationary period determined using *δt* = 3 min with *α* = 0.25, to explain the strategy to obtain stationary intervals associated with similar dynamic processes. This stationary period contains a total of 11 stationary intervals (see details in section 3a), which can be separated into five groups according to their starting and ending points: a group of seven stationary intervals starting from a set of consecutive input samples (0003–0012 UTC) and ending at another set of consecutive input samples (0054–0100 UTC), and each of the other four stationary intervals being an individual group. We select the group of seven stationary intervals to compute turbulence statistics, because they contain the most stationary interval (0006–0054 UTC) of all 11 stationary intervals during 0003–0227 UTC 2 April. These seven stationary intervals are associated with similar dynamic processes characterized by similar velocity integral time-scale profiles (black lines in Fig. 5). Applying the same strategy to each of the eight south-wind reliable stationary periods yields a total of eight groups of stationary intervals shown in Table 2.

For each of the eight south-wind reliable stationary periods determined using *δt* = 3 min with *α* = 0.25 in groups a–e and using *δt* = 6 min with *α* = 0.40 in groups f–h, a group of stationary intervals are determine such that they: 1) contain the most stationary interval, 2) start from a set of consecutive input samples, and 3) end at another set of consecutive input samples. Within an individual resulting group of stationary intervals, mean-flow properties and turbulence statistics computed for individual stationary intervals are averaged after being weighted by the interval duration. The weight-averaged canopy-top mean wind direction is used to define the streamwise direction. The friction velocity is computed as *u*′, *υ*′, and *w*′ are streamwise, spanwise, and vertical velocity fluctuations. The stability regimes are defined following Dupont and Patton (2012a): free convection (FrC;

Mean-flow properties and turbulence statistics during stationary intervals within an individual group in Table 2 are averaged after being weighted by the interval duration. The coordinate system is rotated horizontally to align with the weight-averaged canopy-top mean wind direction (which defines streamwise direction). No coordinate system rotation is performed in the vertical because the originally derived sonic tilt correction coefficients (e.g., Dupont and Patton 2012a,b) involve uncertainties due to arbitrary choice of 5-min-averaged nighttime data without evaluating stationarity (Earth Observing Laboratory 2019), and the newly obtained eight stationary interval groups are insufficient to support accurate sonic tilt correction estimates (see details in section 4d). The data analysis in section 4 primarily focuses on turbulence statistics involving horizontal velocity components only, on which sonic tilt correction influences are negligible. Statistics involving the vertical velocity component are only used to facilitate the discussion of organized ABL-scale structures in section 4c. Figures 10–12 show 3- and 6-min-averaged streamwise, spanwise, and vertical velocity components during these eight stationary interval groups, which will be used to facilitate discussion throughout section 4.

## 4. The response of canopy turbulence to stability and organized ABL-scale structures

Time-averaged statistics during stationary intervals in Table 2 provide ensemble-average estimates representative of statistically steady (first order) and horizontally homogeneous (on scales much larger than canopy elements) turbulent canopy flows. These ensemble-average estimates with categorized stability conditions (section 4a) are used to investigate canopy turbulence response to atmospheric stability, in comparison with LES results and RANS closure models (sections 4b and 4c). The uncertainties in ensemble-average estimates associated with finite averaging time are discussed in section 4d.

### a. Parameters characterizing stability conditions

*z*/

*h*= 1) and tower top (

*z*/

*h*= 2.9),

*z*/

*h*= 0.95),

*g*= 9.81 m s

^{−2}is the gravitational acceleration,

*z*are differences between tower-top and canopy-top mean potential temperature, mean streamwise velocity component, and vertical location, respectively. The mean potential temperature and streamwise velocity gradients at

*z*/

*h*= 0.95 are computed using measurements at

*z*/

*h*= 1 and

*z*/

*h*= 0.9. The Obukhov length,

*u*

_{*}(see full definition in Table 2 caption) and heat flux

Figure 13 shows that both *R*_{b} for 1 ≤ *z*/*h* ≤ 2.9 and *z*/*h* = 1 increase approximately linearly with increasing Ri at *z*/*h* = 0.95, implying that all three parameters equivalently characterize canopy-shear-layer stability conditions for the defoliated phase of CHATS. The only exception is above-canopy *R*_{b} during 1700–1742 UTC 1 April (the annotated red symbol in Fig. 13a), when mean wind is from a direction approximately 50° different from the direction of the tower relative to the sonic boom. We therefore suspect that the tower infrastructure may compromise the observed above-canopy mean winds (see details in section 4b). Stationary intervals for near-neutral *z*/*h* = 0.95 is used to characterize stability conditions, because it is less sensitive to influences of the tower infrastructure, and it does not involve fluxes that are sensitive to sonic tilt angles. For shear-dominated conditions (−1 < Ri < 0.2), the 3- and 6-min-averaged spanwise velocity time series at various heights collapse (Figs. 11c–f), and the 3- and 6-min-averaged vertical velocity time series are confined within

### b. The mean flow field

For each of the stationary intervals belonged to the groups in Table 2, the mean wind magnitude and the streamwise component profiles are approximately on top of each other (not shown), implying that the mean flow fields are approximately two-dimensional (2D). The less than 5° differences between tower-top and canopy-top mean wind directions also suggest that the mean flow fields are approximately 2D (see Table 2). The only exception is 1757–1836 UTC 1 April, a period associated with the most unstable condition, during which the difference between tower-top and canopy-top mean wind directions is about 10°. The approximately 2D mean flow fields allow us to focus on the streamwise direction when performing analysis of mean shear. The 3- and 6-min-averaged streamwise velocity components remain approximately constant with height for *z*/*h* < 0.5 (gray, black, and navy lines in Fig. 10), and increase with increasing height for *z*/*h* > 0.5 (the other lines in Fig. 10).

*κ*is the von Kármán constant, and

*d*is the displacement height. The dimensionless shear,

The within-canopy normalized mean shear values show notable dependence on mean wind directions, which must relate to horizontal spatial variation in leaf area density on row scale (resulting from pruning solely in north–south direction). For all situations with unstable conditions, the canopy-top normalized mean shear decreases by approximately 0.1 with every 10° of increase in the deviation of mean wind direction from south (black plus signs and crosses in Fig. 15a). For situations with similar mean wind directions, varying the stability condition from near neutral (0642–0721 UTC 4 April) to forced convection (0018–0051 UTC 4 April) can reduce canopy-top normalized mean shear by 20% (comparing the black circle and cross around 200° in Fig. 15a). The diminishing of canopy-top mean shear due to unstable stratification is consistent with LES results (Patton et al. 2016). However, the dependence of normalized mean shear on mean wind direction leads to nontrivial dependence of canopy-top normalized mean shear on instability for unstable conditions (black plus signs and crosses in Fig. 15b).

All normalized mean shear profiles show a persistent peak at *z*/*h* ≈ 2/3 (Fig. 14b), consistent with the findings in Dupont and Patton (2012a) when interrogating the same dataset. The normalized mean shear at *z*/*h* ≈ 2/3 also decreases with increasing deviation of mean wind direction from south (red plus signs and crosses in Fig. 15a). The rate of decrease for southwest wind (>180°) is slightly faster than that for southeast wind (<180°), possibly because the row-middle to the west of the tower is slightly denser than the row-middle to the east of the tower. Because forced convection periods are associated with mean wind directions greater than 195° (crosses in Fig. 15a), the corresponding normalized mean shear values at *z*/*h* ≈ 2/3 are smaller than during the other periods. The dependence of normalized mean shear on mean wind direction leads to an unexpected behavior of normalized mean shear at *z*/*h* ≈ 2/3 that first decreases and then increases with increasing atmospheric instability (red symbols in Fig. 15b). Similar dependence of normalized mean shear on atmospheric instability was reported by Dupont and Patton (2012a) when interrogating the same dataset. On the contrary, LES results for horizontally homogeneous canopy (i.e., no dependence of turbulence statistics on mean wind directions) show monotonic decrease of normalized mean shear with increasing atmospheric instability (Patton et al. 2016). These findings suggest that obtaining reliable mean wind direction estimates is an important advantage of applying MSATv2, which enables interrogating the observed canopy turbulence response to atmospheric stability variation in isolation from contamination by dispersive motions [i.e., spatial variation of turbulent statistics with position relative to canopy elements (Raupach and Shaw 1982)].

### c. Turbulence intensity and the influence of organized ABL-scale convective structures

Turbulence intensity is computed as the ratio between the standard deviation *z*/*h* = 0.6, and the turbulence intensity at a given height increases with increasing instability (see cyan, blue, green, and brown lines sequentially in Fig. 16a). Stationary intervals for buoyancy-dominated conditions show maximum turbulence intensity at *z*/*h* = 0.15, but the dependence of turbulence intensity on temperature stratification is nontrivial (see gray, red, black, and orange lines sequentially in Fig. 16a). According to the values of Ri at *z*/*h* = 0.95 (and equivalently *R*_{b} for 1 ≤ *z*/*h* ≤ 2.9 or *z*/*h* = 1), the two midafternoon periods (2048–2248 UTC 1 April and 2106–2342 UTC 3 April) are less unstable than the two morning periods (1700–1742 and 1757–1836 UTC 1 April).

Inspecting the time series of streamwise velocity component suggest that the mean shear remains approximately constant during the two morning periods (Figs. 10a,b), but varies semiperiodically during the two midafternoon periods (Figs. 10g,h). The coincidence between sharp increase in mean-shear and strong updraft during 2103–2115 UTC 1 April, 2139–2151 UTC 1 April, and 2309–2321 UTC 3 April suggests passage of convergence zones (combining Figs. 10g,h and 12g,h). Wide scan (~5 km maximum zonal domain) from REAL (Mayor 2010) during 2305–2311 UTC 3 April show a convergence zone between two convective cells passing the CHATS tower location. The convective cells in the REAL observations are characterized by a horizontal scale of approximately 2 km, consistent with the distance between adjacent troughs in mean shear multiplied by the canopy-top mean velocity (24 min × 1.6 m s^{−1} ≈ 2.3 km). The other two convergence zone passage events (2103–2115 and 2139–2151 UTC 1 April) are likely associated with convective cell structures characterized by a similar horizontal scale (30 min × 1.1 m s^{−1} ≈ 2 km). Unfortunately, REAL observations during these two events are mostly narrow scan (~1-km maximum zonal domain), unable to infer the activity of convective structures. These findings also suggest that the widely used 30-min averaging time [e.g., an empirical estimate in Kaimal and Finnigan (1994)] is insufficient to capture the passage of two or three ABL-scale structures over the CHATS site. The site-specific turbulence characteristics and the corresponding variations in appropriate averaging time confirm the motivation to develop MSATv2, which allows resulting stationary intervals to have flexible durations with no upper limit.

The correlation coefficient for momentum, *R*_{uw} < 0), consistent with LES results (Patton et al. 2016). The suspicious positive momentum fluxes around canopy top for free convection conditions reported by Dupont and Patton (2012a) likely result from uncertainties in mean wind direction estimates for weak wind conditions (e.g., they used an insufficient averaging time of 30 min). Stationary intervals for shear-dominated conditions show similar profiles of *R*_{uw} (see brown, green, cyan, and blue lines in Fig. 16b), consistent with the findings in Dupont and Patton (2012a) when interrogating the same dataset and with LES results in Patton et al. (2016). The momentum transport efficiency decreases dramatically when the stability condition changes from forced convection to free convection, consistent with LES results (Patton et al. 2016).

The normalized friction velocity

### d. The uncertainties in ensemble-average estimates and implications on sonic tilt correction

*τ*

_{u}is the streamwise velocity integral time scale, and

For the stationary intervals considered (Table 2), Fig. 17 shows the fractional error *e* incurred by using an average over a finite time *e* depends primarily on turbulence intensity. Stationary intervals with turbulence intensity below or around 0.5 (red, brown, green, cyan, and blue lines in Fig. 16) yield fractional error estimates below or around 0.05 (Figs. 17a,c–f). Stationary intervals with turbulence intensity above 0.5 (orange, black, and gray lines in Fig. 16) yield fraction error estimates around or above 0.1 (Figs. 17b,g,h).

Comparing Figs. 10 and 12 suggests that 3- and 6-min-averaged turbulent canopy flows are characterized by horizontal velocity scales *L* ~ *H* ~ 1 km; see details in section 4c). Combining this scale analysis, *U*/*L* ~ 10*W*/*H*, and continuity suggests that a 10% fractional error in estimating ensemble average of streamwise velocity component *θ*). However, a sonic lean angle *θ* is typically smaller than about 5°, or it would otherwise have been noticed during the field experiment. Therefore, stationary intervals during buoyancy-dominated conditions involving convective cells are not suitable for sonic tilt correction estimates. A second reason to exclude stationary intervals during buoyancy-dominated conditions from sonic tilt correction estimates is the inspected instances of negative 3- or 6-min-averaged streamwise velocity component (see Figs. 10b,g,h).

Mean velocity components during stationary intervals under shear-dominated conditions are acceptable for sonic tilt correction estimates. However, mean wind directions for groups c–f in Table 2 only span a range of approximately 40°, insufficient for conducting the planar fit technique (Wilczak et al. 2001; with a recommended mean wind direction span over 90° or greater). We expect obtaining additional ensemble-average estimates of velocity components for shear-dominated conditions by running MSATv2 over more of the dataset, which may enable performing planar fit technique.

## 5. Conclusions

A recently proposed statistical technique that determines occurrence and duration of stationary periods across multiple sensors, MSATv1, is improved to eliminate the initial interrogation of time-average wind directions. The new technique, MSATv2, satisfies two basic expectations that are not guaranteed in MSATv1: 1) a nonstationary event should not belong to any stationary intervals identified with a given stringency (*α*), and 2) nonstationary events identified using a relatively weak threshold should continue to be identified as nonstationary when applying increasingly stringent thresholds. The improvement is confirmed by applying MSATv2 to two CHATS case studies that are an order of magnitude longer than those analyzed by Pan and Patton (2017). MSATv2 is capable of determining visually trivial and nontrivial nonstationary transitions in dynamic processes, as well as stationary periods associated with strong afternoon turbulence intensity. The sensitivity of MSATv2 results to the specified stringency (*α*) uncovers details of the time evolution of dynamic processes, supported by visual inspection, turbulence statistics, and empirical knowledge.

An optimal choice of *α* is determined by a balance between interrogating a wide range of atmospheric stability and controlling uncertainties in ensemble-average estimates associated with nonstationarity. In addition to eliminating nonstationary events identified using MSATv2, uncertainties associated with nonstationarity are further controlled using a practical strategy consisting of three steps: 1) identifying reliable stationary periods (see definition in Table 1) during which all stationary intervals show integral time scales less than half of *δt* (the time over which individual input samples are initially chosen to be averaged), 2) grouping stationary intervals associated with distinct dynamic processes within a given reliable stationary period according to their starting and ending points, and 3) selecting the group of stationary intervals containing the most stationary interval within the given reliable stationary period. For the two case studies, combining MSATv2 results using 3- and 6-min-averaged velocity components as inputs (with *α* = 0.25 and 0.40, respectively) yields a total of eight south-wind reliable stationary periods (see definition in Table 1), providing eight groups of stationary intervals characterized by atmospheric stability ranging from near neutral to free convection (see Table 2).

The analysis of turbulence statistics confirm previous findings including 1) a persistent mean-shear peak at *z*/*h* ≈ 2/3 reported by Dupont and Patton (2012a) when interrogating the same observational dataset, and 2) the diminishing of canopy-top mean shear due to unstable stratification reported by Patton et al. (2016) based on LES results. New findings enabled by applying MSATv2 include 1) quantifiable contamination of the observed relationship between normalized mean shear and atmospheric stability by spatial variation with position relative to canopy elements, and 2) the role of organized ABL-scale structures in the observed relationship between bulk drag coefficient and atmospheric stability. MSATv2 results also reveal the connection between the presence of organized ABL-scale structures and semiperiodic variation in mean shear, which enables inferring the occurrence and horizontal scales of organized convective structures using in situ measurements at fixed locations. Future work includes 1) applying MSATv2 to the entire CHATS episode to explore the potential of performing sonic tilt correction, 2) isolating the observed canopy turbulence response to atmospheric stability from contamination by spatial variation with position relative to canopy elements, and 3) separating cases based upon the presence of organized ABL-scale structures for observational data analysis and turbulence parameterization development.

Although both Pan and Patton (2017) and the current work focus on turbulent canopy flows measured during CHATS, the MSATv2 technique should be applicable to any time series dataset as far as the time over which individual input samples are initially chosen to be averaged (*δt*) and the optimal choice of stringency (*α*) are adapted to the scientific questions of interest. Because the reverse arrangement test requires at least 10 input samples to yield reliable measure of mean trend, the upper limit of *δt* is a tenth of the period duration determined by data availability. The lower limit of *δt* is twice the characteristic time scale of interest (e.g., velocity integral time scales for the current study of turbulent flows). The choice of an optimal *α* is determined by the balance of the scientific question of interest and the nonstationarity that one is willing to tolerate.

## Acknowledgments

This material is based upon work supported by the National Center for Atmospheric Research (NCAR), which is a major facility sponsored by the National Science Foundation under Cooperative Agreement 1852977. We would like to acknowledge the operational, technical, and scientific support provided by NCAR’s Earth Observing Laboratory (particularly, Steven Oncley for providing access to the raw data). We also thank the Cilker family for allowing CHATS to be conducted in their orchard. EGP specifically acknowledges support from NCAR’s Geophysical Turbulence Program. We thank the three anonymous reviewers for constructive comments that have helped us improve the manuscript.

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