1. Introduction
Redistribution of snow by wind is an environmental two-phase flow with great spatiotemporal variability. Blowing snow has implications for cold-region hydrology and engineering, avalanche safety, and glaciology, as well as Antarctic and Greenland surface mass balance (Dyunin and Kotlyakov 1980; Pomeroy et al. 1993; Freitag and McFadden 1997; Schweizer et al. 2003; Scarchilli et al. 2010, Lenaerts et al. 2012). Redistribution of snow begins when wind drag and shear stress exceed transport thresholds, and snow grains mobilize on the surface (Li and Pomeroy 1997). Subsecond shear stress peaks can be sufficient to initiate snow transport and may not be reflected in time-averaged mean values (Aksamit and Pomeroy 2016). Once in motion, blowing snow is characterized by a variety of length and time scales, from centimeter hop-lengths and rebound times for grains in saltation at the surface (Nemoto et al. 2004) to turbulent eddy length scales and bursting frequencies for particles in suspension that closely follow wind streamlines (Bintanja 1998, 2000). The choice of representative time scale has considerable effect on the performance of all aeolian transport models in the presence of multiple scales of atmospheric motions (Bisantino et al. 2010). Much is still unknown about the coupling of turbulent wind motions and snow transport in natural terrain, though the importance of turbulent bursts is widely accepted.
Instantaneous wind speed fluctuations and atmospheric structures are vital components for understanding intermittent aeolian behavior (Sterk et al. 1998; Leenders et al. 2005; Wiggs and Weaver 2012). To better represent the role of turbulent motions in snow transport models, researchers are gradually moving away from traditional approaches of modeling blowing snow as a steady-state process driven by time-averaged data (e.g., Nemoto and Nishimura 2004; Groot Zwaaftink et al. 2014), yet models continue to rely on experimentally derived relationships between wind and snow response, largely from low Reynolds number wind tunnel experiments (e.g., Nishimura and Hunt 2000; Clifton et al. 2006).
Modeling wind–snow coupling in truly complex turbulence, such as found in alpine environments, also requires understanding the physical mechanisms generating turbulence at very high Reynolds numbers. Recent research in very high Reynolds number wind tunnels, large-eddy simulations, and some limited results with atmospheric flows have linked the modulation of high-frequency surface turbulence to the passage of large-scale coherent motions (Hutchins and Marusic 2007a; Mathis et al. 2009a,b, 2011, 2013; Anderson 2016; Jacob and Anderson 2017). This has provided new insight into the cascade of turbulent energy and the dissipation of momentum in large coherent structures. It has been shown that the superposition of large-scale motions on local surface turbulence does not behave as a simple shift of mean velocity (Hutchins and Marusic 2007b); rather, the modification of high-frequency turbulence by large-scale motions directly affects turbulence statistics and surface shear stress in high Reynolds number flows. This has not yet been investigated in mountainous terrain, but would imply fundamental differences in the mechanics driving blowing snow in wind tunnel and atmospheric studies and the calculations used to characterize the physical process. This suggests that considerable caution and discretion must be exercised in extrapolating wind tunnel observations to outdoor blowing snow transport phenomena.
Most analysis connecting turbulent motions and saltation has been conducted in the time domain (Liu et al. 2012) and has associated limitations. The temporal lag between the two signals caused by particle inertia and the superposition of a wide variety of scales of atmospheric motions result in better analysis with spectral methods (e.g., Venditti and Bennett 2000; Schönfeldt and von Löwis 2003; Baas 2006; Ellis 2006; Liu et al. 2009, 2012). Quantitative coupling of turbulent structures with sediment response in given frequency ranges is difficult in natural conditions, but some promising progress has been made in wind tunnels (e.g., Liu et al. 2012; Paterna 2016, 2017).
However, rescaling these kind of steady-state wind tunnel relationships to environmental flows is often complicated because the mechanics of transport are a function of turbulence structure, which is in turn influenced by surface topography and mesoscale winds (e.g., Wiggs and Weaver 2012; Chapman et al. 2013). Even mass flux rates and threshold friction velocities vary with turbulence intensity, which is often found to be substantially greater in complex terrain than wind tunnels or horizontally homogenous and flat terrain (Xuan 2004). Exactly how turbulence characteristics in the atmosphere determine the variability and time-averaged properties of saltation remains a critical challenge in aeolian research (Kok et al. 2012). It is still an open question whether top-down or bottom-up motions are more appropriate conceptualizations of the coherent structures driving sediment transport (Bauer et al. 1998; Sterk et al. 1998; Baas 2008).
The objective of this study is to investigate the coherence between intermittent snow transport and turbulent gusts as a coupled nonlinear system to better inform semideterministic models of transport (Bauer et al. 2013). To investigate this, 50-Hz field measurements of wind and blowing snow density were collected at a study site in the Canadian Rockies. Statistically significant times and frequencies of wind–snow coupling were identified with wavelet coherence testing. Through amplitude modulation, the role of large-scale motions on high-frequency wind eddies and blowing snow transport was investigated. Further analysis of the mechanics of significant large motions using quadrant analysis structures was compared to recent blowing snow–coherent structure insights (Aksamit and Pomeroy 2018).
2. Methods
a. Fieldwork
Blowing snow observations were collected during nighttime field experiments between November 2015 and March 2016, at the Fortress Mountain Snow Laboratory (FMSL; 50°49′21″N, 115°11′54″W), Kananaskis Valley, Alberta, Canada (Fig. 1). The experimental site was surrounded by relatively flat terrain on a bench above the main valley at 2000 m MSL. Nearby steep alpine faces rise from valley-bottom elevations of 1500 m to 2900-m ridge tops over distances of less than 5.5 km. The surrounding terrain was snow covered, and shrub vegetation was buried for the duration of the experiment, with snow depths varying from 40 to 80 cm depending on daily erosion and deposition. The height of snow varied no more than 5 cm during any one night of recording. The site was suitable for the observation of the mesoscale influence of topographically driven flows with no immediate (≤200 m) bluff body interference. The same valley was found to have extraordinarily high turbulence intensity and evidence of advection of turbulent bursts from ridgelines to the valley bottom (Helgason and Pomeroy 2012). FMSL can receive 800 mm of water as snowfall each winter and can sustain winds exceeding 35 m s−1. Snow redistribution by wind is substantial at FMSL, with winter snow depths varying from zero on ridgetops to 5 m in gullies where snowdrifts develop.
Blowing snow recording apparatus with nearby wind measurements at FMSL, Kananaskis, Alberta, Canada. The recording plane indicates the location of camera focus inside the illumination plane. The wind is oriented parallel with the plane and toward the continuous wave (CW) laser in this schematic.
Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0179.1
Two Campbell Scientific CSAT3 ultrasonic anemometers were positioned on a single mast to measure wind speed at 50 Hz in three dimensions at two fixed heights above the ground. Measurement heights varied above the snow surface between 0.10–0.40 and 1.40–1.70 m. The location of the lower sonic anemometer, with a pathlength of 15 cm, can result in a certain amount of high-frequency energy that is not measured. Estimates of the high-frequency losses for each anemometer during each night of recording are provided in the online supplementary materials.
Blowing snow observations were made with the laser-video system described by Aksamit and Pomeroy (2016, 2018). The camera was situated 0.30 m from the anemometer mast, in a spanwise orientation to the wind (Fig. 1), and was rotated throughout the observation periods to keep the laser illumination plane parallel to the dominant wind direction. The camera then recorded a plane (~30 mm × 140 mm) of the flow of saltating snow (Fig. 1). To capture the influence of large eddies responsible for the strongest sediment transport fluctuations (Schönfeldt and von Löwis 2003; Baas 2006; Liu et al. 2009), several long, 5–19-min videos of nighttime snow particle transport at 50–100 frames per second were recorded each night. Each blowing snow video recording consisted of 2.7–5.8 × 104 frames, depending on aspect ratio and frame resolution, totaling 4.1 h for 23 blowing snow recordings over five nights.
Descriptions of the snow surface, snow densities from the top 5 cm of the snowpack, average air and snow surface temperatures, surface hand hardness indices (HHIs), mobile grain diameters, and 15-min mean wind speed ranges for the five nights are found in Table 1. Wind and video measurements were synchronized at the onset of nightly recording to minimize datalogger drift. Each night exhibited slightly stable stratification, and wind measurements for each recording were subjected to a dual-axis rotation such that 
Snow surface and meteorological conditions during each night of blowing snow. Missing values are indicated by —.
As amplitude modulation studies have never been conducted during blowing snow storms, it was important to consider the influence of blowing snow particles on the wind as well as the effect of large scale motions. Data from Paterna et al. (2016) were utilized for a low Reynolds number comparison of the role of large turbulent structures modulating surface turbulence and blowing snow. Specifically, 20-Hz directional snow flux and wind speed fluctuations were subjected to the same analysis as the FMSL data.
b. Blowing snow density estimation
Grayscale blowing snow video recordings were binarized following the algorithm of Otsu (1979), to obtain blowing snow particle concentrations per frame. A binarization threshold was determined for each frame to account for varying illumination depending on density of saltation. Similar techniques have been used in wind tunnel sand transport concentration studies (e.g., Liu et al. 2012) and environmental blowing snow studies (e.g., Gordon et al. 2009; Aksamit and Pomeroy 2016, 2018). After binarization, quality controls were implemented to eliminate false-positive snow measurements from camera sensor “hot spots” during periods of low transport as well as to adjust for low-light conditions during periods of high transport when the laser was largely obscured. A flood-fill algorithm was then implemented to identify individual snow particles and estimate their equivalent particle diameters. Assumptions of particle sphericity and constant snow particle density equal to ice (917 kg m−3) allowed estimation of airborne blowing snow density 
c. Signal processing
1) Wavelet Analysis
Wavelet analysis has emerged as the standard technique to detect intermittent behavior in geophysical systems in the time–frequency domain (e.g., Foufoula-Georgiou and Kumar 1994; Torrence and Compo 1998; Grinsted et al. 2004). In comparison to Fourier methods, wavelets are (imperfectly) localized in time and frequency, and wavelet convolution requires only local stationarity under the image of the wavelet. This results in better identification of transient coherent structures when applied to aeolian systems (e.g., Baas 2006; Ellis 2006). To identify statistically significant wind–blowing snow coupling, the wavelet coherence (CH) and statistical significance testing method of Grinsted et al. (2004) is introduced for this blowing snow study. This is a complimentary technique to the wavelet maps of Ellis (2006), wavelet packet decomposition of Liu et al. (2012), and wavelet cospectra of Paterna et al. (2016).
2) Amplitude modulation
One limitation of using Fourier or wavelet coherence tests to characterize the coupling of turbulent bursts with blowing snow is that neither method clearly captures the influence of large-scale motions on high-frequency turbulence or snow transport energy across scales. The role of large-scale motions is increasingly important as strong topographically enhanced turbulent motions in this mountain region have been identified (Helgason and Pomeroy 2012) but the power spectra likely overlap with local processes, often preventing a clear spectral separation of large- and small-scale motions (Sievers et al. 2015). Thus, the relationship of large eddies (duration greater than 30 s) with high-frequency, near-surface turbulence and blowing snow was further investigated with the theory of amplitude modulation.
Consider, for example, representing high-frequency local turbulent motions as a carrier signal 
(a) Synthetic example of amplitude modulation of high-frequency signal, (b) Fourier transforms of modulating and modulated signals showing mirroring of modulating energy, (c) wavelet coherence map of 
Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0179.1
Mathis et al. (2009a) demonstrated a method to identify to what degree passing large-scale motions modulate high-frequency turbulence in sufficiently high Reynolds number flows. For a streamwise wind signal 
d. Turbulence characteristics
Seventy-two turbulence statistics were calculated for each recording, including turbulence intensity, turbulence kinetic energy, friction velocity, drag coefficient (Stull 1988; Lykossov and Wamser 1995), dissipation length, energy flux, and several covariances. Reynolds stress time series were also decomposed following quadrant analysis with a hole size of one (Lu and Willmarth 1973). The percent of Reynolds stress contributed by each quadrant of motion above this threshold was then calculated, as well as the temporal occurrence of each quadrant motion. As these values are compact simplifications of the flow conditions, correlations between them and wavelet coherence or the degree of amplitude modulation across the recordings may provide useful corollaries for semideterministic modeling with lower computational cost. Notable correlations are presented in the results.
3. Results
a. Wavelet coherence
Wavelet coherence was calculated for time series pairs of low-anemometer streamwise wind speed and blowing snow density 
(a) Streamwise wind speed and Reynolds stress for recording 6 on 3 Mar 2016. (b) Blowing snow density for the same recording. (c) Wavelet coherence for blowing snow and streamwise wind and (d) for blowing snow and Reynolds stress. In (c) and (d), bold lines surround 5% significance events. Arrows indicate the phase relationship of the two signals, with in-phase pointing right and antiphase pointing left, and 90° down being the wind leading the snow.
Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0179.1
As was typical for all recordings, there is much greater coherence between blowing snow density and streamwise wind speed in Fig. 3c than that measured with Reynolds stress (Fig. 3d). In Fig. 3, statistically significant regions cover 48% and 4% of the COI for 
Time-averaged 
(a) Frequency-binned 
Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0179.1
In contrast, Fig. 4b displays the mean percent of time (inside the COI) with 
The edge effects associated with the Morlet wavelet prevent resolving coherence of blowing snow with atmospheric motions larger than approximately half the length of a recording, as they are outside the COI (Torrence and Compo 1998). This is a typical issue with time–frequency decompositions and is a manifestation of using finite-length time series and a nonzero minimum area of Heisenberg boxes (Weyl 1950). An analogous problem exists when resolving magnitude-squared coherence with Fourier methods where large fast Fourier transform windows allow lower-frequency coherence measurements but introduce more signal noise (Biltoft and Pardyjak 2009).
b. Amplitude modulation
Wavelet coherence (Figs. 3, 4) indicated that blowing snow responded most strongly to wind energy at long time scales (low frequencies) between 3 s and the limits of the COI. However, there was also coupling of 
The ability to quantify amplitude modulation with Hilbert transforms [Eqs. (3) and (5)] is sensitive to the cutoff frequency separating large-scale and surface motions (Mathis et al. 2009a). The choice of scale separation in the current study is complicated by the topographical influence on turbulence spectra at the study site by the surrounding peaks (Mahrt and Gamage 1987; Sievers et al. 2015). As well, both wind measurements were obtained relatively close to each other and near the snow surface, preventing identification of a distinct outer region peak as found by Hutchins and Marusic (2007a). However, a transition from self-similar behavior to an inertial subrange −5/3 slope was seen near the frequency 
Standardized premultiplied power spectral density of streamwise wind speed for 5 h at two measurement heights over four nights of recording. Self-similar (Hunt and Morrison 2000) and −5/3 slope (Kolmogorov 1941) lines overlaid to show transition and cutoff frequency 
Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0179.1
The large motions defined by the low-pass filter in both the near-surface 
Amplitude modulation coefficients for each recording for blowing snow density by 2-m streamwise wind speed 
While the majority of recordings exhibited strong modulation (Table 2), there were several exceptions that are explained with quadrant analysis (Lu and Willmarth 1973) in section 4. These values are italicized in Table 2. Also of note, in contrast to the results of Hutchins and Marusic (2007a), there was inconsistent amplitude modulation of Reynolds stress signals, and not all recordings showed clear modulation between low-frequency streamwise wind speed and high-frequency Reynolds stress.
Figure 6 displays an example of a time series of low-frequency, upper-anemometer wind speed and high-frequency envelopes for the same recording as in Fig. 3 (
Amplitude modulation of high-frequency wind and blowing snow signals by large-scale events. (a) Raw wind and blowing snow signals. (b) Notice correlation between low-pass wind signal 
Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0179.1
Many local maxima in 
The transient high-frequency coherence between wind and snow signals mentioned in section 3a also appears to be connected to the passage of large-scale motions. To illustrate this, each time step of 
It is worth noting that, like the results of Baas (2006) and Liu et al. (2012), the snow transport signal maintained higher wavelet power spectral density than streamwise wind fluctuations at the upper end of the frequency spectrum, typically >1 Hz, showing a possible inertial effect of snow particles and the presence of a near-surface rebound energy separate from that gained from the wind.
c. Connections with turbulence statistics
Statistical associations were examined between measurements of average coherence and amplitude modulation and turbulence descriptors for each recording to discern if certain atmospheric conditions were more conducive to wind–snow coupling or amplitude modulation. Very few statistically significant correlations were found; for instance, no correlation was found with friction velocity, turbulence intensity, turbulence kinetic energy, or mean wind speed at either height. The strongest correlation discovered was between average high-frequency (>0.35 Hz) coherence and mean sonic temperature during the recordings (
Further characterizing the relevant coherent turbulent structures is helpful for understanding the mechanisms in wind and snow transport modulation. For example, sweeps in Fig. 6b always coincide with local maxima of modulation envelopes or high-frequency coherence. This is further reflected by a fairly good correlation coefficient over the 23 recordings (
A weak positive correlation existed between mean blowing snow density and low-frequency coherence (
As the snowpack deepened over the season and the lower anemometer became closer to the snow surface, the amount of unmeasured high-frequency turbulence energy increased. This is especially true of 3 March, where low anemometer measurements occurred approximately 0.10 m above the snow surface and potentially 30% of high-frequency energy may have not been measured (van Boxel et al. 2004). It is suspected that the inclusion of additional high-frequency energy (e.g., with a hot-wire anemometer) would only enhance the already large high-frequency coherence values between turbulence and blowing snow transport 
d. Wind tunnel comparison
As the degree of amplitude modulation in turbulent boundary layers increases with increasing Reynolds number (Mathis et al. 2009a), understanding the contrasts between atmospheric flows at high Reynolds numbers and analogous wind tunnel experiments at low Reynolds numbers is vital for understanding how and whether to extend laboratory-derived wind–snow transport relationships to natural terrain (e.g., Paterna et al. 2016). To highlight the significant role amplitude modulation played in both high- and low-frequency wind–snow coupling at FMSL, an analogous amplitude modulation analysis of wind tunnel observations of streamwise wind fluctuations and streamwise blowing snow transport (Paterna et al. 2016) was performed. The degree of amplitude modulation was measured in nine 20-Hz time series of 
Streamwise amplitude modulation 
4. Discussion
The coupling of wind and near-surface snow saltation in the atmospheric surface layer proves to be complex, with nonlinear momentum transfer over many time and length scales. Near-surface, high-frequency turbulence was strongly modulated by passing large-scale, low-frequency motions measured 160 cm above. These same large eddies were responsible for modulating the high-frequency components of the blowing snow density signal as seen by the strong correlation between the high-frequency envelopes generated by the streamwise wind and blowing snow signals. In the presence of the large-scale motions, wavelet coherence between wind and snow signals was strongest in the low-frequency range, with significant high-frequency coherence being largely intermittent. However, high-frequency coherence peaks also coincided with passage of the same low-frequency motions. The question remains whether this coupling is the result of snow transport fluctuations responding to modulated high-frequency turbulence over short time scales, or if by the passage of large-scale events, the stochastic dynamics of surface impacts and rebound are amplified, resulting in a synchronous increase in high-frequency snow transport energy. As also noted by Jacob and Anderson (2017), amplitude modulation in nature may indeed play a crucial role in generating turbulent fluctuations sufficient to initiate blowing snow transport.
Two recordings with negligible turbulence modulation occurred on 20 November and 3 March (italicized in Table 2). These recordings coincide with more positive Reynolds stress 
In general, wavelet coherence and amplitude modulation analyses suggest a stronger dependence of snow transport on instantaneous wind speed than Reynolds stress, in agreement with recent aeolian sand studies (Sterk et al. 1998; Schönfeldt and von Löwis 2003; Leenders et al. 2005). While the statistical testing of wavelet coherence does not determine a causal relationship, there is only a 5% probability that coherence peaks occur solely by chance. When limiting analysis to small-scale motions at frequencies greater than 0.35 Hz, mean wavelet coherence (
As modulation of near-surface turbulence by large-scale influences can also be manifested in fluctuations of Reynolds stress (Mathis et al. 2013), the amplitude modulation found indicates that common turbulence statistics, such as friction velocity 
The analysis of turbulence mechanics in a wind tunnel blowing snow study showed negligible amplitude modulation when compared to the alpine observations, indicating an absence of the same magnitude of nonlinear momentum transfer that is found in nature. Thus, adapting wind tunnel–derived scaling relationships (e.g., Nishimura and Hunt 2000; Clifton et al. 2006; Horender et al. 2013) to complex terrain is largely dependent on how topographically influenced motions, and their influence on surface turbulence statistics, can be represented as modifications of simpler cases. This has implications beyond the impact of wind fluctuations on models of steady-state blowing fluxes (e.g., Sørensen 1997). For example, evidence of amplitude modulation has been found in the roughness sublayer (Anderson 2016), the region represented by roughness lengths 
Aksamit and Pomeroy (2018) showed that sweeps are responsible for considerable blowing snow initiation and surface transport, especially during intermittent transport conditions. Through the positive correlation of amplitude modulation coefficients and sweep-generated Reynolds stress, the penetration of these large-scale structures also appears responsible for much of the near-surface turbulence modulation. Separating Reynolds stress into components of large-scale modulation and purely local instability, as by Mathis et al. (2013), is the next step in reconciling scaling relationships derived in wind tunnels and the turbulent structures found in the atmospheric surface layer (ASL). The strong degree of amplitude modulation likely indicates that many of the largest near-surface turbulence events resulted from the passage of large-scale motions and not local surface instabilities.
Further investigation with a wider span of measurement heights may permit implementation of turbulence modulation in driving a blowing snow initiation or transport model. Such an approach would be beneficial to delimit for the necessary resolution of large-eddy simulations. The eddies with length scales of 30–60 m that modulated the amplitude of near-surface motions at FMSL are at the frontier of alpine blowing snow LES models (Vionnet et al. 2017). Further investigations may help realize statistically representative lower-boundary conditions (Anderson 2016) as well as improve wind–snow interactions in the saltation layer in LES.
5. Conclusions
This research examined how the structure of turbulence in the atmospheric surface layer affects near-surface blowing snow fluxes at a mountain site. The effects are multiscale and involve complex interactions between high- and low-frequency motions. Intermittent blowing snow transport responded to large low-frequency streamwise motions that modulated the amplitude of high-frequency turbulence. Because of this amplitude modulation, large-scale motions do not act simply as a shift in mean wind speed on local instabilities, but modify the dynamics of the surface momentum balance by increasing the amplitude of high-frequency turbulence. This modulation effect is important to characterizing turbulence for particle transport calculations, as the coherence between wind speed and snow transport signals also increases under footprint of these structures.
Large-scale sweep motions influenced near-surface turbulence and blowing snow fluxes across all measured frequencies. In light of the evidence of nonlinear momentum transfer across a wide spectrum of scales, determining the appropriate assumptions about the relative magnitude of structures and time scales for blowing snow transport is necessary for transport modeling. Future research to better understand near-surface turbulence amplitude modulation may permit reconciliation of the need for high-frequency, near-surface streamwise velocity measurements in blowing snow models and the more standard and widely available low-frequency micrometeorological observations through near-surface turbulence modulation models. A clearer understanding of how specific turbulent structures affect near-surface turbulence will better characterize the differences between wind tunnel and alpine blowing snow and provide insight for adapting steady-state, low Reynolds number scaling relationships to gusty natural conditions.
Continued observations of amplitude modulation over rough natural surfaces can help suggest improvements to surface interaction models for atmospheric large-eddy simulations (Anderson 2016) and to high-resolution large-eddy simulations of snow transport (e.g., Groot Zwaaftink et al. 2014). Because the amplitude modulation in both the turbulence and blowing snow signals found in this outdoor setting is not present in wind tunnels, considerable advancement in the field of sediment transport in complex terrain will likely only come after appropriately addressing the nonlinear dynamics influencing near-surface turbulent motions and further investigating two-phase flow turbulence in outdoor boundary layers.
Acknowledgments
The authors acknowledge funding from the Canada Foundation for Innovation, the Natural Sciences and Engineering Research Council of Canada, the Changing Cold Regions Network, Canada Research Chairs, the Global Institute for Water Security and Alberta Agriculture and Forestry. The assistance of the Fortress Mountain Resort in logistics is gratefully noted. Data are available at http://dx.doi.org/10.20383/101.010.
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