This phenomenon was described similarly as in a pure amplitude modulation in signal processing where a high-frequency signal is modulated by a low-frequency component. There are a number of attempts in the literature to unambiguously quantify this modulation either through a one-point correlation coefficient or a two-point correlation map.
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These studies focused on different aspects of the modulation phenomenon and contributed towards the understanding of the scale interactions inside the boundary layer. This review study brings together all these different approaches using well-resolved Large Eddy Simulation LES data for a comparably large range of Reynolds numbers.
For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy www. 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 ; Pomeroy et al.
Redistribution of snow begins when wind drag and shear stress exceed transport thresholds, and snow grains mobilize on the surface Li and Pomeroy Subsecond shear stress peaks can be sufficient to initiate snow transport and may not be reflected in time-averaged mean values Aksamit and Pomeroy 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.
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. 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.
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. 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 a ; Mathis et al. 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 b ; 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. 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. 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.
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. 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 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.
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. 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. To investigate this, 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 The experimental site was surrounded by relatively flat terrain on a bench above the main valley at m MSL. Nearby steep alpine faces rise from valley-bottom elevations of m to m ridge tops over distances of less than 5.
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 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 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.
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. 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. 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 , The camera was situated 0. Each blowing snow video recording consisted of 2. 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 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. Table 1. 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. Specifically, Hz directional snow flux and wind speed fluctuations were subjected to the same analysis as the FMSL data.
Grayscale blowing snow video recordings were binarized following the algorithm of Otsu , 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. A flood-fill algorithm was then implemented to identify individual snow particles and estimate their equivalent particle diameters.
Wavelet analysis has emerged as the standard technique to detect intermittent behavior in geophysical systems in the time—frequency domain e. 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. To identify statistically significant wind—blowing snow coupling, the wavelet coherence CH and statistical significance testing method of Grinsted et al. This is a complimentary technique to the wavelet maps of Ellis , wavelet packet decomposition of Liu et al. Let and be the continuous wavelet transforms of the time series and , respectively, where is time and is the scaling of the mother wavelet.
For the present research, the Morlet wavelet was chosen as the mother wavelet as it is well localized in both frequency and time and is useful for event extraction in geophysical time series Grinsted et al. Following Liu and Torrence and Webster , one can derive a measure of the coherence in the cross-wavelet transform in the time—frequency domain: where is a smoothing operator in time and scale, and multiplication by converts values to energy density. The statistical significance testing of Grinsted et al. The cone of influence COI demarcates the regions in the time—frequency domain where zero padding and edge effects from the wavelet transform improperly affect coherence calculations.
Time-averaged , referred to as , also provided a measure of mean coherence in the frequency domain for each recording. Define where is the measure of the length of the time series intersecting the COI at scale. Further background on the theory of wavelets can be found in Daubechies Applications and comparisons of wavelet methods are outlined by Foufoula-Georgiou and Kumar and Torrence and Compo and in the signal processing text of Mallat 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 but the power spectra likely overlap with local processes, often preventing a clear spectral separation of large- and small-scale motions Sievers et al. 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 , and larger atmospheric motions with a higher mean wind speed as a modulating signal. The resultant amplitude-modulated signal and are displayed in Fig. With and being the Fourier transform of and , respectively, in Fig. Because of the broadband overlap of both low- and high-frequency motions in the atmosphere, extracting a signal responsible for modulating high-frequency turbulence with Fourier methods is realistically unfeasible. For clarity, the mirroring effect on the Fourier transforms is proven in more generality in the supplemental material.
Notably, there is no 0. With sufficient frequency separation in the power spectra of the carrier and modulating signals, however, the Hilbert transform can perfectly reconstruct the modulating signal. For a time series , define the Hilbert transform where PV is the Cauchy principal value, and denotes standard convolution.
The Hilbert transform can also be regarded as the imaginary part of the analytic function with. Formally, The modulus of is called the envelope signal of , herein written. Using the same modulating and carrier signals as in Fig. This reconstruction of modulating signals is proven in more generality in the supplemental material. Mathis et al. For a streamwise wind signal , one first separates the signal into low- and high-frequency components, and , respectively, at an appropriate cutoff frequency. The effect of amplitude modulation becomes evident after the same low-pass filter is applied to the Hilbert envelope of the high-frequency signal.
The degree to which the high-frequency turbulence was modulated can be identified when is compared with the low-frequency motions in. This method identifies nonlinear effects of large-scale motions on near-wall turbulence in the atmospheric surface layer that were missed in Fig.
For the present study, it is proposed that the low-frequency component of streamwise wind speed represents large eddies modulating signal that amplify the magnitude of high-frequency surface fluctuations carrier signal. A correlation coefficient can then be defined that measures the degree of amplitude modulation, that is, the influence of large-scale eddies on high-frequency turbulence: where the overbar indicates temporal averaging.
Note that because amplitude modulation inherently involves different scales of motions at different velocities, AM intentionally does not account for a difference in mean or variance in the two time series. Furthermore, all signals were standardized to have zero mean and unit variance prior to calculation of AM coefficients to account for different units and ranges of wind and snow signals.
Background on amplitude modulation and the theory of Hilbert transforms can be found in Bendat and Piersol Seventy-two turbulence statistics were calculated for each recording, including turbulence intensity, turbulence kinetic energy, friction velocity, drag coefficient Stull ; Lykossov and Wamser , 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 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. Wavelet coherence was calculated for time series pairs of low-anemometer streamwise wind speed and blowing snow density as well as low-anemometer Reynolds stress and blowing snow density for all 23 recordings. Examples of raw wind and and signals and their wavelet coherence maps from 3 March are shown in Fig.
Bold outlined regions of the coherence maps in Figs. As was typical for all recordings, there is much greater coherence between blowing snow density and streamwise wind speed in Fig. In Fig. Even the largest coherence measurements indicate highly intermittent structures of natural blowing snow at this site. Time-averaged calculations measured the mean coherence at each scale over an entire time series. The average for the whole experiment is displayed in Fig. One standard deviation is shaded surrounding the mean of both Fig.
The provides similar information to magnitude-squared Fourier coherence with larger values indicating more coherence. Coherence monotonically increased with decreasing frequency between blowing snow and streamwise wind speed, whereas a possible coherence peak was found between Reynolds stress and blowing snow around 0. The measured coherence between the snow and either turbulence signal was equivalent above 0. As is a time average of , the transient nature of intermittent blowing snow is not clearly captured. One standard deviation is shaded for all plots. Notice the higher coherence at lower frequencies for and general agreement of coherence for Hz.
In contrast, Fig. This uses the temporal localization of wavelets to present how often transient but significant coherence occurred. Interestingly, Fig. There was substantial low-frequency coherence below in that was not present in , with significant coherence nearly 4 times more likely with streamwise wind speed. This is to be expected by the relatively low energy in at low frequencies Hunt and Morrison that provides little energy to the cross-wavelet transforms [Eq. Again, there was general agreement in coherence of blowing snow with either wind signal above 0.
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 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 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 Wavelet coherence Figs.
Modulation of the atmospheric turbulence coherent structures by mesoscale motions | SpringerLink
However, there was also coupling of to high-frequency and fluctuations of unknown origin over shorter time scales Figs. The analyses of Baas , Baas and Sherman , and Ellis indicate that the brief coherent structures responsible for intermittent sand transport in their studies did not likely originate from surface instabilities.
Thus, for all recordings, it was investigated whether the significant high-frequency coupling in section 3a could be also be related to large-scale structures. More precisely, was there evidence of amplitude modulation in high-frequency turbulence, blowing snow, and coherence signals during the passage of top-down penetrating structures? The ability to quantify amplitude modulation with Hilbert transforms [Eqs. 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 ; Sievers et al.
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 a. There was also a transition to higher turbulence energy at the lower measurement height for frequencies above this threshold Fig.
Using the mean wind speed at 2 m as a proxy for convective velocity, this frequency corresponded to eddy length scales between 39 and 67 m during the recordings. This is similar to a previous atmospheric boundary layer amplitude-modulation study that found the boundary layer depth m for their experiment to be a suitable separating length scale Mathis et al.
As there was no definitive measurement of the boundary layer depth at this mountain site, the cutoff frequency Hz was chosen as a threshold to separate large-scale and small-scale motions. Varying the choice of threshold in the range from to 0. Standardized premultiplied power spectral density of streamwise wind speed for 5 h at two measurement heights over four nights of recording.
Also note elevated high-frequency energy in ground wind signals above. Date of 7 Feb omitted for plot clarity. The large motions defined by the low-pass filter in both the near-surface and upper anemometer streamwise wind signals were found to strongly modulate high-frequency near-surface turbulence. The coefficients and , as defined by Eq. The consistently large AM values for either pair of turbulent wind signals are convincing evidence of the influence of low-frequency, large-scale events on high-frequency turbulence and suggest that many near-surface turbulence bursts were not a purely local phenomenon.
Amplitude modulation of by upper-anemometer, large-scale motions was also quantified and can be found in Table 2. Table 2. Amplitude modulation coefficients for each recording for blowing snow density by 2-m streamwise wind speed , ground streamwise wind speed by 2-m streamwise wind speed , and ground streamwise wind speed by ground streamwise wind speed. Parameter is covariance derived from the lower anemometer for all recordings. Italicized columns indicate experiments with negligible amplitude modulation as described in section 4.
Missing experiments occurred at higher frame rates and are used in Aksamit and Pomeroy , While the majority of recordings exhibited strong modulation Table 2 , there were several exceptions that are explained with quadrant analysis Lu and Willmarth in section 4. These values are italicized in Table 2.
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Also of note, in contrast to the results of Hutchins and Marusic a , 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.
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The raw wind and snow signals under analysis are shown in Fig. Figure 6b shows the low-pass-filtered upper-anemometer wind signal ; the envelope of high-frequency blowing snow ; and the envelope of high-frequency, near-surface turbulence , as well as purple dots during moments of extreme sweep Reynolds stress from quadrant analysis Lu and Willmarth Marusic Published DOI: View PDF.
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