Edited by: Michael Lehning, École Polytechnique Fédérale de Lausanne, Switzerland
Reviewed by: Vincent Vionnet, University of Saskatchewan, Canada; Ruzica Dadic, Victoria University of Wellington, New Zealand; Stuart John Bartlett, California Institute of Technology, United States
This article was submitted to Cryospheric Sciences, a section of the journal Frontiers in Earth Science
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
Atmospheric pressure changes ranging from high-amplitude, low-frequency events caused by synoptic weather systems to smaller amplitude, high-frequency events caused by turbulence penetrate permeable snow surfaces. Fluxes driven by these pressure changes augment non-radiative processes that filter atmospheric aerosols and drive near-surface vapor flux by sublimation, condensation and deposition. We report on field experiments in which we measured the amplitude of mid-to-high frequency pressure changes as they varied with depth in a seasonal snowpack and on two empirical models that distinguish conditions that promote pressure-driven vapor exchange. We found that the standard deviation of pressure changes poorly characterizes pressure perturbation amplitudes that drive vapor exchange because many low amplitude perturbations mask the influence of less common but more consequential high amplitude perturbations. Spectral analysis of pressure perturbation energy at different snow depths revealed an empirical formula that quantifies perturbation pressure attenuation as a function of frequency and depth in snow. Model results indicated that sublimation enhancement is maximized for perturbation pressure periods between 0.2 and 10 s.
Wintertime sublimation diminishes the capacity of alpine regions to supply runoff during the following dry season. Sublimation rates may be high for windblown snow (
Vapor exchange between snow and the atmosphere is enhanced by solar radiation (
Snow is a porous medium and air can pass between interstitial pore space and the atmosphere. As illustrated in
Increasing pressure compresses atmospheric air into the snowpack and decreasing pressure allows saturated air (denoted in blue) near the snow surface to translate into the atmosphere. Successive pulses of influx and efflux generate an upward flux of water vapor if the air above the snow is subsaturated. As shown, the horizontal axis is spatial but could rather be temporal.
Large amplitude pressure changes caused by synoptic weather patterns are too infrequent to significantly impact vapor flux by bulk air displacement. However, wind flow over surface rugosities and turbulence generate pressure changes at high enough frequencies to impact vapor exchange if their amplitude is sufficient. CB89 and
where M is wind speed (m s–1) at 5 m height and
We deployed the field experiment at three sites: Santiam Pass, OR, United States (1468 m elevation), Dutchman Flat Sno-Park (
Case numbers, associated dates and sensor heights relative to the snow surface for each deployment.
The experimental setup (see
Schematic of the experimental setup. Pressure sensors measure at 4 depths in the snow through flexible tubing. A Campbell Scientific Irgason mounted ∼ 1 m above the snow measures 3D wind components at 20 Hz. Data are stored on a CR-3000 logger mounted on the low-profile tower.
The second CR-3000 logged pressure measurements from four Paroscientific Model 216B pressure sensors. These pressure sensors operated over a 300 hPa (800 hPa to 1100 hPa) range with 0.0001% resolution resulting in 0.03 Pa precision (
As will be described in the section “Model Calculations” we coded two empirical models in MATLAB to obtain order of magnitude estimates of pressure-enhanced sublimation rate based on pressure and wind measurements. The first model, named the simplified Monin-Obukhov (M-O) model, essentially counts water vapor molecules that have sufficient displacement to exceed the aerodynamic roughness length and become candidates for scouring by the atmosphere. An efficiency factor accounts for the stochastic nature of near-surface mixing. The second model, named the Gaussian model, approximates enhanced sublimation from statistical considerations (developed in the manuscript) and is also initialized by field-based measurements.
We formalize definitions for pressure “perturbations” and “fluctuations” as deviations from a defined mean and we refer to successive pressure changes as “excursions” in this paper. As such, “pressure perturbations” refer to a collection of individual pressure fluctuations and “perturbation pressure” is the time-integrated influence of these individual pressure changes. In panel (A) of
Panel
The black line in panel (A) corresponds to values computed using Colbeck’s empirical formula Eq. (1) with wind speed logarithmically corrected from 1.2 to 5 m for a neutrally stable surface layer and assuming a 0.24 mm roughness length (from
We summarize bin-averaged
Ranges for
Rather than performing a curve fit between wind forcing and perturbation pressure response for each case we seek a similarity relationship that produces a generalized curve fit that is valid across cases. Based on Buckingham Pi dimensional analysis (
Where
“
Eq. (2) is supported on physical grounds by the Bernoulli equation, which relates pressure changes to the square of the fluid velocity and theoretically by Eq. (15) in
We next employ spectral analysis to investigate how pressure perturbation energy varies with pressure change frequency. For each case we performed spectral analysis for each pressure sensor to gauge spectral attenuation of pressure gradients. When performing spectral analysis, missing data were gap-filled with data randomly chosen from nearby points in the time series (
Pressure spectra at 1 m above the snow (blue) and 6 cm below the snow surface (green and gold) for case 2 (panel
For the range of snow densities (227–445 kg m–3) and regardless of surface snow topographical differences between cases, relative pressure attenuation below 0.2 Hz was insignificant (
D16 found a decay of high-frequency spectral energy with depth, consistent with the observations presented here. We ascribe the difference in spectral attenuation with depth between D16 and this study to differences in snow layers in D16 that we largely avoided by design in this experiment. Consistent decay of high-frequency pressure perturbations with depth in an approximately homogenous snow layer enabled computation of spectral attenuation of pressure variance with depth. Acoustic attenuation at frequencies below 200 Hz is small and decreases with decreasing frequency (
where
To estimate the spectral attenuation of pressure fluctuations with depth in snow we subset the pressure spectra in
Computation of spectral slope for the 20 cm depth pressure sensor for Case 1 (panel
For cases with a surface pressure measurement, we found an average spectral slope (
where Δ
The empirical formula in Eq. (4) describes the slope of pressure variance due to frequency-dependent attenuation with depth in snow. We integrate Eq. (4) to derive an equation for spectral power as a function of depth in snow:
where
where we have used 10−3
Our experimental results as well as those presented in CW91 and D16 show that a power law function describes attenuation of pressure perturbations with frequency and depth. We therefore reconsider the discrepancy between these experimental results and the theoretical prediction of exponential decay that was addressed in CW91. The authors attributed this discrepancy to either localized pressure disturbance caused by the measurement setup or an over-reliance on Taylor’s Hypothesis. We conclude it is unlikely that a given measurement setup would produce this systematic difference for different snow permeability and over a range of depths spanning up to 60 cm. If, as CW91 suggest, the experimental apparatus artificially produces spectral attenuation that follows power law behavior rather than exponential decay, then we would expect high-frequency attenuation would approach exponential decay with increasing depth and snow density, where disturbance due to the experimental apparatus declines. Our experimental results do not show this tendency. Neither do the results in D16 using a different experimental design and different pressure sensors. This lack of a systematic change in power law behavior with changes in snow depth and density also rules out the theory that an over-reliance on Taylor’s hypothesis has skewed the experimental results. We suggest a third alternative: that the fundamental assumption of the equation that describes the attenuation of pressure perturbations (CW91, Eq. 1) should be modified such that it has a solution that describes attenuation as a power law rather than exponential decay with frequency and depth.
Since Eq. (1) underestimates perturbation pressure and also fails to describe the distribution of pressure fluctuations, we seek an alternative formulation that reproduces the observation that perturbation distribution broadens both with increasing wind speed and averaging time scale. The tails of this distribution would then capture the influence of infrequent, large amplitude pressure fluctuations. We model the range of perturbation pressure as a Gaussian distribution (
for which the peak width is a function of averaging time scale and wind speed. A Gaussian distribution may not be representative for a turbulent regime (
where
A realistic approximation for sublimation rate enhancement accounts for the atmospheric capture efficiency based on frequency and vapor pressure deficit. We propose the following relationship to describe the integrated sublimation rate enhancement as a function of surface area, vapor pressure deficit and vapor capture efficiency:
where
In Monin-Obukhov (M-O) boundary layer theory, displacement length is the height above the surface at which a logarithmic wind profile initiates. Typical applications of displacement length are for winds above uniform agricultural crops and dense forests.
Now that we have an estimate for the distribution, magnitude and attenuation of pressure perturbations as a function of frequency we can attempt to model the wind pumping process. To facilitate comparison between cases we assume a constant vapor pressure deficit of 1% for all cases, which is representative of air very near the snow surface. We approximate the exchange efficiency in Eq. (9), ∈
where we arbitrarily assume coefficients α = 10−6 and β = 1. Unknown coefficients α and β constrain our ability to assign absolute sublimation rates to model results and is a potential topic for future study. This approximation formalizes the M-O assumption that water vapor molecules must traverse the roughness layer to be candidates for capture by the atmosphere. The aerodynamic roughness length stochastically describes the mixing influence of ephemeral turbulent eddies rather than describing the height of a “lid” of air above the snow. For model simulations, we utilize an aerodynamic roughness length of 0.24 mm for fresh snow from
We implemented a surface exchange model to test the functional relationship between perturbation pressure frequency and vapor exchange rate. In this model, vapor exchange caused by turbulent eddies is approximated by hydrostatic pressure changes relative to the aerodynamic roughness length. At each frequency we computed pressure deviations between successive measurements. If the hydrostatic height change exceeded the aerodynamic roughness length then we computed the displacement length above z0. We then computed water vapor flux as a function of the vapor pressure deficit and perturbation pressure frequency. Before summing the flux for each frequency we applied the frequency-dependent efficiency factor described in Eq. (10). We note that the efficiency factor adjusted the magnitude of the vapor exchange rate but the coefficients were tuned such that they did not change the functional shape or the frequency of maximum vapor exchange of the resultant curve.
We also implemented an empirical model utilizing Eqs. (2, 7, and 8) to obtain a distribution of pressure perturbations as a function of wind forcing and snow density. For a given wind speed and for each perturbation pressure frequency we use Eq. 2 to compute σp. As with the simplified M-O model we determine the high frequency attenuation (Eq. 5) for subsurface pressure sensors. We then use the Bienaymé-Chebyshev inequality (Eq. 8) to determine the subset of water molecules that could have sufficient amplitude to generate a hydrostatic adjustment equal to or greater than z0, assuming a normal distribution. We then subdivide pressure perturbations into many narrow ranges of time scale averages and integrate the vapor exchange rate for each frequency and apply the efficiency factor, ∈
We plotted pressure-induced sublimation rate as a function of pressure change period in
Simplified M-O model results in panel
We initialized the Gaussian model using the same average wind speed for cases shown in
We now estimate the magnitude of enhanced vapor exchange by pressure pumping from a sublimation rate equation adapted from
where
In this example we utilize improved values for
In light of these results, we reexamine
The integral of the curve in
where τ is time period in seconds. For case 10, we plotted Eq. (13) as a black dashed line in
Taking the derivative of Eq. (13) and setting it equal to zero yields the peak frequency:
which we solved numerically and found the peak sublimation rate at 0.48 s, which is the same as the empirical result (with peak period of 0.5 s).
Negative feedbacks reduce the rate of vapor exchange by wind pumping and lessen SWE reduction. For example, as winds increase snow grains saltate, become suspended and thereby saturate near surface air, diminishing the vapor pressure gradient needed for wind-pumping driven vapor exchange. Sublimation rounds snow crystals and glazes the snow surface by subsequent deposition. In contrast, saturation vapor pressure is an exponential function of temperature so warmer snow increases vapor exchange by pressure fluctuations relative to colder snow that otherwise has similar microphysical characteristics. A full reckoning of all vapor exchange process are needed to compare the relative roles of these processes and the interplay between them in various environmental conditions.
We find that the
Over short timescales, turbulence combined with dry surface air is needed to generate noteworthy vapor exchange enhancement. In windy conditions, this pressure pumping process is persistent and pervasive so it may have a significant mass balance impact over seasonal time scales. In a natural environment it is difficult to discriminate pressure-induced sublimation enhancement from sublimation that is not spurred by pressure fluctuations. However, it is important to improve our understanding of the different processes by which snow sublimates if we aim to improve our ability in predicting the seasonal evolution of the snowpack.
SD authored the initial manuscript. CH and JS provided insights and edited it.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Thanks to Dr. Noah Molotch and Dr. Michael Durand for their support during the Colorado campaign, Lisa Dilley (USFS) for coordinating access to Dutchman Flat Sno-Park (