Volume 56, Issue 2 e2019WR025307
Research Article
Free Access

A Finite Volume Blowing Snow Model for Use With Variable Resolution Meshes

Christopher B. Marsh

Corresponding Author

Christopher B. Marsh

Centre for Hydrology, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

Global Institute for Water Security, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

Correspondence to: C. B. Marsh,

[email protected]

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John W. Pomeroy

John W. Pomeroy

Centre for Hydrology, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

Global Institute for Water Security, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

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Raymond J. Spiteri

Raymond J. Spiteri

Numerical Simulation Lab, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

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Howard S. Wheater

Howard S. Wheater

Global Institute for Water Security, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

Centre for Hydrology, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

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First published: 23 January 2020
Citations: 27

Abstract

Blowing snow is ubiquitous in cold, windswept environments. In some regions, blowing snow sublimation losses can ablate a notable fraction of the seasonal snowfall. It is advantageous to predict alpine snow regimes at the spatial scale of snowdrifts (≈1 to 100 m) because of the role of snow redistribution in governing the duration and volume of snowmelt. However, blowing snow processes are often neglected due to computational costs. Here, a three-dimensional blowing snow model is presented that is spatially discretized using a variable resolution unstructured mesh. This represents the heterogeneity of the surface explicitly yet, for the case study reported, gained a 62% reduction in computational elements versus a fixed-resolution mesh and resulted in a 44% reduction in total runtime. The model was evaluated for a subarctic mountain basin using transects of measured snow water equivalent (SWE) in a tundra valley. Including blowing snow processes improved the prediction of SWE by capturing inner-annual snowdrift formation, more than halved the total mean bias error, and increased the coefficient of variation of SWE from 0.04 to 0.31 better matching the observed CV (0.41). The use of a variable resolution mesh did not dramatically degrade the model performance. Comparison with a constant resolution mesh showed a similar CV and RMSE as the variable resolution mesh. The constant resolution mesh had a smaller mean bias error. A sensitivity analysis showed that snowdrift locations and immediate up-wind sources of blowing snow are the most sensitive areas of the landscape to wind speed variations.

Key Points

  • A physically based, 3-D blowing snow model is shown to be computationally efficient and suited for multiscale snowpack prediction
  • The variable resolution discretization had a 62% reduction in computational elements and a 44% reduction in total runtime
  • Including blowing snow increased the coefficient of variation of SWE from 0.04 to 0.31 and more than halved the total mean bias error

1 Introduction

Blowing snow redistribution is ubiquitous in cold, windswept environments such as alpine and arctic tundra, grass and croplands, and glaciers and results in spatial patterns of snowpack ablation or deposition, leaving highly heterogeneous snow covers (Fang & Pomeroy, 2009; Freudiger et al., 2017; MacDonald et al., 2010; Marsh, 1999; Mott et al., 2010; Pomeroy et al., 1997; Sturm et al., 2001; Wayand et al., 2018; Winstral et al., 2013). Spatially and temporally variable wind speeds arise due to complex topographic and vegetation interactions with meso- and micro-scale winds and are key drivers of blowing snow redistribution. Wind accelerates over hills and ridge crests (Jackson & Hunt, 1975; Mason & Sykes, 1979; Walmsley et al., 1984; Wood, 2000), resulting in enhanced snowpack erosion (Essery et al., 1999). Subsequent deceleration over lee slopes, in topographic depressions, and behind tall vegetation or snow fences (Pomeroy & Gray, 1995; Tabler et al., 1990) can result in large snowdrift deposits. Spatially variable precipitation fields can further increase this heterogeneity (Lehning et al., 2008). As a result of high rates of ventilation of blowing snow particles, substantial mass loss occurs due to sublimation in undersaturated atmospheres (Bowling et al., 2004; Dyunin, 1954; Mott et al., 2018; Pomeroy et al., 1993; Schmidt, 1972, 1982; Vionnet et al., 2014). The resulting heterogeneity in pre-melt snow covers is an important factor for the timing and magnitude of spring runoff (Marsh & Pomeroy, 1996; Pomeroy et al., 1997; Luce et al., 1998; Woo & Thorne, 2006; Dornes, Pomeroy, Pietroniro, Carey, et al., 2008; Fang et al., 2013). In alpine regions, blowing snow drifts form cornices (Mott et al., 2010) and influence avalanche formation (Bernhardt et al., 2012) that can further impact basin hydrology (de Scally, 1992). Large snowdrift formations on lee slopes can provide important ecosystem services, for example, polar bear den locations (Blix & Lentfer, 1979; Liston et al., 2016). These drifts often persist into summer (Wayand et al., 2018), when they contribute to rain-on-snow events (Pomeroy et al., 2016) and can provide late season runoff for downstream users (Nazemi et al., 2013; Viviroli et al., 2007) and agricultural water supplies (Pomeroy & Male, 1986).

The inclusion in hydrological models of blowing snow transport, via saltation near the ground, suspension diffusing to hundreds of meters above the snow surface, and in-transit sublimation of blowing snow, is critical for simulating the spatial variability and volume of snow accumulation (Pomeroy et al., 1993, 2007; Luce et al., 1998; Dornes, Pomeroy, Pietroniro, Carey, et al., 2008; Fang & Pomeroy, 2009; Fang et al., 2013; Zwaaftink et al., 2013; Freudiger et al., 2017; Mott et al., 2018). The International Network for Alpine Research Catchment Hydrology (INARCH) (Pomeroy et al., 2015), a Global Energy and Water Exchanges (GWEX) project of the World Climate Research Programme, has identified snow-drift resolving scales (≈1 to 100 m) as a key component to predicting alpine catchment hydrology (Pomeroy & Bernhardt, 2017). INARCH has resolved that models that do not explicitly include blowing snow processes at sub-100 m scales or parameterize their impacts on snow cannot adequately represent alpine snow dynamics and hydrological responses.

Despite their importance, blowing snow processes are not often represented in distributed hydrological models (Winstral et al., 2013). Many models compensate for this by relying on calibration of conceptual parameters or adjustments of precipitation to represent the impact of blowing snow on hydrological predictions. Most land surface schemes represent the impacts of blowing snow using a snow water equivalent (SWE) variability parameter that controls the snow cover depletion curve (Pomeroy, Gray, et al., 1998). However, how to set this parameter given its inter-annual variability is uncertain, and sublimation losses are not addressed by this approach. Those land surface schemes that do conceptualize redistribution include it as a sub-grid, inter-tile process (Davison et al., 2016).

The omission of blowing snow processes is a fundamental misrepresentation of alpine snow hydrology and increases model uncertainty; reasons cited for this model inadequacy include computational costs (Winstral et al., 2013), meteorological data requirements, and increased model complexity (Luce et al., 1998). When included, applications are often limited to small research areas (Fang et al., 2013), although large area predictions are possible (Pomeroy et al., 2013). Regardless of scale, the need for alpine hydrological model parameter calibration from streamflow can be reduced through full inclusion of alpine snow redistribution processes such as blowing snow (Pomeroy et al., 2013). Cold-regions hydrological model performance is improved by including spatial heterogeneity—for example, slope, aspect, and vegetation—and representing the full range of hydrological processes operating in these regions (e.g., Bartelt & Lehning, 2002; Bowling et al., 2004; Etchevers et al., 2004; Raderschall et al., 2008; Dornes, Pomeroy, Pietroniro, & Verseghy, 2008; Essery et al., 2009; Essery et al., 2013; Fiddes & Gruber, 2014; Painter, Coon, et al., 2016; Mosier et al., 2016; Musselman et al., 2015). This suggests that an efficient approach for the explicit inclusion of spatially distributed blowing snow processes would be beneficial in advancing cold regions hydrology.

Point-scale blowing snow transport models are challenging to apply in fully distributed hydrological models due to landscape heterogeneity and complex wind fields, although there has been success at large spatial scales in macroscale models without spatial coupling (e.g., Bowling et al., 2004; Yang et al., 2010). Semi-distributed models and terrain parameter-based methods (e.g., Winstral et al., 2013) are either constrained by only capturing the snow cover heterogeneity via a priori knowledge of drift locations (e.g., MacDonald et al., 2009) or require calibration from detailed distributed SWE observations (Schön et al., 2018). Terrain parameter-based models are incapable of calculating blowing snow sublimation losses, which can vary from 1% to 30% of snowfall in alpine catchments (Mott et al., 2018; Musselman et al., 2015). Wind direction variability over large areas (more than a few km) of complex terrain precludes the use of simplifying assumptions about uniform wind direction as was done in some distributed blowing snow models, for example, the Distributed Blowing Snow Model (DBSM; Essery et al., 1999; Fang & Pomeroy, 2009). The use of advection-diffusion equations (scalar-transport), for example, Alpine3D (Lehning et al., 2006), or spatially distributed formulations, for example, DBSM and SnowTran3D (Liston & Sturm, 1998), tend to increase computational demand, parametrization requirements, and model complexity as a result of the use of a fixed-resolution spatial discretization. More complex models consider nonsteady turbulence, for example, the point-scale model PIEKTUK (Déry & Yau, 1999). When distributed, these models couple blowing snow transport and sublimation using three-dimensional wind fields computed from atmospheric models, for example, SURFEX in Meso-NH/Crocus (Vionnet et al., 2014, 2017), Alpine3D (Lehning et al., 2006), and SnowDrift3D (Schneiderbauer & Prokop, 2011), substantially increasing the computational costs.

Aksamit and Pomeroy (2017) note that most models of blowing snow transport have been conceptualized for steady-state conditions based on time-averaged field or wind tunnel observations of snow particle flux and dispersion (Lehning et al., 2006; Pomeroy, 1989; Schmidt, 1986). Thresholds for the initiation or cessation of transport have been based on air temperature, occurrence of melt or rain, and snowpack age (Li & Pomeroy, 1997a, 1997b) as well as surface grain type (Guyomarc'h & Mérindol, 1998). Detailed outdoor field campaigns (Aksamit & Pomeroy, 2016, 2017, 2018) and wind tunnel research (see summary in Paterna et al., 2017) have investigated the mechanisms that erode and entrain surface snowpack into blowing snow. Two-phase flow is governed at short (<1 s) time intervals by particle responses to turbulent gusts and sweep-ejection mechanisms rather than scaling with shear stress (Aksamit & Pomeroy, 2016, 2018). This increase in understanding has not yet led to new turbulence-based blowing snow transport models. For the longer time intervals (15 min to 1 hr) employed by hydrological models, mean shear-stress scaling blowing snow models are used and have been adapted to non-fully developed flow conditions by empirically estimating the length scales necessary for establishment of near-surface two-phase flow via saltation (Comola & Lehning, 2017; Pomeroy & Gray, 1990) and explicitly calculating the upward diffusion of suspended blowing snow using plume dispersion equations (Pomeroy & Male, 1992). These methods have been coupled to complex terrain wind flow estimation procedures and used in distributed models, for example, DBSM, SnowTran3D, Alpine3D, and the Distributed Snow Model (Musselman et al., 2015).

Efficient calculation of spatially distributed blowing snow fluxes is needed to estimate wind redistribution and sublimation of snow at high resolution over large areas. Using a variable-resolution unstructured triangular mesh (unstructured mesh) to represent and discretize terrain and vegetation cover is one approach that can efficiently accomplish this. An unstructured mesh allows for representing important heterogeneities in the surface but can reduce computation elements between 50% to 99% compared to a structured grid (raster) representation (Ivanov et al., 2004; Marsh et al., 2018). The reduction in computation elements can often significantly reduce computation times while at the same time produce more accurate simulations (e.g., Ascher et al., 1995). Furthermore, despite the reduction in computational elements, unstructured meshes can preserve heterogeneity in topography and vegetation (Marsh et al., 2018). There is subjectivity in mesh configuration because the mesh acts as an approximation to the underlying source rasters; however, using numerical guarantees of the approximating accuracy in the meshing tool can alleviate much of this.

This work combines the computational benefits of a variable resolution terrain discretization with a distributed blowing snow model based on rigorous field observations of blowing snow (Pomeroy, 1989; Pomeroy et al., 1993; Pomeroy & Li, 2000). A finite volume method (FVM) discretization is used and does not require the assumptions of uniform wind direction; it is therefore applicable to areas with complex wind flow. DBSM used a lookup table approach for relative wind speed, derived from a Mason and Sykes-based three-dimensional wind flow model MS3DJH/3R (Walmsley et al., 1986). This is extended here to include changes to wind direction and to allow for applicability to an unstructured mesh. The variable resolution blowing snow model is used to simulate snow accumulation patterns in a subarctic catchment and is compared with 5 years of observed snow depth and density transects across a large alpine valley that often contained a snowdrift. An uncertainty analysis uses a 50-member ensemble of perturbed wind speeds. This paper investigates the following questions: (1) Can a variable resolution blowing snow model predict the spatial and temporal variability in SWE and the volume of snow drifts in an alpine catchment? (2) Does a variable resolution discretization produce similar results to a fixed-resolution discretization? and (3) What is the spatial sensitivity of the blowing snow model to wind speed calculation? Application of variable resolution meshes represents a novel and efficient distributed representation of blowing snow processes, suitable for multiscale extents.

2 Model Development

2.1 Overview

A 2-D schematic of the conceptual model for blowing snow transport via saltation and suspension layers is shown in Figure 1. The saltation layer acts to provide a lower boundary condition for diffusion into the suspension layer (Pomeroy & Gray, 1990; Pomeroy & Male, 1992). Diffusion from the saltation layer to the suspension layer is driven by turbulent fluctuations in the instantaneous vertical wind speed exceeding the terminal fall velocities of blowing snow particles (Pomeroy & Male, 1992). Sublimation during transport is modelled as a sink term based on turbulent transfer of sensible and latent heat to the blowing snow particles, which are assumed to cool to the ice-bulb temperature (Pomeroy, 1989; Pomeroy et al., 1993). The nonsteady effects of upwind fetch are represented by a downwind increase with fetch to a fully developed saturation level in the saltation concentrations. This is used to calculate suspended concentrations and the increasing height of the suspended snow layer with fetch.

Details are in the caption following the image
A 2-D schematic of the conceptual model of blowing snow transport. A bottom saltation layer is present over a snowpack, with diffusion into the suspension layers. Sublimation is modelled as a sink term. The suspension layer is discretized with a user defined number of layers.

The steady-state saltation flux parameterizations (Pomeroy & Gray, 1990) are used to calculate the saltation layer mass concentration based on an observed relationship between saltation trajectory height and shear stress. Although this saltation model does not explicitly calculate particle trajectories as do, for example, Doorschot and Lehning (2002) and Clifton and Lehning (2008), it remains in reasonable agreement with these more complex methods and requires fewer parameters and less computational overhead. Its use here does not preclude the use of a different parameterization. Saltation, turbulent suspension (Pomeroy & Male, 1992), sublimation (Pomeroy et al., 1993), threshold shear stress for saltation (Li & Pomeroy, 1997a), shear stress partitioning by vegetation and snow, and a probabilistic upscaling (Pomeroy & Li, 2000) parameterizations comprise the blowing snow model. These were found to be consistent with outdoor observations of blowing snow particle flux made using a snow particle detector (Brown & Pomeroy, 1989; Pomeroy, 1989), changes in blowing snow chemistry due to sublimation (Pomeroy et al., 1991), profile measurements of wind speed, temperature, and humidity in the lowest 3 m of the boundary layer, and snow mass balance measurements (Fang & Pomeroy, 2009; MacDonald et al., 2010; Musselman et al., 2015; Pomeroy et al., 1993; Pomeroy & Li, 2000). The sublimation parameterization of Pomeroy et al. (1993) was found by Pomeroy and Essery et al. (1999) to be consistent with eddy correlation measurements of latent heat flux. There is a strong motivation to use field observation-based parameterizations versus those from wind tunnels because the latter do not have sufficient fetch and have turbulence features that do not represent natural, outdoor conditions (Aksamit & Pomeroy, 2018).

The advection-diffusion equation implicit in the blowing snow model was discretized in space via the finite volume method on a high-quality, spatially variable mesh generated via the mesher code (Marsh et al., 2018). The mesh has well-graded triangles from small to large areas, and all internal angles are greater than 21 degrees. This ensures that there are no sharp transitions between triangles that could cause numerical instabilities. Due to the use of the FVM, neither simplifying assumptions about wind direction nor domain rotations into the wind direction are needed. This allows use where the wind flow is divergent and over large extents. Erosion and deposition are computed as the spatial and temporal divergence of the suspension and saltation fluxes, that is, their rate of change over space and over model time steps.

2.2 Numerical Background

2.2.1 Transport Equation

Suspended blowing snow transport was modelled via the steady-state scalar advection-diffusion equation,
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0001(1)
where K is the diffusion coefficient (m2·s−1), c is the transported scalar, in this case the mass concentration of blowing snow (kg·m−3), urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0002 is the wind vector (m·s−1), and S is a source or sink term. The wind vector is assumed to have zero divergence; therefore, equation 1 may be written as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0003(2)

2.2.2 Numerical Discretization

Equation 2 can be discretized in space using 3-D prisms. To do so, the 2-D triangular mesh is extruded vertically to form a prism, as shown in Figure 2. These prisms are stacked vertically to form multiple discretization layers. A prism in a given layer lies exactly above one below; that is, there is no offset or rotation of triangles with height. This planar extrusion of the triangles into prisms causes prisms to align even over areas of high curvature. Using the divergence theorem, equation 2 may be rewritten as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0004(3)
where Vi is the volume of 3-D prism i, ∂Vi is the boundary surface area of the volume Vi, and urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0005 is an outward pointing unit normal to the volume Vi. Equation 3 is applied for each prism i in the mesh and may be rewritten as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0006(4)
for i ∈ {1,2,…,number of triangles} and for each face j of the prism with area Ai,j. The remaining term, urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0007, is approximated by the directional derivative in the direction urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0008. Let xi be the center of the prism i and xi+ni,j be the center of neighbor j of face i in the direction urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0009. Then equation 4 may be written as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0010(5)
Details are in the caption following the image
(a) Diagram of 3-D prism with face labels. Three outward-pointing face normals ( urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0011) are shown as examples. Face normal 4 is from the top prism face. A surrounding mesh in shown in gray. (b) Diagram of a flattened prism, showing the face labelling. Same orientation as in (a). Face 4 is always assumed to be pointing up, and face 5 is always assumed to be pointing down.

Blowing snow transport is dominated by advection. For numerical stability, a first-order up-winding donor scheme is used to approximate the flux across the prism face in the advection term. For each face of each prism, if urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0012 (angle between vectors is acute), then c is approximated with ci. Alternatively, if urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0013 (angle between vectors is obtuse), then the c value (cj) of the “upwind” neighbor j is used. The inclusion of urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0014 captures the variable contribution to down wind prisms. Prisms that have no neighbor follow one of two cases: (1) If the prism is missing an upwind neighbor, then a ghost cell approach is used, and it is assumed that this upwind cell has the same characteristics as prism i; (2) if the missing prism is downwind, then mass can be removed from the domain. Further, if the prism is in the top layer, then mass can be removed vertically (i.e., c(xi+ni,j) = 0).

2.2.3 Suspension Flux

The particle terminal fall velocity, ω (m·s−1), was calculated using Carrier's drag law following Pomeroy and Male (1986)
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0015(6)
where the mean particle size, r (m), is given as urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0016 (Pomeroy & Male, 1992) based on measurements by Schmidt (1986) in Wyoming and Budd (1966) in Antarctica and confirmed by field observations in Saskatchewan. The boundary condition for the bottom of the suspension layer is given as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0017(7)
where csalt is the blowing snow mass concentration (kg·m−3) of the saltation layer. The divergence of suspended flow results in deposition or erosion of snow from the surface without impacting the saltation layer horizontal fluxes.
The eddy diffusion coefficient, Kz (m2·s−1), for snow particles is assumed proportional to that for momentum (Pomeroy & Male, 1992). Following Rouault et al. (1991) and as used in Déry and Yau (1999) and Michlmayr et al. (2008), this is given as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0018(8)
where u* is the friction velocity (m·s−1) and l(z) is the mixing length given as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0019(9)
and κ is the von Kármán constant (0.4), z is the height above the surface (m), z0 is the roughness length (m), and lmax is a constant generally set to 40 m (Déry & Yau, 1999). The constant of proportionality β has received much debate (Xiao et al., 2000) and generally is taken to range from 0.5 to 1. Herein, β is taken as unity following model intercomparisons (e.g., Xiao et al., 2000) and the field observations of Pomeroy and Male (1992).

2.2.4 Sublimation Flux

In equation 5, the term S represents a height dependent sublimation sink, Sz (kg·s−1), given as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0020(10)
where urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0021 is a sublimation coefficient (s−1) derived by Pomeroy et al. (1993) and cz is the blowing snow mass concentration in the prism at height z. For efficiency, sublimation is calculated for a single (assumed) ice sphere having mean mass urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0022 (kg) at height z
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0023(11)
The formulation for the rate of change of particle mass used herein is the linearized formulation given by (Schmidt, 1972, 1991) based on the investigations of Thorpe and Mason (1966). This formulation was chosen over the newer formulation by Pomeroy and Li (2000) so as to avoid the more computationally expensive iterative solution. Thus,
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0024(12)
where z is the height within the suspension layer, r(z) (m) is the radius of the particle of mean mass at height z, D is the diffusivity of water vapor in the air (m2·s−1), Ls is the latent heat of sublimation (J·kg−1), Qr is the radiative energy absorbed by the particle, σ is the under-saturation of water vapor with respect to ice, Nu is the Nusselt number, R is the universal gas constant (8313 J·mol−1·K−1), T is the ambient air temperature (K), ρs is the saturation density of water vapor at T, Sh is the Sherwood number, and M is the molecular weight of water (18.01 kg·kmol−1). The vertical profile of under-saturation vapor density is given as a function of that specified at a reference as (Pomeroy & Li, 2000):
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0025(13)
where RH (−) is relative humidity with respect to ice. This formulation increases humidity near the surface as would be expected for the water vapor derived from blowing snow sublimation near the surface. However, the sublimation formulation does not include an explicit thermodynamic feedback that reduces under saturation as suggested by Zwaaftink et al. (2013). Pomeroy and Li (2000) and Musselman et al. (2015) have shown that humidity often decreases during blowing snow due to entrainment of warmer, drier air during the strong, very turbulent winds that are characteristic of these events. The thermal conductivity of air, λt (J·m−1·s−1·K−1), is given by (List, 1971)
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0026(14)
where T is air temperature (K). The latent heat of sublimation is Ls = 2.83 × 106 J·kg−1 (Foken, 2018), and the water vapor diffusivity in air is given by (Thorpe & Mason, 1966)
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0027(15)
where T is air temperature (K).

2.2.5 Saltation Flux

The mean mass concentration of snow particles in the saltation layer, csalt (kg·m−3), is given by (Pomeroy & Gray, 1990; Pomeroy & Male, 1992)
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0028(16)
where ρair is the atmospheric density (kg·m−3) and urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0029 (m·s−1) is the friction velocity at the cessation of transport (Li & Pomeroy, 1997b)
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0030(17)
where T is the air temperature (°C) at 2 m height. The shear stress partitioning in equation 16 is via the inclusion of urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0031, the friction velocity of non-erodible elements (e.g., vegetation). This can be estimated using Raupach et al. (1993):
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0032(18)
where β is the ratio of the roughness element to surface drag, λ is the roughness element density (−), and m relates the area averaged surface shear stress to the largest shear stress acting at any point on the surface (Raupach et al., 1993). The roughness element density is given by
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0033(19)
where hv (m) is vegetation height and sdepth (m) is the snow depth (Pomeroy & Li, 2000). Pomeroy et al. (1993) specified
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0034(20)
where N is the vegetation number density (number·m−2) and dv is the vegetation stalk diameter (m). For a subarctic mountain tundra catchment, MacDonald et al. (2009) found m = 0.16 and β = 202, N = 1, and dv = 0.8, and for Arctic tundra, Pomeroy and Li (2000) found N = 30 and dv = 0.8. Due to the deviation of wind profiles from lognormal, as well as the inclusion of exposed vegetation, the surface roughness z0 (m) is given as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0035(21)
where c2 = 1.6, c3 = 0.07519, and c4 = 0.5 (Pomeroy & Li, 2000).
Because equation 16 is a steady-state formulation, applying it where flow is developing is potentially problematic. The impact of fetch was considered by Pomeroy and Male (1986) who developed a hyperbolic fit to observed horizontal profiles of snow transport near the ground in Japan (Takeuchi, 1980). This modified saltation concentration, cf (kg·m−3), is given as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0036(22)
where L is the fetch (m) and f is the equilibrium distance (300 m). This is then reflected in the suspension layer (Pomeroy, 1991). Upwind fetch was calculated as described in Lapen and Martz (1993) with the fetchr algorithm. This was implemented directly on the unstructured mesh.

2.2.6 Total Transport

The total suspension flux, Qsusp (kg·m−1·s−1), is calculated via the integration over all prism layers
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0037(23)
where zb is the height of the boundary layer (m) and hs is the saltation layer height (m) given as (Pomeroy & Male, 1992)
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0038(24)
The saltation flux, Qsalt (kg·m−1·s−1), is given as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0039(25)
where urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0040 is the wind velocity in the saltation layer (Pomeroy & Gray, 1990). The saltation velocity proportionality constant, c, was found equal to 2.8 in experimental observations (Pomeroy & Gray, 1990).

2.2.7 Erosion and Deposition

To compute erosion and deposition rates, the rate of change of the snowpack mass, urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0041 (kg·m−2·s−1), is given as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0042(26)
where urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0043 is a unit vector in the direction of urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0044; this transforms the scalars quantities Qsusp and Qsalt into vectors in the direction of the wind.
Multidimensional transport equations may have spurious oscillations in the solution (Kuzmin, 2010). The urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0045 term is a Laplacian smoothing (Kuzmin, 2010) term, without which oscillations between erosion and deposition may appear in the urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0046 values. The coefficient ε is given as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0047(27)
where α (m) is the largest distance over which oscillations are allowed. The value of α should be a few times the average triangle length scale. Equation 26 is applied for the 2-D case only and is discretized using the FVM as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0048(28)
where urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0049 is the directional derivative as described earlier. The flux at the triangle edge, Qt, is approximated as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0050(29)
where (Qsusp+Qsalt)i,j is the flux in direction xi+ni,j.

Equations 5 and 28 were solved separately as two sparse systems of linear equations using the ViennaCL library (Rupp et al., 2016). ViennaCL can exploit shared memory parallelism or hardware accelerators (e.g., GPU). The parallel ChowPatel ILU pre-conditioner was used (three sweeps and two Jacobi iterations) with the GMRES iterative solver (10−8 tolerance, maximum of 500 iterations, maximum Krylov space dimension of 30). The Forward Euler method is taken for the time integration in equation 26.

3 Methodology

3.1 Study Site

3.1.1 Granger Basin

Wolf Creek Research Basin (WCRB) is a cold, snow-dominated basin located in the subarctic mountains of southern Yukon, Canada, 15 km from Whitehorse (Figure 3) (Janowicz, 1999). It is on the northern edge of the Coast Mountains in a zone of discontinuous permafrost (Pomeroy, Hedstrom, et al., 1998). Janowicz (1999) noted that the regional climate is cold continental and that WCRB has three main landcovers: 22% boreal forest consisting of spruce, pine, and aspen woodlands, 58% subalpine taiga consisting of shrub tundra and sparse spruce, and 20% alpine tundra consisting of short grasses and sparse shrubs at lower elevations and exposed mineral soils with sparse vegetation at the highest elevation. The mean annual temperature of WCRB is approximately −3 °C with a mean annual precipitation between 300 and 400 mm with 40% falling as snow (Janowicz, 1999). The upper elevations of WCRB have substantial and frequent blowing snow redistribution events (Pomeroy, Hedstrom, et al., 1998; MacDonald et al., 2009).

Details are in the caption following the image
The Wolf Creek research basin (top color figure) is located southwest of the city of Whitehorse, YK, Canada (top inset). The meteorological stations used for this study are shown as black crosses. The extent subsection of the Granger Creek sub-basin is shown in the red square and enlarged in the bottom color figure. The snow transects (A–F) used are shown as dark lines that cross the valley. Nearby stations to the Granger Basin subset are shown. Elevation is in meters.

Granger Basin (GB) is a small, 8 km2 (McCartney et al., 2006) sub-basin of WCRB. A subset of GB, about 2 × 2 km in area and shown in red outline in Figure 3 was used in this study. Elevation contour lines are shown for 10 and 25 m intervals. The GB landcover is predominantly exposed mineral soils, grasses, and lichens at high elevations, short birch shrubs (0.3 to 1.5 m) and grasses at intermediate elevations, and tall willow shrubs (1.5 m) in wet valley bottoms (MacDonald et al., 2009; Ménard, Essery, & Pomeroy, 2014). Tall and short shrubs dominate the north face of the valley with tall shrubs in the valley bottom. Short shrubs and grasses are present on the south facing slope, and short grasses and sparse short shrubs on plateaux to the upwind. The subset area landcover is described in detail by Ménard, Essery, and Pomeroy (2014), who previously modelled blowing snow and shrub interactions on snow accumulation and melt.

3.2 Observations

3.2.1 Meteorological

The meteorological stations used for this study are shown in Figure 3, and their operation and instruments are described in Table 1. The stations were controlled by Campbell Scientific 23X data loggers and powered by storage batteries charged by solar panels with monthly site visits over the winter and frequent visits during spring melt. Meteorological observations were recorded every 10 s and were averaged over 30-min periods for all sites. Precipitation data were collected by a stand-pipe gauge and MSC (Meteorological Service of Canada) Nipher-shielded snowfall gauges. These data were interpolated, cleaned, and corrected on a daily basis by Rasouli et al. (2014), with the wind under-catch corrections of Nipher-shielded gauges made to snowfall measurements following Goodison et al. (1998). Precipitation phase was determined via the psychrometric method of Harder and Pomeroy (2013).

Table 1. List of Sites in the Wolf Creek Basin That Were Used
Name Code Northing Easting Elevation (m) Years
Valley GB2 6712350.845 489795.117 1,411 2000–2008
South slope GB3 6712467.725 489845.377 1,451 2000–2008
North slope GB4 6712169.026 489696.727 1,490 2000–2008
Granger Basin 6 GB6 6712347.671 489069.620 1,455 2003
Buckbrush BB 6709522.453 489176.961 1,305 2000–2008
Alpine Alp 6714603.353 489877.381 1,559 2000–2008
Forest For 6717790.285 502583.402 744 2000–2008
  • Note. The years in operation are listed in the last column. Coordinates are given in UTM Zone 8 m.

Missing data can occur due to station power failures during the cold, dark subarctic winters. Shown in Figure 4 is the percentage of missing data for wind speed and direction as a percentage of all time steps in the period between October 1 and May 1 of each winter season. Some of these data gaps spanned all stations and resulted in periods that required the use of the nearby Whitehorse Airport data, for example, in 2007 (Ménard, Essery, & Pomeroy, 2014). The 30-min meteorological data had missing time steps infilled from surrounding stations when possible. To do so, a multivariable linear regression between the missing station and other stations with a measurement at that time step was used to impute the measurement at the missing time step. This procedure was only used for winter (October through April) to ensure that regressions were not biased by the summer climate. The data (not shown) for GB agreed with the results of MacDonald et al. (2009) for WCRB that predominant blowing snow event winds were from the S-SW and W-NW.

Details are in the caption following the image
The percentage of total time steps between October 1 and May 1 that are missing wind data—magnitude or direction. Years with 100% missing data are years the station was not in service.

3.2.2 Snow Transects

Intensive field campaigns sampled snow depth and density along north-south valley transects (named A-F; 400 m long) in GB (Figure 3) for the water years 2000, 2001, 2002, 2003, and 2007 (Janowicz et al., 2002; Pomeroy et al., 2003, 2004; McCartney et al., 2006; Dornes, Pomeroy, Pietroniro, & Verseghy, 2008; MacDonald et al., 2009; Ménard, Essery, & Pomeroy, 2014). These transects traverse a large drift (“Granger drift”) that forms on the north-facing slope. Depth was sampled every meter, and density was sampled every 5 to 10 m (depending on year) using Mt. Rose (for deep snow) and ESC-30 (for shallow snow) samplers. End of winter peak SWE sampling was done between April 14 and April 24, depending on the year. Because the variability in snow depth tends to be greater than that of density (Dickinson & Whiteley, 1972; Elder et al., 1998; Pomeroy & Gray, 1995), representative density measurements were used to estimate transect SWE from a greater number of snow depth observations. The covariance between depth and density was accounted for following Pomeroy and Gray (1995).

Using point data to validate a spatial model is difficult due to a conflict of scales (the incommensurability problem, Beven, 1989). Snow survey transects aim to cover a representative area and provide an areal average that should be approximately at the same spatial scale as the model. However, when individual transect points are compared with a numerical model result, it results in comparing point-scale heterogeneity to an areally averaged model element. This discrepancy is exacerbated with an unstructured mesh because of the variable size, shape, and area of elements. Transect sampling uncertainty was estimated following the bootstrapping method proposed by Harder et al. (2018) based on Steppuhn and Dyck (1974). The 95% confidence interval (CI) for SWE is given as
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0051(30)
where ρsnow is the snow density and hs is snow depth. This method used 10,000 iterations to estimate the 95% CI. The transect start (0-suffix) and end (1-suffix) locations are given in Table 2.
Table 2. Snow Survey Transect Locations
Transect x0 x1 y0 y1
A 489770 489972 6.71214e+06 6.71246e+06
B 489750 489952 6.71216e+06 6.71248e+06
C 489730 489932 6.71218e+06 6.7125e+06
D 489710 489912 6.7122e+06 6.71252e+06
E 489690 489892 6.71222e+06 6.71254e+06
F 489681 489900 6.71218e+06 6.71253e+06
  • Note. Coordinates are given in UTM zone 8 N (WGS84) meters.

3.3 Model

3.3.1 Mesh Generation

A LiDAR derived bare earth DEM and vegetation height at 1 m2 resolution (Chasmer et al., 2008) was used as input for an unstructured, variable resolution triangular mesh, shown in Figure 5. The unstructured mesh was generated using the mesher code (Marsh et al., 2018), shown in Figure 6. The triangle edges are shown as gray lines on top of the DEM. Triangles were created with a maximum triangle area of 300 m2, a minimum area of 10 m2, a vertical error of less than 0.75 m as measured by the root mean square error (RMSE), and a RMSE of no more than 0.25 m for vegetation height. In order to ensure that the variable resolution mesh was correctly producing drift locations and to quantify the error associated with this approximation method, a fixed-resolution triangular mesh was created. This mesh had a constant triangle area of 3 m2 (not shown). The variable resolution mesh has 37,645 triangles, and the fixed-resolution mesh has 99,936 triangles. The vegetation height and wind speed rasters were applied to the triangular mesh by averaging the raster cells that correspond to each triangle and assigning this average to the triangle. No other constraints were used in the generation of the mesh.

Details are in the caption following the image
LiDAR derived elevation (left) and vegetation height (right) for the Granger Basin subset. Transects are shown as lines crossing the valley, A–F going up valley.
Details are in the caption following the image
Unstructured mesh for the Granger Basin subset shown as light-gray triangles, overlain on the elevation raster. Triangles were created with a maximum triangle area of 300 m2, a minimum area of 10 m2, a vertical error of less than 0.75 m as measured by the root mean squared error (RMSE), and an RMSE of 0.25 m to the vegetation height.

3.3.2 Snowpack Model

A physically based snowpack energetics and mass balance model, Snobal (Marks et al., 1999), was used for snow cover simulation. It approximates the snowpack using two layers where the surface fixed-thickness active layer (taken as 0.1 m) is used to estimate surface temperature for outgoing longwave radiation and atmosphere-snow temperature gradients for the turbulent heat flux. Snobal features coupled energy and mass balance, internal energy, and liquid water storage calculations. Turbulent fluxes are explicitly calculated using a bulk transfer approach that includes a Monin-Obukhov stability correction based on Brutsaert (1982; Marks et al., 1992, 2008). The ground heat flux assumes heat flow to a single soil layer of known temperature and thermal conductivity. The model does not consider sub-canopy energetics, snow cover depletion, or the horizontal advection of energy as was done in the model employed in the same region by Ménard, Essery, and Pomeroy (2014). Snow-surface shortwave radiation albedo was estimated an exponential, time-dependent decay with an asymptotic minimum parameterization Verseghy (1991), detailed in Essery and Etchevers (2004), and used in this area by Ménard, Essery, Pomeroy, Marsh, et al. (2014). Snobal was driven using observed or interpolated snowfall, rainfall, incoming solar radiation, incoming longwave radiation, wind speed, air temperature, ground temperature, and relative humidity at a 30-min time interval. A variant of Snobal has been applied successfully in arctic and subarctic regions previously within the Cold Regions Hydrological Model (CRHM) platform (e.g., Pomeroy et al., 2007; Dornes, Pomeroy, Pietroniro, Carey¸et al., 2008; MacDonald et al., 2009; Rasouli et al., 2014; López-Moreno et al., 2016; Krogh et al., 2017).

3.3.3 Meteorological Distribution

The infilled meteorological data were spatially interpolated to the triangle centers using the thin-plate spline with tension. Topographic gradients for air temperature, precipitation, and relative humidity used linear lapse rate adjustments; lapse rates were calculated from the surrounding stations. Incoming shortwave solar radiation was corrected for slope and aspect. Local terrain shadowing and the impact on shortwave radiation were calculated using the algorithm of Marsh et al. (2012). Longwave radiation for each triangle was estimated from measured transmittance and interpolated air temperature and relative humidity following Sicart et al. (2006) and the sky-view factor calculated using the algorithm of Dozier and Frew (1990).

3.3.4 Wind Fields

Due to the non-suitability of terrain-curvature wind models (e.g., Liston & Elder, 2006) for blowing snow (Musselman et al., 2015), the use of a more detailed approach is required. Here, the windflow over the terrain was modelled using WindNinja (Forthofer, Butler, McHugh, et al., 2014; Forthofer, Butler, & Wagenbrenner, 2014). WindNinja is a mass- and momentum-conserving diagnostic wind model that simulates mechanical effects of terrain on windflow (Wagenbrenner et al., 2016). It has been applied to downscale numerical weather prediction models over complex terrain (Wagenbrenner et al., 2016) and is able to produce wind field maps at a variety of scales and spatial extents. In addition, WindNinja is under active development and is a modern code that allows for future extensibility. WindNinja was run as a preprocessing step on the 1 m × 1 m LiDAR derived bare-earth DEM (Chasmer et al., 2008; Hopkinson & Chasmer, 2009) using a spatially constant, bare-earth roughness length (z0 = 0.01 m). This approach neglects the feedbacks associated with shrub bending, burying, and erection over the course of winter and spring (Ménard, Essery, & Pomeroy, 2014) and filling up the topography with snow.

Using the WindNinja model, normalized wind speeds for zonal u and meridional v components as well as wind magnitude ( urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0052) were calculated for eight wind directions (N, N-E, E, … , N-W). These were normalized against the domain mean value. The normalized wind speed maps, for each direction, for the w component are shown in Figure 7. These normalized wind speed rasters were applied to the triangles in the unstructured mesh using the mesher code (Marsh et al., 2018). Thus, for each of the eight directions, each triangle has a normalized wind speed factor for zonal u, meridional v, and w components.

Details are in the caption following the image
Normalized wind speed due to terrain influences. The speedup was calculated for eight directions and used as a lookup table in the model.

For each time step of the blowing snow model, the observed wind speed and wind direction from each meteorological station were converted to zonal u and meridional v components. The observed zonal wind components were then spatially interpolated to the triangle centers using the thin plate spline with tension interpolant (Chang, 2008); this ensures no spurious overshoots in the interpolated values. For each triangle, the wind direction was reconstructed from the interpolated zonal u and bemeridional v components, and the appropriate per-triangle normalized wind speed map was selected. The interpolated zonal u and meridional v components were modified via the normalized wind speed map, thus incorporating terrain influences. Finally, the newly computed wind components were converted to a wind speed and direction for use with the blowing snow model. Wind speeds were then modified to include vegetation interactions using the vegetation heights from the LiDAR data set (via equation 21).

3.3.5 Other Details

The blowing snow model was run for the October 1 water years 2000, 2001, 2002, 2003, and 2007 corresponding to years where suitable transects of SWE and depth were available. The simulation period began on October 1 and was run until May 1. The model time step was 30 min. The blowing snow model discretized a 5 m boundary layer with 10 numerical discretization layers; this depth is representative of fully developed flow in this region (Pomeroy & Li, 2000). Model results were compared to observations using the root mean square error (RMSE), mean bias (MB), and coefficient of variation (CV). All simulated end-of-winter snow covers were rasterized to a 1 m × 1 m raster for computation of error metrics.

3.4 Wind Field Uncertainty

There are uncertainties in the distributed wind field stemming from inaccuracies in the wind fields, vegetation interactions, and in situ observational errors (e.g., Raleigh et al., 2015). These uncertainties are likely dominated by wind speed uncertainty because the topography is relatively simple (for a mountain basin) and there is an observed predominant wind direction throughout the winter (MacDonald et al., 2009). To diagnose the impact of these wind speed uncertainties on the spatial pattern of blowing snow, a linear perturbation approach was used. This quantified the uncertainty associated with the wind fields using 50 realizations of a perturbed wind field. The perturbation magnitude followed the range proposed in Raleigh et al. (2015): A normally distributed perturbation with scaling size 3.0 m·s−1 was applied as a temporally constant bias to the observed wind speed. To diagnose the sensitivity to wind direction, a perturbation of scaling size 45.0° was applied to the wind direction following the same procedure as the perturbation to wind speed. These perturbations in wind speed and direction were added to the observed wind velocities at each meteorological station before interpolation.

Each of these realizations resulted in the calculation of a new, spatially distributed snow cover as a result of the perturbed wind field. The end of winter SWE, from each realization, was rasterized to a 1 m × 1 m raster. For each grid cell i of the raster, the cell CV was computed via
urn:x-wiley:00431397:media:wrcr24400:wrcr24400-math-0053(32)

where σi and μi are standard deviation and mean, respectively, across all ensemble runs for cell i. The larger the variability between the ensemble members at a cell, the larger the CV. This is therefore a quantification of the spatial sensitivity of the blowing snow model to the wind field.

4 Results

4.1 Distributed SWE

Shown in Figure 8 is the simulated spatial distribution of SWE (mm) for the five water years for the GB subset area. Elevation contours are shown in light gray every 10 m. The output from the variable resolution unstructured mesh is shown. The variable resolution mesh resulted in a reduction in computational elements of 62% versus the fixed resolution mesh and a 44% reduction in total runtime.

Details are in the caption following the image
Spatial distribution of SWE for the end of winter snow cover (April 14) for the five water years for the variable resolution unstructured mesh. The large Granger drift is located in approximately the middle of the domain.

Substantial spatial and inter-annual SWE heterogeneity was found. As expected, drifts were the greatest deposition areas, with substantial upwind erosion resulting in shallow upwind snow covers. The large Granger drift is visible along the ridge-line in the lower two thirds of each panel, as are other large drift features. In years 2000, 2001, and 2003, the Granger drift was the most pronounced. This variability in drift was due to fewer wind events from the S-W direction in other years, a requirement for the formation of the drift. The inter-annual variability of this drift is an important feature of the hydrology of this region (Dornes, Pomeroy, Pietroniro, & Verseghy 2008) and has been noted for the Canadian Prairies as well (Pomeroy & Gray, 1995). This inter-annual variability in snow redistribution adds considerable uncertainty to fixed empirical sub-grid estimates of SWE variability that are employed in many land surface hydrological models.

Both the variable resolution mesh and the constant resolution mesh provided similar heterogeneity in simulated SWE. CV values, for all years for both meshes, are shown in Table 3. The constant resolution mesh (not shown) had increased erosion and drift due to a more detailed vegetation height representation. Other differences were due to small-scale topographic features being smoothed out in the variable resolution mesh. The left-most drift in the domain, located at x = 48900,y = 6712250, was more sharply represented in the constant resolution mesh and thus accumulated more snow. The mean CV across the entire domain for all years for the variable resolution mesh was 0.46, slightly lower than the 0.56 for the constant resolution mesh. The increased CV for the constant resolution mesh is as a result of the increased topographic variability and the subsequent filling of small topographic depressions.

Table 3. Coefficient of Variation (CV) for the Entire Simulation Domain for the Variable Resolution Mesh and the Constant Resolution Mesh for the End of Winter (April 14) Snow Cover for Each Water Year
Year CV var. mesh CV const. mesh
2000 0.48 0.54
2001 0.52 0.68
2002 0.36 0.45
2003 0.6 0.68
2007 0.35 0.45

4.2 Transects

Simulated SWE (mm) across the six transects for the five water years is shown in Figure 9 in blue. Blowing snow calculations were disabled for comparison, and this result is shown in green. Shown in red are the results from the constant resolution mesh. Observations are shown as dots with the observation uncertainty bounds shown in gray.

Details are in the caption following the image
Simulated snow water equivalent for end of winter (April 14) across the six transects for the five water years is shown in blue. Observations are shown as dots, and an uncertainty bound is shown in gray. No blowing snow is shown in green and a constant resolution mesh in red. Only transects with observations are shown.

Substantial inter-annual variability was found in observed SWE and CV for all transects. This is most notable for the Granger drift, located on south (far left) side of all transects. This drift varied in SWE from under 100 mm in some years (2002) to over 600 mm (2001). The CV along the transects varied from 0.24 to 0.54. In 2001, there is a notable increase in SWE along the north face (right hand side) of the transects. The high uncertainty for the observed SWE values in transect F for 2001 was due to the limited number of density samples available. The water years 2002 and 2007 had the smallest Granger drift due to fewer S-SW wind events.

The observed inter-annual and spatial variability in SWE was generally well represented by the blowing snow model using the variable resolution mesh. Generally, the Granger drift (far left side of transect) was well represented in both magnitude of SWE and shape. Transect A in 2000 was the most poorly simulated transect due to the magnitude and location of the drift being missed. In other years, the drift tended to be correctly simulated or slightly over estimated (2007). The simulation results for 2007 may have been negatively impacted due to the substantial use of airport data required to infill the missing data. The low drifts of transect A in 2002 were well simulated; however, the short but deep drift on transect F was underestimated. Across all transects and years, the most consistent simulation error was an underrepresentation of the mid-valley SWE, where tall vegetation dominates the landscape along the creek. As this area undergoes minimal snow redistribution, this may indicate an underestimation of seasonal snowfall in the observations.

The constant resolution mesh tends to have similar drift formation as the variable resolution mesh. However, in some years (2001 and 2003), the drift is overestimated by approximately 150 mm. This increase in transport is likely as a result of increased transport from the source plateaux regions, where more small triangles in the constant resolution mesh have greater opportunities for transport initiation. This increase in transport into the drift also manifests as increased transport into the valley. As a result, the mid-valley SWE is better represented by the constant resolution mesh.

Summarized in Table 4 are the RMSE and CV values for SWE versus the observations for each entire transect. For all years and all transects, no blowing snow produced a homogeneous snow cover with a small variability due to spatial differences in over-winter snowpack energetics (CV <0.1). These simulations showed none of the observed inter-annual variability. Landscape-averaged values of RMSE and CV across all years and transects are shown in Table 5. The mean RMSE and MB of SWE were 86 and −20 mm, respectively. On average along the transects, the model SWE CV closely matched observed CV, 0.31 versus 0.41, respectively. The constant resolution mesh had an RMSE of 120 mm, an MB of −1.3 mm, and a CV of 0.37. Without blowing snow, the RMSE and MB were 98 and −49 mm, respectively, with a CV of 0.04. Overall, the use of the variable resolution mesh tends to produce better drift prediction than the constant resolution mesh, but it under predicts mid-valley transport.

Table 4. The RMSE Values for Simulated SWE Versus Observations Are Given Along With the CV Values for Simulated SWE Using the Variable Resolution Mesh and Observed SWE Along the Transects
Year Transect RMSE (mm) MB (mm) CV (−) Obs CV (−)
2000 A 120 −86 0.27 0.46
2000 B 61 −43 0.35 0.24
2000 C 88 −63 0.41 0.38
2000 D 100 −85 0.49 0.31
2000 E 81 −36 0.43 0.38
2000 F 42 −8.1 0.47 0.55
2001 A 95 59 0.37 0.54
2001 B 96 −38 0.41 0.48
2001 C 85 43 0.41 0.4
2001 D 120 −78 0.44 0.32
2001 E 93 −33 0.43 0.4
2001 F 170 −130 0.43 0.38
2002 A 63 −12 0.058 0.55
2002 F 100 −64 0.26 0.5
2003 A 56 19 0.3 0.39
2003 B 61 27 0.43 0.39
2003 C 72 −4.2 0.42 0.3
2003 F 75 2.9 0.47 0.36
2007 A 91 82 0.27 0.46
2007 C 62 6.4 0.26 0.32
2007 F 77 17 0.28 0.5
Table 5. The Values From Table 4 Are Summarized With the Mean RMSE, the Simulated CV, and the Observed CV Values Across All Years and All Transects Are Given Using the Variable Resolution and Constant Resolution Mesh
Simulation type RMSE (mm) MB (mm) CV (−) Obs CV (−)
Const. res. PBSM3D 120 −1.3 0.37 0.41
Var. res. no blowing snow 110 −45 0.036 0.41
Var. res. PBSM3D 99 −20 0.31 0.41

4.3 Sublimation

The model was run with and without blowing snow sublimation, and this difference (sublimation-no sublimation) in the end of winter SWE is shown in Figure 10. The largest differences were found in the sink locations where substantially more snow was deposited when there was no sublimation. With no blowing snow sublimation, large areas of up-wind blowing snow source (e.g., the plateaux) transported substantially more snow into the drift sinks. With sublimation, this mass was lost to the atmosphere resulting in lower drift sizes. As a result, better simulation results are found when including the blowing snow sublimation process, and mean bias and RMSE were reduced by including it. Shown in Table 6 is the domain-averaged sublimation loss as a percentage of total winter time precipitation. The inter-annual variability in total precipitation and blowing snow events resulted in a 6% to 14% loss to sublimation. This demonstrates the importance of including sublimation processes for not only simulating total mass but also for including the feedbacks that lead to spatial heterogeneity in the snow covers.

Details are in the caption following the image
Difference in end of winter SWE (taken as April 14 for all years) with and without blowing snow sublimation.
Table 6. Simulation Domain Mean Simulated Sublimation Loss as a Percentage of Total Winter Precipitation
Year % precip as sublimation
2000 12
2001 14
2002 6
2003 11
2007 10

4.4 Wind Uncertainty

Due to the substantial heterogeneity in wind field and to feedbacks between snow cover, SWE, and blowing snow, there was heterogeneity in the sensitivity of the blowing snow model. Shown in Figures 11 and 12 is the coefficient of variation of SWE, for each grid cell, across the entire 50-member ensemble for the April 15 snow cover of each year for wind speed and wind direction perturbations, respectively. The 2007 year is not shown for clarity due to low sensitivity. Low CV values were associated with limited variability in the ensemble (i.e., the wind field perturbation did not result in much change), and high CV values were associated with large variability in the ensemble, that is, the wind field perturbation dramatically changed the SWE. Therefore, this figure is interpreted as a sensitivity index showing regions of limited and large variability as a result of the wind field perturbation.

Details are in the caption following the image
A 50-member ensemble was run with perturbations to the wind magnitude. The unstructured mesh representing end of winter SWE (April 14), for each member, was rasterized to a 1 m × 1 m raster. The coefficient of variation was computed for the SWE across the ensemble at each grid cell. The larger the variability at a cell, the larger the CV. Therefore, this can be interpreted as a map showing the spatial sensitivity of the model to wind speed.
Details are in the caption following the image
A 50-member ensemble was run with perturbations to the wind direction. The unstructured mesh representing end of winter SWE (April 14), for each member, was rasterized to a 1 m × 1 m raster. The coefficient of variation was computed for the SWE across the ensemble at each grid cell. The larger the variability at a cell, the larger the CV. Therefore, this can be interpreted as a map showing the spatial sensitivity of the model to wind direction.

There was substantial variability in the sensitivity between years, in both degree of sensitivity and spatial distribution of sensitivity to wind speed. The years that produce the large Granger drift (2000, 2001, and 2003) had the greatest areas of increased sensitivity. In all years, the areas with high CV were spatially correlated with drift locations and the immediate upwind scour zones. The further upwind sources for the drifts were generally the least sensitive. This suggests that the drifts are integrating all the blowing snow sensitivity from the upwind areas. That is, even though limited sensitivity was found in the majority of the up-wind sources for the drift, for example, plateaux, all the small sensitivities are compounded and expressed as large uncertainty in the drifts.

The wind direction sensitivity was less spatially variable than wind speed. The areas with greatest sensitivity are the drift (deposition) locations—the upper plateaux tend to be source areas of snow regardless. In the 2000, 2001, and 2003 years, the Granger Creek valley wall had substantial variability in drift locations. In 2002, the drift locations at the north end of the domain were the greatest regions of sensitivity. Overall, the sensitivity was confined to the drift locations changing location due to differences in the gradient in the wind field, and the upper plateaux remained consistent mass sources.

5 Discussion

5.1 Synthesis

As increased attention is focused on hyper-resolution (<1 km) atmospheric modeling (Wood et al., 2011), ensuring that hydrological models adopt snowdrift-resolving scales (Pomeroy & Bernhardt, 2017) is now possible and of substantial importance for cold-regions simulations. Due to the importance of representing landscape heterogeneity on cold regions snow-covered area and runoff calculations (e.g., Dornes, Pomeroy, Pietroniro, Carey, et al., 2008; Dornes, Pomeroy, Pietroniro, & Verseghy, 2008), explicit inclusion of blowing snow and related sublimation losses should be considered a priority. Common assumptions that drifts are the same relative size and in the same location every year and that the CV is the same every year are not supported by field data observations. Simulating this inter-annual variability is an important capability and is critical to improving snow simulations in the alpine.

The model presented herein shows a way toward inclusion of this process in large-extent hydrological models. As described in Marsh et al. (2018), an unstructured meshing algorithm can ensure inclusion of substantial topographic and vegetation heterogeneity while significantly reducing the total number of computational elements. A 62% reduction in computational elements was found when using the variable resolution mesh versus the fixed resolution mesh. Although this did not result in a directly proportional reduction in computational time, there were still substantial computation savings of 44%. This decrease in simulation time is important when applying distributed models to large extents and is a substantial decrease in required computational resources versus fixed-resolution models.

A coarser discretization can be expected to average out variability, and this was observed in the missing small scale topographic variability in the upper plateaux. However, all the major topographic features were represented by the variable resolution mesh. The similarities in SWE heterogeneity (measured by the CV) between the fixed and variable resolution meshes suggest that the variable resolution mesh is capturing essentially as much heterogeneity as the fixed resolution mesh. The larger triangle elements in the variable resolution discretization were predominately in the source-area plateaux. As a result, the constant resolution mesh had more triangles in the source plateaux areas. This resulted in more triangles that could potentially transport snow downwind. Therefore, in some years, there was increased transport into the drift area versus the variable resolution mesh. This increased transport is likely as a result of uncertainties, for example, in the LiDAR derived vegetation heights, impacting threshold saltation initiation. As a result, the variable resolution mesh tends to produce more accurate results, likely as a result of reducing some of this uncertainty via aggregation of the landscape. These results suggest that the use of a spatially variable resolution mesh is appropriate for this domain.

It has been well established by many field and modeling experiments that blowing snow is a required process to consider in order to accurately simulate end of winter snow cover heterogeneity in wind-blown environments (e.g., Pomeroy et al., 1993; Pomeroy et al., 1993; Luce et al., 1998). The results presented here further substantiate this. As measured by the CV, including blowing snow redistribution processes in the model dramatically increases the spatial heterogeneity of snow mass. Indeed, the non-blowing snow simulations produced very low CV values (CV ≈0.04), well below observed values anywhere in the world (Pomeroy, Gray, et al., 1998). The simulated CV values on a per-transect basins agree with the observed CV as well as existing literature values for this domain (e.g., Pomeroy, Gray, et al., 1998). Inclusion of blowing snow sublimation resulted in a domain-averaged 6% to 14% loss of snow mass as a percentage of total snowfall. Reported sublimation losses are highly variable between studies, as summarized by Mott et al. (2018). The simulated sublimation losses herein are somewhat lower than what Pomeroy and Li (2000) reported (22%), lower than MacDonald et al. (2009) (19–81%) (same basin but different domain), and similar in the range reported by Liston and Sturm (1998) (9% to 22%) for other Arctic sites. As a result, this caused feedbacks in the threshold processes that govern blowing snow initiation. This resulted in less SWE in the upwind plateaux source regions and decreased drift sizes due to less snow being transported into these regions. Without blowing snow, there is little reason to expect that snow mass heterogeneity would be accurately captured, and this is observed in these results.

A key component to successfully modeling blowing snow is generating an appropriately accurate wind field (Mott et al., 2010; Musselman et al., 2015; Raderschall et al., 2008; Vionnet et al., 2017). Evaluating point-scale wind speeds alone is insufficient. Musselman et al. (2015) demonstrated that terrain curvature methods are insufficiently detailed to produce the emergent spatial behavior via the accumulation of nonlinear processes of blowing snow despite simulating reasonable wind speeds. The accuracy of a wind field for blowing snow simulation depends heavily on the input wind speed and direction data as well as the impacts of topography and vegetation. Input wind data may be either interpolated weather station observations, such as those found in research basins, or the output of numerical weather prediction models. Due to the harsh conditions of remote arctic research basins, obtaining time-series data without any faults or quality issues can be difficult. The results herein showed that interpolated wind fields from nearby but not in-basin observations (such as the airport data) were insufficient to capture all the blowing snow dynamics. The inclusion of these airport data may have contributed to the weaker results when included. The inclusion of vegetation interactions on the wind field is a major source of uncertainty, and properly capturing the wind flow through protruding vegetation would require even greater detail in the wind model and is an open research question. Previous work in this basin by Ménard, Essery, and Pomeroy (2014) showed modest improvements in snowpack ablation simulation by considering the dynamics of shrub bending, where (some) shrubs are bent over and buried by winter snow covers. Coupling these dynamics with a wind flow model greatly increases uncertainty and computational costs. Such an inclusion in a model requires further work to ensure such a coupling would not further increase uncertainty with no improvements. Lastly, the steady-state assumptions in the wind field are known to be incorrect for at least the small scale processes that govern blowing snow (Aksamit & Pomeroy, 2016, 2018). However, for moderate-complexity models to be applied to large extents in order to improve snow cover heterogeneity estimates for hydrological models, these uncertainties in the wind field are likely outweighed by the improved simulations of SWE heterogeneity. Future work should consider the benefit to including increased detail of vegetation-snow cover feedbacks such as shrub bending and recomputing the wind flow speedup using the snow cover surface.

Previous distributed blowing snow modeling has either focused on distributed models that have used empirical models with substantial errors in the wind field (Liston & Sturm, 1998; Musselman et al., 2015), assumptions regarding wind that limits applicability to large extents due to complex wind flows (e.g., Essery et al., 1999), or coupling with atmospheric models (e.g., Lehning et al., 2008; Vionnet et al., 2014). The look-up table method used here extends the approach of Essery et al. (1999) by incorporating per-element wind flow direction and applying those directions on a per-element basis without any mean-direction assumptions. This approach is functional with sparse or dense point-scale meteorological observations or atmospheric model outputs, is not tied to any particular numerical weather model, and is therefore suitable for a wide range of applications around the world. This method can be extended to large areas without requiring the use of a complex wind flow model run each time step. This model therefore is a pathway for the inclusion of critical cold-regions processes without significant computational overhead.

The ensemble run of various wind field perturbations is a novel way of examining the spatial sensitivity of a distributed blowing snow model. The results showed that there is substantial spatial variability in sensitivity and that this sensitivity varies on a per-year basis. Presumably, there also is mid-winter variability in this sensitivity. Considering these spatial sensitivities is important when adopting snowdrift-resolving scales in hydrological models because they are able to diagnose areas that drive large scale hydrological responses to the cryosphere. This approach allows for intercomparisons of blowing snow transport and sublimation to different wind field models, as per Musselman et al. (2015). The ensemble results showed that the wind speed perturbations had the greatest impact on the results compared to the wind direction. The snowdrifts and immediate up-wind blowing snow source areas are the most sensitive areas of the catchment and that the further upwind plateaux are less sensitive. That is, the drifts are acting to integrate all the upwind uncertainties as suggested by Tabler (1974) and shown in the Canadian Prairies by Fang and Pomeroy (2009). Perturbations to wind direction resulted in changes in drift location; however, the plateaux remained mass sources, regardless of wind direction. Thus, small changes in upwind features, for example, snow model density estimation for snow depth (and subsequently vegetation burial), are likely to have substantial downwind impacts due to wind variability. This suggests that these drift locations are likely some of the most difficult areas to simulate on the landscape.

This result has implications for hydrological simulations in areas undergoing wide-spread vegetation changes, such as the increased shrub cover of the tundra (Sturm et al., 2001). As upland tundra changes to a shrub-dominated landcover, the upwind source areas for blowing snow will hold more snow and so less will be available for downwind transport and sublimation (Essery & Pomeroy, 2004; Ménard, Essery, & Pomeroy, 2014). This would then result in less mass transported into the large drifts. The drift contribution in maintaining summer streamflow during warm dry periods will then have important implications for the region's hydrology as shown by Rasouli et al. (2018) in Wolf Creek and by Krogh and Pomeroy (2018) near Inuvik, NWT. The likely result is less drift formation, unless winter snowfall increases, as is suggested by recent climate modeling in the area (Rasouli et al., 2019). Therefore, this suggests that not only are drift locations difficult to model but they may also be increasingly sensitive areas under climate change induced landscape changes such as shrub expansion that are concomitant with temperature and precipitation increases.

5.2 Uncertainties

There are substantial uncertainties associated with using distributed models in remote cold regions:
  1. Uncertainties associated with meteorological forcing are substantial, even in well-instrumented research basins (Raleigh et al., 2015). Gauge under-catch associated with solid-phase precipitation is an ongoing challenge in cold regions. Operating a research basin in the sub-Arctic is incredibly challenging. Despite many meteorological research stations in the area, some periods had data gaps that necessitated interpolating wind velocity from the Meteorological Service of Canada Whitehorse Airport station (MacDonald et al., 2009; Ménard, Essery, & Pomeroy, 2014). In water year 2000, there was no meteorological data infilling required, while in other years, there was substantial infilling required. This may explain the weaker results in those years. It is possible that uncertainty in infilling resulted in unrealistic wind fields. Uncertainties in interpolation mean that the model cannot rely entirely on observations remote from the area for interest for prediction. This suggests that either collocated stations or high-resolution atmospheric model output is important for applying these types of models.
  2. For computational complexity reasons and due to uncertainties in how to apply horizontal advection over a variable mesh model, the models used here ignored known processes such as shrub-bending under winter snowloads and shrub-erection during spring snowmelt, radiation and snowmelt energetics under and near shrub canopies, and the advection of sensible energy from shrubs and bare ground patches to snow patches during melt (Pomeroy et al., 2006; Bewley et al., 2007; Ménard, Essery, & Pomeroy, 2014; Ménard, Essery, Pomeroy, Marsh, et al., 2014). The purpose of the modeling here was to isolate and show the impact of blowing snow processes, but a more complete snow cover model would include all of those impacts and that of latent heat advection (Harder et al., 2017; Mott et al., 2017; Schlögl et al., 2018).
  3. The wind flow model uses a bare-earth representation year-round that neglects the feedbacks associated with shrub bending, burying, and erection over the course of winter and spring, filling up the topography with snow, and does not include flow separation. However, there is no evidence of flow separation in this environment—plumes of snow have never been observed to form on the ridge tops in the modeling domain subset of the GB.
  4. A potentially important process that was not considered here is preferential deposition (Gerber et al., 2019; Mott et al., 2014; Vionnet et al., 2017; Wang & Huang, 2017). Preferential deposition results in decreased snow deposition on the windward slopes and increased deposition on the leeward slopes, strongly shaping final snow deposition patterns (Mott et al., 2018). It is an important process in regions of sharp mountain topography where snowfall occurs during strong winds. However, this process has not been studied in this region and the degree to which, if any, it impacts snow accumulation this relatively mild mountain topography is uncertain. Despite being a mountainous region, the topography in Granger Basin is less complex than those for where preferential deposition is generally reported as important. Therefore, it is unknown how important preferential deposition is for this region.
  5. Using transects of point data to validate a spatial model is difficult due to a conflict of scales. Snow survey transects aim to cover a representative area and provide an areal average that should be approximately at the same spatial scale as the model. However, for validation of a distributed model against transects like those used here, it is not a perfect match. This scale mismatch can give enormous weight to a single observation. It also means that comparison with a numerical model results in comparing point-scale, sub-grid heterogeneity to an areal averaged model element. This skewed weighting is exacerbated with an unstructured mesh because of the variable size, shape, and area of elements. The use of various remote-sensing data, such as terrestrial laser scanning (TLS) (Grünewald et al., 2010; Prokop, 2008), photogrammetry via unmanned aerial vehicles (UAV) (Harder et al., 2016), and aerial LiDAR (Hopkinson et al., 2011; Deems et al., 2013; Painter, Berisford, et al., 2016) can provide snow-on/snow-off spatial data sets that provide an estimate of snow depth. Such spatially distributed data could partially mitigate the above mentioned scale conflicts. However, in situ density observations must still be taken to quantify SWE. Unfortunately, no repeat multi-year UAV, TLS, or aerial data sets exist for this region. Due to the importance of quantifying whether the model can reproduce observed inter-annual variability in SWE, the transect observations were the best available data. Future work will utilize these spatially distributed data as they become available for this region.

6 Conclusions

A 3-D advection-diffusion blowing snow transport and sublimation model using a finite volume method discretization on a variable resolution unstructured mesh was presented. The use of an unstructured mesh provided a 62% reduction in computational elements versus a fixed-resolution mesh and a 44% decrease in computation time. Transects of a large drift formation in a subarctic valley were used to gauge the accuracy of the model. The model generally captures the drift dynamics; over all transects, an RMSE and MB of 86 and −20 mm, respectively, were found. Falsifying the model by removing the blowing snow processes increased the SWE RMSE to 98 mm for SWE and more than doubled the MB to −49 mm. Further, the removal of blowing snow resulted in a CV of approximately 0.04, well below the observed CV (0.41). Inclusion of blowing snow resulted in transect and basin-wide CVs that agreed well with observed values and previously published results (0.36 vs. 0.41). The SWE from the variable resolution mesh was similar to the more computationally expensive fixed resolution mesh, including the CV. The inclusion of this blowing snow model dramatically increased the spatial heterogeneity of SWE and highlights the critical importance of including this process in cold-regions hydrological models.

The windflow model used here extends previous work by incorporating a lookup table approach that uses per-element wind directions. The blowing snow model presented here demonstrates a computationally efficient method to include blowing snow over large areas without requiring the use of a complex wind flow model run each time step. Poorer predictions of SWE occurred in years with increased infilling from the nearby airport weather station. This suggests that this method should be applied using either collocated weather stations such as are found in research basins or high-resolution atmospheric model outputs. Future research will need to examine how predictive efficiency relies on atmospheric model resolution.

An ensemble of model runs with various perturbations to the wind field were used to establish a spatial sensitivity measure. This showed that SWE is most sensitive in the snowdrifts and the immediate up-wind source areas and that the SWE further upwind on the high plateaux are less sensitive to wind perturbations. Therefore, the snowdrifts appear to be integrating all the up-wind uncertainties in blowing snow processes, as proposed by Tabler (1974). As a result, the drift locations are the most difficult areas of the landscape to simulate as small errors across the landscape are accumulated. However, these areas are critical sources for summer runoff generation, and so their correct simulation is extremely important in alpine catchments.

The importance of blowing snow processes on shaping the seasonal snow cover and their resulting impact on spring snowmelt volume and timing are well documented throughout the cold regions literature. Despite their importance, these critical processes are often neglected in hydrological models, especially in large-extent models because of challenges from discretizing the landscape, interpolating wind fields, and associated computational costs. The new 3-D blowing snow model presented here utilizes a variable resolution discretization to dramatically reduce the total number of computational elements and produces heterogeneous snow covers without the need for calibration. This presents a promising way forward for including blowing snow processes at snowdrift-resolving scales in large-extent hydrological models.

Acknowledgments

The authors would like to acknowledge the tremendous contribution to cold regions hydrology made by J. Richard Janowicz of Whitehorse, who maintained Wolf Creek Research Basin for 25 years, graciously hosted and worked with researchers from southern Canada, and whose thesis data contributed to the gridded snow survey data used in this study. Mr. Janowicz passed away during the preparation of this manuscript. Funding received from the Canada Excellence Research Chairs and Canada Research Chairs programmes, NSERC's Discovery Grants, Alexander Graham Bell PhD Scholarships and Changing Cold Regions Network, Yukon Environment, and the Global Water Futures programme is gratefully acknowledged. The code is open source under the GPLv3 license and is available from github.com/Chrismarsh/CHM under src/modules/PBSM3D.cpp.