## 1. Introduction

In the Northern Hemisphere, a seasonal snowpack can cover over 50% of the land area with the snow surface often the interface between the atmosphere and the earth [

1]. The roughness of a snow surface is an important control on air-snow heat transfer [

2], and changes in the snow surface can have substantial effects on the energy balance at this interface. Snow is a complicated surface with rapidly evolving physical roughness characteristics due to changing atmospheric conditions, the metamorphism of snow crystals, melting and freezing processes and redistribution by wind, especially in open areas [

3]. Roughness characteristics also influence the air-surface momentum transfer on the snowpack due to wind [

4]. The changes in wind momentum can reduce the energy budget, influence the formation of roughness features, and affect the aeolian movement of snow [

4]. Heat flux modeling has typically used the aerodynamic roughness length (

z_{0}) as a static parameter, in hydrologic, snowpack, and climate models [

5,

6], with

z_{0} only varying as a function of land cover type. For example, the Community Land Model version 4.0 (CLM4;

http://www.cesm.ucar.edu/models/ccsm4.0/clm/) applies a single

z_{0} value of 2.4 × 10

^{−3} m to all snow-covered surfaces. However,

z_{0} varies both spatially [

7] and temporally [

8], which may result in variable estimates of turbulent heat fluxes not captured by most models [

9]. Wind velocity profile measurements are often used to calculate

z_{0} estimates [

6], but there are a limited number of sites that measure the wind profile over a snowpack surface, making the spatio-temporal representation of

z_{0} challenging.

Millimeter-scale variations in snow-surface roughness features have been estimated from a black plate pushed partially into the snow [

10,

11,

12], using two-dimensional photography, digital processing, and automated post-processing software [

13,

14,

15,

16]. Snow surface elevation data are now available over large areas at the resolution (±80 mm) of airborne light detection and ranging (lidar) [

17,

18,

19], terrestrial laser scanner (TLS) (resolution of ±5 mm) [

20,

21,

22,

23,

24,

25,

26], and photogrammetry [

25]. Although most lidar and photogrammetry efforts have only focused on snow depth [

26], only a few datasets have been used to evaluate snow surface roughness at the meter-scale or sub-meter scale [

27,

28]. However, few of these datasets have been applied to interpolate

z_{0} and create a digital elevation model of the snowpack surface for evaluating surface roughness [

27]. Aerodynamic roughness length (

z_{0}) has been estimated from the geometry of the snow surface [

2,

7,

29,

30,

31]. However, this method is time consuming and typically only applicable over smaller scales [

13]. Also, Fassnacht et al. [

27] have identified potential errors with the different methods of computing

z_{0} from the geometry of the surface that result in values varying over 1–3 orders of magnitude and have suggested these methods need to be evaluated for varying scales, resolutions, and environments.

This study used TLS-derived surface geometry and vertical wind profile measurements to compare concurrent z_{0} estimates for changing snow surface features of shallow snowpacks. Here, we asked the following questions: (1) How does the aerodynamic roughness length (z_{0}) vary spatially and temporally for a shallow snow environment? (2) How does z_{0} estimated from geometric measurements (z_{0G}) compare to z_{0} estimated from anemometric measurements (z_{0A}), and (3) How does z_{0} vary with snow-covered area based on the underlying terrain?

## 2. Materials and Methods

The capability of a rough surface to absorb momentum from a turbulent boundary layer can be quantified by

z_{0}, which is a measure of the vertical turbulence that occurs when a horizontal wind flows over a rough surface [

32]. In general,

z_{0} is a quantity that is computed from the Reynolds number and the roughness geometry of the surface [

29]. For rough, turbulent regimes occurring in the atmospheric boundary layer, dependence on the Reynolds number vanishes and

z_{0} is only a function of roughness geometry [

33]. Various relations have been found to relate the geometry of roughness elements with

z_{0} [

2,

29]. For example, the dependence of

z_{0} on the size, shape, density, and distribution of surface elements has been studied using wind tunnels, analytical investigations, numerical modeling, and field observations [

34,

35]. Smith [

36] provides a comprehensive review of the different approaches and models developed to analyze surface roughness and highlights that almost all models were developed for simplistic natural surfaces (i.e., regular arrays of roughness elements).The lack of a clear method for calculating

z_{0} as a function of surface roughness is due to the complexity of surfaces that exists in nature and the direction, spatial, and temporal dependence.

The most robust approach for estimating

z_{0} is from the anemometric method used to generate a logarithmic wind profile and solve for

z_{0} [

32]. The anemometric method can be used for any surface with any arrangement of roughness elements but requires a meteorological tower of at least two vertically spaced wind, temperature, and humidity measurements that can be used to approximate the respective gradients. The measurements integrate over a footprint area rather than the single-point location of the sensors based on the distance from measurement source, elevation of sensor, meteorological conditions, turbulent boundary layer, and atmospheric stability. All of these factors can potentially create turbulent fluctuations affecting the downwind measurements of the wind profile [

37,

38]. The anemometric method is also very sensitive to the wind measurement heights; Munro [

2] found that adding 0.1 m to any of the heights can alter

z_{0} by an order of magnitude. In contrast, the geometric method uses algorithms relating

z_{0} to characteristics of surface roughness elements and thus does not require tower instrumentation but only a measure of the geometry of the surface [

29].

Anemometric data are used to estimate

z_{0} from the logarithmic wind profile through an empirical relation that describes the vertical distribution of horizontal wind speeds within the lowest portion of the planetary boundary layer [

39]. The wind speed (

U_{z} in m/s) at height

z (in m) above a surface is given by:

where

U* is the friction velocity (m/s), k is the Von Kármán constant (~0.40), and

ψ is a stability term, and

L is the Monin-Obukhov stability parameter. This equation is only valid through the hypothesis of stationarity and horizontal homogeneity. Under neutral stability conditions,

z/L tends towards zero, and

ψ can be neglected.

The most common geometric method for estimating

z_{0} is simply a function of the height of the elements:

where

z_{h} is the mean height of roughness elements in meters, and

f_{0} is an empirical coefficient derived from observation [

28]. The frontal area index, which combines mean height and breadth (all in meters), and density of the roughness elements, is defined as roughness area density given by:

where

Ly is the mean breadth of the roughness elements perpendicular to the wind direction, and

$\rho el$ is the density or number (

n) of roughness elements per unit area [

40]. Lettau [

29] developed a formula for

z_{0} based on the geometry of the surface for irregular arrays of reasonably homogenous elements:

In the Lettau formula, the coefficient 0.5 represents an average drag coefficient for the roughness elements, which was determined experimentally. Other geometric methods have been developed, especially to consider more regularly-shaped and distributed roughness elements, such as buildings in an urban setting [

41,

42]. The Counihan equation provides a geometric estimate of

z_{0} as:

where

A_{f} is the total area in square meters silhouetted by the roughness elements, and

A_{d} is the total area covered by roughness elements.

A meteorological tower was erected at Colorado State University Agricultural Research, Development and Education Center (ARDEC) South (

http://aes-ardec.agsci.colostate.edu/), (40.629680, −104.99699) on a flat field that had no obstructions at least 100 m in the prevailing wind direction. The fetch was 40 m wide with the tower placed in the middle, leaving 100 unobstructed, homogenous meters upwind. Ten anemometers and five temperature and relative humidity sensors were placed vertically at different heights on the tower. The accuracy of the air temperature and relative humidity sensors (METER VP-3) was variable across a range of ±0.25–0.50 °C and ±4%, respectively (see

http://manuals.decagon.com/Manuals/14053_VP-3_Web.pdf for more information). The METER Davis Cup Anemometers have a wind direction accuracy of ±7° and a speed accuracy within ±5% (see

http://manuals.decagon.com/Manuals/). Data were collected from February 2014 through March 2015. In mid-March 2014, the flat field was plowed to create additional underlying roughness, specifically furrows and troughs were formed perpendicular to the dominant wind direction at an approximate spacing of two meters. The approximate amplitude of the troughs and furrows was 25 cm deep and 50 cm wide.

Meteorological data were recorded every five minutes based on the average of one-minute observations. Anemometric data were evaluated for 153 instances when wind speeds were faster than 4 m/s to ensure neutral stability [

8] and when the log-linear fit had an r

^{2} greater than 0.95. The height of the instruments was calculated based on the depth of snow, which did not exceed 10 cm.

This study estimated

z_{0} values from anemometric measurements and used them as a reference to evaluate concurrent geometric methods. The

z_{0A} values were calculated using Equation (1) from logarithmic anemometer wind profile data. Surface elevation was measured using a FARO Focus3D X 130 model TLS (

https://www.faro.com/products/). This lidar tool generates a point cloud scan of a given area with an error of ±2 mm and a resolution of approximately 7.5 mm. The TLS was set up in 2–3 locations around the area of interest with 6 reference spheres to match the images using the FARO Scene Software. The data were generated into a point cloud and interpolated to a solid surface with 10 mm resolution with the kriging method using the Golden Software’s Surfer 8 (

https://www.goldensoftware.com/products/surfer). The gridded data were de-trended in the x-y plane to remove the bias in slope of the field or the angle of the lidar. Gaps in the scans tended to be small (<100 mm), and the kriging interpolation eliminated them. Individual roughness elements were identified and for each element the silhouette lot area and obstacle height were determined using a MATLAB code (

https://www.mathworks.com/products/matlab.html). This was required to compute the Lettau formula (Equation (4)). The 1000-m

^{2} area around the tower was scanned on 12 occasions when the concurrent anemometric and geometric measurements were acquired. One concurrent measurement set was made with no snow cover for each of the unplowed and plowed scenarios; seven concurrent measurement sets were made with partial snow-covered area (SCA) and three with complete snow cover. SCA was determined from digital photos taken from the TLS unit.

Both the Counihan and Lettau methods were used to calculate z_{0G} (Equations (4) and (5), respectively). The Counihan method was appropriate for this study because the roughness elements (furrows) in this study site were semi-regular. During each concurrent anemometric and geometric measurement set, the percentage of the area covered in snow, or SCA, was estimated from photographs.

## 3. Results

The unplowed versus plowed field yielded different

z_{0A} values (

Figure 1). On average, the plowed field was almost 20 times as rough as the unplowed field, yet the coefficient of variation (COV) was essentially the same (0.67 and 0.62, respectively) (

Figure 1). The smallest

z_{0A} values for the plowed field were of the same magnitude as some of the largest

z_{0A} values for the unplowed field, in the range of 1 to 3 × 10

^{−3} m.

The Counihan method estimated

z_{0G} values that were 1.39 times larger and had greater variation than the estimated

z_{0A} values (

Figure 2). We used the Nash-Sutcliffe coefficient of efficiency (NSCE), which is a performance statistic based on a comparison of the data fit to the 1:1 line, to evaluate how estimates of

z_{0G} compared with

z_{0A} [

43]. The NSCE of the Counihan

z_{0G} was −1.18, and the Lettau

z_{0G} was 0.14, indicating the Lettau method compared more favorably with the

z_{0A}. A linear regression between both

z_{0G} estimates (Counihan

z_{0G} and Lettau

z_{0G}) and

z_{0A} was fit through the data origin to evaluate if the bias between the two methods could be removed through simple linear scaling (

Figure 2). When the Counihan

z_{0G} values were scaled by 0.721 (1/1.39), the NSCE value only increased to 0.07. However, the NSCE increased to 0.88 when the Lettau

z_{0G} values were scaled by 2.34 (1/0.428).

The estimated

z_{0} values were found to vary as a function of the amount of SCA present (

Figure 3). As SCA increases,

z_{0} decreases, with variability based on the calculation method (

Figure 3a). A linear regression between SCA and each of the

z_{0} estimates showed r

^{2} values that were 0.01, 0.7, and 0.88 for the Counihan, anemometric, and Lettau methods of

z_{0} calculation, respectively. There were noticeable differences in

z_{0} depending whether SCA was increasing because snow was accumulating versus when SCA was decreasing because the snow was melting. For periods of snow accumulation, removing snow measurements that were not immediately following a snow event (the yellow boxes in

Figure 3b that represent non-accumulation values) improved the linear relation between accumulating SCA and

z_{0} (R

^{2} = 0.94).

## 4. Discussion

Geometrically estimated

z_{0}, although easier to measure, produced different values when compared to the anemometric derived values. The Counihan method overestimated by a factor of 0.721, whereas the Lettau method underestimated anemometric

z_{0} (

Figure 2) by a factor of 2.34. The Lettau method (Equation (4)) has a constant of 0.5 based on the average drag coefficient of the roughness characteristic of the silhouetted area of the average obstacle. By dividing the Lettau based

z_{0} values by the 0.5 and thus eliminating the drag coefficient from the equation, we get a new NSCE of 0.856, with scatter in the data much closer to the 1:1 line (

Figure 2). The removal of the drag coefficient suggests that the geometric data generated from the lidar point cloud appears to account for spatial and temporal variability in the roughness of a snow surface.

Lidar-based snow data are becoming more readily available [

19,

26]. The accuracy of the scans from about 1 mm for terrestrial lidar to 10 cm for airborne lidar can account for fine-scale changes of the snowpack [

26], which enables the computation of

z_{0} at any scale. Although anemometric data can yield reliable estimates of

z_{0}, meteorological towers are expensive to set up and operate. In addition, data from a single tower does not consider spatial variability as well as the geometric method [

44]. Comparing the two methods does not consider the scale of the study area; the geometric measurement is taken over the entire area near the meteorological tower whereas the anemometric measurement is only influenced by the fetch area upwind of the sensors [

29].

The roughness of the snowpack can vary substantially both spatially and temporally creating many implications [

13,

14,

45]. Roughness variations can be caused by heterogeneities in land cover, vegetation, and meteorological conditions [

46]; non-uniform distribution of snow cover during accumulation and melt [

45,

46]; snow-canopy interactions [

47]; and snow redistribution by wind [

48]. This was apparent in differences between the estimated

z_{0} for the plowed versus unplowed field (

Figure 1). Land cover varies throughout regions particularly those with a shallow snow environment, and this creates variations that depend on the underlying topography [

13,

14,

46]. Thus, there are many different values of

z_{0} in the literature [

7] that are broader than our observed mean range of 0.2 to 10 × 10

^{−3} m (

Figure 1). For example, Miles et al. [

31] found the

z_{0} of a hummocky glacier (a particularly rough underlying surface) to range between 5 to 500 × 10

^{−3} m, whereas Brock et al. [

7] reported

z_{0} values for fresh snow and older snow of 0.2 × 10

^{−3} and 3.56 × 10

^{−3} m, respectively. Our results show that change in roughness between a plowed and unplowed field yielded a 20-fold difference in

z_{0}. The results of this study can be applied to areas of similar climate and land cover, which included flat, bare soil, and bare soil with small furrows (<1 m); and therefore, the results of this study may not scale appropriately to different land cover types. Further studies of a shallow snowpack in sagebrush steppe [

49], farmland, or non-densely forested environments may be able to replicate our study results and scale from 1000 m

^{2} to a larger area. The

z_{0} values observed here had a notable change between flat soil and small furrows, so the changes in

z_{0} values in different environments with even minimal vegetation will have much larger effects on the

z_{0} values.

The inverse relation of SCA and

z_{0} (

Figure 3) [

50] is affected by the underlying terrain and size of the roughness features. As the snow accumulation increases, the roughness elements become buried, and the topography appears to be smooth [

50,

51]. This relation indicates that as snow accumulates over topographic features the snow will begin to level out at a

z_{0} height dependent scale. A hysteresis can be noted, and it has been found that a single snowfall event on a hummocky glacier can alter the micro-topography by up to 75% due to the shallow snowpack over the small scaled features [

45,

52]. The CLM4 uses a

z_{0} value of 2.4 × 10

^{−3} m, a value that falls near the mean of the unplowed field, which is applicable for deep, flat snowpack surface with minimal influences from underlying terrain. However, this is not typical for shallower snowpacks or in complex terrains.

Relations between

z_{0} and SCA (

Figure 3) can be used to improve snow-energy balance modeling by estimating the percentage of SCA via remote sensing and applying

z_{0} only to the portion of area it accurately describes [

46,

53]. Currently, most models use 100% SCA even though many areas will remain snow free due to complex terrain and can drastically change during periods of melt and accumulation [

13,

53]. Aerodynamic roughness length is incorporated into many climate and energy models, which require sub-grid snow distribution [

54] and are still inadequate at representing SCA [

46,

48]. A dynamic

z_{0} based on SCA and land cover type can improve these on a sub-grid scale. Another complication with these models is the lack of accountability for snowpack variability throughout accumulation and melt [

48,

53,

54].

Resolution is an important factor to consider when discussing both SCA and

z_{0G}. The higher the resolution of the measurements (lidar, satellite, etc.), the higher the

z_{0-G} accuracy. However, lidar datasets are often large, especially those acquired with TLS, making them difficult and time consuming to process. Lower resolution data from remote sensing or airborne lidar systems (ALS) can cause problems when scaling [

53]. Quincey et al. [

52] found that

z_{0G} is typically underestimated with a small area and coarse resolution and overestimated with a large area and fine resolution when compared to anemometric data. Nonetheless, even with lower resolution, applying dynamic

z_{0} values may greatly improve models. Scaling can be an effective way to incorporate both an anemometric and geometric

z_{0} value. Based on a specific land cover type, a scaling factor can be applied to areas with the same land cover. This can help to improve modeled

z_{0} accuracy, once preliminary

z_{0} values have been established.