MODELLING SNOW REDISTRIBUTION BY WIND – LOW TATRAS, SLOVAKIA

Anna Seres1*

1 University of Miskolc, Miskolc, Hungary

ABSTRACT: A submodule of AvalMap, a snowpack model developed by the author (Seres, 2018),models snow redistribution by wind to incorporate it in the final avalanche danger maps it produces. As a first step 10 minute wind measurements are averaged as vectors to get one speed, direction and duration value at each timestep in AvalMap. These values are then modified by a factor based on macro (general shape of the mountain and distance from ridge) and micro terrain features (curvature, slope and aspect-wind direction ratio) to spatially distribute the point data. Then, as a second step snow transport is calcu-lated based on the ratio of shear velocity (from logarithmic wind profile), to threshold shear velocity (equa-tion modified from Liston et al. (2007)). Flux, the potential carrying capacity of the wind is calculated by an equation from Kind (1981), which is also based on shear and threshold shear velocity. Difference of the fluxes is then determined for each pixel from the direction of the wind, to derive the amount of snow that could actually be carried away or deposited. Sublimation and densification in the carried snow are also considered. The amount of snow actually transported depends on the temperature, height and density of the available snow layers. By putting these all together we determine the spatial distribution of wind deposited snow by considering the deposition areas indicated by the difference of the fluxes, the distance from the ridges and the time the air spends at each pixel by the given wind speed as well.

KEYWORDS: wind, snow, transport, redistribution.

1. INTRODUCTIONOne of the main causes of spatial inhomogeneity of snowpacks in alpine environments is snow re-distribution by wind. Wind can add a huge amount of snow to lee sides, even without new snowfall, increasing the avalanche danger and contributing to its spatial variability. The modification of wind speed and direction by topographic features and the amount of snow carried by wind is difficult to model, so snow redistribution by wind is usually not included in snowpack models. A submodule of Av-alMap, a snowpack model developed by the au-thor, models this phenomenon to incorporate it in the final avalanche hazard maps it produces.

The current paper discusses the major working principles of the wind transport submodule

2. STUDY AREAThe study site is located in the central part of the Low Tatras mountain range in Slovakia, which is a popular skiing area with an east to west oriented, long, grassy ridge of about 2000 meters. The north facing slopes are steep, rocky, the south facing ones are rather gentle sloping. It has a continen-tal/alpine climate with prevailing winds from the

west, distorted to the north around the ridgeline. At the time of the study (2009/2010), there were 5 weather stations at the study area, from which only the one at the peak Chopok was an automatic weather station with reliable wind data (called peak AWS in the following).

3. METHODS

3.1 Averaging of the data AvalMap is run twice daily, but wind is measured in every 10 minutes by automatic weather stations (AWS). To overcome this difference in the tem-poral frequencies of wind data, in AvalMap wind measurements above 4,7 m/s are vector aver-aged over the time period of the running frequency of AvalMap. The value 4,7 m/s was chosen, as speeds around 5 m/s proved to be lowest speed with which the wind is able to move significant amounts of snow by both Liston et al (2007) and Kind (1981).

3.2 Extrapolation of wind measurements The wind modification effect of terrain in scientific literature is either studied on a macro scale, in case of mountains or on a mezo or micro scale, in case of gently undulating hills or flat areas. As even small terrain features can already largely al-ter the area and outcome of snow deposition by the wind, we consider both the macro and the mi-cro scale together in our model.

The macro scale wind modification factor,when the wind is modified by the general shape of

* Corresponding author address:Anna Seres, University of Miskolc, Institute of Geography and Geoinformatics, Miskolc, Hun-gary;tel: +36 30 4026115;email: anna.seres@gmail.com

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the mountain, is calculated based on a method,originally developed for measuring the wind load of buildings at Purdue University, USA (ASCE 2010) (Equations 1-6). After evaluating that all the condi-tions described in the method are met for the spe-cific study area, the equation parameters are also adapted to the study site. At our study site in the LowTatras this specifically means the following

H: height of the mountain from the surrounding large valley floors

LH/2: width of the mountain at the elevation halfway between the valley floors and the ridge

x: distance from the ridge. As the Low Tatras has several side ridges beside the main ridge and the steepness of the slopes are different on the south-ern and northern side of the mountain, we changed this into a factor dependent on elevation and slope angle with a tangential formula.

μ: horizontal reducing factor. For elongated moun-tains, its value is 1,5 according to ASCE (2010), but it was modified to 0,4 based on wind measure-ments in the Low Tatras

γ: vertical reducing factor. For elongated moun-tains, its value is 3 according to ASCE (2010)

z: height above snow surface in meters. The ane-mometer is at the height of 3 m, so z was taken to be 3, not considering the snowpack height.

The macro scale wind modification factor (S1) is described by Equations 1-6.

21 1 2 3(1 )S K K K (1)

K1 accounts for the general shape of the mountain and the land cover according to Equations 2-4. If the mountain is relatively steep, meaning that its height (from the surrounding valleys) to its width (at the height halfway between the ridge and the sur-rounding valleys) ratio is higher than 0,5, as in case the of the Low Tatras, /2HH L is considered to be 0,5 in case of K1 and LH/2 is considered to be 2H in case of K2 and K3 (ASCE 2010).

1/2

1,55grassH

HKL

(2)

1/2

1, 45dwarfpineH

HKL

(3)

1/2

1,3forestH

HKL

(4)

K2 accounts for the distance from the ridges. In the case of the Low Tatras x, the distance from the ridge was changed with a tangential formula into slope angle and an elevation dependent factor,

where α is the slope angle and h is the elevation of the given pixel (Equation 5).

2/2 /2

1 1H H

H hx tgK

L L (5)

K3 accounts for the wind speed change at the height above the snow surface (Equation 6).

/23

Hz LK e (6)

The micro scale modification factor, which mod-ifies the wind speed and direction according to the smaller terrain forms is calculated by a method de-veloped by Liston et al. (2007). It depends on the slope angle, the relation of wind direction and as-pect and the convexity and is calculated by Equa-tions 7-12.

The wind speed modification due to the convexity (Ωci) is determined by dividing the curvature (Ωc) (calculated by ArcGIS) with the twice of the maxi-mum curvature (Liston et al 2007) (Equation 7).

2 maxc

cic

(7)

The wind speed modification factor due to the slope angle and the relation of wind direction and aspect (Ωsi) is calculated by Equations 8-9, where α is the slope angle, θ0 is the starting wind direction and ε is the aspect (Liston et al 2007).

0coss (8)

2 maxs

sis

(9)

We modified this to take the turbulence causing ef-fect of the very steep hills into consideration. Ac-cording to Shiau et al (2002) the isolines of the wind start to diverge at slope angles of 15°. We be-lieve that this value is too low, so changed it in a way that till the slope angle of 30° the factor Ωs will increase like described above in Equations 8-9,but above this, Ωs will decrease proportionally to the slope 0(60 ) cos . 60 was used as the maximum slope angle in the area is 58°. This mod-ification only refers to the luv side, Ωs on the lee side will be calculated by the original equations.The micro scale wind speed modification factor is calculated as Equation 10 (Liston et al 2007).

2 1 si ciS (10)

The terrain also modifies the wind direction. The wind direction modification effect of the terrain (θd)

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is described by several authors (Liston et al. 2007, Gugolj 2005).

0sin 2d si (11)

Equation 11 is from the SnowTran-3D model, de-scribed by Liston et al. (2007) and it returns a mod-ification value of -0,2° to 0,2°. This value is too low for the Low Tatras. The wind direction modification method by Gugolj (2005) results in modification values between -15° and 15°. These values are closer to the measured values in the Low Tatras, but the spatial distribution of the results is much better in Liston et al’s (2007) method. If we change the constant -1 to -25 in Equation 11 we get much better values for the Low Tatras.

The wind direction modification factor is then added to the starting wind direction to yield the fi-nal, modified wind direction (θ) (Equation 12).

0 d (12)

The modified wind direction is also used in the wind speed calculations as follows. First Ωsi is cal-culated, using the starting wind speed at the peak AWS (Equations 8-9). Then θd is calculated with this Ωsi value (Equation 11). This θd is then used in equations 8-9 to calculate the final, spatially distrib-uted Ωsi.

The final, combined wind modification factor (S) is received by multiplying the macro and micro scale wind modification factors and bringing the micro scale factor to square to enhance the micro scale effect (Equation 13).

21 2S S S (13)

After S is calculated, the spatially distributed, mod-ified wind speed is determined by the relation in Equation 14, where ui is the modified wind speed at point i, uChopok is the measured wind speed at the peak AWS, Chopok and SChopok is the combined wind modification factor at the same peak AWS.

i ChopokChopok

Su uS

(14)

This refers for wind speeds at the height of 3m. Inorder to get the wind at the snow surface, a loga-rithmic wind profile is taken, see Section 3.3.

3.3 Snow transport by wind Snow transport by wind mainly depends on the rate of shear velocity (u*) to threshold shear veloc-ity (u*th). Shear velocity is calculated from the mod-ified wind speeds by the logarithmic wind profile (Equation 15), where u is the wind speed at 3 m,

is the Karman constant (0,4), z is the height at which u is calculated (3m), z0 is the roughness length of snow, 0,0005m.

0

ln

uuzz

(15)

To calculate the threshold shear, AvalMap’s wind module uses the equations by Liston et al (2007), as this gives the best results for all kinds of snow densities (Equation 16-17), where ρs is the density of snow (kg/m3).

0,005exp(0,013 ) _ _300 450th s su ha (16)

0,1exp(0,003 ) _ _50 300th s su ha (17)

Flux, the snow carrying capacity of the wind is de-fined by the equation of Pomeroy and Grey (1990)(Equation18), where Q is the flux in (kg m-1s-1), ρa is the density of air (~1,29 kg/m3) and g is the grav-itational acceleration (9,81 m/s2).

2 20,68 a thth

uQ u ugu

(18)

Flux however only shows us how much snow the wind is able to pick up until it reaches its maximum carrying capacity. In reality the wind arrives at apixel already carrying some snow. This means that it is not able to pick up all the snow defined by the flux of a certain pixel, but it will only pick up the amount corresponding to the difference of the fluxes between the two pixels. If the flux of the ac-tual pixel is less than that of the pixel from the di-rection of the wind, the amount of snow, corre-sponding to the difference of the fluxes will be de-posited. With this, we assume that the wind/air ar-rives to the actual pixel saturated with snow. So the amount of wind transported snow can best be de-scribed with the difference of fluxes, dQ. By multi-plying dQ with the time the wind is blowing the given day in seconds (t) we get the potentially car-ried amount of snow. The height of the wind trans-ported snow (h) is then received by dividing this with the density of the snow layer (ρs) (Equation 19).

luvs

Qth (19)

Part of the snow sublimates while being trans-ported and its density increases. We have defined the amount of sublimation in 10% and the density is multiplied by 1.66. Thus, to receive the amount of snow deposited on the lee side, the amount of snow picked up from the luv side is decreased ac-cording to this. Altogether the potentially depos-ited snow height (hlee) is defined by Equation 20.

0,91,66lee

s

Qth (20)

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In AvalMap wind transport can only occur if the snow surface temperature is below freezing point. The amount of snow actually transported depends on the height and density of the available snow lay-ers. If the wind has blown away the top layer, and it did not need all the time it was blowing that day for it, dQ is recalculated from the density of the next layer and the wind will start picking up the snow from that layer as well. This process continues until all the time or all the layers are used up. In order to determine the spatial distribution of wind deposited snow, first the pickup areas are defined. These are pixels with luv aspect, where dQ from the direction of the wind is positive. Deposition areas are either luv or lee pixels, where dQ from the direction of the wind is negative. Potentially carried and deposited snow heights are defined for both the pickup and the deposition areas respectively. The amount of snow actually transported from the pickup areas is summed for the whole area. This amount - re-duced with the sublimation and the densification-- is then distributed on the deposition areas accord-ing to the potentially deposited layer height raster.

4. RESULTS AND DISCUSSIONCurrently we are working on validating the model against new wind data received from AWSs in the area, while the preliminary model runs show sen-sible results both on spatial and temporal scale.

As an example the model results for the 13th of February 2010 are discussed here. The spatial dis-tribution of the wind direction, the result of Equa-tion 12, can be found at Figure 1. It can be seen from the map that the starting 10° (yellow) changednorthward (brown) in case of east facing slopes, and eastward in case of west facing slopes. The extent of the change was about 10°.

Figure 1. Spatial distribution of wind direction with starting peak wind direction of 10° at Chopok.

The spatial distribution and the modified values of the wind speed at 3m height, the result of Equation 14 is shown on Figure 2. That day at the peak AWS at Chopok the starting wind speed was 14,8 m/s and its direction was 10°. According to our model

the wind was accelerated up to about 20 m/s (30 m/s outliers are shown blue in the map) and the highest wind speeds are found along the ridges and at high elevations on the luv side, especially where the slopes are gentle. Lowest wind speeds are located in the valleys of the lee side.

Figure 2. Spatial distribution of wind speed at 3 m height with starting wind speed of 14,8 m/s and wind direction of 10° at the peak AWS.

The spatial distribution of the snowpack height, due to the wind transport is depicted on Figure 3.The uniform 50 cm fresh snow that had fallen onthe previous day was carried away and redepos-ited by winds with the vector average of 14 m/s in speed, 10° in direction and 11 hours in duration. The only factor changing the height of the snow-pack is wind transport in this case. One can see that the main deposition of snow took place at the areas behind the ridges from the direction of the wind and in the narrow valleys and couloirs of the lee slopes. The usual maximum snow heights be-came to be about 100 cm, meaning a 50 cm of snow redeposition (outliers of up to 270 cm are marked blue). The minimum snow height found was zero, meaning that there were places, espe-cially in the immediate luv side of the ridge, where the snow was completely blown away.

Figure 3. Spatial distribution of the snowpack height due to wind transport. 50 cm of snow was redeposited by winds of 14 m/s in speed, 10° in

direction and 11 hours in duration.

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The temporal distribution of snow layer heights also give information about wind transport (Figure 4). The appearance of new layers are either caused by new snowfall or a new layer of wind deposited snow or both (arrows 1,2,3). If there is significant wind during snow-fall, AvalMap first calculates a uniform layer of new snow and then modifies this by the wind factor. The wind deposited snow in this case is added as a new layer, as it will have different properties (like increased density).The sudden disappearance of layers, when the snow surface temperature was below 0°C, was caused by wind carrying away the snow (arrow 3 and 4). In both cases, the top layer was completely carried away and some snow of the second layer was also picked up.

Figure 4: Temporal distribution of snow layer heights at Chopok for the season 2009/2010. Bands of differ-ent colors show the heights of snow layers, red line

shows the snow surface temperature.

All these preliminary results show sensible and ex-pected outcomes both with respect to the spatial re-distribution of the snowpack, as well as the temporal change in the snowpack due to the wind. By validat-ing these results against past wind data, we hope to achieve a wind transport module that can accurately account for the snowpack modifying effect of the wind.

5. CONCLUSION AND FUTURE PLANSWind is a very important factor in avalanche formation, thus it is crucial to incorporate it into the snowpack models used for avalanche forecasting. AvalMap is a unique snowpack model that does contain such a de-tailed snow transport module. This module shows very promising results both on the spatial and the temporal scale. However due to the complexity of the problem careful validation and a number of small further im-provements are necessary to make the results even more accurate.

Plans for the future include: (1) Validating the results with measured AWS data. (2) Validating the spatial distribution of the results by measuring snowpack heights before and after windy days. (3) Running the model for different locations to see how the module is

applicable to different shaped mountains. (4) Deter-mining K1 in a way that

/2HH L is calculated from the DEM and not considered a constant.

6. ACKNOWLEDGEMENTI would like to thank COST Action ES1404 for support-ing and providing opportunities on collaboration within the COST network.

I would also like to thank for the assistance given by the Slovakian Avalanche Forecast and Mountain Rescue Service (HZS-SLP), especially Marek Biskupič and Filip Kyzek for their support and for providing the meteorological, snowpack and ava-lanche data. I would also like to extend my thanks to the Slovakian Hydrometeorological Institute (SHMÚ)for providing the data of the official meteorological station at Chopok.

7. REFERENCESAmerican Society of Civil Engineers 2010: ASCE Standard

ASCE/SEI 7-10. www.pubs.asce.org ISBN 978-0-7844-1085-1

Gugolj, D. 2005: Three-dimensional Snow Distribution Modeling in GIS. GEOG 647, Department of Geography, University of Cal-gary

Kind, R.J. 1981: Snow Drifting in Handbook of Snow: Principles, Pro-cesses, Management and Use, (Grey, D.M., Male, D.H. eds.). The Blackburn Press, Cadwell, New Jersey, USA, pp. 338-359

Liston, G.E., Haehnell, R.B., Sturm, M., Hiemstra, C.A., Bere-zovskaya, S., Tabler, R.D. 2007: Instruments and Methods - Simulating complex snow distributions in windy environments using SnowTran-3D in Journal of Glaciology Vol. 53., No. 181. pp. 241-256

Pomeroy, J.W., Grey, D.M. 1990: Saltation of snow. Water Re-sources Research, Vol. 26, No. 7, pp. 1583–1594

Seres, A. 2018: AvalMap – a snowpack, weather, terrain and land cover based avalanche hazard model for the Low Tatras, Slo-vakia in International in International Snow Science Workshop, conference proceedings in preparation

Shiau, B.-S., Hsieh, C.-T. 2002: Wind flow characteristics and Reyn-olds stress structure around the two-dimensional embankmentof trapezoidal shape with different slope gradients. Journal of Wind Engineering and Industrial Aerodynamics Vol. 90, pp. 1646-165

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