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Water vapour diffusion at the boundary between snow and groundPermafrost, Phillips, Springman & Arenson (eds)© 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 582 7N. Osokin, R. Samoilov, A. Sosnovskiy, V. ZhidkovInstitute of Geography, Russian Academy of Sciences, Moscow, RussiaABSTRACT: We evaluated the water vapour diffusion flux within the snow layer at the soil surface and fromthe upper layer of ground into the snow. We also determined approximate values of thinning of the snow layer atsoil surface, taking into account various thermal and physical properties of snow and conditions of snow accumulation,including water vapour super-saturation in snow.1 INTRODUCTIONSnow cover significantly influences cryogenicprocesses in the ground. On the other hand, the thermaland physical states of the uppermost layer of thesoil influence the thermal regime of the snow layer atsoil surface and heat and mass transfer fluxes.In this paper, we develop a mathematical model todescribe the connections between the atmosphere, thesnow cover and the ground. The model includes: (1)variability in meteorological parameters and snowaccumulation conditions (a lag between the momentwhen negative mean daily temperature is reached – d ,and the beginning of snow accumulation and a regimeof snow accumulation); (2) changes in the density,thermal and physical characteristics of snow; (3) variabilityin soil properties (thermal conductivity, specificheat and latent heat) due to freezing; (4)availability of a phase transformation layer and moisturemigration to the freezing plane.2 THEORETICAL BASIS OF THE MODELNote: The following symbols are used in the equations:– T temperature, °K;– t temperature, °C;– z, x, vertical coordinates in snow cover andsoil, and a boundary of phase separation;– h soil layer thickness;– time;– thermal conductivity;– density;– w total moisture content;– c specific heat capacity;– moisture diffusion coefficient.The mathematical model uses the followingequations:Fourier’s equation for heat conduction in snow,thawed and frozen ground with time-dependent thermaland physical parameters for snow and ground:c rjj(1)An equation for moisture movement within thethawed part of the ground:wTjTjlj ⎛ ⎞x⎝⎜ x⎠⎟th ⎛ x k w w th ⎞th( )⎝⎜x⎠⎟(2)in which index “j” is substituted for “s” in snow withthickness h s (), f – in a frozen zone (0 x ) and“th” in a thawed zone of the soil (x h). Moistureflow and heat flow at the bottom boundary of thethawed zone are assumed to be equal to zero.Stefan condition on the boundary between thawedand frozen ground:⎛ T ⎞l rLw w d j ⎛ L k w ⎞th w⎝⎜x⎠⎟ ( )d ⎝⎜x⎠⎟t(3)The dependence of effective volumetric heat capacityof the frozen ground on total moisture:c , ( T, w) c ( w) Lre f f fww( T)T(4)where the unfrozen moisture in the ground w w m(273 T) n at m 9.8325 and n 0.3221 (Osokinet al. 2000).To solve equations (1)–(4), the following conditionsare set: boundary conditions at the surface and in theupper layer of the soil; the initial distribution of temperatureand moisture in the thawed soil; the dynamicsof meteorological elements, snow accumulation;and the thermal and physical parameters of snow.857

Heat exchange with the atmosphere occurs at the soilsurface when h 0 (or on snow cover, when z h s ):l ms ( )Tms ( )xwhere: Q th ,Q e ,Q r , Q sn are heat flows resulting fromsensible heat transfer, evapouration, effective radiationand solar radiation, respectively.Heat exchange as a result of effective radiation isdefined as:Q r T f 4 (0.254 5 10 5 e f )(1 nc r 4T f 3 (T m(s),0 T f )where: is the emissivity of the surface; is theStefan–Boltzmann constant; T f ,T m(s) are the temperaturesof the air and soil surface, respectively, °K; n iscloud cover; e f is water vapour elasticity in air; c r is aconstant linearly dependent on latitude that equals 0.8and 0.7 at latitudes 70° and 45°, respectively.Total heat flow is calculated from the equations:Q e,th (T m(s),0 T e,f ) e,th th (1 1.95 10 2 a 1 ) 0.205(T f /100) 3T e,f ( th (T f 1.95 10 2 (b 1 e f f ) 19.9(T f /100) 4 Q sn )/ e,thwhere: the coefficient of heat exchange in the ground th is calculated according to Pavlov’s (1979) formula: th v 0.5 (7 7.2v 2 )and in snow according to the formula of Kuzmin(1961): th 3.4 2.2vwhere: v is wind speed; f is the humidity of air.The model calculates changes in temperature in snowand ground during freezing. This allows us to estimatethe intensity of water vapour diffusion in snow and theupper layer of the ground. The water vapour diffusionflux I i at a depth h i is calculated using the formula:I i D(t i )(e i e i 1 )/h Q Q Q Qth e r snwhere: e i is water vapour density in the layer “i ”; his a distance between nearby layers of snow; D is thetemperature-dependent coefficient of water vapourdiffusion in snow derived by Pavlov (1979):D(t) 0.92 10 4 0.29 10 5 t 0.56 10 7 t 2 (5)The coefficient of diffusion in the soil is also calculatedfrom this equation.To define the water vapour concentration in snowwe normally use the temperature-dependent saturationvapour density formula of Magnus:e h (t) e o exp[17t/(t 235)]where: e o is the saturation water vapour density, equalto 0.0049 kg/m 3 at t 0°C. However, experimentalresearch has shown that super-saturation of watervapour can occur within snow (Golubev et al. 1997).Thus the absolute value of super-saturation remainspractically constant when the temperature changes, andthe relative super-saturation increases when the temperaturefalls. The experimental curve of relativesuper-saturation K c e/e h (Golubev et al. 1997), weapproximate by:K c 1.0167 0.0067t (6)The opposite phenomenon, water vapour undersaturation,occurs in the ground. Thus, when soil moistureincreases, water vapour amounts also increase,achieving saturation when humidity is 30%. Experimentalresults concerning water vapour relative undersaturationin pores of the ground, when the soilmoisture “w” changes in the range from 10 up to 30%(Golubev et al., 1997), are approximated using theformula:K c 0.925 0.0025w (7)It is presumed that if water vapour concentrationexceeds the saturation value, condensation occurs onice crystals and snow density increases. At watervapour under-saturation, ice sublimation and thinningof the snow occurs.The change of snow density in the layer “i” during time is estimated with well-known values of theintensity and the direction of water vapour diffusionemploying the formula: t (I i I i 1 )/h3 MODEL USENumerical experiments were carried out with themodel to estimate the influence of snow thermal andphysical properties and snow accumulation on thethermal regime and mass transfer in the snow layer atthe soil surface and in the upper layer of the soil.Snow and climate conditions were parameterized asfollows:The dependence of snow thickness h s on time s ,was defined with h s /h max ( s / max ) 0.5 , where h max 70 cm and max 180 days. The dependence of snow858

Table 1. Coefficients used to relate snow density to snowthickness.CoefficientsRegion a b Density valuesNorth West Siberia 83.06 0.355 HighChukotka 46.00 0.360 MeanYakutsk 59.20 0.302 Lowdensity s on the thickness h s was calculated from theformula s (h s ) ah s b , where the coefficients a, b aretaken from different regions of Russia. “High”, “mean”and “low” values of density were employed based onTable 1.The coefficient of effective thermal conductivity ofsnow was calculated from formulae derived from morethan 20 published empirical relationships (Osokinet al. 2000). The resulting curve of “mean” values isapproximated with: s 9.165 10 2 3.814 10 4 s 2.905 10 6 s2“High” and “low” experimental curves s ( s ) F( s )are calculated from:Freezing depth, m1,61,41,210,80,60,40,200 20 40 60 80 100 120 140 160 180 200Freezing time, daysFigure 1. Modeled snow cover thickness (m) (line 1) andsoil freezing depth (m) (line 2) against time (days).Temperature,ºC0-5-10-15-20-25-30-351 20 50 100 150 200Freezing time, daysFigure 2. Modeled soil surface temperature (line 1) andair temperature (line 2) against time (days).12 s h 1.36 10 2 1.110 3 s s 2 , s l 2.96 10 2 3 10 4 s 2 10 6 s 2 .In the calculations, “high”, “low” or “mean” thermaland physical parameters of snow are used to establishranges of results. They are determined from correspondingpairs of relationships of density and thermalconductivity (for example sh II sh ).The change of mean daily air temperature duringthe cold period is calculated from a sine-wave functionused for the annual cycle.The temperature of falling snow is considered to beequal to the temperature of the air, wind speed is 5 m/s,moisture of the air is 70% and cloud cover is 0.65.The delay before the beginning of snow coverestablishment is equal to d 0, 8 and 16 days.Soil conditions were parameterized as follows:The ground is loamy soil with a density of1600 kg/m 3 , a moisture content of 25%, and a saturatedwater capacity of 35%. The maximum temperaturein the thawed soil is at the 1.5 m depth. Whenfreezing starts, the surface temperature is 4°C, andthe freezing plane has a temperature of 0°C.The values of heat capacity and thermal conductivityin thawed and frozen ground were calculateddepending on moisture content according to equationsin Osokin et al. (2000).The results of the calculations are shown in Figures1–4 and Table 2. For these calculations we used meanFigure 3. Modelled water vapour diffusion flux againsttime (days): from snow to soil (line 1), from the bottom snowlayer to upper snow layer (line 2), from soil to snow (line 3).Figure 4. Mean snow density (line 1) and the thinning of10 cm-thick snow layer at soil surface, taking into accountwater vapour super-saturation in snow and under-saturationin the soil (line 2); without these factors (line 3).859

Table 2. The influence of various parameters on thinningof 10 cm-thick snow layer at soil surface at the time of maximumfreezing depth.TimeDepth Thinning ofSnow shift t min Moisture hoar snow layer*parameters (d) (°C) migration (cm) (kg/m 3 )M 8 30 137 377/137M 8 30 – 141 378/138M 8 15 77 315/97M 8 15 – 85 320/99M 0 30 108 401/160M 16 30 160 351/118L 8 30 33 407/215L 8 30 – 42 424/226L 0 30 8 477/299L 0 30 – 18 507/318L 16 30 69 401/211L 15 20 282/114Notes: M – mean; L – low; : taking moisture migration intoaccount; : without accounting for moisture migration; * numeratorshows results with water vapour super-saturation in snow andunder-saturation in soil; denominator shows results without thesefactors.thermal and physical parameters for the snow, t shf was8 days, and the minimum air temperature was set to30°C. Figure 1 shows modelled snow accumulationand corresponding soil freezing which reaches a maximumdepth of 136 cm.Mean daily air and surface temperatures are shownin Figure 2.The minimum surface temperature, about 6°C,occurs on the 8th day, when snow cover appears. Thetemperature then rises to 2.4°C over 3 days and subsequentlyfalls to 5.2°C (on the 108th day). At the endof the freezing period the temperature of soil surfaceincreases to 2.5°C. Temperature gradients in the upperlayer of the soil average 5.5°C/m for the first half ofthe cold season and fall to 2°C/m during the secondhalf (on the 160th day). Temperature gradients in thesnow near the boundary with the soil are ten timesgreater, changing from 77°C/m at the beginning ofwinter (when snow thickness is small) to 18°C/m on the160th day.The water vapour diffusion flux in snow at theboundary with the freezing soil is shown in Figure 3.The modeling showed that water vapour diffusioncould occur in either direction, from snow to soilbecause of water vapour super-saturation in snow andunder-saturation in soil), and from soil to snow (withoutthese factors). In the former case, the diffusion fluxfrom snow to soil averages 14–16 10 7 kg/(m 2 c), andin the latter case, diffusion from soil to snow is1 10 7 kg/(m 2 c).In both cases, water vapour also transfers from thewarmer bottom snow layer to the colder upper snowlayer. This diffusion flux is 18 10 7 kg/(m 2 c) at thebeginning of winter and drops to 1 10 7 kg/(m 2 c) atthe end of winter (line 2 in Figure 3). In both caseswater vapour transition is accompanied by a thinningof snow layers at the soil surface (in the second casebecause of the diffusion flux from snow to snow isgreater than from soil to snow).According to experimental research (Golubev &Sokratov 1991), intensive water vapour transition generallyoccurs when snow thicknesses are greater than10 cm. From our calculations, the intensity of watervapour diffusion in the bottom (5 and 10 centimeters)snow layers differs from the value in 1 centimeterlayer at the soil surface by 5% and 12%, respectively.This allows us to estimate the intensity of the thinningof various thicknesses of snow layers at the soil surface.It should be noted that settling of the snowpackunder the influence of structural reorganization andpressure from the upper layers is not considered inthese calculations.The results of modeling for a 10 cm-thick snowlayer at the soil surface are shown in Figure 4.The increase in mean snow density, depending onthe snow layer height is shown by line 1 (Figure 4).The reduction of the density of the bottom 10 cmlayer () without water vapour super-saturation insnow and under-saturation in soil is shown in line 3(Figure 4). If we take into account these factors in ourcalculations (line 1), this leads to an increase in thevalue of by 2–3 times, which can exceed the realamount of water in the calculated layer.It is known that water vapour diffusion promotesrecrystallization and the formation of depth hoar.According to published data (Golubev et al., 1991) thedevelopment of the edge and vertex of crystal-formspromotes an increase in vapour pressure above thecrystal. However, in our opinion, a reduction prevailsbecause of the increase in the distance between crystalcolumnar structures in the depth hoar. That is whydiffusion to the soil can lead to an increase of watervapour concentration in it and to a reduction anda closing of some part of the pores that complicateswater vapour diffusion to soil. Probably, water vapourdiffusion from snow to soil occurs at the beginning ofwinter and diffusion changes direction during the coldseason. In our opinion, the results of real diffusion arelikely to be closer to the calculated curve, modeledwithout water vapour super-saturation in snow andunder-saturation in soil.In some model runs, the coefficient of water vapourdiffusion in soil was not calculated from equation (5),but was set equal to the coefficient of water vapourdiffusion in air in relation to soil porosity –0.14 10 4 m 2 c. In this case, the 10 cm thick snowlayer at soil surface with water vapour super-saturationreduces from 380 kg/m 3 to 180 kg/m 3 by the end of860

winter (line 2 in Figure 4). During field research onSpitsbergen, the Hibini mountains, Caucasus and theRussian Plain, we have repeatedly observed the thinningof the snow layer at soil surface associated with theformation of depth hoar. The process of the reducingthe snow density, corresponding to curve 3 in fig.4, hasbeen observed in many different geographical areas.The consequences of water vapour diffusion onmodel runs with different initial parameters are shownin Table 2.The results show that increasing the delay of thestart of snow accumulation from 0 to 16 days reducesthe value by 12 or 26% (with super-saturation/without it) using mean snow parameters, and by 16 or29% using low parameters. Taking account of moisturemigration in soil essentially reduces the value.When the minimum temperature of the air is increasedfrom 30°C to 15°C, the value reduces by 16 or29% at mean snow parameters and by 31 or 47% atlow parametersA higher air temperature or a longer snowfall leadsto a reduction in the temperature gradient and consequentlyto a reduction in water vapour diffusion.Consequently, the temperature rising from 50th till80th day up to 3°C reduces the value by approximately10–12%. When the maximum thickness ofsnow cover (70 cm) is doubled (or halved), the valuereduces (increases) by approximately 20% (15%), dueto a change in temperature gradients in the basal partof the snow cover.We determined approximate values of thinning ofthe snow layer at soil surface for various thermal andphysical properties of snow cover and conditions ofsnow accumulation.We have shown that water vapour super-saturationin snow and under-saturation in the upper layer of soil,frequently leads to increasing snow thinning as aresult of diffusion. However, further research is neededthat incorporates all the factors which influence thisprocess.REFERENCESGolubev V.N., Seliverstov Yu.G., Sokratov S.A. 1997.Peculiarities of water vapour migration on the frozenground-snow interface. Cryosphere of the Earth, vol.I,N 3; 39–43.Golubev V.N., Sokratov S.A. 1991. Evapouration of snowunder isothermal conditions. Data of glaciologicalstudies, N71, p. 27–31.Kuzmin P.P. 1961. Protses tayaniya snezhnogo pokrova. L.Gidrometeoizdat,.345p.Osokin N.I., Samoilov R.S., Sosnovskii A.V., Sokratov S.A.,Zhidkov V.A. 1999. On estimation the influence ofsnow cover characteristics variability on soils freezing.Cryosphere of the Earth, vol.III, N1; 3–10.Osokin N.I., Samoilov R.S., Sosnovskiy A.V., Sokratov S.A.,Zhidkov V.A. 2000. Model of the influence of snowcover on soil freezing. Annals of Glaciology, 31,p. 417–421.Pavlov A.V. 1979. Teplophizika landshaftov. Novosibirsk,izd-vo “Nauka”, 286p.4 CONCLUSIONSWe estimated water vapour diffusion in the snow layerat soil surface and the upper layer of soil using a mathematicalmodel that we developed.861