1. GEOtop: some of the dynamics Segantini - Mezzogiono sulle Alpi Riccardo Rigon, Stefano Endrizzi, Matteo DallAmico, Stephan GruberWednesday, June 29, 2011

2. Yes, still the snow ... What will be of the snow of the garden, what will be of free will and of destiny and of those who their way in the snow have lost suddenly .... Andrea Zanzotto (La belt, 1968)Wednesday, June 29, 2011 3. Energy and Snow BudgetsObjectives Talking about the mass an energy equations of snow And especially the snowpack evolution 3Rigon et Al.Wednesday, June 29, 2011 4. Energy and Snow Budgets The control volume4Rigon et Al.Wednesday, June 29, 2011 5. Energy and Snow Budgets Mass, Energy and Entropy of Snow There are various layersSnowUnsaturated soil Water tableFor the moment we take care of the snow layers 5Rigon et Al.Wednesday, June 29, 2011 6. Energy and Snow Budgets A snow modelAs input it has precipitation and meteorological data (temperature,relative humidity, pressure and windspeed at the ground) These are parametrized boundary conditions6Rigon et Al.Wednesday, June 29, 2011 7. Energy and Snow Budgets A snow modelIt also parameterizes atmospheric radiation and its components, andturbulence.7Rigon et Al.Wednesday, June 29, 2011 8. Energy and Snow Budgets A snow model: the real dynamics Is in the transfer of fluxes (the internal layers) 8Rigon et Al.Wednesday, June 29, 2011 9. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snow mass of snow 9Rigon et Al.Wednesday, June 29, 2011 10. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snow mass of snow 9Rigon et Al.Wednesday, June 29, 2011 11. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snowmass of ice mass of snow10Rigon et Al.Wednesday, June 29, 2011 12. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snowmass of ice mass of snow10Rigon et Al.Wednesday, June 29, 2011 13. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snow mass of ice mass of snowmass of liquid water11Rigon et Al.Wednesday, June 29, 2011 14. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snow mass of ice mass of snowmass of liquid water11Rigon et Al.Wednesday, June 29, 2011 15. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snow mass of air mass of ice mass of snowmass of liquid water12Rigon, Endrizzi, DallAmicoWednesday, June 29, 2011 16. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snowflux of liquid water phase transition13Rigon et Al.Wednesday, June 29, 2011 17. Snow Budgets The snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snow flux of liquid waterphase transitionvariation of mass per unit time 13Rigon et Al.Wednesday, June 29, 2011 18. Snow Budgets The snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snow flux of liquid waterphase transitionvariation of mass per unit time 13Rigon et Al.Wednesday, June 29, 2011 19. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snowflux of liquid water phase transition14Rigon et Al.Wednesday, June 29, 2011 20. Snow Budgets The snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snow flux of liquid waterphase transitionvariation of mass per unit time 14Rigon et Al.Wednesday, June 29, 2011 21. Snow Budgets The snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snow flux of liquid waterphase transitionvariation of mass per unit time 14Rigon et Al.Wednesday, June 29, 2011 22. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snowflux of liquid water phase transition15Rigon et Al.Wednesday, June 29, 2011 23. Snow Budgets The snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snow flux of liquid waterphase transitionvariation of mass per unit time 15Rigon et Al.Wednesday, June 29, 2011 24. Snow Budgets Mass BalanceAs in any budget, a surface layer must be implemented to set up boundaryconditions, and an internal layer to account for water transfer inside snow Snow surface layer Snow internal layers 16Rigon et Al.Wednesday, June 29, 2011 25. Snow Budgets Mass Balance of the surface layerThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snowNeve surface layermass conservation of snow17Rigon et Al.Wednesday, June 29, 2011 26. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snowSnow surface layer18Rigon, Endrizzi, DallAmicoWednesday, June 29, 2011 27. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snow Snow surface layer variation of mass per unit time18Rigon, Endrizzi, DallAmicoWednesday, June 29, 2011 28. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snow Snow surface layer total precipitation variation of mass per unit time 18Rigon, Endrizzi, DallAmicoWednesday, June 29, 2011 29. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snowevaporation/sublimation Snow surface layer total precipitation variation of mass per unit time18Rigon, Endrizzi, DallAmicoWednesday, June 29, 2011 30. Snow BudgetsThe snow modelThe evolution of the water equivalent of snow is obtained from the mass balanceequation: solid and liquid precipitation less the water flows due to melting andsublimation is equal to the variation in the water equivalent of the snowevaporation/sublimation Snow surface layer percolation total precipitation variation of mass per unit time 18Rigon, Endrizzi, DallAmicoWednesday, June 29, 2011 31. Snow Budgets The snow modelSnow internal layersis considered negligible. ThenOr, after dividing by the liquid water density and reference volume: 19Rigon et Al.Wednesday, June 29, 2011 32. Darcian flowThe snow modelSnow internal layers where kw and w are the intrinsic permeability of the snow to liquid water (m2) and the dynamic viscosity of liquid water (kg m1 s1) As normally, in a snowpack, capillary forces are two or three orders of magnitude less than those of gravity, the capillary pressure gradient can be neglected 20Rigon et Al.Wednesday, June 29, 2011 33. Darcian flow The snow model Snow internal layersColbeck (1972) related kl and ks to the effective water saturation (S)by means of this expression (Brooks and Corey, 1964):where S is defined as So: 21Rigon et Al.Wednesday, June 29, 2011 34. Darcian flow The snow modelSnow internal layersThe intrinsic permeability of snow at saturation is a function of many physicalproperties of a snow cover, including its density and grain size, and thedistribution, continuity, size, shapes and number of its pores (Male and Gray,1981). Shimizu (1970) proposed the following relationship:where d is the grain diameter (m), which is normally in the range of0.04-0.2 mm for new snow, 0.2-0.6 mm for fine-grained older snow and2.0-3.0 mm for older wet snow (Jordan, 1991) ).22Rigon et Al.Wednesday, June 29, 2011 35. Energy budgetsThe energy balance of snow at the surface dU = Rn lw + Rn sw H s E v + G + Pedt23Rigon et Al.Wednesday, June 29, 2011 36. Energy budgetsThe energy balance of snow at the surface dU = Rn lw + Rn sw H s E v + G + PedtRadiation budget23Rigon et Al.Wednesday, June 29, 2011 37. Energy budgetsThe energy balance of snow at the surface dU = Rn lw + Rn sw H s E v + G + PedtRadiation budgetR(i, t) = (1 ) Rsw (i, t) + Rlw (i, Ta (t)) Rlw (i, Ts (t))23Rigon et Al.Wednesday, June 29, 2011 38. Energy budgetsThe energy balance of snow at the surface dU = Rn lw + Rn sw H s E v + G + PedtRadiation budgetR(i, t) = (1 ) Rsw (i, t) + Rlw (i, Ta (t)) Rlw (i, Ts (t))23Rigon et Al.Wednesday, June 29, 2011 39. Energy budgets The energy balance of snow dU = Rn lw + Rn sw H s E v + G + Pedt24Rigon et Al.Wednesday, June 29, 2011 40. Energy budgets The energy balance of snow dU = Rn lw + Rn sw H s E v + G + Pedt Variation in energy24Rigon et Al.Wednesday, June 29, 2011 41. Energy budgets The energy balance of snow dU = Rn lw + Rn sw H s E v + G + Pedt Variation in energy Radiation balance24Rigon et Al.Wednesday, June 29, 2011 42. Energy budgets The energy balance of snow dU = Rn lw + Rn sw H s E v + G + Pedt Variation in energy Radiation balance Energy transfers due toturbulent fluxes24Rigon et Al.Wednesday, June 29, 2011 43. Energy budgets The energy balance of snow dU = Rn lw + Rn sw H s E v + G + Pedt Variation in energy Radiation balance Energy transfers due toturbulent fluxes24Rigon et Al.Wednesday, June 29, 2011 44. Energy budgets The energy balance of snow dU = Rn lw + Rn sw H s E v + G + Pedt Variation in energy Radiation balance Energy transfers due toturbulent fluxes24Rigon et Al.Wednesday, June 29, 2011 45. Energy budgets The energy balance of snow dU = Rn lw + Rn sw H s E v + G + Pedt25Rigon et Al.Wednesday, June 29, 2011 46. Energy budgets The energy balance of snow dU = Rn lw + Rn sw H s E v + G + Pedt Variation in energy25Rigon et Al.Wednesday, June 29, 2011 47. Energy budgets The energy balance of snow dU = Rn lw + Rn sw H s E v + G + Pedt Variation in energy Radiation balance25Rigon et Al.Wednesday, June 29, 2011 48. Energy budgets The energy balance of snow dU = Rn lw + Rn sw H s E v + G + Pedt Variation in energy Radiation balance Energy transfers due toturbulent fluxes25Rigon et Al.Wednesday, June 29, 2011 49. Energy budgets The energy balance of snow dU = Rn lw + Rn sw H s E v + G + Pedt Variation in energy Radiation balance Energy transfers due toturbulent fluxesConduction of heattowards the ground 25Rigon et Al.Wednesday, June 29, 2011 50. Energy budgets The energy balance of snow dU = Rn lw + Rn sw H s E v + G + Pedt Variation in energy Radiation balance Energy transfers due toturbulent fluxesConduction of heattowards the ground Energy brought from precipitation 25Rigon et Al.Wednesday, June 29, 2011 51. Energy budgets The energy balance of snowHThe flow of sensible heat depends on the surface temperature, it beingproportional to the temperature gradient between the surface and the height atwhich the sensor is measuring the air temperature.The coefficient of proportionality is greater when there is more turbulence.Therefore, the coefficient is reduced in the presence of thermal stratification andincreased in conditions of de-stratification.It is calculated by applying the similarity theory of Monin-Obukhov, which,however, is only strictly valid in flat terrains and quasi-stationary atmosphericconditions. 26Rigon et Al.Wednesday, June 29, 2011 52. Energy budgets The energy balance of snow ETSimilarly, the latent heat flux depends on the specific humidity at the interfacebetween snow and atmosphere (by assuming saturated conditions the specifichumidity is a function solely of the surface temperature) in that it isproportional to the humidity gradient between the surface and the height atwhich the sensor is measuring the air humidity.27Rigon et Al.Wednesday, June 29, 2011 53. Energy budgets The Snow Energy Budget in the internallayers variation of the energy of snowenergy fluxes at the boundary phase transition28Rigon et Al.Wednesday, June 29, 2011 54. Energy budgets The Snow Energy Budget in the internallayers variation of the energy of snowenergy fluxes at the boundary phase transition28Rigon et Al.Wednesday, June 29, 2011 55. Energy budgets The Snow Energy Budget energy of snow energy fluxesat the boundaryphase transition 29Rigon et Al.Wednesday, June 29, 2011 56. Energy budgets The Snow Energy Budget energy of snow energy fluxesat the boundaryphase transition 29Rigon et Al.Wednesday, June 29, 2011 57. Energy budgets The Snow Energy Budget energy of snow energy fluxesat the boundaryphase transition 30Rigon et Al.Wednesday, June 29, 2011 58. Energy budgets The Snow Energy Budgetheating/cooling by conductionheating/cooling by advection (mainly of liquid water)31Wednesday, June 29, 2011 59. Energy budgetsThe Snow Internal Energy variation on the energy of snow 32Rigon et Al.Wednesday, June 29, 2011 60. Energy budgetsThe Snow Internal Energy variation on the energy of snow 32Rigon et Al.Wednesday, June 29, 2011 61. Energy budgetsThe Snow Internal Energy variation on the energy of snowA part depends on temperature33Rigon et Al.Wednesday, June 29, 2011 62. Energy budgetsThe Snow Internal Energy variation on the energy of snowA part depends on temperature33Rigon et Al.Wednesday, June 29, 2011 63. Energy budgetsThe Snow Internal Energy variation on the A part depends on the energy of snow substanceA part depends on temperature34Rigon et Al.Wednesday, June 29, 2011 64. Energy budgets You can believe me that the energy hasthe previous form. Or try to get it by yourself from thebasic definitions ;-)35Rigon et Al.Wednesday, June 29, 2011 65. Energy budgets You can believe me that the energy hasthe previous form. Or try to get it by yourself from thebasic definitions ;-) Then you get in troubles!35Rigon et Al.Wednesday, June 29, 2011 66. Energy budgets revisited How does it relates with?36 Rigon et Al.Wednesday, June 29, 2011 67. Energy budgets revisitedIn fact, the formula takes a different routethrough the definition of entalphy which is an equivalent of the energy (for details, DallAmico, 2010), and 37 Rigon et Al.Wednesday, June 29, 2011 68. Energy budgets revisited And the thing complicates a little more if you take the time variation of it: just because of the Gibbs-Duhem identity38 Rigon et Al.Wednesday, June 29, 2011 69. Energy budgets revisitedFinallyOne discovers that hentalphy can be approximated as a functionoftemperature (and pressure actually) as: where the derivative of hentalphy is used as quite often that has the name of thermal capacity (at constant pressure) 39 Rigon et Al.Wednesday, June 29, 2011 70. Energy budgets revisitedWe are not there but lets stop this story for a moment and look at the other terms of the energy budget. The heating/ cooling by conduction heating or cooling or the heat flux40 Rigon et Al.Wednesday, June 29, 2011 71. Energy budgets revisitedWe are not there but lets stop this story for a moment and look at the other terms of the energy budget. The heating/ cooling by conduction heating or cooling or the heat flux40 Rigon et Al.Wednesday, June 29, 2011 72. Energy budgets revisitedWe are not there but lets stop this story for a moment and look at the other terms of the energy budget. The heating/ cooling by conductiontemperaturegradient heating or cooling or the heat flux41 Rigon et Al.Wednesday, June 29, 2011 73. Energy budgets revisitedWe are not there but lets stop this story for a moment and look at the other terms of the energy budget. The heating/ cooling by conductiontemperaturegradient heating or cooling or the heat flux41 Rigon et Al.Wednesday, June 29, 2011 74. Energy budgets revisitedWe are not there but lets stop this story for a moment and look at the other terms of the energy budget. The heating/ cooling by conductiontemperaturegradient This is Osangers theory that brings to heating orFouriers law! cooling or the heat flux thermal conductivity42 Rigon et Al.Wednesday, June 29, 2011 75. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature43Rigon et Al.Wednesday, June 29, 2011 76. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature43Rigon et Al.Wednesday, June 29, 2011 77. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature43Rigon et Al.Wednesday, June 29, 2011 78. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature43Rigon et Al.Wednesday, June 29, 2011 79. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature43Rigon et Al.Wednesday, June 29, 2011 80. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature43Rigon et Al.Wednesday, June 29, 2011 81. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature43Rigon et Al.Wednesday, June 29, 2011 82. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature43Rigon et Al.Wednesday, June 29, 2011 83. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature43Rigon et Al.Wednesday, June 29, 2011 84. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature43Rigon et Al.Wednesday, June 29, 2011 85. MetamorphismsAccumulation periodTranslated in terms of energy balance. For T < 0 C, at the top layer Variation of the internal energy of the snow44Rigon et Al.Wednesday, June 29, 2011 86. Metamorphisms Accumulation periodTranslated in terms of energy balance. For T < 0 C, at the top layer in the other layers45Rigon et Al.Wednesday, June 29, 2011 87. MetamorphismsMelting of the snowpackThe accumulation phase is followed by the snow melting phase.At the beginning of the snow melting phase the snowpack is generallymade up of layers of varying density. The melting process is obviouslylinked to the radiative input.However, given the elevated albedo of snow, the direct importance ofradiation can be of limited importance.While melting, the density of the snowpack increases and the verticalvariation tends to disappear. During the melting process the density canfluctuate on an hourly and daily basis. 46Rigon et Al.Wednesday, June 29, 2011 88. MetamorphismsMelting of the snowpack Schematically, three phases of the melting period are distinguished: heating maturation flow generation47Rigon et Al.Wednesday, June 29, 2011 89. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature48Rigon et Al.Wednesday, June 29, 2011 90. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature48Rigon et Al.Wednesday, June 29, 2011 91. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature48Rigon et Al.Wednesday, June 29, 2011 92. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature48Rigon et Al.Wednesday, June 29, 2011 93. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature48Rigon et Al.Wednesday, June 29, 2011 94. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature48Rigon et Al.Wednesday, June 29, 2011 95. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature48Rigon et Al.Wednesday, June 29, 2011 96. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature48Rigon et Al.Wednesday, June 29, 2011 97. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature48Rigon et Al.Wednesday, June 29, 2011 98. Metamorphisms Snowpack dynamics at mid and higher latitudesMeltingAccumulation MaturationMelting Snow water equivalent RunoffTemperature48Rigon et Al.Wednesday, June 29, 2011 99. MetamorphismsMelting of the snowpackthe maturation phase (T = 0 C) The maturation phase of the melting process occurs when the snowpack is an isotherm at T = 0 C. From this point on, any further increase in energy produces meltwater, which is initially trapped in the pores by surface tension. 49Rigon et Al.Wednesday, June 29, 2011 100. Metamorphisms Melting of the snowpackThe snowpack does not advance linearly through these three phases: rather itfollows the daily temperature trends, and typically the melting takes place atthe surface layers in contact with the warm air.The water then percolates downwards and recondenses, releasing latent heat,and so contributes to raising the temperature of the snowpack.During the night the melting snow can refreeze and so the process can carry onfor various days in a row. 50Rigon et Al.Wednesday, June 29, 2011 101. Metamorphisms Melting of the snowpackWhich of the two phases exists is solely a function of pressure andtemperature, and it depends on the chemical potential of water and ice.The phase that is present is (with very high probability) the phase withthe lower chemical potential: this is a consequence of the first and secondlaws of thermodynamics.51Rigon et Al.Wednesday, June 29, 2011 102. MetamorphismsMelting of the snowpack The equivalence of chemical potentials i (T, p) = w (T, p)identifies, in the (T,p) plane, the separation curve between phases (solid andliquid) which is given by a Clausius-Clapeyron relationship 52Rigon et Al.Wednesday, June 29, 2011 103. MetamorphismsMelting of the snowpackFurthermore, there remains the equilibrium case, which is not well defined bythermodynamics, when:T = 0 Cat this temperature (with p ~ 105 Pa), according to the scholastic view, phasechange occurs. This means that at this temperature both phases can co-exist in arbitrary proportions. 53Rigon et Al.Wednesday, June 29, 2011 104. MetamorphismsMelting of the snowpackLet us suppose, however, that the temperature of the system with which thesnow is in contact is slightly greater than zero. In these circumstances thesnow is: slightly heated transformed to waterthe thermal energy supplied by the system is, during this process, storedas internal potential energy of the water and the temperature of theremaining snow stays:T = 0 Cuntil all of the snow has melted. Only after this can the temperature rise. 54Rigon et Al.Wednesday, June 29, 2011 105. Metamorphisms Melting of the snowpack energywise Lets assume that the pressure is constant. Then: 55Rigon et Al.Wednesday, June 29, 2011 106. Metamorphisms And But T=0 at the phase transition. Then 56Rigon et Al.Wednesday, June 29, 2011 107. Metamorphisms Which can be understood if and therefore57Rigon et Al.Wednesday, June 29, 2011 108. Metamorphisms Furthermorethe difference if the entalphies of water and ice are definend to be the entalphyof fusion of ice: Usually the specific entalphy of ice is taken as a reference and to be null. So And therefore: 58Rigon et Al.Wednesday, June 29, 2011 109. Metamorphisms To sum upWhere we are now able to express the flux of advected energy in terms of theentalphy (i.e. the internal energy at constant pressure) of water expendable inthe process.59Rigon et Al.Wednesday, June 29, 2011 110. Metamorphisms Equations to Solve60Rigon et Al.Wednesday, June 29, 2011 111. Phase transitions complexities Do we forgot something ? Capillary forces in the snow cause however a fraction of liquid water to be retained in the snowpack and to be prevented from draining away. Colbeck (1972) defined the irreducible water saturation (sr) as the minimum liquid level (expressed as a fraction of porosity) to which a snow cover can be drained at the atmospheric pressure. In a literary review, Kattelmann (1986) showed that the irreducible water content is highly variable, ranging from 0 to 0.4, which corresponds for the relative saturation to ranging from 0.014 and 0.069 for a snow of density 250 kg m3. 61 Rigon et Al.Wednesday, June 29, 2011 112. Phase transitions complexitiesCapillary water ? It should be noted that once liquid water is present, in the form of capillary water, it refreezes with difficulty because of freezing pointdepression, which is due to the capillary forces (surface tension) thatalter the energy balance values that lead to an estimate of the chemicalpotential. free watercapillary water 62 Rigon et Al.Wednesday, June 29, 2011 113. Phase transitions complexities Solutes A similar effect is observed when, for any reason, there are solutes presentin the water. free waterwater with solute63 Rigon et Al.Wednesday, June 29, 2011 114. Phase transitions complexities Freezing point depression It can be calculated by generalising the Clausius-Clapeyron equation. Freezing point Specific volume Specific volume depressionand pressureand pressureof the iceof the waterFreezing point 64 Rigon et Al.Wednesday, June 29, 2011 115. Numerics Numerics65Rigon et Al.Wednesday, June 29, 2011 116. Numerics Top Boundary Conditions Energy66Rigon et Al.Wednesday, June 29, 2011 117. Numerics Bottom Boundary Conditions Energy 67Rigon et Al.Wednesday, June 29, 2011 118. Numerics Top and Bottom Boundary Conditions Mass68Rigon et Al.Wednesday, June 29, 2011 119. Parameters Energy balance parameters- the air temperature above which all precipitations are liquid (2 C)- the air temperature below which all precipitations are snow (0 C)- the radiative emissivity of snow, which is close to 1 (0.98)- the water content that the snow can retain by capillary action, expressed as a fraction of the porosity (0.05) 69Rigon et Al.Wednesday, June 29, 2011 120. Parameters Energy balance parameters- the saturated hydraulic conductivity of snow (~ 5.55 kg/(m2*s))- the surface thermal conductivity of snow (~ 5.55*10^-5 m/s)- the depth of albedo extinction (50 mm water equivalent): the albedo iscalculated with an algorithm which is a function of the age of the snow.However, when the snow cover is less than this value it is assumed that the snow cover is not continuous, but rather distributed in zones. In these cases the albedo that is used is calculated as the average of the albedo calculated on the basis of the age of the snow and the albedo of the bare soil, which must be considered as another parameter.70Rigon et Al.Wednesday, June 29, 2011 121. Parameters Energy balance parameters- the roughness length for temperature (0.05 m): the vertical temperatureprofile in the atmosphere, in turbulent conditions, is logarithmic; therefore it is necessary to define an altitude, said roughness length, so that the logarithmic profile can be considered valid for altitudes greater than this length. The roughness length is a function of the surface roughness. It can be demonstrated that if this parameter diminishes then there is an increase in the proportionality coefficient between the sensible and latent heat fluxes and their respective gradients.71Rigon et Al.Wednesday, June 29, 2011 122. ParametersEnergy balance parameters- the roughness length for the windspeed (0.5 m*): that described for the temperature is also valid for windspeed. The two roughness lengths are correlated: normally the windspeed roughness length is between 7 and 10 times greater than the temperature roughness length* the roughness length is effectively very high with respect to real physical conditions. In reality it should be of the order of 0.0001 m, but the parameter serves to take account of the existence of a small but important sub-layer of atmosphere where the dynamics are laminar. 72Rigon et Al.Wednesday, June 29, 2011 123. ParametersEnergy balance parameters - soil density (1600 kg/m3) - the thickness of thermally active soil (0.4 m), that is, the thickness of soil that, in the absence of snow, is subject to an appreciable dailythermal excursion - the heat capacity of the soil (890 J/(kg * K)), considered constant, butin reality it is highly variable in function of soil properties - the albedo of the soil not covered by snow (0.2), variable in function of land use 73Rigon et Al.Wednesday, June 29, 2011 124. ParametersRun it! ParameterInitialBoundary estimation conditions ConditionsRun the code!Print the result74Rigon et Al.Wednesday, June 29, 2011 125. Parameters Thank you for your attention. G.Ulrici - 2000 ? 75Wednesday, June 29, 2011