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2 geotop-summer-school2011
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2 geotop-summer-school2011

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Some of the dynamics of the GEOtop model revealed. Especially regarding snow modeling.

Some of the dynamics of the GEOtop model revealed. Especially regarding snow modeling.

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  • 1. GEOtop: some of the dynamics Segantini - Mezzogiono sulle Alpi Riccardo Rigon, Stefano Endrizzi, Matteo Dall’Amico, 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 beltà, 1968)Wednesday, June 29, 2011
  • 3. Energy and Snow Budgets Objectives •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 volume 4Rigon et Al.Wednesday, June 29, 2011
  • 5. Energy and Snow Budgets Mass, Energy and Entropy of Snow There are various layers Snow Unsaturated soil Water table For the moment we take care of the snow layers 5Rigon et Al.Wednesday, June 29, 2011
  • 6. Energy and Snow Budgets A snow model As input it has precipitation and meteorological data (temperature, relative humidity, pressure and windspeed at the ground) These are parametrized boundary conditions 6Rigon et Al.Wednesday, June 29, 2011
  • 7. Energy and Snow Budgets A snow model It also parameterizes atmospheric radiation and its components, and turbulence. 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 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 mass of snow 9Rigon et Al.Wednesday, June 29, 2011
  • 10. 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 mass of snow 9Rigon et Al.Wednesday, June 29, 2011
  • 11. 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 mass of ice mass of snow 10Rigon et Al.Wednesday, June 29, 2011
  • 12. 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 mass of ice mass of snow 10Rigon et Al.Wednesday, June 29, 2011
  • 13. 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 mass of ice mass of snow mass of liquid water 11Rigon et Al.Wednesday, June 29, 2011
  • 14. 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 mass of ice mass of snow mass of liquid water 11Rigon et Al.Wednesday, June 29, 2011
  • 15. 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 mass of air mass of ice mass of snow mass of liquid water 12Rigon, Endrizzi, Dall’AmicoWednesday, June 29, 2011
  • 16. 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 water phase transition 13Rigon 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 water phase transition variation 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 water phase transition variation of mass per unit time 13Rigon et Al.Wednesday, June 29, 2011
  • 19. 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 water phase transition 14Rigon 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 water phase transition variation 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 water phase transition variation of mass per unit time 14Rigon et Al.Wednesday, June 29, 2011
  • 22. 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 water phase transition 15Rigon 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 water phase transition variation 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 snow Neve surface layer mass conservation of snow 17Rigon et Al.Wednesday, June 29, 2011
  • 26. 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 Snow surface layer 18Rigon, Endrizzi, Dall’AmicoWednesday, June 29, 2011
  • 27. 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 Snow surface layer variation of mass per unit time 18Rigon, Endrizzi, Dall’AmicoWednesday, June 29, 2011
  • 28. 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 Snow surface layer total precipitation variation of mass per unit time 18Rigon, Endrizzi, Dall’AmicoWednesday, June 29, 2011
  • 29. 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 evaporation/sublimation Snow surface layer total precipitation variation of mass per unit time 18Rigon, Endrizzi, Dall’AmicoWednesday, June 29, 2011
  • 30. 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 evaporation/sublimation Snow surface layer percolation total precipitation variation of mass per unit time 18Rigon, Endrizzi, Dall’AmicoWednesday, June 29, 2011
  • 31. Snow Budgets The snow model Snow internal layers is considered negligible. Then Or, after dividing by the liquid water density and reference volume: 19Rigon et Al.Wednesday, June 29, 2011
  • 32. Darcian flow The snow model Snow internal layers where kw and μw are the intrinsic permeability of the snow to liquid water (m2) and the dynamic viscosity of liquid water (kg m−1 s−1) 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 layers Colbeck (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 model Snow internal layers The intrinsic permeability of snow at saturation is a function of many physical properties of a snow cover, including its density and grain size, and the distribution, 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 of 0.04-0.2 mm for new snow, 0.2-0.6 mm for fine-grained older snow and 2.0-3.0 mm for older wet snow (Jordan, 1991) ). 22Rigon et Al.Wednesday, June 29, 2011
  • 35. Energy budgets The energy balance of snow at the surface dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt 23Rigon et Al.Wednesday, June 29, 2011
  • 36. Energy budgets The energy balance of snow at the surface dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Radiation budget 23Rigon et Al.Wednesday, June 29, 2011
  • 37. Energy budgets The energy balance of snow at the surface dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Radiation budget ∆R(i, t) = (1 − α∗ ) R↓sw (i, t) + R↓lw (i, Ta (t)) − R↑lw (i, Ts (t)) 23Rigon et Al.Wednesday, June 29, 2011
  • 38. Energy budgets The energy balance of snow at the surface dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Radiation budget ∆R(i, t) = (1 − α∗ ) R↓sw (i, t) + R↓lw (i, Ta (t)) − R↑lw (i, Ts (t)) 23Rigon et Al.Wednesday, June 29, 2011
  • 39. Energy budgets The energy balance of snow dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt 24Rigon et Al.Wednesday, June 29, 2011
  • 40. Energy budgets The energy balance of snow dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Variation in energy 24Rigon et Al.Wednesday, June 29, 2011
  • 41. Energy budgets The energy balance of snow dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Variation in energy Radiation balance 24Rigon et Al.Wednesday, June 29, 2011
  • 42. Energy budgets The energy balance of snow dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Variation in energy Radiation balance Energy transfers due to turbulent fluxes 24Rigon et Al.Wednesday, June 29, 2011
  • 43. Energy budgets The energy balance of snow dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Variation in energy Radiation balance Energy transfers due to turbulent fluxes 24Rigon et Al.Wednesday, June 29, 2011
  • 44. Energy budgets The energy balance of snow dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Variation in energy Radiation balance Energy transfers due to turbulent fluxes 24Rigon et Al.Wednesday, June 29, 2011
  • 45. Energy budgets The energy balance of snow dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt 25Rigon et Al.Wednesday, June 29, 2011
  • 46. Energy budgets The energy balance of snow dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Variation in energy 25Rigon et Al.Wednesday, June 29, 2011
  • 47. Energy budgets The energy balance of snow dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Variation in energy Radiation balance 25Rigon et Al.Wednesday, June 29, 2011
  • 48. Energy budgets The energy balance of snow dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Variation in energy Radiation balance Energy transfers due to turbulent fluxes 25Rigon et Al.Wednesday, June 29, 2011
  • 49. Energy budgets The energy balance of snow dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Variation in energy Radiation balance Energy transfers due to turbulent fluxes Conduction of heat towards the ground 25Rigon et Al.Wednesday, June 29, 2011
  • 50. Energy budgets The energy balance of snow dU∗ = Rn lw + Rn sw − H − λ s E v + G + Pe dt Variation in energy Radiation balance Energy transfers due to turbulent fluxes Conduction of heat towards the ground Energy brought from precipitation 25Rigon et Al.Wednesday, June 29, 2011
  • 51. Energy budgets The energy balance of snow H∗The 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 λET Similarly, the latent heat flux depends on the specific humidity at the interface between snow and atmosphere (by assuming saturated conditions the specific humidity is a function solely of the surface temperature) in that it is proportional to the humidity gradient between the surface and the height at which the sensor is measuring the air humidity. 27Rigon et Al.Wednesday, June 29, 2011
  • 53. Energy budgets The Snow Energy Budget in the internal layers variation of the energy of snow energy fluxes at the boundary phase transition 28Rigon et Al.Wednesday, June 29, 2011
  • 54. Energy budgets The Snow Energy Budget in the internal layers variation of the energy of snow energy fluxes at the boundary phase transition 28Rigon et Al.Wednesday, June 29, 2011
  • 55. Energy budgets The Snow Energy Budget energy of snow energy fluxes at the boundary phase transition 29Rigon et Al.Wednesday, June 29, 2011
  • 56. Energy budgets The Snow Energy Budget energy of snow energy fluxes at the boundary phase transition 29Rigon et Al.Wednesday, June 29, 2011
  • 57. Energy budgets The Snow Energy Budget energy of snow energy fluxes at the boundary phase transition 30Rigon et Al.Wednesday, June 29, 2011
  • 58. Energy budgets The Snow Energy Budget heating/cooling by conduction heating/cooling by advection (mainly of liquid water) 31Wednesday, June 29, 2011
  • 59. Energy budgets The Snow Internal Energy variation on the energy of snow 32Rigon et Al.Wednesday, June 29, 2011
  • 60. Energy budgets The Snow Internal Energy variation on the energy of snow 32Rigon et Al.Wednesday, June 29, 2011
  • 61. Energy budgets The Snow Internal Energy variation on the energy of snow A part depends on temperature 33Rigon et Al.Wednesday, June 29, 2011
  • 62. Energy budgets The Snow Internal Energy variation on the energy of snow A part depends on temperature 33Rigon et Al.Wednesday, June 29, 2011
  • 63. Energy budgets The Snow Internal Energy variation on the A part depends on the energy of snow substance A part depends on temperature 34Rigon et Al.Wednesday, June 29, 2011
  • 64. Energy budgets You can believe me that the energy has the previous form. Or try to get it by yourself from the basic definitions ;-) 35Rigon et Al.Wednesday, June 29, 2011
  • 65. Energy budgets You can believe me that the energy has the previous form. Or try to get it by yourself from the basic 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 revisited In fact, the formula takes a different route through the definition of entalphy which is an equivalent of the energy (for details, Dall’Amico, 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 identity 38 Rigon et Al.Wednesday, June 29, 2011
  • 69. Energy budgets revisited Finally One discovers that hentalphy can be approximated as a function of temperature (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 revisited We are not there but let’s 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 flux” 40 Rigon et Al.Wednesday, June 29, 2011
  • 71. Energy budgets revisited We are not there but let’s 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 flux” 40 Rigon et Al.Wednesday, June 29, 2011
  • 72. Energy budgets revisited We are not there but let’s stop this story for a moment and look at the other terms of the energy budget. The heating/ cooling by conduction temperature gradient heating or cooling or the “heat flux” 41 Rigon et Al.Wednesday, June 29, 2011
  • 73. Energy budgets revisited We are not there but let’s stop this story for a moment and look at the other terms of the energy budget. The heating/ cooling by conduction temperature gradient heating or cooling or the “heat flux” 41 Rigon et Al.Wednesday, June 29, 2011
  • 74. Energy budgets revisited We are not there but let’s stop this story for a moment and look at the other terms of the energy budget. The heating/ cooling by conduction temperature gradient This is Osanger’s theory that brings to heating or Fourier’s law! cooling or the “heat flux” thermal conductivity 42 Rigon et Al.Wednesday, June 29, 2011
  • 75. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 43Rigon et Al.Wednesday, June 29, 2011
  • 76. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 43Rigon et Al.Wednesday, June 29, 2011
  • 77. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 43Rigon et Al.Wednesday, June 29, 2011
  • 78. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 43Rigon et Al.Wednesday, June 29, 2011
  • 79. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 43Rigon et Al.Wednesday, June 29, 2011
  • 80. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 43Rigon et Al.Wednesday, June 29, 2011
  • 81. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 43Rigon et Al.Wednesday, June 29, 2011
  • 82. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 43Rigon et Al.Wednesday, June 29, 2011
  • 83. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 43Rigon et Al.Wednesday, June 29, 2011
  • 84. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 43Rigon et Al.Wednesday, June 29, 2011
  • 85. Metamorphisms Accumulation period Translated in terms of energy balance. For T < 0 ºC, at the top layer Variation of the internal energy of the snow 44Rigon et Al.Wednesday, June 29, 2011
  • 86. Metamorphisms Accumulation period Translated in terms of energy balance. For T < 0 ºC, at the top layer in the other layers 45Rigon et Al.Wednesday, June 29, 2011
  • 87. Metamorphisms Melting of the snowpack The accumulation phase is followed by the snow melting phase. At the beginning of the snow melting phase the snowpack is generally made up of layers of varying density. The melting process is obviously linked to the radiative input. However, given the elevated albedo of snow, the direct importance of radiation can be of limited importance. While melting, the density of the snowpack increases and the vertical variation tends to disappear. During the melting process the density can fluctuate on an hourly and daily basis. 46Rigon et Al.Wednesday, June 29, 2011
  • 88. Metamorphisms Melting of the snowpack Schematically, three phases of the melting period are distinguished: •heating •maturation •flow generation 47Rigon et Al.Wednesday, June 29, 2011
  • 89. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 48Rigon et Al.Wednesday, June 29, 2011
  • 90. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 48Rigon et Al.Wednesday, June 29, 2011
  • 91. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 48Rigon et Al.Wednesday, June 29, 2011
  • 92. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 48Rigon et Al.Wednesday, June 29, 2011
  • 93. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 48Rigon et Al.Wednesday, June 29, 2011
  • 94. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 48Rigon et Al.Wednesday, June 29, 2011
  • 95. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 48Rigon et Al.Wednesday, June 29, 2011
  • 96. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 48Rigon et Al.Wednesday, June 29, 2011
  • 97. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 48Rigon et Al.Wednesday, June 29, 2011
  • 98. Metamorphisms Snowpack dynamics at mid and higher latitudes Melting Accumulation Maturation Melting Snow water equivalent Runoff Temperature 48Rigon et Al.Wednesday, June 29, 2011
  • 99. Metamorphisms Melting of the snowpack the maturation phase (T = 0 ºC) The maturation phase of the melting process occurs when the snowpack is an isotherm at T = 0 ºC. 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 snowpack The snowpack does not advance linearly through these three phases: rather it follows the daily temperature trends, and typically the melting takes place at the 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 on for various days in a row. 50Rigon et Al.Wednesday, June 29, 2011
  • 101. Metamorphisms Melting of the snowpack Which of the two phases exists is solely a function of pressure and temperature, and it depends on the chemical potential of water and ice. The phase that is present is (with very high probability) the phase with the lower chemical potential: this is a consequence of the first and second laws of thermodynamics. 51Rigon et Al.Wednesday, June 29, 2011
  • 102. Metamorphisms Melting of the snowpack The equivalence of chemical potentials µi (T, p) = µw (T, p) identifies, in the (T,p) plane, the separation curve between phases (solid and liquid) which is given by a Clausius-Clapeyron relationship 52Rigon et Al.Wednesday, June 29, 2011
  • 103. Metamorphisms Melting of the snowpack Furthermore, there remains the equilibrium case, which is not well defined by thermodynamics, when: •T = 0 ºC at this temperature (with p ~ 105 Pa), according to the scholastic view, phase change occurs. This means that at this temperature both phases can co- exist in arbitrary proportions. 53Rigon et Al.Wednesday, June 29, 2011
  • 104. Metamorphisms Melting of the snowpack Let us suppose, however, that the temperature of the system with which the snow is in contact is slightly greater than zero. In these circumstances the snow is: • slightly heated • transformed to water the thermal energy supplied by the system is, during this process, stored as internal potential energy of the water and the temperature of the remaining snow stays: •T = 0 ºC until 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 Let’s 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 therefore 57Rigon et Al.Wednesday, June 29, 2011
  • 108. Metamorphisms Furthermore the difference if the entalphies of water and ice are definend to be the entalphy of 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 up Where we are now able to express the flux of advected energy in terms of the entalphy (i.e. the internal energy at constant pressure) of water expendable in the process. 59Rigon et Al.Wednesday, June 29, 2011
  • 110. Metamorphisms Equations to Solve 60Rigon 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 m−3. 61 Rigon et Al.Wednesday, June 29, 2011
  • 112. Phase transitions complexities Capillary water ? It should be noted that once liquid water is present, in the form of capillary water, it refreezes with difficulty because of freezing point depression, which is due to the capillary forces (surface tension) that alter the energy balance values that lead to an estimate of the chemical potential. free water capillary 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 present in the water. free water water with solute 63 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 depression and pressure and pressure of the ice of the water Freezing point 64 Rigon et Al.Wednesday, June 29, 2011
  • 115. Numerics Numerics 65Rigon et Al.Wednesday, June 29, 2011
  • 116. Numerics Top Boundary Conditions Energy 66Rigon 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 Mass 68Rigon 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 is calculated 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 temperature profile 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. Parameters Energy 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. Parameters Energy 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 daily thermal excursion - the heat capacity of the soil (890 J/(kg * K)), considered constant, but in 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. Parameters Run it! Parameter Initial Boundary estimation conditions Conditions Run the code! Print the result 74Rigon et Al.Wednesday, June 29, 2011
  • 125. Parameters Thank you for your attention. G.Ulrici - 2000 ? 75Wednesday, June 29, 2011

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