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GEOtop 2.0 - The snow and freezing soil modelling

The document discusses a simulation framework, geotop 2.0, used to model energy and water balance dynamics in alpine environments, specifically considering the cryosphere's impact on hydrology. It includes equations for snow metamorphism and energy budgets, as well as various parameterizations essential for modeling. The work emphasizes the integration of traditional hydrology with cryospheric processes and outlines a new snow model's structure and capabilities.

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Gino Severini, Blue Dancer, 1912 - Gugghenaim museum, Venice GEOtop 2.0 Simulating the combined energy and water balance at and below the land surface accounting for soil freezing, snow cover and terrain effects S. Endrizzi, S. Gruber, M. Dall’Amico and R. Rigon Dec. 12, 2013 - AGU Fall Meeting S. Francisco
Thermodynamics today is a blend of statements from most of the founders: Gibbs, Planck, Boltzmann, even from information theory. Confusion is nearly universal. Constitutive properties are not delimited, just pulled out from under the table as needed. ! C. Truesdell
I am just a poor hydrologists Well, Do not blame me, I am not a cryospheric guy my co-authors are ;-) Just pretending to become it sometimes My story starts instead from the more traditional warm hydrology, realising that in alpine areas, the cryosphere cannot be neglected. !3 Endrizzi et al.
Already talked about GEOtop in this AGU Warm hydrology in GEOtop http://abouthydrology.blogspot.com/2013/12/geotop-20-at-agu-2013.html Endrizzi et al. !4
Equations (Monin - Obukov) Snow metamorphism (with some assumptions) Energy budget Radiation Flux-gradient relationship Double layer vegetation Diffusive approximation to shallow water equation Richards equation + van Genuchten parameterization + Mualem derived conductivity !5 Endrizzi et al.
Equations The Cryospheric processes parameterisation enter mainly Snow metamorphism (with some assumptions) Energy budget here Richards equation + van Genuchten parameterization + Mualem derived conductivity !6 Endrizzi et al.
Grid size based A snow model It also parameterizes atmospheric radiation and its components, and turbulence. !7 Endrizzi et al.
Grid size based The control volume *New multi-layering with dynamic grid Endrizzi et al. !8
Equations again The new* snow model (the energy budget) Chapter 10 <latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit> Snow <latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit> 10.1 Introduction <latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit> Figure 10.1: Snow stratigraphy *For the old one see Zanotti et al., 2004 Endrizzi et al. 10.2 Input !9
Also The new* snow model The snow compaction, grain size (after Anderson, 1976 and Jordan et al., 1999) Water percolation in snow (after Colbeck, 1972) http://www.its.caltech.edu/~atomic/snowcrystals/photos/w031224a130.jpg *For the old one see Zanotti et al., 2004 Endrizzi et al. !10
Endrizzi et al. 2013 The whole story here see also http://abouthydrology.blogspot.com/search/label/GEOtop Endrizzi et al. !11
Very well tested  Gubler S., Endrizzi S., Gruber S., Purves R. S., Sensitivity and uncertainty of modeled ground temperatures and related variables in mountain environments, Geosci. 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Use it ! So well tested that is confidently used for real-time forecasting (driven by ground data) !13 Endrizzi et al.
The long story of soil freezing What about soil freezing ? The Cryosphere, 5, 469–484, 2011 www.the-cryosphere.net/5/469/2011/ doi:10.5194/tc-5-469-2011 © Author(s) 2011. CC Attribution 3.0 License. The Cryosphere A robust and energy-conserving model of freezing variably-saturated soil M. Dall’Amico1,* , S. Endrizzi2 , S. Gruber2 , and R. Rigon1 1 Department of Civil and Environmental Engineering, University of Trento, Trento, Italy 2 Department of Geography, University of Zurich, Winterthurerstrasse 190, Zurich, Switzerland * now at: Mountain-eering srl, Via Siemens 19, Bolzano, Italy Received: 29 June 2010 – Published in The Cryosphere Discuss.: 11 August 2010 Revised: 18 May 2011 – Accepted: 19 May 2011 – Published: 1 June 2011 Abstract. Phenomena involving frozen soil or rock are imand numerical physically-based (Zhang et al., 2008). Emportant in many natural Ph.D thesis: a consequence, there pirical and semiempirical algorithms relate ground thawing- 4 see also Dall’Amico systems and, ashttp://eprints-phd.biblio.unitn.it/335/ !1 is a great interest in the modeling of their behavior. Few freezing depth to some aspect of surface forcing by one or Endrizzimodels exist that describe this process for both saturated and more experimentally established coefficients (e.g. Anisimov et al. unsaturated soil and in conditions of freezing and thawing, et al., 2002). Analytical algorithms are specific solutions to
in vadose and saturated conditions Two cases is hydraulic head [L] of water <latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit> Endrizzi et al. !15
Back to the basics Two equations (just one here) internal kinetic potential energy fluxes at energy energy energy the boundaries first principle second principle more details onhttp://abouthydrology.blogspot.com/2013/04/beyond-and-side-by-side-with-numerics.html Endrizzi et al. !16
Back to the basics Four phases !17 Endrizzi et al.
Back to the basics Water is •in pores •often in unsaturated conditions •it is known that it does not freeze until very negative temperatures are obtained •a relationship (the Soil Water Retention Curves needs to be invoked between water head and water content to close the equations) !18 Endrizzi et al.
Capillarity (and other stuff) Unsaturated conditions means that capillary forces acts, i.e. we have to account for the tension forces that accumulate in curves surfaces !19 Endrizzi et al.
Capillarity (and other stuff) Unsaturated conditions Young-Laplace equation pw = pa wa ⇤Awa (r) = pa ⇤Vw (r) wa ⇤Awa /⇤r = pa ⇤Vw /⇤r wa 2 := pa r pwa (r) !20 Endrizzi et al.
Capillarity (and other stuff) In unsaturated conditions the equilibrium condition: !21 Endrizzi et al.
Capillarity (and other stuff) In unsaturated conditions the equilibrium condition becomes !22 Endrizzi et al.
Capillarity (and other stuff) So, skipping a few passages The situation at the freezing point is the opposite, and represented by the blue arrow Freezing point depression !23 Endrizzi et al.
Capillarity (and other stuff) Because, the smaller the pores, the larger the freezing point depression ! larger pores freezes before than smaller pores !24 Endrizzi et al.
Beyond the Stefan problem Because by means of the Clausius-Clapeyron equation there is a one-to-one relations between the size of the pores and the temperature depression, and because there is also a one-to-one relationship between the size of the pores and the pressure there is a one-one relation among T and !25 Endrizzi et al.
Capillarity (and other stuff) Unsaturated unfrozen Unsaturated Frozen Freezing starts Freezing procedes !26 Endrizzi et al.
Four phases … well interfaces are phases too, indeed Curved interfaces with three phases pw0 = pa wa ⇥Awa (r0 ) = pa ⇥Vw pw1 = pa pwa (r0 ) ia p i = pa ⇥Aia r(0) ⇥Vw iw ia ⇥Aia (r0 ) := pa ⇥Vw ⇥Aiw (r1 ) ⇥Vw Two interfaces (air-ice and water- ice) should be considered!!! Endrizzi et al. pia (r0 ) !27
Making it short Now we have enough information to write the right equations ! Perhaps If we do not get lost in simplifications !28 Endrizzi et al.
Freezing=Drying A further assuption To make it manageable, we do a further assumption. Mainly the freezing=drying one. Considering the assumption “freezing=drying” (Miller, 1963) the ice “behaves like air” and does not add further pressure terms Endrizzi et al. !29
Freezing=Drying How this reflects on pressure head pressure head: pw ⇥w = w g Unfrozen water content w (T ) = w [⇥w (T )] soil water + retention curve Endrizzi et al. thermodynamic equilibrium (Clausius Clapeyron) !30
Freezing=Drying Soil water retention curves Unsaturated unfrozen Unsaturated Frozen Freezing starts Freezing procedes !31 Endrizzi et al.
Freezing=Drying Soil water retention curves !32 Endrizzi et al.
Freezing=Drying Soil water retention curves !33 Endrizzi et al.
Water and ice mass budget Modified Richards equations = ⇥r + (⇥s Total water content: liquid water content: ⇥w = ⇥r + (⇥s ice content: depressed melting point i ⇥w = ⇥i T := T0 + ⇤ ⇥r ) · {1 + [ ⇥r ) · 1 + w g T0 Lf ⇥ ⇤w0 n · ⇤w0 ] } Lf (T g T0 m T ⇥ ) · H(T ⇥n ⌅ T ⇥) m w0 !34 Endrizzi et al.
Water and ice energy budget in soil Energy Equation ph (pw Vw + pi Vi ) + µw Mw + µi Miph U = hg Mg + hw Mw + hi Mi 0 assuming equilibrium thermodynamics: µw=µi and Mwph = -Miph 0 assuming freezing=drying 0 assuming: Eventually: U = Cg (1 s) T + ⇥w cw w U ⌥ ⌥ ⌥ + ⇥ • (G + J) + Sen = 0 t T + ⇥i ci i no expansion: ρw=ρi T + ⇥w Lf ⌃ G= J= no flux during phase change w T (⇥w0 , T ) w ⌃ · ⇤T · Jw (⇥w0 , T ) · [Lf + cw T ] conduction advection !35 Endrizzi et al.
Water and ice mass and energy budget together Finally the “right” equations 1D representation: ⇤ ⌃ ⇧ ⌃ ⌅ ⇤U ( w0 ,T ) ⇤t ⇤ ( w0 ) ⇤t ⇤ ⇤z ⇤ ⇤z ⌥ ⇥T (⇤w0 , T ) · KH (⇤w0 , T ) · ⇤T ⇤z ⇤ w1 ( ⇥ J(⇤w0 , T ) + Sen = 0 w0 ,T ) ⇤z KH cos + Sw = 0 !36 Endrizzi et al.
No numerics, no party The right numerical methods !37 Endrizzi et al.
You need to have the right equations, and the right numerics • Finite difference discretization, semi-implicit Crank-Nicholson method; ! • Conservative linearization of the conserved quantity (Celia et al, 1990); ! • Linearization of the system through Newton-Raphson method; ! • when passing from positive to negative temperature, NewtonRaphson method is subject to big oscillations (Hansson et al, 2004) !38 Endrizzi et al.
You need to have the right equations, and the right numerics Globally convergent Newton Method if ||⌅ (⇥)m+1 || > ||⌅ (⇥)m || ⌅ ⌅ m+1 ⇤ ⌅ m ⇥ ⇥ ⌅ ⇥⇥ · reduction factor δ with 0 ≤ δ ≤ 1. If δ = 1 the scheme is the normal NewtonRaphson scheme !39 Endrizzi et al.
Going to an end So far, so good but: does it works ? <latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXN !40 Endrizzi et al.
Verifications Some verifications in Dall’Amico et al., 2011 paper !41 Endrizzi et al.
Verifications Some verifications in Endrizzi and Marsh, 2010 !42 Endrizzi et al.
Verifications Some verifications Discussion Paper in Endrizzi et al., TCD, 2011 TCD 5, 367–400, 201 | Discussion Paper Modelling the sp pattern of grou thaw in a small b in the arctic tun S. Endrizzi et a Title Page | Discussion Paper Abstract Introd Conclusions Refere Tables Figu J I J I | Endrizzi et al. | 399 Discussion Paper Fig. 8. End-of-summer thaw depth [cm] excluding lateral subsurface flow (a), excluding subsurface flow only in partially frozen soil (b), considering uniform ground heat flux and spatially variable thermal conductivity (c), and considering spatially variable ground heat flux and uniform thermal conductivity (d). Back Clo Full Screen / Esc Printer-friendly Vers Interactive Discussi !43
Verifications Some verifications in Endrizzi et al., TCD, 2011 0 −20 −40 −60 distance from the surface [cm] 0 −20 −40 measured −80 −80 measured modelled 0 10 20 30 40 50 60 0 70 10 20 30 40 50 (c) 70 (d) −20 −40 measured −80 10 20 30 40 50 60 70 relative frequency [%] 0 10 20 30 40 50 60 70 relative frequency [%] Fig. 9. Comparison of the weekly averaged frequency distributions of the thaw depth measurements performed in 1993 in the 3 plots (dashed line) and the modeled thaw depth in the strip along the west bank of the main channel reported in Fig. 1 (continuous line) for 4 characteristic weeks during the summer: (a) 24–31 May, (b) 17–23 June, (c) 15–22 July, (d) 9–13 September. Endrizzi et al. | 400 Discussion Paper 0 TCD 5, 367–400, 2011 Modelling the spatial pattern of ground thaw in a small basin in the arctic tundra S. Endrizzi et al. Title Page Abstract Introduction Conclusions References Tables Figures J I J I Back Close | −80 measured modelled −60 −20 −40 −60 modelled Discussion Paper distance from the surface [cm] 0 | distance from the surface [cm] 60 relative frequency [%] 0 relative frequency [%] Discussion Paper −60 modelled | distance from the surface [cm] Discussion Paper (b) (a) Full Screen / Esc Printer-friendly Version Interactive Discussion !44
Ideal tests Discussion Paper Some other tests in Endrizzi et al., 2013 | S a more 22 sun ore shad e m 0 gra nt die 6 3 m 1 et er s 2 | 5 ters e 0m 16 Co | Fig. 2. Synthetic catchment and location of points analysed. The converging topography has a sun-exposed and a more shaded side. It is varied with respect to the inclination angle of its hillslopes . Channel inclination, in the direction from point 3 to 4, is always 5 . Discussion Paper 4 Discussion Pap Endrizzi et al. Discussion Paper cell size: 20 meters !45
Ideal tests Some other tests in Endrizzi et al., 2013 !46 Endrizzi et al.
Ideal tests Some other tests in Endrizzi et al., 2013 !47 Endrizzi et al.
Ideal tests Some other tests in Endrizzi et al., 2013 !48 Endrizzi et al.
Conclusions This is not the end is not even the end of the beginning Lot of work was done t built a consistent framework for simulating the cryospheric processes at high resolution. Lot of testing was done for the verification of snow height, density, and and SWE against CROCUS and real data Lot of work was done for implementing sound numerical methods for both snow and freezing soil (permafrost) modelling An operational system is actually working and daily checked The thermodynamics of freezing soil was completely checked and its simplifications understood Interaction with vegetation in Tundra sites was analysed with promising results The dynamic of freezing and thawing derived is consistent with data Temperature and active layers evolution measurements are constant with observations ….. !49 Endrizzi et al.
Conclusions So we are ready to go on If you want to join the GEOtop crew, contact us: ! stefano.end@gmail.com stephan.gruber@carleton.ca matteo@mountain-eering.com riccardo.rigon@unitn.it ! Code is distribute along with GPL v 3 and available at https://code.google.com/p/geotop/ Endrizzi et al. !50
Epilogue Ulrici, 2000 ? Thank you This presentation is available at: http://abouthydrology.blogspot.com/2013/12/geotop-20-at-agu-2013-ii-cryosphere.html !51 Endrizzi et al.

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GEOtop 2.0 - The snow and freezing soil modelling

  • 1.
    Gino Severini, BlueDancer, 1912 - Gugghenaim museum, Venice GEOtop 2.0 Simulating the combined energy and water balance at and below the land surface accounting for soil freezing, snow cover and terrain effects S. Endrizzi, S. Gruber, M. Dall’Amico and R. Rigon Dec. 12, 2013 - AGU Fall Meeting S. Francisco
  • 2.
    Thermodynamics today isa blend of statements from most of the founders: Gibbs, Planck, Boltzmann, even from information theory. Confusion is nearly universal. Constitutive properties are not delimited, just pulled out from under the table as needed. ! C. Truesdell
  • 3.
    I am justa poor hydrologists Well, Do not blame me, I am not a cryospheric guy my co-authors are ;-) Just pretending to become it sometimes My story starts instead from the more traditional warm hydrology, realising that in alpine areas, the cryosphere cannot be neglected. !3 Endrizzi et al.
  • 4.
    Already talked aboutGEOtop in this AGU Warm hydrology in GEOtop http://abouthydrology.blogspot.com/2013/12/geotop-20-at-agu-2013.html Endrizzi et al. !4
  • 5.
    Equations (Monin - Obukov) Snowmetamorphism (with some assumptions) Energy budget Radiation Flux-gradient relationship Double layer vegetation Diffusive approximation to shallow water equation Richards equation + van Genuchten parameterization + Mualem derived conductivity !5 Endrizzi et al.
  • 6.
    Equations The Cryospheric processesparameterisation enter mainly Snow metamorphism (with some assumptions) Energy budget here Richards equation + van Genuchten parameterization + Mualem derived conductivity !6 Endrizzi et al.
  • 7.
    Grid size based Asnow model It also parameterizes atmospheric radiation and its components, and turbulence. !7 Endrizzi et al.
  • 8.
    Grid size based Thecontrol volume *New multi-layering with dynamic grid Endrizzi et al. !8
  • 9.
    Equations again The new*snow model (the energy budget) Chapter 10 <latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit> Snow <latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit> 10.1 Introduction <latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit> Figure 10.1: Snow stratigraphy *For the old one see Zanotti et al., 2004 Endrizzi et al. 10.2 Input !9
  • 10.
    Also The new* snowmodel The snow compaction, grain size (after Anderson, 1976 and Jordan et al., 1999) Water percolation in snow (after Colbeck, 1972) http://www.its.caltech.edu/~atomic/snowcrystals/photos/w031224a130.jpg *For the old one see Zanotti et al., 2004 Endrizzi et al. !10
  • 11.
    Endrizzi et al.2013 The whole story here see also http://abouthydrology.blogspot.com/search/label/GEOtop Endrizzi et al. !11
  • 12.
    Very well tested  GublerS., Endrizzi S., Gruber S., Purves R. S., Sensitivity and uncertainty of modeled ground temperatures and related variables in mountain environments, Geosci. Model Dev., 6, 1319–1336, 2013. <late xit sha1_ base6 4="SR 4="nY 4="ip 4="ZT E2VSo y6NJQ LknMd 4qZix /QTxH OsxPY ZL1j8 YS4gq T1F+9 3EObH MPzSO c8cRA sp22X OBo/s sUfkM hI2B6 aqlL4 yi34Q Z42A8 uI+68 =">AA <lat cfVFN cfVHL ShxBF SxxBE SgMxF SiNBF K1pY6 K0dTT E3HV6 K20zu ITjY9 RrEj9 3vbt0 hkHj5 sBDdF yCeTS Ei+Cq mI7gp BsHV0 uARyW zLjQr DAOzC O0icR mZy0J kRwoz t0u1J VERdz yCkBB sFq0J WIIzI FwHQU yiwFH nkEzm 7I9gq phupr hZ1Be tg3mM pBupr r4zU1 npqdx SR3lB r5JCu iPpuq v7Y+i LmB9z vRVN1 2MhT9 uUZZm 6df6N WQtHf DdnGT fkOuy aS34q MFvn0 /NvrJ U/z39 HghcD +ZvrM npRXT i7Lhh j3nPv ReTIx 7pFJY uiqGi yNaWl beopH XDOH7 U9pel 5ZXWu AYx/8 qRCsf NL1lb utzY2 60crq Vj9+W f/FyY W1vbO h49r6 lvvft /NVf+ 7t7d8 5+6n7 98/ba 98/ba 98/ba 98/ba 98/ba 7Y/LK v1m7f v1m7f 5UlkO 98/ba 98/ba 98/ba 98/ba 98/ba 7Y/LK v1m7f v1m7f 5UlkO 98/ba 98/ba 98/ba 1vbP7 bO7t7 fG2ns xubV9 x5nac p962T Q84cS 7UxtO ki5kc mAurL KGhjw aTcSG beFMy LuvOI IlPJQ NvC+Z BFBpS elTSY WmMol ACg0p FCjhp kySF5 3OdPF CpRwW rLAVC 41Dri 9P8/T 1lgqp Hhurg uFZ9X NYJ4y BwU9z 7m+Wv lt3n+ +xUkJ /muVv 78E6Y 7Aqdl mWIjL HsA6Y fQApx 9ac0K YaCMw fQVTi Xkio2 zBUvO zU9eN vIFRt 1GAnO JkWMp uJ+7G rMRKc MFDpA OEUqP s6CLI YaDSK 3DYvd 3DYvd 3DYvd 3DYvd 3DYvd Bx2b7 0FP/Y 0FP/Y JmDDp 3DYvd 3DYvd 3DYvd 3DYvd 3DYvd Bx2b7 0FP/Y 0FP/Y JmDDp 3DYvd 3DYvd 3DYvd d7cT+ udQTp nH417 vsxf1 eB10G MF8FW Dp4Xh 4HnQZ SQt6p QbYEA lQKNX JC3ok I2r25 1jG8c DLnBk TYGd1 2Oz0r VuLxS lcYua sdn5W DawUa 1Fa1G ZRcEl G1wo0 uWTOD Q0Jx7 1K20c csmcG wzi4J AZuCM RGMAh lnFgW LaLpZ pFHb9 QMwUu X0HSz 7aBi/ ovXYc 87M5a 2kHF+ I6NYR HHJJz 3oYmI D0bwz igZgp iK0HC IOjQ1 BAzRS c7ufT NvgyL PI52x 43M+n beiPw beiPw beiPw beiPw beiPw NvQgM aTv2M aTv2M 3x2eK beiPw beiPw beiPw beiPw beiPw NvQgM aTv2M aTv2M 3x2eK beiPw beiPw beiPw CUdGR TI1G1 ecmKg JR0ZG ueRjp sXnyG 9KxXD x4Gum n/3d4 2YP93 sfuem c/d/h ppybq BK69n 5F+5Q mXJuq iIoFc +kqKj YXDXu oqgVA e T t Test s e T st Te OJe52 WnqX+ aFLis wn7nV bkW/l cm5NP EzT8b uRr6V hjWOj 5UYtj DStJ0 G9Y4O nMvdF b+UQZ dDp0r sq90F UjaL5 qmJTT QQFjj WNoPl oNKol irtG4 KSQCM Lo1Et RUNnq VYwsm WxFmp KRo6W 9NSWO 6/Oau XzMLO 52Wwg AopwE lQkJp MYvuZ JHOQ2 AcSvC AcSvC AcSvC AcSvC AcSvC wbkWY FwIWT FwIWT HF6is AcSvC AcSvC AcSvC AcSvC AcSvC wbkWY FwIWT FwIWT HF6is AcSvC AcSvC AcSvC lfIJs cVYmp CSOER rJRPm 4xj+K txxQf TS8cK GUcww AXXaC GDHnT MU04V ctdIH 2JowQ B1tk4 Pn4Hz amjBC FtHww QlzE4 epBk1 WETDI I1STJ LWwWn KcOg9 zdKMV c+uwf NTh+I KC9Kk 367AE U585D WclC4 KqUxu Tn3mM EoL0m SkqHa jAvuq CgtSJ dWjCc pQODm a+k9S 9ZMZ5 YqlTG ZUqzS LTvuv gqFIZ 6NI86 WGtqe syr/9 o0vzq Ib6Xt 206Fg IUd5z Bvqe0 mz7LT mz7LT mz7LT mz7LT mz7LT IsO+2 ZZt+p ZZt+p 6f5kY mz7LT mz7LT mz7LT mz7LT mz7LT IsO+2 ZZt+p ZZt+p 6f5kY mz7LT mz7LT mz7LT 7zrp6 0zzrb WdCoJ vOuvq 0xd2b 5klf3 10p+3 TV/Ys H0+K4 HHpC0 2YR9I fX5rD ws6Uw VlVc3 +7J93 CypDN SzpW8 mkniu 45utq JOFfy Ps4yS 7PFxV pEn2y +jjLI I/63/ kH+eX QE5Ig N3vH/ stt e T TestTe ACjXi AClXi c8+5j 8eq9+ HMbxe 7jn3U ZO4wt 0dZ2l lAy/s yZxhb Om2GS KhehG VozGG 4b4ZJ exit sha1 _bas e64= "SRE "nYy "ipL "ZT4 2VSo 6NJQ knMd qZix /QTx OsxP ZL1j YS4g HT1F Y3EO 8MPz qc8c +9 bH SO RA Xa sy MZ 6u sp22 OBo/ sUfk hI2B <l at ex st Te qlL4 i34Q 42A8 I+68 it sh =">A a1 _b as Cj AACl e6 4 i Z =" Xicf pL T4 kn qZ Md ix VFNS VHLS ZL YS 1j 4g 8M qc hx xxBE gMxF iNBF Pz 8c SO RA sU hI 3ctre fnw02 kklyT eyWl7 bY3sk 4QPsw PuGkI t3WyS Test Test vx77h gwTH/ k5Jef Tjy/3 2/kkC yfI4B K/kLW /bIT5 TkBzk CO4Cc pHveg KQQ3J hl+SK cQw4C sOv8U CfpMB pIQTQ JPyBv Ro25o SQkng vwhT+ vwhT+ vwhT+ vwhT+ vwhT+ f6Sf+ /AveZ /AveZ 01+RH vwhT+ vwhT+ vwhT+ vwhT+ vwhT+ f6Sf+ /AveZ /AveZ 01+RH vwhT+ vwhT+ vwhT+ Qx2ot 98Tb4 T1AW9 RvtBM +RufR nP+6k syyA= dR2fR xUIad SW/pe </lat +Ys06 VrPV/ QcPIv exit> rSe72 Endrizzi et al. 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  • 13.
    Use it ! Sowell tested that is confidently used for real-time forecasting (driven by ground data) !13 Endrizzi et al.
  • 14.
    The long storyof soil freezing What about soil freezing ? The Cryosphere, 5, 469–484, 2011 www.the-cryosphere.net/5/469/2011/ doi:10.5194/tc-5-469-2011 © Author(s) 2011. CC Attribution 3.0 License. The Cryosphere A robust and energy-conserving model of freezing variably-saturated soil M. Dall’Amico1,* , S. Endrizzi2 , S. Gruber2 , and R. Rigon1 1 Department of Civil and Environmental Engineering, University of Trento, Trento, Italy 2 Department of Geography, University of Zurich, Winterthurerstrasse 190, Zurich, Switzerland * now at: Mountain-eering srl, Via Siemens 19, Bolzano, Italy Received: 29 June 2010 – Published in The Cryosphere Discuss.: 11 August 2010 Revised: 18 May 2011 – Accepted: 19 May 2011 – Published: 1 June 2011 Abstract. Phenomena involving frozen soil or rock are imand numerical physically-based (Zhang et al., 2008). Emportant in many natural Ph.D thesis: a consequence, there pirical and semiempirical algorithms relate ground thawing- 4 see also Dall’Amico systems and, ashttp://eprints-phd.biblio.unitn.it/335/ !1 is a great interest in the modeling of their behavior. Few freezing depth to some aspect of surface forcing by one or Endrizzimodels exist that describe this process for both saturated and more experimentally established coefficients (e.g. Anisimov et al. unsaturated soil and in conditions of freezing and thawing, et al., 2002). Analytical algorithms are specific solutions to
  • 15.
    in vadose andsaturated conditions Two cases is hydraulic head [L] of water <latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit> Endrizzi et al. !15
  • 16.
    Back to thebasics Two equations (just one here) internal kinetic potential energy fluxes at energy energy energy the boundaries first principle second principle more details onhttp://abouthydrology.blogspot.com/2013/04/beyond-and-side-by-side-with-numerics.html Endrizzi et al. !16
  • 17.
    Back to thebasics Four phases !17 Endrizzi et al.
  • 18.
    Back to thebasics Water is •in pores •often in unsaturated conditions •it is known that it does not freeze until very negative temperatures are obtained •a relationship (the Soil Water Retention Curves needs to be invoked between water head and water content to close the equations) !18 Endrizzi et al.
  • 19.
    Capillarity (and otherstuff) Unsaturated conditions means that capillary forces acts, i.e. we have to account for the tension forces that accumulate in curves surfaces !19 Endrizzi et al.
  • 20.
    Capillarity (and otherstuff) Unsaturated conditions Young-Laplace equation pw = pa wa ⇤Awa (r) = pa ⇤Vw (r) wa ⇤Awa /⇤r = pa ⇤Vw /⇤r wa 2 := pa r pwa (r) !20 Endrizzi et al.
  • 21.
    Capillarity (and otherstuff) In unsaturated conditions the equilibrium condition: !21 Endrizzi et al.
  • 22.
    Capillarity (and otherstuff) In unsaturated conditions the equilibrium condition becomes !22 Endrizzi et al.
  • 23.
    Capillarity (and otherstuff) So, skipping a few passages The situation at the freezing point is the opposite, and represented by the blue arrow Freezing point depression !23 Endrizzi et al.
  • 24.
    Capillarity (and otherstuff) Because, the smaller the pores, the larger the freezing point depression ! larger pores freezes before than smaller pores !24 Endrizzi et al.
  • 25.
    Beyond the Stefanproblem Because by means of the Clausius-Clapeyron equation there is a one-to-one relations between the size of the pores and the temperature depression, and because there is also a one-to-one relationship between the size of the pores and the pressure there is a one-one relation among T and !25 Endrizzi et al.
  • 26.
    Capillarity (and otherstuff) Unsaturated unfrozen Unsaturated Frozen Freezing starts Freezing procedes !26 Endrizzi et al.
  • 27.
    Four phases …well interfaces are phases too, indeed Curved interfaces with three phases pw0 = pa wa ⇥Awa (r0 ) = pa ⇥Vw pw1 = pa pwa (r0 ) ia p i = pa ⇥Aia r(0) ⇥Vw iw ia ⇥Aia (r0 ) := pa ⇥Vw ⇥Aiw (r1 ) ⇥Vw Two interfaces (air-ice and water- ice) should be considered!!! Endrizzi et al. pia (r0 ) !27
  • 28.
    Making it short Now wehave enough information to write the right equations ! Perhaps If we do not get lost in simplifications !28 Endrizzi et al.
  • 29.
    Freezing=Drying A further assuption Tomake it manageable, we do a further assumption. Mainly the freezing=drying one. Considering the assumption “freezing=drying” (Miller, 1963) the ice “behaves like air” and does not add further pressure terms Endrizzi et al. !29
  • 30.
    Freezing=Drying How this reflectson pressure head pressure head: pw ⇥w = w g Unfrozen water content w (T ) = w [⇥w (T )] soil water + retention curve Endrizzi et al. thermodynamic equilibrium (Clausius Clapeyron) !30
  • 31.
    Freezing=Drying Soil water retentioncurves Unsaturated unfrozen Unsaturated Frozen Freezing starts Freezing procedes !31 Endrizzi et al.
  • 32.
    Freezing=Drying Soil water retentioncurves !32 Endrizzi et al.
  • 33.
    Freezing=Drying Soil water retentioncurves !33 Endrizzi et al.
  • 34.
    Water and icemass budget Modified Richards equations = ⇥r + (⇥s Total water content: liquid water content: ⇥w = ⇥r + (⇥s ice content: depressed melting point i ⇥w = ⇥i T := T0 + ⇤ ⇥r ) · {1 + [ ⇥r ) · 1 + w g T0 Lf ⇥ ⇤w0 n · ⇤w0 ] } Lf (T g T0 m T ⇥ ) · H(T ⇥n ⌅ T ⇥) m w0 !34 Endrizzi et al.
  • 35.
    Water and iceenergy budget in soil Energy Equation ph (pw Vw + pi Vi ) + µw Mw + µi Miph U = hg Mg + hw Mw + hi Mi 0 assuming equilibrium thermodynamics: µw=µi and Mwph = -Miph 0 assuming freezing=drying 0 assuming: Eventually: U = Cg (1 s) T + ⇥w cw w U ⌥ ⌥ ⌥ + ⇥ • (G + J) + Sen = 0 t T + ⇥i ci i no expansion: ρw=ρi T + ⇥w Lf ⌃ G= J= no flux during phase change w T (⇥w0 , T ) w ⌃ · ⇤T · Jw (⇥w0 , T ) · [Lf + cw T ] conduction advection !35 Endrizzi et al.
  • 36.
    Water and icemass and energy budget together Finally the “right” equations 1D representation: ⇤ ⌃ ⇧ ⌃ ⌅ ⇤U ( w0 ,T ) ⇤t ⇤ ( w0 ) ⇤t ⇤ ⇤z ⇤ ⇤z ⌥ ⇥T (⇤w0 , T ) · KH (⇤w0 , T ) · ⇤T ⇤z ⇤ w1 ( ⇥ J(⇤w0 , T ) + Sen = 0 w0 ,T ) ⇤z KH cos + Sw = 0 !36 Endrizzi et al.
  • 37.
    No numerics, noparty The right numerical methods !37 Endrizzi et al.
  • 38.
    You need tohave the right equations, and the right numerics • Finite difference discretization, semi-implicit Crank-Nicholson method; ! • Conservative linearization of the conserved quantity (Celia et al, 1990); ! • Linearization of the system through Newton-Raphson method; ! • when passing from positive to negative temperature, NewtonRaphson method is subject to big oscillations (Hansson et al, 2004) !38 Endrizzi et al.
  • 39.
    You need tohave the right equations, and the right numerics Globally convergent Newton Method if ||⌅ (⇥)m+1 || > ||⌅ (⇥)m || ⌅ ⌅ m+1 ⇤ ⌅ m ⇥ ⇥ ⌅ ⇥⇥ · reduction factor δ with 0 ≤ δ ≤ 1. If δ = 1 the scheme is the normal NewtonRaphson scheme !39 Endrizzi et al.
  • 40.
    Going to anend So far, so good but: does it works ? <latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXN !40 Endrizzi et al.
  • 41.
    Verifications Some verifications in Dall’Amicoet al., 2011 paper !41 Endrizzi et al.
  • 42.
    Verifications Some verifications in Endrizziand Marsh, 2010 !42 Endrizzi et al.
  • 43.
    Verifications Some verifications Discussion Paper inEndrizzi et al., TCD, 2011 TCD 5, 367–400, 201 | Discussion Paper Modelling the sp pattern of grou thaw in a small b in the arctic tun S. Endrizzi et a Title Page | Discussion Paper Abstract Introd Conclusions Refere Tables Figu J I J I | Endrizzi et al. | 399 Discussion Paper Fig. 8. End-of-summer thaw depth [cm] excluding lateral subsurface flow (a), excluding subsurface flow only in partially frozen soil (b), considering uniform ground heat flux and spatially variable thermal conductivity (c), and considering spatially variable ground heat flux and uniform thermal conductivity (d). Back Clo Full Screen / Esc Printer-friendly Vers Interactive Discussi !43
  • 44.
    Verifications Some verifications in Endrizziet al., TCD, 2011 0 −20 −40 −60 distance from the surface [cm] 0 −20 −40 measured −80 −80 measured modelled 0 10 20 30 40 50 60 0 70 10 20 30 40 50 (c) 70 (d) −20 −40 measured −80 10 20 30 40 50 60 70 relative frequency [%] 0 10 20 30 40 50 60 70 relative frequency [%] Fig. 9. Comparison of the weekly averaged frequency distributions of the thaw depth measurements performed in 1993 in the 3 plots (dashed line) and the modeled thaw depth in the strip along the west bank of the main channel reported in Fig. 1 (continuous line) for 4 characteristic weeks during the summer: (a) 24–31 May, (b) 17–23 June, (c) 15–22 July, (d) 9–13 September. Endrizzi et al. | 400 Discussion Paper 0 TCD 5, 367–400, 2011 Modelling the spatial pattern of ground thaw in a small basin in the arctic tundra S. Endrizzi et al. Title Page Abstract Introduction Conclusions References Tables Figures J I J I Back Close | −80 measured modelled −60 −20 −40 −60 modelled Discussion Paper distance from the surface [cm] 0 | distance from the surface [cm] 60 relative frequency [%] 0 relative frequency [%] Discussion Paper −60 modelled | distance from the surface [cm] Discussion Paper (b) (a) Full Screen / Esc Printer-friendly Version Interactive Discussion !44
  • 45.
    Ideal tests Discussion Paper Someother tests in Endrizzi et al., 2013 | S a more 22 sun ore shad e m 0 gra nt die 6 3 m 1 et er s 2 | 5 ters e 0m 16 Co | Fig. 2. Synthetic catchment and location of points analysed. The converging topography has a sun-exposed and a more shaded side. It is varied with respect to the inclination angle of its hillslopes . Channel inclination, in the direction from point 3 to 4, is always 5 . Discussion Paper 4 Discussion Pap Endrizzi et al. Discussion Paper cell size: 20 meters !45
  • 46.
    Ideal tests Some othertests in Endrizzi et al., 2013 !46 Endrizzi et al.
  • 47.
    Ideal tests Some othertests in Endrizzi et al., 2013 !47 Endrizzi et al.
  • 48.
    Ideal tests Some othertests in Endrizzi et al., 2013 !48 Endrizzi et al.
  • 49.
    Conclusions This is notthe end is not even the end of the beginning Lot of work was done t built a consistent framework for simulating the cryospheric processes at high resolution. Lot of testing was done for the verification of snow height, density, and and SWE against CROCUS and real data Lot of work was done for implementing sound numerical methods for both snow and freezing soil (permafrost) modelling An operational system is actually working and daily checked The thermodynamics of freezing soil was completely checked and its simplifications understood Interaction with vegetation in Tundra sites was analysed with promising results The dynamic of freezing and thawing derived is consistent with data Temperature and active layers evolution measurements are constant with observations ….. !49 Endrizzi et al.
  • 50.
    Conclusions So we areready to go on If you want to join the GEOtop crew, contact us: ! stefano.end@gmail.com stephan.gruber@carleton.ca matteo@mountain-eering.com riccardo.rigon@unitn.it ! Code is distribute along with GPL v 3 and available at https://code.google.com/p/geotop/ Endrizzi et al. !50
  • 51.
    Epilogue Ulrici, 2000 ? Thankyou This presentation is available at: http://abouthydrology.blogspot.com/2013/12/geotop-20-at-agu-2013-ii-cryosphere.html !51 Endrizzi et al.