GEOtop 2.0 - The snow and freezing soil modelling

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

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19

12

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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

!3

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.

Endrizzi et al.

I am just a poor hydrologists

!4

Warm hydrology in GEOtop

http://abouthydrology.blogspot.com/2013/12/geotop-20-at-agu-2013.html

Endrizzi et al.

Already talked about GEOtop in this AGU

!5

Richards equation +

van Genuchten parameterization +

Mualem derived conductivityEner

gy

bu

dget

(wit

h s

om

e as

sum

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s)

Flux-gradient relationship

(Monin - Obukov)

Diffusive approximation to shallow

water equation

Double layer vegetation

Rad

iati

on

Snow

met

amorp

his

m

Equations

Endrizzi et al.

!6

The Cryospheric processes parameterisation enter mainly

Richards equation +

van Genuchten parameterization +

Mualem derived conductivityEner

gy

bu

dget

(wit

h s

om

e as

sum

pti

on

s)

here

Endrizzi et al.

Equations

Snow

met

amorp

his

m

!7

A snow model

It also parameterizes atmospheric radiation and its components, and

turbulence.

Endrizzi et al.

Grid size based

!8

The control volume

Endrizzi et al.

Grid size based

*New multi-layering with dynamic grid

!9

The new* snow model

*For the old one see Zanotti et al., 2004

Chapter 10

Snow

10.1 Introduction

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Figure 10.1: Snow stratigraphy

10.2 Input10.2.1 Parameters

Keyword Description M. U. range DefaultValue

Sca /Vec

Str / Num/ Opt

ThresSnowSoilRough Threshold on snow depth to changeroughness to snow roughness valueswith d0 set at 0, for bare soil fraction

mm 0,1000

10 sca num

ThresSnowVegUp Threshold on snow depth abovewhich the roughness is snow rough-ness, for vegetation fraction

mm 0,20000

1000 sca num

ThresSnowVegDown Threshold on snow depth belowwhich the roughness is vegetationroughness, for vegetation fraction

mm 0,20000

1000 sca num

RoughElemXUnitArea Number of roughness elements(=vegetation) per unit area - usedonly for blowing snow subroutines

Numberm�2

0, inf 0 sca num

continued on next page

37

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(the energy budget)

Endrizzi et al.

Equations again

!10

The new* snow model

*For the old one see Zanotti et al., 2004

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

Endrizzi et al.

Also

!11

Endrizzi et al.

Endrizzi et al. 2013

see also http://abouthydrology.blogspot.com/search/label/GEOtop

The whole story here

http://abouthydrology.blogspot.com/search/label/GEOtop

!12

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. Model Dev., 6, 1319–1336, 2013.

Test

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Test






















Test






















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latexit>

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Test






















Test






















Endrizzi et al.

Test it!

!13

So well tested that is confidently used for real-time

forecasting (driven by ground data)

Endrizzi et al.

Use it !

!14

The Cryosphere, 5, 469–484, 2011www.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 freezingvariably-saturated soilM. Dall’Amico1,*, S. Endrizzi2, S. Gruber2, and R. Rigon11Department of Civil and Environmental Engineering, University of Trento, Trento, Italy2Department 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 2010Revised: 18 May 2011 – Accepted: 19 May 2011 – Published: 1 June 2011

Abstract. Phenomena involving frozen soil or rock are im-portant in many natural systems and, as a consequence, thereis a great interest in the modeling of their behavior. Fewmodels exist that describe this process for both saturated andunsaturated soil and in conditions of freezing and thawing,as the energy equation shows strongly non-linear character-istics and is often difficult to handle with normal methodsof iterative integration. Therefore in this paper we proposea method for solving the energy equation in freezing soil.The solver is linked with the solution of Richards equation,and is able to approximate water movement in unsaturatedsoils and near the liquid-solid phase transition. A globally-convergent Newton method has been implemented to achieverobust convergence of this scheme. The method is tested bycomparison with an analytical solution to the Stefan problemand by comparison with experimental data derived from theliterature.

1 Introduction

The analysis of freezing/thawing processes and phenomenain the ground is important for hydrological and other landsurface and climate model simulations (e.g. Viterbo et al.,1999; Smirnova et al., 2000). For example, comparisonsof results from the Project for Intercomparison of Land Sur-face Parameterization Schemes have shown that the modelswith an explicit frozen soil scheme provide more realisticsoil temperature simulation during winter than those without(Luo et al., 2003). Freezing soil models may be divided intothree categories: empirical and semiempirical, analytical,

Correspondence to: M. Dall’Amico([email protected])

and numerical physically-based (Zhang et al., 2008). Em-pirical and semiempirical algorithms relate ground thawing-freezing depth to some aspect of surface forcing by one ormore experimentally established coefficients (e.g. Anisimovet al., 2002). Analytical algorithms are specific solutions toheat conduction problems under certain assumptions. Themost widely applied analytical solution is Stefan’s formula-tion, which simulates the freezing/thawing front using ac-cumulated ground surface degree-days (either a freezing orthawing index) (Lunardini, 1981). Numerical physically-based algorithms simulate ground freezing by numericallysolving the complete energy equation, and in natural condi-tions they are expected to provide the best accuracy in sim-ulating ground thawing and freezing (Zhang et al., 2008).However, this approach has difficulties, especially regardingthe treatment of phase change, which is strongest in a narrowrange of temperatures near the melting point, and thus rep-resents a discontinuity that may create numerical oscillations(Hansson et al., 2004). Furthermore, the freezing processhas a profound effect also to the water fluxes in the soil, asit changes the soil hydraulic conductivity and induces pres-sure gradients driving water movements. Therefore, a cou-pled mass and energy system is needed to simulate both thethermal and hydraulic characteristics of the soil.The objectives of the paper are: (1) to revisit the theory

of the freezing soil in order to provide the formulation forthe unfrozen water pressure, which can accomodate variably-saturated soils; (2) to outline and describe a numerical ap-proach for solving coupled mass and energy balance equa-tions in variably-saturated freezing soils, based on the split-ting method; (3) to provide an improved numerical schemethat: (i) is written in conservative way, (ii) is based on theglobally convergent Newton scheme, and (iii) can handlethe high non-linearities typical of the freezing/thawing pro-cesses.

Published by Copernicus Publications on behalf of the European Geosciences Union.

What about soil freezing ?

see also Dall’Amico Ph.D thesis: http://eprints-phd.biblio.unitn.it/335/

Endrizzi et al.

The long story of soil freezing

http://eprints-phd.biblio.unitn.it/335/

!15

Two cases

is hydraulic head [L] of water <latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit>

Endrizzi et al.

in vadose and saturated conditions

!16

Two equations (just one here)

first principle

potential energy

kinetic energy

internal energy

energy fluxes at the boundaries

second principle

more details onhttp://abouthydrology.blogspot.com/2013/04/beyond-and-side-by-side-with-numerics.html

Endrizzi et al.

Back to the basics

onhttp://abouthydrology.blogspot.com/2013/04/beyond-and-side-by-side-with-numerics.html

!17

Four phases

Endrizzi et al.

Back to the basics

!18

Water is

•often in unsaturated conditions

•in pores

•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)

Endrizzi et al.

Back to the basics

!19

Unsaturated conditions

means that capillary forces acts, i.e. we have to account for the tension forces that accumulate in curves surfaces

Endrizzi et al.

Capillarity (and other stuff)

!20

Unsaturated conditionsYoung-Laplace equation

pw = pa � �wa⇤Awa(r)⇤Vw(r)

= pa � �wa⇤Awa/⇤r

⇤Vw/⇤r= pa � �wa

2r

:= pa � pwa(r)

Endrizzi et al.


!21

In unsaturated conditionsthe equilibrium condition:

Endrizzi et al.


!22

In unsaturated conditionsthe equilibrium condition becomes

Endrizzi et al.


!23

So, skipping a few passages

The situation at the freezing point is the opposite, and represented by the

blue arrowFreezing point depression

Endrizzi et al.


!24

Because, the smaller the pores,

the larger the freezing point depression !

larger pores freezes before than

smaller pores

Endrizzi et al.


!25

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

Beyond the Stefan problem

Endrizzi et al.

!26

Unsaturated unfrozen

Unsaturated Frozen

Freezing starts

Freezing procedes

Endrizzi et al.


!27

pw0 = pa � �wa⇥Awa(r0)

⇥Vw= pa � pwa(r0) pi = pa � �ia

⇥Aia(r0)⇥Vw

:= pa � pia(r0)

pw1 = pa � �ia⇥Aiar(0)

⇥Vw� �iw

⇥Aiw(r1)⇥Vw

Two interfaces (air-ice and water- ice) should be considered!!!

Curved interfaces with three phases

Endrizzi et al.

Four phases … well interfaces are phases too, indeed

!28

Now

we have enough information to write the right equations

!Perhaps

If we do not get lost in simplifications

Endrizzi et al.

Making it short

!29

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.

Freezing=Drying

!30

Unfrozen water content

soil water retention curve

thermodynamic equilibrium (Clausius Clapeyron)

+

⇥w =pw

�w gpressure head:

�w(T ) = �w [⇥w(T )]

How this reflects on pressure head

Endrizzi et al.

Freezing=Drying

!31

Unsaturated unfrozen

Unsaturated Frozen

Freezing starts

Freezing procedes

Soil water retention curves

Endrizzi et al.

Freezing=Drying

!32


Endrizzi et al.

Freezing=Drying

!33


Endrizzi et al.

Freezing=Drying

!34

T � := T0 +g T0

Lf�w0

ice content: �i =⇥w

⇥i

�� w

⇥

⇥w = ⇥r + (⇥s � ⇥r) ·⇤

1 +��⇤w0 � �

Lf

g T0(T � T ⇥) · H(T � T ⇥)

⇥n⌅�m

liquid water content:

Total water content:

depressed melting point

Modified Richards equations

� = ⇥r + (⇥s � ⇥r) · {1 + [�� · ⇤w0]n}�m

Endrizzi et al.

Water and ice mass budget

!35

U = Cg(1� �s) T + ⇥wcw�w T + ⇥ici�i T + ⇥wLf�w

�U

�t+ ⌥⇥ • (⌥G + ⌥J) + Sen = 0

⌃G = ��T (⇥w0, T ) · ⌃⇤T

�J = �w · �Jw(⇥w0, T ) · [Lf + cw T ]

0 assuming freezing=drying

U = hgMg + hwMw + hiMi � (pwVw + piVi) + µwMphw + µiM

phi

no expansion: ρw=ρi

assuming:0 no flux during phase change

Eventually:

0 assuming equilibrium thermodynamics: µw=µi and Mw

ph = -Miph

conduction

advection

Energy Equation

Endrizzi et al.

Water and ice energy budget in soil

!36

⇤⌃⇧

⌃⌅

⇤U(�w0,T )⇤t � ⇤

⇤z

�⇥T (⇤w0, T ) · ⇤T

⇤z � J(⇤w0, T )⇥+ Sen = 0

⇤�(�w0)⇤t � ⇤

⇤z

⌥KH(⇤w0, T ) · ⇤�w1(�w0,T )

⇤z �KH cos ��

+ Sw = 0

1D representation:

Finally the “right” equations

Water and ice mass and energy budget together

Endrizzi et al.

!37

The right numerical methods

Endrizzi et al.

No numerics, no party

!38

• 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, Newton-Raphson method is subject to big oscillations (Hansson et al, 2004)

Endrizzi et al.

You need to have the right equations, and the right numerics

!39

if ||⌅�(⇥)m+1|| > ||⌅�(⇥)m|| ⌅ ⌅⇥m+1 ⇤ ⌅⇥m � ⌅⇥⇥ · �

reduction factor δ with 0 ≤ δ ≤ 1. If δ = 1 the scheme is the normal Newton-Raphson scheme

Globally convergent Newton Method

Endrizzi et al.

You need to have the right equations, and the right numerics

!40

So far, so good but: does it works ?


Endrizzi et al.

Going to an end

!41

Some verifications in Dall’Amico et al., 2011 paper

Endrizzi et al.

Verifications

!42

Some verifications in Endrizzi and Marsh, 2010

Endrizzi et al.

Verifications

!43

TCD5, 367–400, 2011

Modelling the spatialpattern of ground

thaw in a small basinin the arctic tundra

S. Endrizzi et al.

Title Page

Abstract Introduction

Conclusions References

Tables Figures

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Fig. 8. End-of-summer thaw depth [cm] excluding lateral subsurface flow (a), excluding sub-surface flow only in partially frozen soil (b), considering uniform ground heat flux and spatiallyvariable thermal conductivity (c), and considering spatially variable ground heat flux and uni-form thermal conductivity (d).

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Some verifications in Endrizzi et al., TCD, 2011

TCD5, 367–400, 2011

Modelling the spatialpattern of ground

thaw in a small basinin the arctic tundra

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Fig. 9. Comparison of the weekly averaged frequency distributions of the thaw depth measure-ments performed in 1993 in the 3 plots (dashed line) and the modeled thaw depth in the stripalong the west bank of the main channel reported in Fig. 1 (continuous line) for 4 characteristicweeks during the summer: (a) 24–31 May, (b) 17–23 June, (c) 15–22 July, (d) 9–13 September.

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GMDD6, 1–62, 2013

Simulating energyand water balancewith freezing and

terrain e◆ects

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160 meters

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cell size: 20 meters

more sunmore shade

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Fig. 2. Synthetic catchment and location of points analysed. The converging topography hasa sun-exposed and a more shaded side. It is varied with respect to the inclination angle of itshillslopes �. Channel inclination, in the direction from point 3 to 4, is always 5�.

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Some other tests in Endrizzi et al., 2013

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GMDD6, 1–62, 2013


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Fig. 5. Time evolution of temperature, and liquid, solid, and total water content for soil andsnow in Point 1, as shown in Fig. 2, for the simulation for the 5� topography 3-D. The blue lineindicates the water table, while the lower and upper borders of the thawed layer are shown inred. Negative depths correspond to soil, and positive depths to the snow cover.

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GMDD6, 1–62, 2013


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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

…..

Endrizzi et al.

Conclusions

!50

So we are ready to go on

If you want to join the GEOtop crew, contact us: [email protected] [email protected] [email protected] [email protected] !

Endrizzi et al.

Conclusions

https://code.google.com/p/geotop/

Code is distribute along with GPL v 3 and available at

mailto:[email protected]



!51

Thank you

Ulr

ici, 2

00

0 ?

Endrizzi et al.

http://abouthydrology.blogspot.com/2013/12/geotop-20-at-agu-2013-ii-cryosphere.html

This presentation is available at:

Epilogue

Education

GEOtop 2.0 - The snow and freezing soil modelling