This is the second talk given at AGU Fall Meeting 2013. It complements the first talk by presenting something of the new snow modelling, and freezing soil algorithms
1. 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
2. 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. 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.
4. 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
5. 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.
6. 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.
7. Grid size based
A snow model
It also parameterizes atmospheric radiation and its components, and
turbulence.
!7
Endrizzi et al.
8. Grid size based
The control 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* 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
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
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.
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<latexit sha1_base64="ZT4qZixYS4gqc8cRAhI2B6uI+68=">AAAClXicfVHLSiNBFK20zuhkHj5mI7gpDAOzCt0u1JWIIzI7I9gqpBuprr5JCuvRVN1WQtHfMFvn0+ZvrMReTIx4oeBw7jn3UbeopHAYx/860crqh49r65+6n798/baxubV97UxtOaTcSGNvC+ZACg0pCpRwW1lgqpBwU9z/muVvHsA6YfQVTivIFRtrMRKcYaDSK3DYvdvsxf14HnQZJC3okTYGd1sdn
sha1_base64="SRE2VSo/QTxHT1F+9sp22XaqlL4=">AAAClXicfVHLShxBFK1pY6ITjY9sBDdFBsHV0O0icRVERdzFwHQUphuprr4zU1iPpuq2MhT9DdnGT/NvrJnpRXTECwWHc8+5j7pFJYXDOH7qRCsfVj9+Wlvvft7Y/LK1vbP7x5nacki5kcbeFMyBFBpSFCjhprLAVCHhurg7m+Wv78E6YfQApxXkio21GAnOMFDpABx2b7d7cT+eB10GSQt6pI2r252Oz
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</lat
cg==<
13. Use it !
So well tested that is confidently used for real-time
forecasting (driven by ground data)
!13
Endrizzi et al.
15. 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
16. 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
17. Back to the basics
Four phases
!17
Endrizzi et al.
18. 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.
19. 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.
20. 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.
21. Capillarity (and other stuff)
In unsaturated conditions
the equilibrium condition:
!21
Endrizzi et al.
22. Capillarity (and other stuff)
In unsaturated conditions
the equilibrium condition becomes
!22
Endrizzi et al.
23. 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.
24. 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.
25. 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.
26. Capillarity (and other stuff)
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
we have 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
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
30. 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
34. 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.
35. 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.
36. 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.
37. No numerics, no party
The right numerical methods
!37
Endrizzi et al.
38. 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.
39. 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.
40. Going to an end
So far, so good
but: does it works ?
<latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXN
!40
Endrizzi et al.
43. 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
44. 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
45. 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
|
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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
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49. 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
…..
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Endrizzi et al.
50. 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.
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51. Epilogue
Ulrici, 2000 ?
Thank you
This presentation is available at:
http://abouthydrology.blogspot.com/2013/12/geotop-20-at-agu-2013-ii-cryosphere.html
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Endrizzi et al.