Download - Diffusion equationhydrology2014
![Page 1: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/1.jpg)
R. RigonR. Rigon
Ubiquitous Diffusion
Jack
son
Poll
ok, F
ree
Form
, 19
49
, Mom
a
![Page 2: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/2.jpg)
!2
Objectives
•See where diffusion equations appears in
hydrology:
•The case of Richards’ equation and its
extensions
•The case of snow thermodynamics
!All presented in a somewhat confuse way ...
!Sorry for the mixed English and Italian ….
R. Rigon
Introduction
![Page 3: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/3.jpg)
R. Rigon
Richards ++
R. Rigon
![Page 4: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/4.jpg)
!4
Four phases
Back to the basics
R. Rigon
![Page 5: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/5.jpg)
!5
Four phases
However, we neglect, at the moment, ice.
Soil
Water
Air
Massa Volume
VagMag
La colonna di neve
MwVw
M⇤ V⇤
R. Rigon
Back to the basics
![Page 6: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/6.jpg)
!6
V a r i a z i o n e d i contenuto d’acqua nel suolo nell’unità di tempo
Divergenza del flusso volumetrico attraverso il contorno del volume infinitesimo
Ric
har
ds,
19
31
!6
⇤�w
⇤t= ⇥ · ⌃Jv(⇥)
L’equazione di continuità
Back to the basics
R. Rigon
![Page 7: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/7.jpg)
!7
Legge di Darcy-Buckingham
Flusso volumetrico attraverso il contorno del volume infinitesimo
Conducibilità idraulica x gradiente del caricoB
uck
ingh
am, 1
90
7, R
ich
ard
s, 1
93
1
!7
~Jv = K(✓w)~r h
]Back to the basics
R. Rigon
![Page 8: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/8.jpg)
!8
Il carico idraulico è una energia per unità di volume e si misura in unità di lunghezza
h = z + �
Carico idraulico
campo gravitazionale
forze capillari - pressione
Ric
har
ds,
19
31
!8
Legge di Darcy-Buckingham
Back to the basics
R. Rigon
![Page 9: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/9.jpg)
!9
Per semplificare possiamo pensare vi sia una relazione biunivoca tra pressione e contenuto d’acqua del suolo
⇤�(⇥)⇤t
=⇤�(⇥)⇤⇥
⇤⇥
⇤t� C(⇥)
⇤⇥
⇤t
Capacità idraulica dei suoli
!9
Back to the basics
R. Rigon
![Page 10: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/10.jpg)
!10
La capacità idraulica è proporzionale alla distribuzione dei pori
!10
Interpretations
R. Rigon
![Page 11: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/11.jpg)
!11
FORME PARAMETRICHE DELLA SWRC:
QUELLA PIU’ USATA E’ QUELLA di van Genucthen
Che ha cinque parametri
!11
Se ⌘✓w � ✓r
�s � ✓r=
h 11 + (↵ )n
im
✓r
�s
↵nm
Parameterisations
R. Rigon
![Page 12: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/12.jpg)
!12
si ottiene:
K(Se) = KsSve
⇤1�
�1� S1/m
e
⇥m⌅2
(m = 1� 1/n)
o, esprimendo il tutto in funzione del potenziale di suzione:
K(⇥) =Ks
�1� (�⇥)mn [1 + (�⇥)n]�m
⇥2
[1 + (�⇥)n]mv (m = 1� 1/n)
FORME PARAMETRICHE DELLA CONDUCIBILITA’ IDRAULICA
!12
Parameterisations
R. Rigon
![Page 13: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/13.jpg)
!13
What I mean with Richards ++
First, I would say, it means that it would be better to call it, for
instance: Richards-Mualem-vanGenuchten equation, since it is:
Se = [1 + (��⇥)m)]�n
Se :=�w � �r
⇥s � �r
C(⇥)⇤⇥
⇤t= ⇥ ·
�K(�w) �⇥ (z + ⇥)
⇥
K(�w) = Ks
⇧Se
⇤�1� (1� Se)1/m
⇥m⌅2
Water balance
Parametric Mualem
Parametric van Genuchten
C(⇥) :=⇤�w()⇤⇥
R. Rigon
To sum-up
![Page 14: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/14.jpg)
!14
What I mean with Richards ++
Extending Richards to treat the transition saturated to unsaturated zone. Which means:
R. Rigon
Extensions
![Page 15: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/15.jpg)
!15
Richards equation is part of a more general equation
that can be obtained by considering also saturated soil/aquifers*
* see for instance, Lu and Godt, 2012, Chapter 4 - Freeze and Cherry, 1979, pg 51
Since the rate of gain/loss of water mass is in general:
<latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit>
where:
<latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit>
R. Rigon
Extensions
![Page 16: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/16.jpg)
!16
Allora l’equazione generale è ancora
C(⇥)⇤⇥
⇤t= ⇥ ·
�K(�w) �⇥ (z + ⇥)
⇥
purchè (usando la parametrizzazione di van Genuchten):
R. Rigon
Extensions
![Page 17: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/17.jpg)
!17
Le equazioni sono non lineari e richiedono il metodo di
Newton con doppia iterazione interna (nested Newton) per essere risolte
R. Rigon
Phase Transition
![Page 18: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/18.jpg)
R. Rigon
Freezing soil
Gin
o S
ever
ini, B
lue
Dan
cer,
19
12
- G
uggh
enai
m m
use
um
, Ven
ice
R. Rigon
![Page 19: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/19.jpg)
!19
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/
The long story of soil freezing
R. Rigon
![Page 20: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/20.jpg)
!20
Two cases
is hydraulic head [L] of water <latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit>
R. Rigon
in vadose and saturated conditions
![Page 21: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/21.jpg)
!21
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
Back to the basics
R. Rigon
![Page 22: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/22.jpg)
!22
Four phases
Back to the basics
R. Rigon
![Page 23: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/23.jpg)
!23
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)
Back to the basics
R. Rigon
![Page 24: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/24.jpg)
!24
Unsaturated conditions
means that capillary forces acts, i.e. we have to account for the tension forces that accumulate in curves surfaces
Capillarity (and other stuff)
R. Rigon
![Page 25: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/25.jpg)
!25
Unsaturated conditionsYoung-Laplace equation
pw = pa � �wa⇤Awa(r)⇤Vw(r)
= pa � �wa⇤Awa/⇤r
⇤Vw/⇤r= pa � �wa
2r
:= pa � pwa(r)
Capillarity (and other stuff)
R. Rigon
![Page 26: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/26.jpg)
!26
In unsaturated conditionsthe equilibrium condition:
Capillarity (and other stuff)
R. Rigon
![Page 27: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/27.jpg)
!27
In unsaturated conditionsthe equilibrium condition becomes
Capillarity (and other stuff)
R. Rigon
![Page 28: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/28.jpg)
!28
So, skipping a few passages
The situation at the freezing point is the opposite, and represented by the
blue arrowFreezing point depression
Capillarity (and other stuff)
R. Rigon
![Page 29: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/29.jpg)
!29
Because, the smaller the pores,
the larger the freezing point depression !
larger pores freezes before than
smaller pores
Capillarity (and other stuff)
R. Rigon
![Page 30: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/30.jpg)
!30
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
R. Rigon
![Page 31: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/31.jpg)
!31
Unsaturated unfrozen
Unsaturated Frozen
Freezing starts
Freezing procedes
Capillarity (and other stuff)
R. Rigon
![Page 32: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/32.jpg)
!32
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
Four phases … well interfaces are phases too, indeed
R. Rigon
![Page 33: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/33.jpg)
!33
Now
we have enough information to write the right equations
!Perhaps
If we do not get lost in simplifications
Making it short
R. Rigon
![Page 34: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/34.jpg)
!34
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
Freezing=Drying
R. Rigon
![Page 35: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/35.jpg)
!35
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
Freezing=Drying
R. Rigon
![Page 36: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/36.jpg)
!36
Unsaturated unfrozen
Unsaturated Frozen
Freezing starts
Freezing procedes
Soil water retention curvesFreezing=Drying
R. Rigon
![Page 37: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/37.jpg)
!37
Soil water retention curvesFreezing=Drying
R. Rigon
![Page 38: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/38.jpg)
!38
Soil water retention curvesFreezing=Drying
R. Rigon
![Page 39: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/39.jpg)
!39
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
Water and ice mass budget
R. Rigon
![Page 40: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/40.jpg)
!40
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
Water and ice energy budget in soil
R. Rigon
![Page 41: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/41.jpg)
!41
⇤⌃⇧
⌃⌅
⇤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
R. Rigon
![Page 42: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/42.jpg)
R. RigonR. Rigon
Tu
rner
, Sn
ow
Sto
rm, 1
84
2
Snow
![Page 43: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/43.jpg)
!43
Il manto nevosoNeve, Ghiaccio, Permafrost
Acqua (Liquida)
Ghiaccio
Aria
Massa Volume
Vag
ViMi
Mag
La colonna di neve
Mw Vw
M⇤ V⇤
R. Rigon
Introduction
![Page 44: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/44.jpg)
!44
Il manto nevoso
Il manto nevoso (snow-pack) è:
! - un mezzo poroso (come mostrato nella slide precedente)
!Generalmente composto da strati, più o meno omogenei, di differente
spessore e da tipi differenti di neve
!Gli strati sono composti da cristalli e grani che sono, di solito, legati da
qualche tipo di coesione.
R. Rigon
Introduction
![Page 45: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/45.jpg)
!45
Massa dell’acqua liquida
Massa del vapore Massa del ghiaccio
Massa della neve
Massa dell’aria
Notazione di base
M⇤ = Mag + Mw + Mi
M⇤ = Mv + Mw + Mi
R. Rigon
Introduction
![Page 46: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/46.jpg)
!46
Notazione di baseI volumi con gli stessi indici delle masse
V⇤ = Vag + Vw + Vi
Vtw = Vv + Vw + Vi
R. Rigon
Introduction
![Page 47: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/47.jpg)
Densità del ghiaccio ice density
!47
Densità apparente della neve snow bulk density
Notazione di base
⇢i :=Mi
Vi
⇢⇤ :=M⇤V⇤
=M⇤
Vag + Vw + Vi
R. Rigon
Introduction
![Page 48: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/48.jpg)
!48
Notazione di base
Contenuto volumetrico d’acqua nella neve(adimensionale)
Volume fraction of liquid water in snow pores
✓w :=Vw
Vag + Vw + Vi
Contenuto volumetrico adimensionale di ghiaccio nella neve
Volume fraction of frozen water (ice) in snow
✓i :=Vi
Vag + Vw + Vi
R. Rigon
Introduction
![Page 49: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/49.jpg)
!49
Porosità della neve
Notazione di base
Saturazione (relativa) della neve
�⇤ :=Vag + Vw
Vag + Vw + Vi
S⇤ :=✓w
�⇤
R. Rigon
Introduction
![Page 50: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/50.jpg)
!50
Notazione di base
Equivalente in acqua della neve
Volume dell’acqua derivante dalla completa fusione della neve su un area orizzontale corispondente.
h⇤ =✓
✓w + (1� �⇤)⇢i
⇢w
◆V⇤A
=✓
✓w + (1� �⇤)⇢i
⇢w
◆hsn
hsn :=V⇤A
h⇤ :=Vw(A) + ⇢i
⇢wVi(A)
A
R. Rigon
Introduction
![Page 51: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/51.jpg)
!51
Proprietà termiche della neve
Si assume che il flusso di calore segua la legge di Fourier:
~Jh = Kh~rT
Flusso di calore W m-2
Conducibilità termica
W m-1 K-1
Gradiente di Temperatura
K m-1
R. Rigon
Thermal conductivity
![Page 52: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/52.jpg)
!52
Proprietà termiche della neve
La conducibilità termica Kh è una misura della abilità di un materiale di trasmettere
calore. un buon conduttore di calore ha un alto valore di K, un isolante ha un basso
valore di K (in W/m K).
Neve Fresca 0.03 (meglio della lana di vetro!)
Neve vecchia 0.4
Ghiaccio 2.1
~Jh = Kh~rT
La neve attenua i cambiamenti termici dell’atmosfera. Per esempio un cambio
di 1 grado di temperatura dell’aria in 15 minuti, cambia la temperatura a 20
cm di profondità nella neve di soli 0.1 gradi e di 0.01 gradi ad un metro.
R. Rigon
Thermal conductivity
![Page 53: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/53.jpg)
!53
Proprietà termiche della neve
~Jh = Kh~rT
Kh cresce con il metamorfismo della neve. Ad esempio, Sturm, 1997 fornisce
questa formula parametrica:
Kh = 0.138� 1.01 ⇢ ⇤+3.233 ⇢2⇤
R. Rigon
Thermal conductivity
![Page 54: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/54.jpg)
!54
Temperatura
Generalmente nel manto nevoso si presentano due situazioni:
!
- E’ presente una variazione di temperatura tra la sommità della neve
e il il terreno su cui si posa: la temperatura è normalmente dominata
dalla temperatura in superficie e il terreno si trova generalmente a 0
C .... a meno che non si sia in presenza di permafrost.
!
- Non è presente alcun gradiente: la neve si trova in uno stato isotermo
R. Rigon
Thermal conductivity
![Page 55: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/55.jpg)
!55
Temperatura
La neve è un buon isolante termico. Si generano gradienti di temperatura anche molto elevati in prossimità della superficie.
R. Rigon
Phenomenology
![Page 56: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/56.jpg)
!56
050
100
150
Snow
Dep
th [c
m]
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●
●●
●
●
●
●●●●●●●●●●●
●●●
●●●●
●
●●●●
●●●●●
●●
●●●
●●
●
●●
●●●●
●●●●●●●
●
●●●●●●●●●●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●
●
●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●
●
●
●
●●
●●
●●
●
●●●●●
●
●●●●●
●
●●●●●●●●●●●●●
●
●
●
●●
●●●●●●●●●●●●●●●
●
●●
●
●
●
●
●
●●●●●
●
●●●●●●●●●●●●●
●●
●
●
●
●
●
●
●●●●
●●●●●●●●●●●●●●●●●●●●●●●●
●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●
●●●
●●●●●●●●●●●●●●●
●
●
●●
●●
●
●●●●●●●
●●●●●
●●●●●●●●●
●●●●●●●●●●
●●●●●●●●●●●●●
●
●●
●●●
●●●
●
●●●
●
●
●
●●●●●
SnowD simFlux to ground
Nov 97 Feb 98 May 98 Aug 98 Nov 98
030
6090
120
150
Flux
to g
roun
d [W
/m^2
]
● SnowD meas
estateinverno
circa 50 W/m2circa 5 W/m2
Temperaturawith and without
R. Rigon
Phenomenology
![Page 57: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/57.jpg)
!57
Scambi di energia attraverso i flussi
turbolenti
Conduzione di calore verso il terreno
Percolazione di acqua verso il terreno
Bilancio di radiazione
dU⇤dt
= Rn lw + Rn sw �H � �s Ev + G + Pe
Variazione di energia
dU⇤dt
= CpdT⇤dt
Capacità termica della neveVariazione di temperatura
della neve
Il bilancio di energia della neve
R. Rigon
Energy budget
![Page 58: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/58.jpg)
!58!58
energy fluxes at the boundary
phase transition
Variazione di energia della neve
Il bilancio di energia interno
Neve
R. Rigon
Energy budget
![Page 59: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/59.jpg)
!59!59
phase transition
Variazione di energia della neve
Flussi di energia al contorno
Il bilancio di energia interno
Neve
Energy budget
![Page 60: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/60.jpg)
!60!60
Transizioni di fase
Variazione di energia della neve
Flussi di energia
Il bilancio di energia interno
Neve
Energy budget
![Page 61: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/61.jpg)
!61
Il flusso di calore trasportato dalla precipitazione è calcolato
supponendo che la precipitazione abbia la stessa temperatura
dell’aria e quello trasportato dall’acqua di fusione supponendo che
questa sia alla temperatura di 0°C.
Una nota sulla precipitazione
R. Rigon
Precipitation and Energy
![Page 62: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/62.jpg)
!62!62
raffreddamento/riscaldamento per conduzione
raffreddamento/riscaldamento per avvezione (principalmente di
acqua liquida)
Il bilancio di energia interno
Neve
Energy budget
![Page 63: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/63.jpg)
!63!63
Dove il termine di flusso
è dato dal termine riscaldamento/raffreddamento per
conduzione:
raffreddamento/riscaldamento:
il flusso di calore
Neve
Energy budget
![Page 64: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/64.jpg)
!64!64
gradiente di temperatura
raffreddamento/riscaldamento:
il flusso di calore
Dove il termine di flusso
è dato dal termine riscaldamento/raffreddamento per
conduzione:
Neve
Energy budget
![Page 65: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/65.jpg)
!65!65
conducibilità termica
Questa è la teoria di
Onsager che porta
alla legge di Fourier
gradiente di temperatura
raffreddamento/riscaldamento:
il flusso di calore
Dove il termine di flusso
è dato dal termine riscaldamento/raffreddamento per
conduzione:
Neve
Energy budget
![Page 66: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/66.jpg)
!66!66
L’energia interna della neve
variazione dell’energia interna della neve
Neve
Energy budget
![Page 67: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/67.jpg)
!67!67
Una parte dipende dalla temperatura
variazione dell’energia interna della neve
L’energia interna della neve nelle sue parti
Energy budget
![Page 68: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/68.jpg)
!68!68
Una parte dipende dalla quantità della sostanza
Una parte dipende dalla temperatura
variazione dell’energia interna della neve
L’energia interna della neve nelle sue parti
Energy budget
![Page 69: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/69.jpg)
!69
Lo scioglimento del manto nevoso
Tradotto in termini del bilancio di energia. Per T < 0
Variazione dell’energia interna del
ghiaccio
<latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit>
R. Rigon
Energy budget
![Page 70: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/70.jpg)
!70
Lo scioglimento del manto nevoso
Tradotto in termini del bilancio di energia. Per T > 0
Variazione dell’energia interna dell’acqua ... ma in questo caso bisognerebbe
fare dei distinguo ...
<latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit>
R. Rigon
Energy budget
![Page 71: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/71.jpg)
!71
Lo scioglimento del manto nevoso
`
Variazione dell’energia interna
del sistema complessivo
acqua + ghiaccio Variazione di entalpia del sistema
acqua + ghiaccio
dT
dt= 0T = 0 dp
dt= 0
<latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit>
<latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit>
R. Rigon
Energy budget
![Page 72: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/72.jpg)
!72
Lo scioglimento del manto nevosoLo scioglimento del manto nevoso
Le porzioni relative nel volume di controllo di ghiaccio e neve sono determinate
dai rispettivi volumi.
!Questi ultimi sono, evidentemente funzione della storia energetica della neve.
dT
dt= 0T = 0 dp
dt= 0
dU⇤dt
=dH
dt
R. Rigon
Phase Transition
![Page 73: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/73.jpg)
!73
Lo scioglimento del manto nevoso
Nei layer non superficiali, si ha conduzione del calore secondo la legge di
Fourier (se si trascura la percolazione)
<latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit>
<latexit sha1_base64="tYHCApFiY8slQcKMwQxwGacE74A=">AAAA+3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8XFy8cvEFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKBcQLKBvoGYCBAibDEMpQZoACoHJDdElMRqiRnpmeQSBCG4e0koahuYNHQGhyStfknfsPQoQZGaHyggyo4BQAVIE48g==</latexit>
R. Rigon
Phase Transition
![Page 74: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/74.jpg)
!74!74
le due equazioni di conservazione della massa e dell’energia vengono risolte
congiuntamente per la massa di neve
Poichè le equazioni sono accoppiate
dM⇤dt
= P � Ev �Gp in superficie
all’interno della neve
in superficie
all’interno della neve
R. Rigon
Equations
![Page 75: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/75.jpg)
!75
Ma i metodi di soluzione
rimangono i medesimi
Le equazioni sono non lineari e discontinue alla transizione di fase richiedono il metodo di
Newton con doppia iterazione interna
R. Rigon
Phase Transition
![Page 76: Diffusion equationhydrology2014](https://reader034.vdocument.in/reader034/viewer/2022051412/54857828b47959f10c8b4e71/html5/thumbnails/76.jpg)
!76
Grazie per l’attenzione!
G.U
lric
i -
, 20
00
?
R. Rigon
The End