This is similar to the lecture Niccolò gave in Ottawa during his staying in Carleton University. This also contains further results from his Ph.D. thesis
25 CHUYÊN ĐỀ ÔN THI TỐT NGHIỆP THPT 2023 – BÀI TẬP PHÁT TRIỂN TỪ ĐỀ MINH HỌA...
Freezing Soil for the class of Environmental Modelling
1. Department of Civil, Environmental and Mechanical Engineering
Master of Science in
Environmental and Land Engineering
THEORETICAL PROGRESS IN
FREEZING – THAWING PROCESSES STUDY
Supervisor Student
Prof. Riccardo Rigon Niccolò Tubini
Co-supervisors
Prof. Stephan Gruber
Dr. Francesco Serafin
2. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
This work is licensed under a Creative
Commons “Attribution-ShareAlike 4.0
International” license.
Niccolò Tubini Theoretical progress in freezing – thawing processes study
3. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
What is the purpose?
The aim of my Master’s thesis is to develop a new
interpretation of modeling freezing soils.
Niccolò Tubini Theoretical progress in freezing – thawing processes study 2 / 34
4. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Why studing the influence of coupled heat and water flow
in soils?
Freezing – thawing processes entail a huge
exchange of heat;
Niccolò Tubini Theoretical progress in freezing – thawing processes study 3 / 34
5. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Why studing the influence of coupled heat and water flow
in soils?
To simulate more realistic soil temperature
(Luo et al., 2003).
Niccolò Tubini Theoretical progress in freezing – thawing processes study 3 / 34
6. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Why studing the influence of coupled heat and water flow
in soils?
Studies have shown that proper frozen soil
schemes help improve climate model simulation
(Viterbo et al., 1999 and Smirnova et al., 2000).
Niccolò Tubini Theoretical progress in freezing – thawing processes study 3 / 34
7. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soils
Some definitions
Air gas
Liquid water
Soil particle
Va
Vw
Vs
Vc
Niccolò Tubini Theoretical progress in freezing – thawing processes study 4 / 34
8. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soils
Some definitions
Soil porosity
φ :=
Vs
Vc
Water content
θ :=
Vw
Vc
Niccolò Tubini Theoretical progress in freezing – thawing processes study 5 / 34
9. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soils
Some definitions
Assuming the rigid soil scheme
θs := φ
0 < θr ≤ θ ≤ θs < 1
Niccolò Tubini Theoretical progress in freezing – thawing processes study 6 / 34
10. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soils
Young – Laplace equation
r γaw
α pw = pa −
2γaw cos α
r
Niccolò Tubini Theoretical progress in freezing – thawing processes study 7 / 34
11. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soils
Young – Laplace equation
pa ← 0
Let us define suction as
ψ :=
pw
gρw
Niccolò Tubini Theoretical progress in freezing – thawing processes study 8 / 34
12. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Mualem’s assumption
Wetting and drying processes are assumed to be
selective processes.
Niccolò Tubini Theoretical progress in freezing – thawing processes study 9 / 34
13. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Water – retention – hydraulic – conductivity models
Dealing with unsaturated soils requires the
definition of the relationship between
θ–ψ and K–ψ
Niccolò Tubini Theoretical progress in freezing – thawing processes study 10 / 34
14. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Empirical curve-fitting models
Parameters of these models have been related to
the soil texture and other soil properties
Niccolò Tubini Theoretical progress in freezing – thawing processes study 11 / 34
15. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Empirical curve-fitting models
Parameters of these models have been related to
the soil texture and other soil properties
Despite their usfulness they do not emphasize the
physical significance of their empirical parameters
Niccolò Tubini Theoretical progress in freezing – thawing processes study 11 / 34
16. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Lognormal distribution model (Kosugi, 1996)
The idea is to derive the water retention curve from
the pore-size distribution:
f (r) :=
dθ
dr
Niccolò Tubini Theoretical progress in freezing – thawing processes study 12 / 34
17. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Lognormal distribution model (Kosugi, 1996)
r
0
50
100
150
f(r)
R
Water
f (r) =
θs − θr
√
2π σr
exp
−
ln
r
rm
2
2σ2
θ(R) = θr +
R
0
f (r)dr
Niccolò Tubini Theoretical progress in freezing – thawing processes study 13 / 34
18. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Lognormal distribution model (Kosugi, 1996)
Young-Laplace equation allows to transform the
pore-size distribution into the capillary pressure
distribution function
g(ψ) = f (r)
dr
dψ
Niccolò Tubini Theoretical progress in freezing – thawing processes study 14 / 34
19. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Lognormal distribution model (Kosugi, 1996)
θ(Ψ) = θr +
Ψ
−∞
g(ψ)dψ
Ψ = −
2γaw cos α
g ρw R
Niccolò Tubini Theoretical progress in freezing – thawing processes study 15 / 34
20. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Some definitions
Air gas
Ice
Liquid water
Particle soil
Va
Vi
Vw
Vs
Vc
Niccolò Tubini Theoretical progress in freezing – thawing processes study 16 / 34
21. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Some definitions
Liquid water content
θw :=
Vw
Vc
Ice content
θi :=
Vi
Vc
Niccolò Tubini Theoretical progress in freezing – thawing processes study 17 / 34
22. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Some definitions
Total water content
θ := θw + θi
0 < θr ≤ θ ≤ θs < 1
Niccolò Tubini Theoretical progress in freezing – thawing processes study 18 / 34
23. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Model assumptions
Model assumptions
rigid soil scheme
Niccolò Tubini Theoretical progress in freezing – thawing processes study 19 / 34
24. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Model assumptions
Model assumptions
rigid soil scheme
freezing = drying (Miller, 1965; Spaans and Baker, 1996)
Niccolò Tubini Theoretical progress in freezing – thawing processes study 19 / 34
25. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Model assumptions
Model assumptions
rigid soil scheme
freezing = drying (Miller, 1965; Spaans and Baker, 1996)
the phase change is assumed to occur at the
thermodynamic equilibrium
Niccolò Tubini Theoretical progress in freezing – thawing processes study 19 / 34
26. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Freezing point depression
Gibbs-Thomson equation (Acker et al., 2001)
Tm − T∗
=
2 γaw Tm cos α
ρw r
+
πw Tm
ρw
Capillary effect
Dissolved solutes
Niccolò Tubini Theoretical progress in freezing – thawing processes study 20 / 34
27. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Freezing point depression
Gibbs-Thomson equation (Acker et al., 2001)
The ice-water interface occurs at:
ˆr(T) := −
2 γaw Tm cos α
ρw (T − Tm)
for T < Tm∗
Niccolò Tubini Theoretical progress in freezing – thawing processes study 21 / 34
28. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Water and ice content
Let us define
r∗
:=
R if ˆr ≥ R or T ≥ Tm
ˆr otherwise
∂r∗
∂t
:=
∂R
∂t
if ˆr ≥ R or T ≥ Tm
∂ˆr
∂t
otherwise
Niccolò Tubini Theoretical progress in freezing – thawing processes study 22 / 34
29. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Water and ice content
r
0
50
100
150
f(r)
Rˆr = r∗
Water
Ice
θw = θr +
r∗
0
f (r)dr
θi =
R
r∗
f (r)dr
Niccolò Tubini Theoretical progress in freezing – thawing processes study 23 / 34
30. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Water and ice content
The phase change rate
θi =
R
r∗
f (r)dr
∂θi
∂t
=
∂R
∂t
f (R) −
∂r∗
∂t
f (r∗
)
Niccolò Tubini Theoretical progress in freezing – thawing processes study 24 / 34
31. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Water and ice content
The phase change rate
r
0
20
40
60
80
100
120
f(r)
R(t) R(t + δt)ˆr
Water
Ice at time t
Ice at time t + δt
Niccolò Tubini Theoretical progress in freezing – thawing processes study 25 / 34
32. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Water and ice content
The phase change rate
r
0
20
40
60
80
100
120
f(r)
Rˆr(t)ˆr(t + δt)
Water
Ice formed in δt
Ice at time t
Niccolò Tubini Theoretical progress in freezing – thawing processes study 26 / 34
33. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Water and ice content
The phase change rate
∂θi
∂t
=
∂Ψ
∂t
g(Ψ) −
∂ψ∗
∂t
g(ψ∗
)
Niccolò Tubini Theoretical progress in freezing – thawing processes study 27 / 34
34. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Mass conservation equation
θ
J ET
Jw
∂
∂t
(ρw θw + ρiθi) = −ρw · Jw
Niccolò Tubini Theoretical progress in freezing – thawing processes study 28 / 34
35. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Mass conservation equation
θ
J ET
Jw
∂
∂t
(ρw θw + ρiθi) = −ρw · Jw
Water flux:
Jw = −K(ψ∗
) (ψ∗
+ z)
Niccolò Tubini Theoretical progress in freezing – thawing processes study 28 / 34
36. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Mass conservation equation
Setting ρw = ρi
∂θ
∂t
=
∂Ψ
∂t
g(Ψ) = · [K(ψ∗
) (ψ∗
+ z)]
∂θi
∂t
=
∂Ψ
∂t
g(Ψ) −
∂ψ∗
∂t
g(ψ∗
)
∂θw
∂t
=
∂θ
∂t
−
∂θi
∂t
Niccolò Tubini Theoretical progress in freezing – thawing processes study 29 / 34
37. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation
ε
J
HRn
ET
JwJg
∂ε
∂t
= − · (Jw + Jg)
Niccolò Tubini Theoretical progress in freezing – thawing processes study 30 / 34
38. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation
ε
J
HRn
ET
JwJg
∂ε
∂t
= − · (Jw + Jg)
Advective flux:
Jw = Jw ρw [ + cw (T − Tm)]
Heat conduction:
Jg = −λ T
Niccolò Tubini Theoretical progress in freezing – thawing processes study 30 / 34
39. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation
Setting ρw = ρi
CT
∂T
∂t
− ρi
∂Ψ
∂t
g(Ψ) −
∂ψ∗
∂t
g(ψ∗
)
−ρi(cw − ci)(T − Tm)
∂Ψ
∂t
g(Ψ) −
∂ψ∗
∂t
g(ψ∗
)
+ρicw Jw · T + ρigz · Jw − · Jg = 0
Niccolò Tubini Theoretical progress in freezing – thawing processes study 31 / 34
40. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation: if ice occurs
ψ∗
:= ˆψ
∂ψ∗
∂t
:=
g Tm
∂T
∂t
Cph
∂T
∂t
− ρi[ + (cw − ci)(T − Tm)]
∂Ψ
∂t
g(Ψ)
+ ρicw Jw · T + ρigz · Jw − · Jg = 0
Niccolò Tubini Theoretical progress in freezing – thawing processes study 32 / 34
41. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
The apparent heat capacity
Cph := CT + ρi[ + (cw − ci)(T − Tm)]
g Tm
Niccolò Tubini Theoretical progress in freezing – thawing processes study 33 / 34
42. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
The apparent heat capacity
Cph := CT + ρi[ + (cw − ci)(T − Tm)]
g Tm
CT := ρscs(1 − θs) + ρiciθi + ρw cw θw
Niccolò Tubini Theoretical progress in freezing – thawing processes study 33 / 34
43. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
To take home
Freezing=drying and rigid soil scheme
assumptions are useful when freezing-induced
mechanical deformations are not considered;
Niccolò Tubini Theoretical progress in freezing – thawing processes study 34 / 34
44. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
To take home
Freezing/thawing processes do not occur at the
thermodynamic equilibrium (Kurylyk, 2013).
Niccolò Tubini Theoretical progress in freezing – thawing processes study 34 / 34
45. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
To take home
Kosugi retention model has the benefit to be
straightforward extended to freezing soils case
by making use of Gibbs – Thomson equation;
Niccolò Tubini Theoretical progress in freezing – thawing processes study 34 / 34
46. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
To take home
This formulation allows to take into account of
dissolved solutes;
Niccolò Tubini Theoretical progress in freezing – thawing processes study 35 / 34
47. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
To take home
It is possible to solve the mass and energy
equation in a decoupled way;
Niccolò Tubini Theoretical progress in freezing – thawing processes study 35 / 34
49. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
References
K. Kosugi, Lognormal distribution model for unsaturated
soil hydraulic properties, Water Resources Research, vol. 32,
no. 9, pp. 2697–2703, 1996.
J. T. Acker et al., Intercellular ice propagation:
experimental evidence for ice growth through membrane
pores, Biophysical journal, vol. 81, no. 3, pp. 1389–1397,
2001.
M. Dall’Amico et al., A robust and energy-conserving
model of freezing variably-saturated soil, The Cryosphere,
vol. 5, no. 2, p. 469, 2011.
M. Dall’Amico, Coupled water and heat transfer in
permafrost modeling, Ph.D. dissertation, University of
Trento, 2010.
Niccolò Tubini Theoretical progress in freezing – thawing processes study
50. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
References
E. J. Spaans and J. M. Baker, The soil freezing
characteristic: Its measurement and similarity to the soil
moisture characteristic, Soil Science Society of America
Journal, vol. 60, no. 1, pp. 13–19, 1996.
R. D. Miller, Phase equilibria and soil freezing, vol. 287, pp.
193–197, 1965.
B. L. Kurylyk and K. Watanabe, The mathematical
representation of freezing and thawing processes in
variably-saturated, non-deformable soils, Advances in Water
Resources, vol. 60, pp. 160–177, 2013.
Niccolò Tubini Theoretical progress in freezing – thawing processes study
51. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
References
L. Luo et al., Effects of frozen soil on soil temperature,
spring infiltration, and runoff: Results from the PILPS 2 (d)
experiment at Valdai, Russia, Journal of Hydrometeorology,
vol. 4, no. 2, pp. 334–351, 2003.
T. G. Smirnova, J. M. Brown, S. G. Benjamin, and D. Kim,
Parameterization of cold-season processes in the maps
land-surface scheme, Journal of Geophysical Research:
Atmospheres, vol. 105, no. D3, pp. 4077– 4086, 2000.
P. Viterbo, A. Beljaars, J.-F. Mahfouf, and J. Teixeira, The
representation of soil moisture freezing and its impact on
the stable boundary layer, Quarterly Journal of the Royal
Meteorological Society, vol. 125, no. 559, pp. 2401–2426,
1999.
Niccolò Tubini Theoretical progress in freezing – thawing processes study
52. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Freezing=drying assumptionDall’Amico,2010
pa
pw (R)
R
r
Air-water interface
pw (R) = pa −
2 γaw cos α
R
Niccolò Tubini Theoretical progress in freezing – thawing processes study
53. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Freezing=drying assumptionDall’Amico,2010
pa
pi
pw (r)
R
r
Air-ice interface
pi = pa −
2 γai cos α
R
Ice-water interface
pw (r) = pi −
2 γiw cos α
r
Niccolò Tubini Theoretical progress in freezing – thawing processes study
54. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Freezing=drying assumptionDall’Amico,2010
pa
pi ≡ pa
pw (r)
R
r
Air-water interface
pw (r) = pa −
2 γaw cos α
r
Niccolò Tubini Theoretical progress in freezing – thawing processes study
55. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
r
0
20
40
60
80
100
120f(r)
R = r∗
ˆr
Water
Niccolò Tubini Theoretical progress in freezing – thawing processes study
56. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
The phase change rate
Thanks to the Young-Laplace equation
ψ∗
:=
Ψ if ˆr ≥
2 γaw cos α
ρw g Ψ
or T ≥ Tm
ˆψ = ψ(ˆr) otherwise
Niccolò Tubini Theoretical progress in freezing – thawing processes study
57. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
The phase change rate
Thanks to the Young-Laplace equation
∂ψ∗
∂t
:=
∂Ψ
∂t
if ˆr ≥
2γaw cos α
ρw gΨ
or T ≥ Tm
∂ ˆψ
∂t
otherwise
Niccolò Tubini Theoretical progress in freezing – thawing processes study
58. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Comparison with Dall’Amico model (Dall’Amico et al., 2011)
By making use of Clausius – Clapeyron equation:
dT
dpw
=
T
ρw
Niccolò Tubini Theoretical progress in freezing – thawing processes study
59. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Comparison with Dall’Amico model (Dall’Amico et al., 2011)
By making use of Clausius – Clapeyron equation:
dT
dpw
=
T
ρw
T∗
= Tm +
g Tm
ψw0
ψ(T) = ψw0 +
g T∗
(T − T∗
)H(T∗
− T)
Niccolò Tubini Theoretical progress in freezing – thawing processes study
60. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Mass conservation equation: if there is no ice
ψ∗
:= Ψ
∂ψ∗
∂t
:=
∂Ψ
∂t
∂θ
∂t
=
∂Ψ
∂t
g(Ψ) = · [K(Ψ) (Ψ + z)]
∂θi
∂t
=
:0
∂Ψ
∂t
g(Ψ) −
∂ψ∗
∂t
g(ψ∗
)
∂θw
∂t
=
∂θ
∂t
Niccolò Tubini Theoretical progress in freezing – thawing processes study
61. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Mass conservation equation: if ice occurs
ψ∗
:= ˆψ
∂ψ∗
∂t
:=
g Tm
∂T
∂t
∂θ
∂t
=
∂ ˆψ
∂t
g(Ψ) = · [K( ˆψ) ( ˆψ + z)]
∂θi
∂t
=
∂Ψ
∂t
g(Ψ) −
∂ ˆψ
∂t
g( ˆψ)
∂θw
∂t
=
∂θ
∂t
−
∂θi
∂t
Niccolò Tubini Theoretical progress in freezing – thawing processes study
62. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation: if there is no ice
ψ∗
:= Ψ
∂ψ∗
∂t
:=
∂Ψ
∂t
CT
∂T
∂t
− ρil
:0
∂Ψ
∂t
g(Ψ) −
∂ψ∗
∂t
g(ψ∗
)
− ρi(cw − ci)(T − Tm)
:0
∂Ψ
∂t
g(Ψ) −
∂ψ∗
∂t
g(ψ∗
)
+ ρicw Jw · T + ρigz · Jw − · Jg = 0
Niccolò Tubini Theoretical progress in freezing – thawing processes study
63. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation: if there is no ice
CT
∂T
∂t
+ ρw cw Jw · T + ρw gz · Jw − · Jg = 0
Niccolò Tubini Theoretical progress in freezing – thawing processes study
64. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation: if there is no ice
CT
∂T
∂t
+ ρw cw Jw · T + ρw gz · Jw − · Jg = 0
CT := ρscs(1 − θs) + ρiciθi + ρw cw θw
Niccolò Tubini Theoretical progress in freezing – thawing processes study
65. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Numerical scheme for unfrozen soils
The mass conservation equation ⇒ Nested Newton
method (Casulli and Zanolli, 2010).
The energy consevation equation ⇒ Implicit upwind
method
Niccolò Tubini Theoretical progress in freezing – thawing processes study
66. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Numerical scheme for frozen soils
The mass conservation equation becomes
∂θ
∂t
= · K( ˆψ) ( ˆψ + z)
Niccolò Tubini Theoretical progress in freezing – thawing processes study
67. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Numerical scheme for frozen soils
The mass conservation equation becomes
∂θ
∂t
= · K( ˆψ) ( ˆψ + z)
Nested Newton method (Casulli and Zanolli, 2010).
should be extended for equations of two variables
Niccolò Tubini Theoretical progress in freezing – thawing processes study
68. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Numerical scheme for frozen soils
The mass conservation equation becomes
∂θ
∂t
= · K( ˆψ) ( ˆψ + z)
Nested Newton method (Casulli and Zanolli, 2010).
should be extended for equations of two variables
The energy consevation equation ⇒ Implicit upwind
method
Niccolò Tubini Theoretical progress in freezing – thawing processes study