Ph.D. Thesis Defense

McGill University
Department of Physics

Transcript

1. Rafael Fuentes Andrew Cumming McGill University Hydrodynamics of convection with composition gradients: Salty water and Jupiter PhD Thesis Defense - Oct. 17th 2022
2. Rafael Fuentes Andrew Cumming McGill University Hydrodynamics of convection with composition gradients: Salty water and Jupiter PhD Thesis Defense - Oct. 17th 2022
3. Z - core H-He envelope Classic picture of Jupiter (Mizuno et al., 1978; Bodenheimer and Pollack, 1986; Pollack et al., 1996) Large - scale convection !! Jupiter’s interior Rafael Fuentes PhD Thesis Defense 2/15
4. Z - core H-He envelope Classic picture of Jupiter (Mizuno et al., 1978; Bodenheimer and Pollack, 1986; Pollack et al., 1996) Large - scale convection !! Jupiter’s interior Rafael Fuentes PhD Thesis Defense H-He envelope Jupiter after Juno (Wahl et al. 2017, Debras & Chabrier 2019) Di ff use core 2/15
5. Z - core H-He envelope Classic picture of Jupiter (Mizuno et al., 1978; Bodenheimer and Pollack, 1986; Pollack et al., 1996) Large - scale convection !! Jupiter’s interior Rafael Fuentes PhD Thesis Defense H-He envelope Jupiter after Juno (Wahl et al. 2017, Debras & Chabrier 2019) Di ff use core 2/15 Jupiter is not fully-mixed, it has a large composition gradient
6. Why this is important? Rafael Fuentes PhD Thesis Defense How compositional gradients form? How they survive over long-time scales? Composition gradients can stabilize fl uids against convection. A ff ects cooling evolution of the planet! 3/15
7. Why this is important? Rafael Fuentes PhD Thesis Defense How compositional gradients form? How they survive over long-time scales? Composition gradients can stabilize fl uids against convection. A ff ects cooling evolution of the planet! 3/15 Di ff usive interface Convective layer Multiple layers convective layers can form in the fl uid! (Garaud et al. 2018)
8. Why this is important? ∇μ = d ln μ d ln P > 0 Rafael Fuentes PhD Thesis Defense How compositional gradients form? How they survive over long-time scales? Composition gradients can stabilize fl uids against convection. A ff ects cooling evolution of the planet! 3/15 Di ff usive interface Convective layer Multiple layers convective layers can form in the fl uid! (Garaud et al. 2018)
9. Jupiter evolution with composition gradients (Vazan et al. 2018) 1D evolution models Rafael Fuentes PhD Thesis Defense 4/15 Di ff usive interface Convective layer
10. Jupiter evolution with composition gradients (Vazan et al. 2018) 1D evolution models Rafael Fuentes PhD Thesis Defense In general models predicts that compositional gradient can persist in time 4/15 Di ff usive interface Convective layer
11. Jupiter evolution with composition gradients (Vazan et al. 2018) 1D evolution models Rafael Fuentes PhD Thesis Defense In general models predicts that compositional gradient can persist in time 4/15 Di ff usive interface Convective layer
12. Jupiter evolution with composition gradients (Vazan et al. 2018) 1D evolution models Rafael Fuentes PhD Thesis Defense (This could be arti fi cial because treatment of compositional mixing is wrong in many ways) In general models predicts that compositional gradient can persist in time 4/15 Di ff usive interface Convective layer
13. Jupiter evolution with composition gradients (Vazan et al. 2018) 1D evolution models Rafael Fuentes PhD Thesis Defense (This could be arti fi cial because treatment of compositional mixing is wrong in many ways) In general models predicts that compositional gradient can persist in time We need better understanding of transport across layers (multi D simulations and experiments) 4/15 Di ff usive interface Convective layer
14. Fluid dynamics perspective Rafael Fuentes PhD Thesis Defense Experiments with Salty water (Turner 1965, 1968, Fernando 1987) 5/15
15. Fluid dynamics perspective Rafael Fuentes PhD Thesis Defense Experiments with Salty water (Turner 1965, 1968, Fernando 1987) Pr = ν κT In salty water Pr ≈ 7 In giant planets Pr ≈ 0.01 − 1 (Chabrier and Bara ff e 2007) But cannot be extrapolated to Gas giants 5/15
16. Fluid dynamics perspective Rafael Fuentes PhD Thesis Defense Experiments with Salty water (Turner 1965, 1968, Fernando 1987) Pr = ν κT In salty water Pr ≈ 7 In giant planets Pr ≈ 0.01 − 1 (Chabrier and Bara ff e 2007) But cannot be extrapolated to Gas giants This thesis: Experiments of penetrative convection in more than 1D and at low Pr (for the fi rst time) 5/15
17. Fluid dynamics perspective Rafael Fuentes PhD Thesis Defense Experiments with Salty water (Turner 1965, 1968, Fernando 1987) Pr = ν κT In salty water Pr ≈ 7 In giant planets Pr ≈ 0.01 − 1 (Chabrier and Bara ff e 2007) But cannot be extrapolated to Gas giants This thesis: Experiments of penetrative convection in more than 1D and at low Pr (for the fi rst time) 5/15
18. Numerical model Rafael Fuentes PhD Thesis Defense Temperature Solute • Dedalus code, Open source, python based spectral code for solving PDEs Burns et al (2020) http://dedalus-project.org • 2D Boussinesq (incompressible fl ow) • Initial conditions: Uniform temperature, linear solute gradient • Drive convection by applying a constant heat fl ux at the top F0 6/15
19. Numerical model Rafael Fuentes PhD Thesis Defense Temperature Solute • Dedalus code, Open source, python based spectral code for solving PDEs Burns et al (2020) http://dedalus-project.org • 2D Boussinesq (incompressible fl ow) • Initial conditions: Uniform temperature, linear solute gradient • Drive convection by applying a constant heat fl ux at the top F0 6/15
20. Numerical model Rafael Fuentes PhD Thesis Defense Temperature Solute • Dedalus code, Open source, python based spectral code for solving PDEs Burns et al (2020) http://dedalus-project.org • 2D Boussinesq (incompressible fl ow) • Initial conditions: Uniform temperature, linear solute gradient • Drive convection by applying a constant heat fl ux at the top F0 Vary in 0.1 - 7 and cooling fl ux ! Pr 6/15
21. Some questions to answer Rafael Fuentes PhD Thesis Defense 7/15
22. Some questions to answer Rafael Fuentes PhD Thesis Defense 1) How quickly does the outer convection layer move inwards? 2) Do multiple convective layers form in the fluid? 3) Does the fluid become fully mixed? 7/15
23. Rafael Fuentes PhD Thesis Defense Some answers How quickly does the outer convection layer move inwards? 8/15
24. Rafael Fuentes PhD Thesis Defense Some answers How quickly does the outer convection layer move inwards? Convection zone grows faster at low Pr ! Why? 0.000 0.002 0.004 0.006 0.008 0.010 t/tdiÆ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 h/H Pr = 0.1, F0 = 5.4Fcrit Pr = 7, F0 = 5.4Fcrit C = 1/3 “Toy Jupiter” C = 1 Salty-water 8/15
25. Rafael Fuentes PhD Thesis Defense Some answers How quickly does the outer convection layer move inwards? Convection zone grows faster at low Pr ! Why? 0.000 0.002 0.004 0.006 0.008 0.010 t/tdiÆ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 h/H Pr = 0.1, F0 = 5.4Fcrit Pr = 7, F0 = 5.4Fcrit C = 1/3 “Toy Jupiter” C = 1 Salty-water 8/15 h(t) F0 Fa
26. Rafael Fuentes PhD Thesis Defense Some answers How quickly does the outer convection layer move inwards? Convection zone grows faster at low Pr ! Why? 0.000 0.002 0.004 0.006 0.008 0.010 t/tdiÆ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 h/H Pr = 0.1, F0 = 5.4Fcrit Pr = 7, F0 = 5.4Fcrit C = 1/3 “Toy Jupiter” C = 1 Salty-water 8/15 h(t) F0 Fa Higher mixing e ff i ciency at low Pr!!
27. Rafael Fuentes PhD Thesis Defense Some answers How quickly does the outer convection layer move inwards? 9/15
28. Rafael Fuentes PhD Thesis Defense Some answers How quickly does the outer convection layer move inwards? h(t) = 2 C ( F0 Fcrit ) 1/2 (κT t) 1/2 Fcrit = k ( β α dS0 dz ) 9/15
29. Rafael Fuentes PhD Thesis Defense Some answers How quickly does the outer convection layer move inwards? C = 1 1 − ε + 2γ This is new! h(t) = 2 C ( F0 Fcrit ) 1/2 (κT t) 1/2 Fcrit = k ( β α dS0 dz ) 9/15
30. 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 t/tdiÆ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 h/H Pr = 0.1, F0 /Fcrit = 5.4 Pr = 1, F0 /Fcrit = 5.4 Pr = 7, F0 /Fcrit = 5.4 C = 1/3 C = 1 Rafael Fuentes PhD Thesis Defense Some answers How quickly does the outer convection layer move inwards? C = 1 1 − ε + 2γ This is new! h(t) = 2 C ( F0 Fcrit ) 1/2 (κT t) 1/2 Fcrit = k ( β α dS0 dz ) 9/15
31. Some answers Rafael Fuentes PhD Thesis Defense Does the fl uid become fully mixed? 10/15
32. Some answers Rafael Fuentes PhD Thesis Defense Does the fl uid become fully mixed? 10/15
33. Finn (1993)
 Rosenbluth (1994)
 Goluskin et al. (2014), Shear fl ows previously observed in 2D convection Some answers Rafael Fuentes PhD Thesis Defense Does the fl uid become fully mixed? 10/15
34. Finn (1993)
 Rosenbluth (1994)
 Goluskin et al. (2014), Shear fl ows previously observed in 2D convection Some answers Rafael Fuentes PhD Thesis Defense Does the fl uid become fully mixed? No, because shear fl ows suppress vertical transport 10/15
35. Finn (1993)
 Rosenbluth (1994)
 Goluskin et al. (2014), Shear fl ows previously observed in 2D convection Some answers Rafael Fuentes PhD Thesis Defense Does the fl uid become fully mixed? No, because shear fl ows suppress vertical transport But shear fl ows do not arise in wider boxes ! 10/15
36. Rafael Fuentes PhD Thesis Defense Some answers Do multiple convective layers form in the fl uid? 11/15
37. Rafael Fuentes PhD Thesis Defense Some answers No staircases at low Pr They form at Pr = 7, Do multiple convective layers form in the fl uid? 11/15
38. Rafael Fuentes PhD Thesis Defense Some answers Mechanism for layer formation 12/15
39. Rafael Fuentes PhD Thesis Defense Some answers Mechanism for layer formation 12/15
40. Rafael Fuentes PhD Thesis Defense Some answers Mechanism for layer formation 2nd layer forms deeper in the interface when
 
 
 
 (our local version of the Ledoux criterion) α dT dz > β dS dz 12/15
41. Some answers Rafael Fuentes PhD Thesis Defense Why secondary layers do not form at low Pr ? 13/15
42. Some answers Rafael Fuentes PhD Thesis Defense Why secondary layers do not form at low Pr ? Low Pr “Toy Jupiter” Large Pr Salty-water 13/15
43. Some answers Rafael Fuentes PhD Thesis Defense Why secondary layers do not form at low Pr ? Temperature gradient is smaller at low !!! Pr Low Pr “Toy Jupiter” Large Pr Salty-water 13/15
44. Some answers Rafael Fuentes PhD Thesis Defense Why secondary layers do not form at low Pr ? Temperature gradient is smaller at low !!! Pr Low Pr “Toy Jupiter” Large Pr Salty-water Convective layers can form at low Pr by other mechanisms Double-di ff usive instabilities - (the initial gradients need to have the right value!) 13/15
45. Some answers Rafael Fuentes PhD Thesis Defense Does the fl uid become fully mixed? 14/15
46. Some answers Rafael Fuentes PhD Thesis Defense Does the fl uid become fully mixed? 0.000 0.025 0.050 0.075 0.100 0.125 t/tdiÆ 0.0 0.2 0.4 0.6 0.8 1.0 h/H Pr = 0.1, F0 /Fcrit = 5.4 Pr = 1, F0 /Fcrit = 5.4 Pr = 7, F0 /Fcrit = 5.4 It does ! 14/15
47. Rafael Fuentes PhD Thesis Defense Conclusions and final thoughts 15/15
48. Boussinesq Hydrodynamical simulations predict strong mixing at low Pr - Secondary layers are hard to form and the fl uid fully-mixes (this is di ff erent from 1D models !!! ). Rafael Fuentes PhD Thesis Defense Conclusions and final thoughts 15/15
49. Boussinesq Hydrodynamical simulations predict strong mixing at low Pr - Secondary layers are hard to form and the fl uid fully-mixes (this is di ff erent from 1D models !!! ). Rafael Fuentes PhD Thesis Defense Conclusions and final thoughts To approach Jupiter we must improve/do: 
 • Use more real distribution of heavy elements (real distribution is far varying linearly with depth)
 • Relax assumption of 2D (working on this)
 • Add rotation (working in this) • Adding density strati fi cation 15/15
50. Extra slides Rafael Fuentes PhD Thesis Defense
51. Rafael Fuentes PhD Thesis Defense Juno and gravity measurements is the planet’s equatorial radius a
52. Rafael Fuentes PhD Thesis Defense Juno and gravity measurements is the planet’s equatorial radius a
53. Rafael Fuentes PhD Thesis Defense Juno and gravity measurements is the planet’s equatorial radius a Measured
54. Rafael Fuentes PhD Thesis Defense Juno and gravity measurements is the planet’s equatorial radius a Measured Unknown function of X, T, P
55. Rafael Fuentes PhD Thesis Defense Some answers Why staircases do not form at low Pr ?
56. Rafael Fuentes PhD Thesis Defense Some answers Why staircases do not form at low Pr ? Large Pr, Salty-water Low Pr, Toy Jupiter
57. Rafael Fuentes PhD Thesis Defense Some answers Why staircases do not form at low Pr ? smaller at low , decreases the T gradient ΔT Pr Large Pr, Salty-water Low Pr, Toy Jupiter
58. Rafael Fuentes PhD Thesis Defense Some answers Why staircases do not form at low Pr ? smaller at low , decreases the T gradient ΔT Pr Large Pr, Salty-water Low Pr, Toy Jupiter
59. Rafael Fuentes PhD Thesis Defense Some answers Why staircases do not form at low Pr ? smaller at low , decreases the T gradient ΔT Pr smaller at low , increases the T gradient δT Pr Large Pr, Salty-water Low Pr, Toy Jupiter
60. Rafael Fuentes PhD Thesis Defense Some answers Why staircases do not form at low Pr ? smaller at low , decreases the T gradient ΔT Pr smaller at low , increases the T gradient δT Pr Large Pr, Salty-water Low Pr, Toy Jupiter
61. Rafael Fuentes PhD Thesis Defense Some answers Why staircases do not form at low Pr ? smaller at low , decreases the T gradient ΔT Pr smaller at low , increases the T gradient δT Pr Large Pr, Salty-water Low Pr, Toy Jupiter Nonetheless, the gradient is smaller at low Pr
62. Layered convection Rafael Fuentes PhD Thesis Defense
63. • Layered convection reduces the internal heat fl ux of the planet. It can explain in fl ated radii of some exoplanets (Chabrier and Bara ff e 2007) Layered convection Rafael Fuentes PhD Thesis Defense
64. • Layered convection reduces the internal heat fl ux of the planet. It can explain in fl ated radii of some exoplanets (Chabrier and Bara ff e 2007) Layered convection Rafael Fuentes PhD Thesis Defense
65. • Layered convection reduces the internal heat fl ux of the planet. It can explain in fl ated radii of some exoplanets (Chabrier and Bara ff e 2007) Adiabatic models Layered convection Rafael Fuentes PhD Thesis Defense
66. • Layered convection reduces the internal heat fl ux of the planet. It can explain in fl ated radii of some exoplanets (Chabrier and Bara ff e 2007) Adiabatic models Models with layered convection Layered convection Rafael Fuentes PhD Thesis Defense
67. • Layered convection reduces the internal heat fl ux of the planet. It can explain in fl ated radii of some exoplanets (Chabrier and Bara ff e 2007) • Reduction in cooling rate due to staircases could also explain Saturn’s higher luminosity (Leconte and Chabrier 2013) Adiabatic models Models with layered convection Layered convection Rafael Fuentes PhD Thesis Defense
68. • Layered convection reduces the internal heat fl ux of the planet. It can explain in fl ated radii of some exoplanets (Chabrier and Bara ff e 2007) • Reduction in cooling rate due to staircases could also explain Saturn’s higher luminosity (Leconte and Chabrier 2013) Adiabatic models Models with layered convection Composition gradients need to be considered in theoretical/numerical models Layered convection Rafael Fuentes PhD Thesis Defense
69. (Garaud 2018 and collaborators) ∝ dS/dz dT/dz Rafael Fuentes PhD Thesis Defense Double-di ff usive instabilities
70. Rafael Fuentes PhD Thesis Defense Double-di ff usive instabilities
71. Rafael Fuentes PhD Thesis Defense Double-di ff usive instabilities
72. Rafael Fuentes PhD Thesis Defense Aliasing errors
73. Rafael Fuentes PhD Thesis Defense Aliasing errors
74. Rafael Fuentes PhD Thesis Defense Aliasing errors On this low resolution grid cos(4x) is indistinguishable from cos(6x) !!
75. Rafael Fuentes PhD Thesis Defense Aliasing errors On this low resolution grid cos(4x) is indistinguishable from cos(6x) !! cos(2x)cos(4x)
76. Rafael Fuentes PhD Thesis Defense Aliasing errors On this low resolution grid cos(4x) is indistinguishable from cos(6x) !! cos(2x)cos(4x) = 1 2 cos(2x) + 1 2 cos(6x)
77. Rafael Fuentes PhD Thesis Defense How quickly does the outer convection layer move inwards? 9/17
78. Entrainment (Fernando 1987, Molemaker and Dijkstra 1997) Rafael Fuentes PhD Thesis Defense How quickly does the outer convection layer move inwards? 9/17
79. Entrainment (Fernando 1987, Molemaker and Dijkstra 1997) Δρg dh dt h = γρ0 v3 conv Mixing e ff i ciency! Rafael Fuentes PhD Thesis Defense How quickly does the outer convection layer move inwards? 9/17
80. Entrainment (Fernando 1987, Molemaker and Dijkstra 1997) Δρg dh dt h = γρ0 v3 conv Mixing e ff i ciency! Rafael Fuentes PhD Thesis Defense How quickly does the outer convection layer move inwards? 0.002 0.004 0.006 0.008 0.010 t/tdiÆ 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ∞ Pr = 0.1, F0 = 5.4Fcrit Pr = 0.1, F0 = 10.8Fcrit Pr = 1, F0 = 5.4Fcrit Pr = 1, F0 = 10.8Fcrit Pr = 7, F0 = 5.4Fcrit Pr = 7, F0 = 10.8Fcrit Pr = 0.1 Pr = 1 Pr = 7 F0 /Fcrit = 5.4 F0 /Fcrit = 10.8 “Toy Jupiter” Salty-water 9/17
81. Entrainment (Fernando 1987, Molemaker and Dijkstra 1997) Δρg dh dt h = γρ0 v3 conv Mixing e ff i ciency! Larger for large and for lower γ F0 Pr Rafael Fuentes PhD Thesis Defense How quickly does the outer convection layer move inwards? 0.002 0.004 0.006 0.008 0.010 t/tdiÆ 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ∞ Pr = 0.1, F0 = 5.4Fcrit Pr = 0.1, F0 = 10.8Fcrit Pr = 1, F0 = 5.4Fcrit Pr = 1, F0 = 10.8Fcrit Pr = 7, F0 = 5.4Fcrit Pr = 7, F0 = 10.8Fcrit Pr = 0.1 Pr = 1 Pr = 7 F0 /Fcrit = 5.4 F0 /Fcrit = 10.8 “Toy Jupiter” Salty-water 9/17
82. Rafael Fuentes PhD Thesis Defense Some answers How quickly does the outer convection layer move inwards? 10/17
83. h(t) F0 Fa Rafael Fuentes PhD Thesis Defense Some answers How quickly does the outer convection layer move inwards? 10/17
84. h(t) F0 Fa Change in the heat content within the layer determines ΔT d dt (ρ0 cP ΔTh) = F0 − Fa Rafael Fuentes PhD Thesis Defense Some answers How quickly does the outer convection layer move inwards? 10/17
85. h(t) F0 Fa Change in the heat content within the layer determines ΔT d dt (ρ0 cP ΔTh) = F0 − Fa Fa = εF0 ε ≈ 0.25 − 0.5 Rafael Fuentes PhD Thesis Defense Some answers How quickly does the outer convection layer move inwards? 10/17
86. Rafael Fuentes PhD Thesis Defense Standard model : core accretion
87. ( Pollack et al. 1996, Helled et al. 2013, Mordasini et al. 2012 Mgas MZ Mtot Rafael Fuentes PhD Thesis Defense Standard model : core accretion
88. ( Pollack et al. 1996, Helled et al. 2013, Mordasini et al. 2012 Mgas MZ Mtot Phase I : Solid accretion (build- up of a “core” made of heavy elements) I Rafael Fuentes PhD Thesis Defense Standard model : core accretion
89. ( Pollack et al. 1996, Helled et al. 2013, Mordasini et al. 2012 Mgas MZ Mtot Phase I : Solid accretion (build- up of a “core” made of heavy elements) I Phase II: Gas accretion (H - He) II Rafael Fuentes PhD Thesis Defense Standard model : core accretion
90. ( Pollack et al. 1996, Helled et al. 2013, Mordasini et al. 2012 Mgas MZ Mtot Phase I : Solid accretion (build- up of a “core” made of heavy elements) I Phase II: Gas accretion (H - He) II III Phase III: when , runaway accretion Mgas ∼ MZ Rafael Fuentes PhD Thesis Defense Standard model : core accretion
91. ( Pollack et al. 1996, Helled et al. 2013, Mordasini et al. 2012 Mgas MZ Mtot Phase I : Solid accretion (build- up of a “core” made of heavy elements) I Phase II: Gas accretion (H - He) II III Phase III: when , runaway accretion Mgas ∼ MZ Gas giants form with “dense cores” Rafael Fuentes PhD Thesis Defense Standard model : core accretion
92. Some answers Rafael Fuentes PhD Thesis Defense Does the fl uid become fully mixed?
93. Some answers Rafael Fuentes PhD Thesis Defense Does the fl uid become fully mixed?
94. Finn (1993)
 Rosenbluth (1994)
 Goluskin et al. (2014), Shear fl ows previously observed in 2D convection Some answers Rafael Fuentes PhD Thesis Defense Does the fl uid become fully mixed?
95. Finn (1993)
 Rosenbluth (1994)
 Goluskin et al. (2014), Shear fl ows previously observed in 2D convection Some answers Rafael Fuentes PhD Thesis Defense Does the fl uid become fully mixed? 0.00 0.01 0.02 0.03 0.04 t/tdiÆ 102 103 104 105 vrms /vdiÆ hu2i1/2, L = H hw2i1/2, L = H
96. Finn (1993)
 Rosenbluth (1994)
 Goluskin et al. (2014), Shear fl ows previously observed in 2D convection Some answers Rafael Fuentes PhD Thesis Defense Does the fl uid become fully mixed? 0.00 0.01 0.02 0.03 0.04 t/tdiÆ 102 103 104 105 vrms /vdiÆ hu2i1/2, L = H hw2i1/2, L = H 0.00 0.01 0.02 0.03 0.04 t/tdiÆ 102 103 104 105 vrms /vdiÆ hu2i1/2, L = H hw2i1/2, L = H hu2i1/2, L = 2H hw2i1/2, L = 2H hu2i1/2, L = 3H hw2i1/2, L = 3H hu2i1/2, L = 4H hw2i1/2, L = 4H hu2i1/2, L = 5H hw2i1/2, L = 5H
97. (Guillot et al. 2004) Rafael Fuentes PhD Thesis Defense Some numbers Jupiter Salty water κT ≈ 1.4 × 10−3 cm2 s−1 ν ≈ 10−2 cm2 s−1
98. Rafael Fuentes PhD Thesis Defense Numerics Advection equation, explicitly Can be numerically stable if instead
99. Rafael Fuentes PhD Thesis Defense Numerics Di ff usion equation, explicitly Di ff usion equation, Implicitly