1.3plus1.3minus1.3101.3generation and transport of liquid ......generation and transport of liquid...

Generation and transport of liquidwater in the high-pressure icelayers of Ganymede and Titan

Klára Kalousová1, Christophe Sotin2

1 Department of Geophysics, Charles University, Prague2 Jet Propulsion Laboratory - California Institute of Technology, Pasadena

Ada Lovelace Workshop on Numerical Modellingof Mantle and Lithosphere Dynamics,

La Certosa di Pontignano, Italy,August 26, 2019

Is there life somewhere else in the Universe?

How do we find it?

NASA: Follow the water

Follow the waterI life might have started at the bottom of Earth’s oceans

I water-rock interactions → rich communities of organismsI Galileo & Cassini: deep oceans within the icy moons → ocean worlds

I Europa:- induced magnetic field- geologic evidenceI Ganymede:- induced magnetic field- auroral ovals oscillationI Titan:- electric signal variations- large Love numberI Enceladus:- geysers- libration of outer layer

Follow the waterI life might have started at the bottom of Earth’s oceansI water-rock interactions → rich communities of organisms

I Galileo & Cassini: deep oceans within the icy moons → ocean worlds


Follow the waterI life might have started at the bottom of Earth’s oceansI water-rock interactions → rich communities of organismsI Galileo & Cassini: deep oceans within the icy moons → ocean worlds



I Europa:- induced magnetic field- geologic evidence

I Ganymede:- induced magnetic field- auroral ovals oscillationI Titan:- electric signal variations- large Love numberI Enceladus:- geysers- libration of outer layer


I Europa:- induced magnetic field- geologic evidenceI Ganymede:- induced magnetic field- auroral ovals oscillation

I Titan:- electric signal variations- large Love numberI Enceladus:- geysers- libration of outer layer


I Europa:- induced magnetic field- geologic evidenceI Ganymede:- induced magnetic field- auroral ovals oscillationI Titan:- electric signal variations- large Love number

I Enceladus:- geysers- libration of outer layer

Europa & EnceladusI direct contact of silicates with

the oceanI conditions similar to the Earth

oceans - emergence of life?

Ganymede & TitanI thick HP ice layer between

silicates and the oceanI is water & material exchange

possible?

GanymedeI full differentiation:

MoI=0.310I tenuous O2 exosphereI intrinsic magnetic field

TitanI partial differentiation:

MoI=0.3414I dense N2-CH4-Ar atmosphere→ possibly ongoing exchange?

GanymedeI dehydrated mantle

+ thick H2O layerI H&400 km: only ice VI

TitanI core of hydrated silicates

+ thinner H2O layerI H∼50–300 km: ices VI, V, III?

Heat and material exchange

I no direct contact between the silicates and the oceanI BUT: material exchange possible through convectionI HP ice dynamics governs heat extraction from the interior

I 1d thermo(chemical) evolution of the whole hydrosphere:e.g. Kirk & Stevenson (1987), Grasset & Sotin (1996), Showman &Malhotra (1997), Tobie + (2005), Fortes + (2007), Grindrod +(2008), Bland + (2009), Noack + (2016)

a) T=Tm, heat transfer through HP ice not treatedb) use of scaling laws based on the hot TBL instability

I effect of ice phases transitions on convection:a) linear stability analysis (Bercovici +, 1986; Sotin & Parmentier,1989)b) 2d convection in ice VI and VII mantle (Journaux & Noack, 2015)


I Choblet + (2017): solid-state thermal convection in 3d sphericalI Tmax and Tavrg > Tmelt in the upper part

→ cold thermal boundary layer cannot exist in a solid state→ two-phase mixture model (Bercovici +, 2001; Šrámek +, 2007)


I Choblet + (2017): solid-state thermal convection in 3d sphericalI Tmax and Tavrg > Tmelt in the upper part→ cold thermal boundary layer cannot exist in a solid state

→ two-phase mixture model (Bercovici +, 2001; Šrámek +, 2007)


I Choblet + (2017): solid-state thermal convection in 3d sphericalI Tmax and Tavrg > Tmelt in the upper part→ cold thermal boundary layer cannot exist in a solid state→ two-phase mixture model (Bercovici +, 2001; Šrámek +, 2007)

Model setting

I pure H2O system (solid ice & liquid water):

a) TiTm: one-phase material (water only)

I porosity φ= VwVi+Vw (vol. fraction of water in the mixture)I amount of ice: 1−φ

I different material densities → 2 mass balancesI phase separation → 2 linear momentum balancesI thermal equilibrium between the two phases → 1 energy balance

Governing equations1. Thermal convection in ice - incompressible extended Boussinesq approximation

2. Mixture of two incompressible components: ξi → ξ := (1−φ)ξi + φξw3. µw�µi: σ → (1−φ)σi , Tw=Tm: ρα→ (1−φ)ρiαi , ρcp → (1−φ)ρicpi → ρic

pi

4. Melting/freezing → compressible mixture → ∇ · vi 6= 0 → ζi∼µiφ→ latent heat Lm consumption/release ; solid phase change

5. Melt buoyancy: ∆ρ=ρi−ρw 6= 0 → water percolation through ice

I mass balance of ice∇ · vi = 0

I linear momentum balance of ice0 = −∇Π− ρiαiTg +∇ · σi

I energy balance of ice

ρicpi∂T∂t

+ ρicpi vi · ∇T − ρiTαivi · g = ∇ · (ki∇T ) + σi : ∇vi

I linear momentum balance of water ∼ Darcy lawvw − vi = −

Kφµwφ

(1−φ)∆ρg

I mass balance of water → advection of porosity∂φ

∂t+∇ · (φvw) =

Γm

ρw

Governing equations1. Thermal convection in ice - incompressible extended Boussinesq approximation2. Mixture of two incompressible components: ξi → ξ := (1−φ)ξi + φξw

3. µw�µi: σ → (1−φ)σi , Tw=Tm: ρα→ (1−φ)ρiαi , ρcp → (1−φ)ρicpi → ρicpi



I mass balance of ice∇ · vi = 0



ρicpi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw





I mass balance of mixture∇ · v = 0



ρicpi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw






I linear momentum balance of ice0 = −∇Π− ραTg +∇ · σ


ρicpi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw







I energy balance of mixture

ρcp∂T∂t

+ ρcpv · ∇T − ρiTαv · g = ∇ · (k∇T ) + σ : ∇v


Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw

Governing equations1. Thermal convection in ice - incompressible extended Boussinesq approximation2. Mixture of two incompressible components: ξi → ξ := (1−φ)ξi + φξw3. µw�µi: σ → (1−φ)σi

, Tw=Tm: ρα→ (1−φ)ρiαi , ρcp → (1−φ)ρicpi → ρicpi






ρcp∂T∂t

+ ρcpv · ∇T − ρiTαv · g = ∇ · (k∇T ) + σ : ∇v


Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw

Governing equations1. Thermal convection in ice - incompressible extended Boussinesq approximation2. Mixture of two incompressible components: ξi → ξ := (1−φ)ξi + φξw3. µw�µi: σ → (1−φ)σi

, Tw=Tm: ρα→ (1−φ)ρiαi , ρcp → (1−φ)ρicpi → ρicpi




I linear momentum balance of ice0 = −∇Π− ραTg +∇ · (1−φ)σi


ρcp∂T∂t

+ ρcpv · ∇T − ρiTαv · g = ∇ · (k∇T ) + (1−φ)σi : ∇vi


Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw

Governing equations1. Thermal convection in ice - incompressible extended Boussinesq approximation2. Mixture of two incompressible components: ξi → ξ := (1−φ)ξi + φξw3. µw�µi: σ → (1−φ)σi , Tw=Tm: ρα→ (1−φ)ρiαi

, ρcp → (1−φ)ρicpi → ρicpi




I linear momentum balance of ice0 = −∇Π− ραTg +∇ · (1−φ)σi


ρcp∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw

Governing equations1. Thermal convection in ice - incompressible extended Boussinesq approximation2. Mixture of two incompressible components: ξi → ξ := (1−φ)ξi + φξw3. µw�µi: σ → (1−φ)σi , Tw=Tm: ρα→ (1−φ)ρiαi

, ρcp → (1−φ)ρicpi → ρicpi




I linear momentum balance of ice0 = −∇Π− (1−φ)ρiαiTg +∇ · (1−φ)σi


ρcp∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw

Governing equations1. Thermal convection in ice - incompressible extended Boussinesq approximation2. Mixture of two incompressible components: ξi → ξ := (1−φ)ξi + φξw3. µw�µi: σ → (1−φ)σi , Tw=Tm: ρα→ (1−φ)ρiαi , ρcp → (1−φ)ρicpi

→ ρicpi4. Melting/freezing → compressible mixture → ∇ · vi 6= 0 → ζi∼µiφ

→ latent heat Lm consumption/release ; solid phase change5. Melt buoyancy: ∆ρ=ρi−ρw 6= 0 → water percolation through ice




ρcp∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw

Governing equations1. Thermal convection in ice - incompressible extended Boussinesq approximation2. Mixture of two incompressible components: ξi → ξ := (1−φ)ξi + φξw3. µw�µi: σ → (1−φ)σi , Tw=Tm: ρα→ (1−φ)ρiαi , ρcp → (1−φ)ρicpi → ρic

pi






ρcp∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi






ρicpi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi

4. Melting/freezing

→ compressible mixture → ∇ · vi 6= 0 → ζi∼µiφ→ latent heat Lm consumption/release ; solid phase change





ρicpi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi

4. Melting/freezing → compressible mixture

→ ∇ · vi 6= 0 → ζi∼µiφ→ latent heat Lm consumption/release ; solid phase change


I mass balance of mixture∇ · v =

∆ρ

ρiρwΓm


I energy balance of mixtureρic

pi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi

4. Melting/freezing → compressible mixture

→ ∇ · vi 6= 0 → ζi∼µiφ→ latent heat Lm consumption/release ; solid phase change


I mass balance of mixture∇ · vi +∇ · [φ(vw−vi)] =

∆ρ

ρiρwΓm



pi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi

4. Melting/freezing → compressible mixture → ∇ · vi 6= 0

→ ζi∼µiφ→ latent heat Lm consumption/release ; solid phase change



∆ρ

ρiρwΓm



pi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi

4. Melting/freezing → compressible mixture → ∇ · vi 6= 0 → ζi∼µiφ



∆ρ

ρiρwΓm



pi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi




∆ρ

ρiρwΓm

I linear momentum balance of ice0 = −∇Π− (1−φ)ρiαiTg +∇ · (1−φ)σi +∇

[(1−φ)µi

φ(∇ · vi)

]I energy balance of mixture

ρicpi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi




∆ρ

ρiρwΓm


[(1−φ)µi

φ(∇ · vi)

]I energy balance of mixtureρic

pi∂T∂t

+ ρcpv · ∇T − ρiTαv · g = ∇ · (k∇T ) + (1−φ)σi : ∇vi +1− φφ

µi(∇ · vi)2


Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi




∆ρ

ρiρwΓm


[(1−φ)µi

φ(∇ · vi)

]I energy balance of mixture (Ψ̃ = (1−φ)σi:∇vi+ 1−φφ µi(∇·vi)2)

ρicpi∂T∂t

+ ρcpv · ∇T − ρiTαv · g = ∇ · (k∇T ) + Ψ̃


Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi

4. Melting/freezing → compressible mixture → ∇ · vi 6= 0 → ζi∼µiφ→ latent heat Lm consumption/release

; solid phase change5. Melt buoyancy: ∆ρ=ρi−ρw 6= 0 → water percolation through ice


∆ρ

ρiρwΓm


[(1−φ)µi

φ(∇ · vi)


ρicpi∂T∂t

+ ρcpv · ∇T − ρiTαv · g = ∇ · (k∇T ) + Ψ̃


Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi

4. Melting/freezing → compressible mixture → ∇ · vi 6= 0 → ζi∼µiφ→ latent heat Lm consumption/release

; solid phase change5. Melt buoyancy: ∆ρ=ρi−ρw 6= 0 → water percolation through ice


∆ρ

ρiρwΓm


[(1−φ)µi

φ(∇ · vi)


ρicpi∂T∂t

+ ρcpv · ∇T − ρiTαv · g + Γm Lm = ∇ · (k∇T ) + Ψ̃


Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi




∆ρ

ρiρwΓm


[(1−φ)µi

φ(∇ · vi)


ρicpi∂T∂t

+ ρcpv · ∇T − ρiTαv · g + Γm Lm + Γs Ls = ∇ · (k∇T ) + Ψ̃


Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi


5. Melt buoyancy: ∆ρ=ρi−ρw 6= 0

→ water percolation through ice


∆ρ

ρiρwΓm


[(1−φ)µi

φ(∇ · vi)


ρicpi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi


5. Melt buoyancy: ∆ρ=ρi−ρw 6= 0

→ water percolation through ice


∆ρ

ρiρwΓm

I linear momentum balance of ice0 = −∇Π− (1−φ)ρiαiTg − φ∆ρg +∇ · (1−φ)σi +∇

[(1−φ)µi

φ(∇ · vi)


ρicpi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi


5. Melt buoyancy: ∆ρ=ρi−ρw 6= 0 → water percolation through iceI mass balance of mixture

∇ · vi +∇ · [φ(vw−vi)] =∆ρ

ρiρwΓm


[(1−φ)µi

φ(∇ · vi)


ρicpi∂T∂t



Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw


pi


5. Melt buoyancy: ∆ρ=ρi−ρw 6= 0 → water percolation through iceI mass balance of mixture

∇ · vi +∇ · [φ(vw−vi)] =∆ρ

ρiρwΓm


[(1−φ)µi

φ(∇ · vi)

]I energy balance of mixture (Ψ = (1−φ)σi:∇vi+ 1−φφ µi(∇·vi)2+

µwφ2Kφ

(vw−vi)2)

ρicpi∂T∂t

+ ρcpv · ∇T − ρiTαv · g + Γm Lm + Γs Ls = ∇ · (k∇T ) + Ψ


Kφµwφ

(1−φ)∆ρg


∂t+∇ · (φvw) =

Γm

ρw

Material parametersI ice permeable to water transport above a threshold porosity φc

Kφ ={

k0(φ−φc)2 φ ≥ φc0 φ < φc

I temperature and depth dependent ice viscosity

µi = µ0 exp[

QR

(1T−

1Tm(z)

)]

Implementation

I FEniCS (fenicsproject.org, Logg +, 2012; Alnaes +, 2015), MPI

Reference simulation for Ganymede

H=200 km, µi=1015 Pa s, qs=20 mW m−2

I silicates: T=Tm, φ∼few %: volatiles leachingI interior: T av


H=200 km, µi=1015 Pa s, qs=20 mW m−2

I silicates: T=Tm, φ∼few %: volatiles leaching

I interior: T av


H=200 km, µi=1015 Pa s, qs=20 mW m−2

I silicates: T=Tm, φ∼few %: volatiles leachingI interior: T av

Increasing HP ice layer thickness H (∼ increasing Ra)


(1) Direct exchange ∼ small Rayleigh number



(2) Indirect exchange ∼ medium Rayleigh number




(3) Limited exchange ∼ medium Rayleigh number




(3) Limited exchange ∼ medium Rayleigh number

(4) No exchange ∼ large Rayleigh number

Results of parametric study

I degree of exchange decreases withI increasing ice layer thickness H (∼ time)

I decreasing ice viscosity µiI decreasing heat flux from silicates qs


I degree of exchange decreases withI increasing ice layer thickness H (∼ time)I decreasing ice viscosity µi

I decreasing heat flux from silicates qs


I degree of exchange decreases withI increasing ice layer thickness H (∼ time)I decreasing ice viscosity µiI decreasing heat flux from silicates qs

Long-term evolution modeling

I 2d model: strong dependence on input parameters:- HP ice layer thickness H- ice viscosity µi- silicates heat flux qs

I 1d thermo(-chemical) evolution model:- scaling laws for rocky interior, HP ice layer, ocean, ice I crust- qs and H are part of solution- µi remains an input parameter

Critical heat flux for bottom melting

I fixed viscosity µi, range of qs and H

→ critical qcs for bottom melting: qcs = αH − β

Present-day bottom melting: Ganymede vs TitanI Titan, µi = 1015 Pa s: qcs ∝ H

I µi = 1014–1016 Pa sI scaling for Ganymede

I qs∼5–15 mW m−2

I Ganymede: H>400 kmI Titan: H∼50–300 km

→ Titan: melting at silicates interface and leaching of 40Ar?

Present-day bottom melting: Ganymede vs TitanI Titan, µi = 1015 Pa s: qcs ∝ HI µi = 1014–1016 Pa s

I scaling for Ganymede

I qs∼5–15 mW m−2



Present-day bottom melting: Ganymede vs TitanI Titan, µi = 1015 Pa s: qcs ∝ HI µi = 1014–1016 Pa sI scaling for Ganymede

I qs∼5–15 mW m−2




I qs∼5–15 mW m−2

I Ganymede: H>400 km

I Titan: H∼50–300 km



I qs∼5–15 mW m−2



Conclusions

I two-phase model necessary to address melt evolution

I melting at the interface with silicates → volatiles leachingI small amount of water (φ∼few %)I volatiles transfered by upwelling plumes → extraction into ocean

I Ganymede: exchange more likely early in its evolutionI Titan: exchange possibly ongoing: transport of 40Ar into the ocean?

I further development: effect of salt & ammonia on Tm and ∆ρI thermo-chemical evolution model → Cassini, JUICE & Dragonfly

Thank you for yourattention!

Conclusions



I Ganymede: exchange more likely early in its evolution

I Titan: exchange possibly ongoing: transport of 40Ar into the ocean?



Conclusions






Conclusions & perspectives




I further development: effect of salt & ammonia on Tm and ∆ρ

I thermo-chemical evolution model → Cassini, JUICE & Dragonfly






I further development: effect of salt & ammonia on Tm and ∆ρI thermal evolution model → Cassini, JUICE & Dragonfly


References

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- Alnaes + (2015), University of Cambridge.

- Bercovici + (1986), Geophys. Res. Lett., 13, 448–451.

- Bercovici + (2001), J. Geophys. Res., 106, 8887–8906.

- Bland + (2009), Icarus, 200, 207–221.

- Choblet + (2017), Icarus, 285, 252–262.

- Fortes + (2007), Icarus, 188, 139–153.

- Grasset & Sotin (1996), Icarus, 123, 101–112.

- Grindrod + (2008), Icarus, 197, 137–151.

- Jarvis (1984), Phys. Earth Planet. Inter., 36, 305–327.

- Journaux & Noack (2015), EPSC, EPSC2015-774.

- Kirk & Stevenson (1987), Icarus, 69, 91–134.

- Logg + (2012), Springer-Verlag, Berlin Heidelberg.

- Mitrovica & Jarvis (1987), Geophys. Astro. Fluid, 38, 193–224.

- Noack + (2016), Icarus, 277, 215–236.

- Showman & Malhotra (1997), Icarus, 127, 93–111.

- Sotin & Parmentier (1989), Phys. Earth Planet. Inter., 55, 10–25.

- Šrámek + (2007), Geophys. J. Int., 168, 964–982.

- Tobie + (2005), Icarus, 175, 496–502.

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1.3plus1.3minus1.3101.3generation and transport of liquid ......generation and transport of liquid...

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