The thermo-chemical evolution of Mars with a compositionally stratified mantle
Henri Samuel 1 , Maxim Ballmer 2 , Sebastiano Padovan 3 , Ana-Catalina Plesa 3 , Nicola Tosi 3 , Francis Nimmo 4 , Doris Breuer 3 , Attilio Rivoldini 5 , Mark Panning 6 , Philippe Lognonné 1
1 IPGP, CNRS, Université Paris Sorbonne Cité, Paris, France, 2 UCL, London, United Kingdom, 3 DLR, Berlin, Germany,
4 University of Santa Cruz, CA, USA, 5 Royal Observatory of Belgium, Brussels, Belgium, 6 JPL, CA, USA
References
L. Elkins-Tanton, M. Parmentier, P. Hess, MPS, 38, 2003
M. Ballmer et al., G-cubed, doi: 10.1002/2017GC006917, 2017
M. Maurice et al., JGR-Planets, doi: 10.1002/2016JE005250, 2017 C.-E. Boukaré, M. Parmentier, S. Parman, EPSL, 491, 2018
G. Zeff & Q. Williams, GRL, doi: 10.1029/2019GL084810, 2019
H. Samuel, P. Lognonné, M. Panning, V. Lainey, Nature, 569, 2019
7. Conclusions
The (likely) presence of a dense, enriched (stratified) layer mostly influences:
✔ Core temperature (increase with decreasing mantle viscosity)
✔ k 2 increase ( proportional to the layer thickness)
✔ Crustal thickness (increase or decrease)
✔ Surface heat flux (decrease)
Accounting for the possibility of a strong, early layering may be important:
✔ For seismic data interpretation: Mars’ structure, presence of shadow zones etc..
✔ For geodetic data interpretation: Core size, composition, thermal state
✔ To understand Mars’ early history and formation/differentiation
1. Motivations
To better constrain Mars' evolution, the InSight mission has recently landed on the surface of Mars to record its seismic activity, surface heat flow, magnetic field, and to refine the current estimates of Mars' response to tidally induced gravitational changes. Combining such data with geodynamic considerations provides a way to better understand and constrain the thermo-chemical history and the present-day structure of Mars.
Despite our relatively poor knowledge of Mars' early history, several indications suggest that Mars' mantle went through a global magma ocean stage.
The crystallization and fractionation of such a magma ocean is likely to have led to the presence of a compositionally distinct material at the bottom of the mantle. Such a layer would have been heavily enriched in iron and Heat-Producing Elements (HPE) - with either a homogeneous or a depth- dependent enrichment - with respect to the overlying mantle. The significant iron enrichment may have led to a strongly stratified mantle with a flat interface.
Using a parameterized convection approach, we modeled the thermochemical evolution of Mars' main envelopes: a liquid convecting core, a dense silicate layer enriched in HPE and sitting atop of the core-mantle boundary, overlaid by a less dense and more depleted silicate mantle, convecting under a stagnant lithospheric lid. The latter includes a crust building up with time and enriched in HPE with respect to the underlying silicate mantle.
The dense layer is assumed to be convecting separately if its composition is homogeneous, or purely diffusing heat if its iron enrichment decreases with depth (stable stratification). Our efficient approach allows exploring a wide range of parameter space including the dense layer thickness, Mars mantle rheological parameters, Mars’ initial thermal state and core size. For each case considered, we predict the obtained present-day thermal structure, heat flux, crustal thickness, degree two Love number k 2 in order to compare with available and upcoming observations from the InSight mission. This will allow interpreting InSight data to place constraints on Mars thermal evolution, but also on the initial conditions as a function of planetary accretion and differentiation.
0 0.2 0.4 0.6 0.8
Temperature
0 0.2 0.4 0.6 0.8 1
Height
0 0.2 0.4 0.6 0.8 1
Temperature
0 0.2 0.4 0.6 0.8 1
Height
0 0.2 0.4 0.6 0.8 1
Temperature
0 0.2 0.4 0.6 0.8 1
Height
2. Thermo-chemical evolution: Parametrised convection model
500 1000 1500 2000 Present−day temperature [K]
500 1000 1500 2000 2500 3000
Depth [km]
D cr
D l
T l
T m
T b T c
1800 1900 2000 2100
Temperature [K]
0 1 2 3 4
Time [Gyr]
Core (T c ) Mantle (T m )
50 100 150 200 250 300 350
Thickness [km]
0 1 2 3 4
Time [Gyr]
Lithosphere (D l ) Crust (D cr )
Model layers at present−day
(a) (b)
(c) (d)
Dense layer
✔ No layer: homogeneous mantle
✔ Layer with homogeneous composition ➪ layered convection develops
✔ Stably stratified layer (compositional gradient dFe#/dz<0) ➪ purely diffusive layer
Influence of the presence of a dense, HPE-enriched layer atop of the core:
➪ Increase of the deep mantle temperature ➪ melting at present-day
➪ Hotter present-day core
➪ Effects more pronounced for stably stratified dense layer 1000 2000 3000
Temperature [K]
0
500 1000 1500 2000 2500 3000
Depth [km]
1000 2000 3000 Temperature [K]
0
500 1000 1500 2000 2500 3000
Depth [km]
1000 2000 3000 Temperature [K]
0
500 1000 1500 2000 2500 3000
Depth [km]
Solidus Liquidus
Stably stratified layer, D d =300km R c =1700km V*=5cm 3 /mol
100 200 300
E * [kJ mol − 1 K − 1 ]
20 21 22 23
Log 10 (η 0 [Pa s])
T planet
20 21 22 23
Log 10 (η 0 [Pa s])
1400 1500 1600 1700 [K]
20 21 22 23
Log 10 (η 0 [Pa s])
Surface flux
13 14 15 16 17 18
[mW/m 2 ]
20 21 22 23
Log 10 (η 0 [Pa s])
Crustal thickness
300 400 500 600 700 800 900 1000
E * disl
[kJ mol − 1 K − 1 ]
20 21 22 23
Log 10 (η 0 [Pa s])
20 30 40 50 60 70 80 [km]
100 200 300
E * [kJ mol − 1 K − 1 ]
20 21 22 23
Log 10 (η 0 [Pa s])
D molten
160 180 200 220 240 260 280 [km]
20 21 22 23
Log 10 (η 0 [Pa s])
k 2
0.16 0.17 0.18 0.19 0.20 0.21 0.22
20 21 22 23
Log 10 (η 0 [Pa s])
Q
300 400 500 600 700 800 900 1000
E * disl
[kJ mol − 1 K − 1 ]
20 21 22 23
Log 10 (η 0 [Pa s])
100 200 300 400 500
✔ Thickness of molten part of the dense layer (D molten ) correlate with avrg. planet temperature (T planet )
✔ k 2 (Andrade rheology) correlates with D molten
✔ Present-day surface heat flux and Q correlate with 1/η d (with η d = η(T d ), and T d ~ correlates with T planet )
✔ Present-day D l correlates positively with T planet
✔ Surface heat flux anticorrelates with D l
4. Systematic correlations
➪ Despite present-day k 2 and Q constraints (k 2 =0.169±0.02, Q=95±10), various possibilities for thermo-chemical structure remain, but the plausible range for E* can be reduced
Two distinct interpretations for k 2 :
-Homogeneous mantle with a large core -A layered mantle with a small core
6. Consequences : interpretations of geodetic & seismic data
1500 1600 1700 1800
Rc [km]
200 400 600
Dd [km]
Elastic k 2
0.15 0.20 0.25 0.30 0.35
k 2
200 400 600
Dd [km]
Normalised MoI
0.360 0.363 0.366
I/(M R p 2 )
1500 1600 1700 1800
Rc [km]
200 400 600
Dd [km]
k 2 and MoI
Further MoI constraints:
- Core size
- Core composition
k 2
4000 4200 4400
Vs [km/s]
0
100
200
300
400
Depth [km]
No layer
Unstratified layer Stratified layer
4000 4200 4400
Vs [km/s]
0
100
200
300
400
Depth [km]
No layer
Unstratified layer Stratified layer
✔ Different crustal thicknesses
✔ Distinct shape/extent of LVZ (?)
Numerical (dynamic) modeling
Parameterized convection modeling
100 200 300
E * [kJ mol − 1 K − 1 ]
20 21 22 23
Log 10 (η 0 [Pa s])
∆(Surface flux)
20 21 22 23
Log 10 (η 0 [Pa s])
−28 −24 −20 −16 −12
[%]
20 21 22 23
Log 10 (η 0 [Pa s])
∆(T core )
20 21 22 23
Log 10 (η 0 [Pa s])
40 45 50 55 60
[%]
20 21 22 23
Log 10 (η 0 [Pa s])
∆(k 2 )
20 21 22 23
Log 10 (η 0 [Pa s])
32 40 48 56 64
%
20 21 22 23
Log 10 (η 0 [Pa s])
∆(Crustal thickness)
300 400 500 600 700 800 900 1000
E * disl
[kJ mol − 1 K − 1 ]
20 21 22 23
Log 10 (η 0 [Pa s])
−60 −40 −20 0
[%]
100 200 300
E * [kJ mol − 1 K − 1 ]
20 21 22 23
Log 10 (η 0 [Pa s]) ∆(T avrg. planet )
20 21 22 23
Log 10 (η 0 [Pa s])
3 4 5 6 7
[%]
20 21 22 23
Log 10 (η 0 [Pa s])
∆(T convective mantle )
20 21 22 23
Log 10 (η 0 [Pa s])
5 6 7 8 9 10
[%]
20 21 22 23
Log 10 (η 0 [Pa s])
∆(Q)
20 21 22 23
Log 10 (η 0 [Pa s])
−28 −21 −14 −7 0
%
20 21 22 23
Log 10 (η 0 [Pa s])
∆(Lithospheric thickness)
300 400 500 600 700 800 900 1000
E * disl
[kJ mol − 1 K − 1 ]
20 21 22 23
Log 10 (η 0 [Pa s])
−10 −5 0 5 10 15 [%]
Relative change with respect to the homogeneous case: ∆(X)= (X layered -X homogeneous )/ X homogeneous
5. Differences between homogeneous and layered cases
The presence of a dense, enriched (stratified) layer mostly influences:
✔ Core temperature (increase with decreasing mantle viscosity)
✔ k 2 increase (proportional to the layer thickness)
✔ Crustal thickness (increase or decrease)
✔ Surface heat flux (decrease)
Str ong influence W eaker influence
Layered convection
Homogeneous mantle Stably stratified layer
?
? ? ?
?
3. End-member scenarios
1764 L. T. Elkins-Tanton et al.
pyroxene-olivine mineralogy for the analysis of stability, as discussed earlier. The pre-overturn profile for the garnet segregation model, therefore, has a very dense, highly aluminous layer between 2220 and 2440 that, during overturn, falls into the deepest layers of the mantle.
Idealizing overturn as described above, the pre- and post- overturn density profiles for the 2 models are shown in Fig. 7.
The larger range of densities in the garnet segregation model
is evident when compared to the simple fractional crystallization model. The garnet segregation model, thus, has a higher driving force for overturn; the most dense layer in this model is at a higher temperature and, therefore, lower viscosity than the most dense layer in the simple fractional crystallization model, allowing easier overturn. The final density profile has a far more stable, dense lowest mantle than that in the simple fractional crystallization model. The resulting compositional Fig. 5. Compositional stratification resulting from simple fractional crystallization of the magma ocean as shown in Fig. 2. In (a), before overturn, the deepest majorite + g-olivine sequesters aluminum in the lower mantle. In (b), after overturn, the highest alumina and lowest silica are distributed heterogeneously in the mid-mantle. The lowest mantle has no alumina, and the most shallow mantle has the lowest FeO, but otherwise the most shallow and most deep mantles are compositionally similar. The greatest heterogeneity is produced in the mid-mantle.
a
[Elkins-Tanton, 2003; Ballmer et al. 2017; Maurice et al., 2018; Boukaré et al., 2018; Zeff & Williams, 2019]… b
⇢ d C m V m " m (1 + St m ) dT m
dt = q d A d q m A m + H m V m
<latexit sha1_base64="QAbjCJv5HmEPzV+1+sTY34vUxKw=">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</latexit><latexit sha1_base64="QAbjCJv5HmEPzV+1+sTY34vUxKw=">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</latexit><latexit sha1_base64="QAbjCJv5HmEPzV+1+sTY34vUxKw=">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</latexit><latexit sha1_base64="QAbjCJv5HmEPzV+1+sTY34vUxKw=">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</latexit>
➪ Accounts for crustal and lithospheric evolution and HPE partitioning (crust/lithosphere/mantle)
➪ Accounts for latent heat effects
➪ Accounts for different types of stratification (stably stratified or homogeneous layer)
⇢ d C c V c " c dT c
dt = q c A c
<latexit sha1_base64="vQ73U6XEQRiTXzIld3zhiH9QhyU=">AAACI3icbZDLSsNAFIYnXmu9VV26GSyCG0sigiII1W5cVugNmhAmJ5N26OTizKRQQt7Fja/ixoVS3LjwXZxeFtp64MDH/5/DzPm9hDOpTPPLWFldW9/YLGwVt3d29/ZLB4ctGacCaBNiHouORyTlLKJNxRSnnURQEnqctr1BbeK3h1RIFkcNNUqoE5JexAIGRGnJLd3Yoh+7Pq65gFu67SERNJGMaxMwtgNBIPNxw4U881WOb/E5ftLOnQtuqWxWzGnhZbDmUEbzqrulse3HkIY0UsCJlF3LTJSTEaEYcJoX7VTShMCA9GhXY0RCKp1semOOT7Xi4yAWuiOFp+rvjYyEUo5CT0+GRPXlojcR//O6qQqunYxFSapoBLOHgpRjFeNJYNhngoLiIw0EBNN/xdAnOhalYy3qEKzFk5ehdVGxND9elqv38zgK6BidoDNkoStURQ+ojpoI0DN6Re/ow3gx3oyx8TkbXTHmO0foTxnfP4wzou8=</latexit><latexit sha1_base64="vQ73U6XEQRiTXzIld3zhiH9QhyU=">AAACI3icbZDLSsNAFIYnXmu9VV26GSyCG0sigiII1W5cVugNmhAmJ5N26OTizKRQQt7Fja/ixoVS3LjwXZxeFtp64MDH/5/DzPm9hDOpTPPLWFldW9/YLGwVt3d29/ZLB4ctGacCaBNiHouORyTlLKJNxRSnnURQEnqctr1BbeK3h1RIFkcNNUqoE5JexAIGRGnJLd3Yoh+7Pq65gFu67SERNJGMaxMwtgNBIPNxw4U881WOb/E5ftLOnQtuqWxWzGnhZbDmUEbzqrulse3HkIY0UsCJlF3LTJSTEaEYcJoX7VTShMCA9GhXY0RCKp1semOOT7Xi4yAWuiOFp+rvjYyEUo5CT0+GRPXlojcR//O6qQqunYxFSapoBLOHgpRjFeNJYNhngoLiIw0EBNN/xdAnOhalYy3qEKzFk5ehdVGxND9elqv38zgK6BidoDNkoStURQ+ojpoI0DN6Re/ow3gx3oyx8TkbXTHmO0foTxnfP4wzou8=</latexit><latexit sha1_base64="vQ73U6XEQRiTXzIld3zhiH9QhyU=">AAACI3icbZDLSsNAFIYnXmu9VV26GSyCG0sigiII1W5cVugNmhAmJ5N26OTizKRQQt7Fja/ixoVS3LjwXZxeFtp64MDH/5/DzPm9hDOpTPPLWFldW9/YLGwVt3d29/ZLB4ctGacCaBNiHouORyTlLKJNxRSnnURQEnqctr1BbeK3h1RIFkcNNUqoE5JexAIGRGnJLd3Yoh+7Pq65gFu67SERNJGMaxMwtgNBIPNxw4U881WOb/E5ftLOnQtuqWxWzGnhZbDmUEbzqrulse3HkIY0UsCJlF3LTJSTEaEYcJoX7VTShMCA9GhXY0RCKp1semOOT7Xi4yAWuiOFp+rvjYyEUo5CT0+GRPXlojcR//O6qQqunYxFSapoBLOHgpRjFeNJYNhngoLiIw0EBNN/xdAnOhalYy3qEKzFk5ehdVGxND9elqv38zgK6BidoDNkoStURQ+ojpoI0DN6Re/ow3gx3oyx8TkbXTHmO0foTxnfP4wzou8=</latexit><latexit sha1_base64="vQ73U6XEQRiTXzIld3zhiH9QhyU=">AAACI3icbZDLSsNAFIYnXmu9VV26GSyCG0sigiII1W5cVugNmhAmJ5N26OTizKRQQt7Fja/ixoVS3LjwXZxeFtp64MDH/5/DzPm9hDOpTPPLWFldW9/YLGwVt3d29/ZLB4ctGacCaBNiHouORyTlLKJNxRSnnURQEnqctr1BbeK3h1RIFkcNNUqoE5JexAIGRGnJLd3Yoh+7Pq65gFu67SERNJGMaxMwtgNBIPNxw4U881WOb/E5ftLOnQtuqWxWzGnhZbDmUEbzqrulse3HkIY0UsCJlF3LTJSTEaEYcJoX7VTShMCA9GhXY0RCKp1semOOT7Xi4yAWuiOFp+rvjYyEUo5CT0+GRPXlojcR//O6qQqunYxFSapoBLOHgpRjFeNJYNhngoLiIw0EBNN/xdAnOhalYy3qEKzFk5ehdVGxND9elqv38zgK6BidoDNkoStURQ+ojpoI0DN6Re/ow3gx3oyx8TkbXTHmO0foTxnfP4wzou8=</latexit>
(if homogeneous) (if stably stratified)
⇢ d C d V d " d (1 + St d ) dT d
dt = q c A c q d A d + H d V d
<latexit sha1_base64="2dnaTaXCHqiyP4M2vnj3ESyrI/Q=">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</latexit><latexit sha1_base64="2dnaTaXCHqiyP4M2vnj3ESyrI/Q=">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</latexit><latexit sha1_base64="2dnaTaXCHqiyP4M2vnj3ESyrI/Q=">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</latexit><latexit sha1_base64="2dnaTaXCHqiyP4M2vnj3ESyrI/Q=">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</latexit>
⇢ d C d @T
@t = 1 r 2
@
@r
✓
r 2 k d @T
@r
◆
+ H d (r) ⇢ d L @
@t
<latexit sha1_base64="ol4lBYjqpmHWYZk0BykHN0NjSXM=">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</latexit>
