Scaling Laws and the Problem of the Prediction of Tectonic Modes

Transcription

1 Scaling Laws and the Problem of the Prediction of Tectonic Modes D. Breuer 1 and V. Stamenkovic 2 1 DLR, Institute of Planetary Research, Berlin 2 MIT, Earth, Atmospheric and Planetary Sciences

2 PLATE TECTONICS Many parameters influence the propensity of plate tectonics One necessary but not sufficient criterion is that stresses in the lithosphere D exceed the yield stress y for plate deformation

3 Why Using Scaling Laws? - To make predictions about the ratio between lithospheric and yield stress when varying relevant parameters (T surf, T m, E, volatiles, Q rad, M ) What we want to address in the present talk - show the sensibility of the propensity of plate tectonics with the scaling parameters - explain why there are diverging results in the literature - make some own pediction related to propensity of plate tectonics

4 4 What are the stresses in the lithosphere where plates subduct? - Considering two extremes that cause the stress in lithosphere lid F: lithospheric stress is equivalent to imposed basal convective shear stress S: lithospheric stress is equivalent to normal stress lid Λ c C F= > 1 Y * N =C δ u * N S= > 1 Y Λ

5 tbl δ u L tbl δ u

6 6 What are the stresses in the lithosphere where plates subduct? - Considering two extremes that cause the stress in lithosphere lid F: lithospheric stress is equivalent to imposed basal convective shear stress S: lithospheric stress is equivalent to normal stress lid Λ c C F= > 1 Y = * N C * N S= > 1 Y L Λ +δ u

7 The classical question: How does the convective stress vary with temperature? η C dv dr D 2 ηra γ η 1 γ T γ v D 1 ( Ra / Ra ) γ c Ra = αgρ TD κη 3 increase with T E exp RT η=η0 decrease with T

8 The classical question: How does the convective stress vary with temperature? η C dv dr D 2 ηra γ η 1 γ T γ 1 γcrit γcrit c η ( T ) = = T T 0

9 The classical question: How does the convective stress vary with temperature? η C dv dr D 2 ηra γ η 1 γ T γ 1 γcrit γcrit c η ( T ) = = T T 0 γ crit = 1 + R(Ts + T) T E 2 1 γ<γ crit γ>γ crit convective stress decreases with temperature convective stress increases with temperature

10 γ crit = 1 + R(Ts + T) T E 2 1 γ crit Stamenković and Breuer 2014 for γ 2/3 c decreases with increasing temperature (Rayleigh number) decreasing viscosity dominates the influence on convective stress not increasing velocity

11 Next Step: Comparison lithospheric stress and yield stress - Lithospheric stress = lid c lid Λ =C δ u - Yield stress Y = gρcfricδu X = lid Y > 1 plate tectonics more likely < 1 plate tectonics less likely

12 Scaling parameters β, γ and ε δ u Nusselt-Rayleigh (β), convective velocity (γ) and aspect ratio (ε) scaling. DRa β 2 γ c D ηra Λ DRa ε lid XMt (,) = Y χ=β+γ χ= 2β+γ ε shear stress 3 1 ( gd C fric ) ( ηra χ ) normal stress

13 The failure function X M t gd C Ra χ 3 1 (,) ( fric ) ( η ) χ 1 3( χ 1) χ χ 1 χ χ Cfricg D ( ρα m ) κ ( η( T)) T static dynamic χ=β+γ χ= 2β+γ ε shear stress normal stress

14 14 Plate tectonics and heat: (β,ɣ,ε,x) χ crit = 1 + R(Ts + T) T E 2 1 for E = 300 kj/mol, T S = 288 K χ χ crit crit (max) = 0.96 (min) = 0.9 PLATE TECTONICS & HEAT F S (ε=0) S (ε=1/6) Χ crit (max) Χ crit (min) χ=β+γ χ= 2β+γ χ= 2β+γ ε Stamenković and Breuer 2014 shear stress normal stress

15 15 Plate tectonics and heat: (β,ɣ,ε,x) tectonics is more or less likely with increasing interior heat. - Plate tectonics more likely with increasing inner heat only for white zone! PLATE TECTONICS & HEAT F S (ε=0) S (ε=1/6) Χ crit (max) Χ crit (min) χ=β+γ χ= 2β+γ χ= 2β+γ ε Stamenković and Breuer 2014 shear stress normal stress

16 Static contribution χ 1 3( χ 1) χ χ 1 χ χ Cfricg D ( ρα m ) κ ( η( T)) T static Using g(m)~m 0.46 D(M)~M ρ m (M)~M ( ) X(stat) = M χ χ crit(stat) = 0.89

17 17 Plate tectonics, heat and mass: (β,ɣ,ε,x) tectonics is more or less likely with increasing interior heat. - Plate tectonics more likely with increasing inner heat and mass only for white zone! PLATE TECTONICS & HEAT F S (ε=0) S (ε=1/6) Χ crit (max) Χ crit (min) χ=β+γ χ= 2β+γ χ= 2β+γ ε shear stress normal stress Stamenković and Breuer 2014

18 Evolution models show that initial conditions result in a variation of failure function with time but general trend remains the same F S (ε=0) S (ε=1/6)

19 Results for plate tectonics maintenance

20 2-3D convection models suggest - diverging results for β, γ and ε - For isoviscous viscosity (free slip) β ~ γ ~ <0.5 2/3 ε ~ 0 1/6 preferred β ~ 0.3 γ ~ <0.5 Spherical symmetry, mixed heating Deschamps et al. (2010)

21 Results for plate tectonics maintenance

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23 Variation of surface temperature η 1 χ χ X ( dyn) ( ( T )) T Stamenkovic (2015) Maintance: increase of surface temperature likelihood for PT decreases Initiation: increase of surface temperature likelihood for PT slightly increases

24 Conclusions - Likelihood of plate tectonics depends strongly on correct scaling parameters and on dominant stress in lithosphere (shear or normal stresses) - The difference in scaling parameters can explain the contradicting findings in PT propensity for exoplanets - 2 D convection models suggest χ < χ crit and shear stress in lithosphere - PT less likely with increasing T, volatiles in the mantle and planetary mass - Influence of surface temperature differs between maintanance and initiation