Subduction II Fundamentals of Mantle Dynamics
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1 Subduction II Fundamentals of Mantle Dynamics Thorsten W Becker University of Southern California Short course at Universita di Roma TRE April 18 20, 2011
2
3 Rheology Elasticity vs. viscous deformation η = O (1021) Pa s = viscosity µ= O (1011) Pa = shear modulus = rigidity τ = η / µ = O(1010) sec = O(103) years = Maxwell time
4 Elastic deformation In general: σij = Cijkl εkl (in 3-D 81 degrees of freedom, in general 21 independent) For isotropic body this reduces to Hooke s law: σij = λεkkδij + 2µεij with λ and µ Lame s parameters, εkk Taking shear components ( σ12 = 2µε12 = ε11 + ε22 + ε33 i j ) gives definition of rigidity: Adding the normal components ( i=j ) for all i=1,2,3 gives: σkk = (3λ + 2µ) εkk = 3κεkk with κ = λ + 2µ/3 = bulk modulus
5 Linear viscous deformation (1) Total stress field = static + dynamic part: σ ij = pδ ij + τ ij Analogous to elasticity General case: τ ij = C 'ijkl ε kl Isotropic case: τ ij = λ ' ε kk δ ij + 2ηε ij
6 Linear viscous deformation (2) Split in isotropic and deviatoric part (latter causes deformation): 1 σ 'ij = σ ij σ kk δ ij = σ ij + pδ ij 3 1 ε 'ij = ε ij ε kk δ ij 3 which gives the following stress: σ 'ij = ( p p )δ ij + ςε kkδ ij + 2ηε 'ij With compressibility term assumed 0 (Stokes condition 2 ( ς = λ ' + η = bulk viscosity) 3 τ Now τ = 2ηε or η = ij ij ij 2ε ij In general η = f (T, d, p, H 2O ) ε kk = 0 )
7 Non-linear (or non-newtonian) deformation General stress-strain relation: n = 1 : Newtonian n > 1 : non-newtonian n : pseudo-brittle Effective viscosity ηeff ε = Aτ n n 1 1 / n (1 n ) / n = A τ = A ε 2 2 Application: different viscosities under oceans with different absolute plate motion, anisotropic viscosities by means of superposition (Schmeling, 1987)
8 Microphysical theory and observations Maximum strength of materials (1) Strength is maximum stress that material can resist In principle, viscous fluid has zero strength. In reality, all materials have finite strength. (Ranalli, 1995) Elastic deformation until atom jumps to next equilibrium position. So theoretical strength σ = O( µ )
9 Microphysical theory and observations Maximum strength of materials (2) However, from laboratory measurements: Shear strength = O (10 Structural flaws Cracks Vacancies Dislocations Subgrain boundaries 4 µ) due to: (Ranalli, 1995) Stress concentration makes deformation possible under smaller stresses (compare to breaking/tearing of sheet of paper)
10 Diffusion creep (Schubert, Turcotte & Olson, 2001)
11 Dislocation creep (Ranalli, 1995)
12 Steady state creep models Theoretically many different models: only a few relevant for Earth (Climb-controlled) dislocation creep or powerlaw creep: gliding of dislocations controlled by the climb rate around n impurities/obstacles: E p V D SD D SD =D 0 exp RT Diffusion creep (Newtonian, linear creep): Nabarro-Herring creep (diffusion through grain) Coble creep (diffusion along grain boundary) D SD m d m=2-3 Peierl s stress mechanism (low-t plasticity): dislocation glide Grain-boundary sliding (superplasticity) Pressure-solution Brittle deformation, Byerlee s law
13 Strength of the Lithosphere and Mantle Experimental data for olivine
14 Strength of the Lithosphere and Mantle (Turcotte and Schubert, 2002) Collection of rheological data for different materials
15 See Hirth & Kohlstedt (2005) for olivine in the upper mantle
16 Laboratory experiments: large temperature, strain-rate & grain-size dependence.
17 Dislocation creep decreases viscosity where the strain-rate is more than the transition value.
18 Non-deforming regions remain highly viscous. Yielding concentrates deformation.
19 Deformation maps (Ranalli, 1995)
20 Strength of the Lithosphere and Mantle (3) (Kohlstedt et al., 1995) Strength curves for different materials: lithosphere
21 Slab rheology Figure courtesy of M. Billen
22 Governing equations Conservation of mass: continuity Conservation of momentum: Stokes equation Equation of state: density constitutive equation: rheology Conservation of energy: temperature
23 Continuity (from Turcotte and Schubert, 2002) This gives: u = 0
24 Derivation of Stokes equation Static force balance f = 0 ij f i =0 xj Incompressible constitutive relationship and separation into deviatoric stress ij = ij p ij =2 ij p ij Constant viscosity ij p 2 f i =0 x j xi
25 Derivation of Stokes equation (2) 2 ij vi vi v j vj 2 = = x j x j x j xi x j x j xi x j 2 ij vi 2 2 = = v i x j x j x j p v i f i =0 xi Because for incompressible vj =0 xj 2 2 vi p f i =0 x j x j xi p f = v 0 2
26 Incompressible, Newtonian flow
27
28
29 Simplified equations
30 Simple 1-D fluid dynamics examples Couette channel flow: Channel flow with horizontal pressure gradient (Hagen Poiseuille):
31 Stokes sinker solution
32 Inferences based on Stokes flow u u L L
33 Scaling: Dimensional equations: u = 0 p + η 2u = ρ 0α (T T0 ) gδ i 3 ρ 0C p dt = k 2T ρc pu T dt Scaling parameters x' = x / h t ' = t /(h 2 / κ ) T ' = (T T0 ) / T η ' = η / η0 Scaled equations (with primes left out): u = 0 p + η u = RaTδ i 3 2 dt = 2T u T dt αρ 0 g Th 3 with Rayleigh number Ra = ηκ
34 Rayleigh numbers Bottom heated Internal heating αρ0 g Th Ra = ηκ H h =k T h =k T h 2 Hh T = k 5 0 g H h Ra H = k
35 Effect of phase transitions Clapeyron slope dp dt = 0 g T h Ra = c 3 Q latent 1 2 T a g T h Rb= = 3 Buoyancy parameter Rb P 2 g h Ra = = c = 0 g h c T = e.g. Schubert et al. (1975); Christensen (1985); Schubert et al. (2001), p. 466f
36 Linear stability analysis (1) for small T (or Ra): conduction for larger Ra: convection sets in so minimum Ra = Rac exists below which no convection occurs T = Tconductive + T1 = T0 + T1 For small velocity and T1 we can linearize system: linear stability analysis Energy equation in terms of T1 and Ψ Remove small terms Solve with separation of variables: Ψ = Ψ * cos(nπz ) sin(kx ) exp(αt ) T1 = T1 cos(nπz ) cos(kx ) exp(αt ) * α<0: stable α>0: instable α=0: marginal stability Rac = (k n π ) k2 See Schubert et al. (2001) p. 288ff
37 Linear stability analysis (2) minimum Rac = for free-slip, bottom heated, c = 2.8 Rac as function of horizontal wavelength k=2πb/λ, b = box height, i.e. one wavelength of counter-rotating cells with aspect ratio 2.8 x 1 for = c (from Turcotte and Schubert, 2002)
38 Simple convection model for high Ra (from (Davies, 1999)) B = gddρα T R = 4ηv R+B=0 v~14 cm/yr relates physical quantities
39 Boundary layer theory u Ra h 2 3 Q T Ra Nu Ra See Schubert et al. (2001) p. 353ff Grigne et al. (2005)
40 Example: Boussinesq convection, isoviscous, no internal heating Ra=1 Nu=1 Figure courtesy of P. van Keken
41 Ra=103 Nu>1 Figure courtesy of P. van Keken
42 Ra=104 Nu=3.5 Figure courtesy of P. van Keken
43 Ra=105 Nu=8 Figure courtesy of P. van Keken
44 Ra=106 <Nu>=16 Figure courtesy of P. van Keken
45 Ra=2x106 <Nu>=18 Figure courtesy of P. van Keken
46 Ra=5x106 <Nu>=20 Figure courtesy of P. van Keken
47 Logistical equation (May, Nature, 1976): xn+1 = r xn ( 1 xn )
48 Ra > Rac modeling results higher Ra gives thinner thermal boundary layers For larger Ra flow usually not steady state
49 2D convection experiments, isoviscous case Ra = 107 H=0 '(T)= 1 lm/ um = 1 Courtesy of Allen McNamara
50 Temperature dependent viscosity Ra = 107 H=0 '(T) = 1000 lm/ um = 1 Courtesy of Allen McNamara
51 Thermal convection in the mantle Ra = 107 H=0 '(T) = 1000 lm/ um = 50 Courtesy of Allen McNamara
52 Increase of internal heating Ra = 2.4e5
53 Heating mode from (Davies, 1999) bottom/internal heating passive/active upwellings (MOR?) time dependence bottom/top boundary layer independent (plumes vs. plates)
54 Convection with η(t) (1) large aspect ratios large viscosity variations in top 200 km asymmetry between up- & downwellings from Bercovici et al., (1996)
55 Convection with η(t) (2) Stagnant lid regime on Earth? missing rheology! from Bercovici et al. (1996)
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