Fluid Dynamics and Mantle Convection

Subduction II Fundamentals of Mantle Dynamics Thorsten W Becker University of Southern California Short course at Universita di Roma TRE April 18 – 20, 2011 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 Elastic deformation In general: σ ε ij = Cijkl kl (in 3-D 81 degrees of freedom, in general 21 independent) For isotropic body this reduces to Hooke’s law: σ λε δ µε ij = kk ij + 2 ij λ µ ε ε ε ε with and Lame’s parameters, kk = 11 + 22 + 33 Taking shear components ( i ≠ j ) gives definition of rigidity: σ µε 12 = 2 12 Adding the normal components ( i=j ) for all i=1,2,3 gives: σ λ µ ε κε kk = (3 + 2 ) kk = 3 kk with κ = λ + 2µ/3 = bulk modulus Linear viscous deformation (1) Total stress field = static + dynamic part: σ = − δ +τ ij p ij ij Analogous to elasticity … General case: τ = ε ij C'ijkl kl Isotropic case: τ = λ ε δ + ηε ij ' kk ij 2 ij Linear viscous deformation (2) Split in isotropic and deviatoric part (latter causes deformation): 1 σ ' = σ − σ δ = σ + pδ ij ij 3 kk ij ij ij 1 ε' = ε − ε δ ij ij 3 kk ij which gives the following stress: σ = − δ + ςε δ + ηε 'ij ( p p) ij kk ij 2 'ij ε = With compressibility term assumed 0 (Stokes condition kk 0 ) 2 ( ς = λ ' + η = bulk viscosity) 3 τ Now τ = 2 η ε or η = ij ij ij ε 2 ij η = In general f (T,d, p, H2O) Non-linear (or non-Newtonian) deformation General stress-strain relation: ε = Aτ n n = 1 : Newtonian n > 1 : non-Newtonian n → ∞ : pseudo-brittle 1 −1 1−n 1 −1/ n (1−n) / n Effective viscosity η = A τ = A ε eff 2 2 Application: different viscosities under oceans with different absolute plate motion, anisotropic viscosities by means of superposition (Schmeling, 1987) 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) Microphysical theory and observations Maximum strength of materials (2) However, from laboratory measurements: −4 Shear strength = O ( 10 µ ) due to: •Structural flaws •Cracks •Vacancies •Dislocations •Subgrain boundaries (Ranalli, 1995) Stress concentration makes deformation possible under smaller stresses (compare to breaking/tearing of sheet of paper) Diffusion creep (Schubert, Turcotte & Olson, 2001) Dislocation creep (Ranalli, 1995) 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 impurities/obstacles: n E¿ pV ¿ ∝ DSD D =D exp − SD 0 RT • Diffusion creep (Newtonian, linear creep): • Nabarro-Herring creep (diffusion through grain) • Coble creep (diffusion along grain boundary) DSD m=2-3 ∝ d m • Peierl’s stress mechanism (low-T plasticity): dislocation glide • Grain-boundary sliding (superplasticity) • Pressure-solution • Brittle deformation, Byerlee’s ‘law’ Strength of the Lithosphere and Mantle Experimental data for olivine Strength of the Lithosphere and Mantle (Turcotte and Schubert, 2002) Collection of rheological data for different materials See Hirth & Kohlstedt (2005) for olivine in the upper mantle Laboratory experiments: large temperature, strain-rate & grain-size dependence. Dislocation creep decreases viscosity where the strain-rate is more than the transition value. • Non-deforming regions remain highly viscous. • Yielding concentrates deformation. Deformation maps (Ranalli, 1995) Strength of the Lithosphere and Mantle (3) (Kohlstedt et al., 1995) Strength curves for different materials: lithosphere Slab rheology Figure courtesy of M. Billen Governing equations • Conservation of mass: continuity • Conservation of momentum: Stokes’ equation • Equation of state: density • constitutive equation: rheology • Conservation of energy: temperature Continuity (from Turcotte and Schubert, 2002) This gives: ∇ ⋅u = 0 Derivation of Stokes equation Static force balance ∂ ij ∇⋅ f =0 f i=0 ∂ x j Incompressible constitutive relationship and separation into deviatoric stress ij=ij− pij=2ij− pij Constant viscosity ∂ij ∂ p 2 − f i=0 ∂ x j ∂ xi Derivation of Stokes equation (2) 2 ∂ ∂ ∂ v ∂ v ∂ v ∂ ∂ v 2 ij = i j = i j ∂ x j ∂ x j ∂ x j ∂ xi ∂ x j x j ∂ xi ∂ x j 2 ∂ij ∂ vi 2 Because for incompressible 2 = =∇ vi ∂ ∂ ∂ ∂ v j x j x j x j =0 ∂ x j 2 ∂ p ∇ vi− f i=0 ∂ 2 xi ∇ v−∇ pf =0 2 ∂ vi ∂ p − f i=0 ∂ x j ∂ x j ∂ xi Incompressible, Newtonian flow Simplified equations Simple 1-D fluid dynamics examples Couette channel flow: Channel flow with horizontal pressure gradient (Hagen Poiseuille): Stokes sinker solution Inferences based on Stokes flow u∝ u ∝ ∝ ∝ L L Scaling: Dimensional equations: ∇ ⋅u = 0 − ∇∆ +η∇2 = ρ α − δ p u 0 (T T0 )g i3 dT ρ C = k∇2T − ρC u ⋅∇T 0 p dt p Scaling parameters x'= x / h t'= t /(h2 /κ ) = − ∆ T ' (T T0 ) / T η =η η ' / 0 Scaled equations (with primes left out): ∇ ⋅u = 0 αρ g∆Th3 − ∇∆p +η∇2u = RaTδ Ra = 0 i3 with Rayleigh number ηκ dT = ∇2T − u ⋅ ∇T dt Rayleigh numbers Bottom heated αρ g∆Th3 Ra = 0 ηκ 3 2 Internal heating H h =k ∇ T h =k T h H h2 T = k g H h5 Ra = 0 H k Effect of phase transitions Clapeyron slope dp Qlatent 1 2 = = dT c T a 3 3 0 g T h g T h Ra= Rb= Buoyancy parameter Rb 0 g h P = = = 2 c= g h Ra c T e.g. Schubert et al. (1975); Christensen (1985); Schubert et al. (2001), p. 466f 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’ Ψ • Energy equation in terms of T1 and • Remove ‘small’ terms • Solve with separation of variables: Ψ = Ψ* cos(nπz)sin(kx)exp(αt) = * π α T1 T1 cos(n z)cos(kx)exp( t) α 3 • <0: stable (k 2n2π 2 ) • α = >0: instable Rac 2 • α=0: marginal stability k See Schubert et al. (2001) p. 288ff Linear stability analysis (2) minimum Rac = 657.5 for free-slip, bottom heated, = 2.8 lc 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 = l lc (from Turcotte and Schubert, 2002) Simple convection model for high Ra (from (Davies, 1999)) • B = gDdρα∆T • R = 4ηv • R + B = 0 • v~14 cm/yr • relates physical quantities Boundary layer theory 2 u∝ Ra 3 h 1 Q∝T Ra 3 1 Nu∝ Ra 3 See Schubert et al. (2001) p. 353ff Grigne et al. (2005) Example: Boussinesq convection, isoviscous, no internal heating Ra=1 Nu=1 Figure courtesy of P. van Keken Ra=103 Nu>1 Figure courtesy of P. van Keken Ra=104 Nu=3.5 Figure courtesy of P. van Keken Ra=105 Nu=8 Figure courtesy of P. van Keken Ra=106 <Nu>=16 Figure courtesy of P. van Keken Ra=2x106 <Nu>=18 Figure courtesy of P. van Keken Ra=5x106 <Nu>=20 Figure courtesy of P. van Keken Logistical equation (May, Nature, 1976): xn+1 = r xn ( 1 – xn ) Ra > Rac modeling results • higher Ra gives thinner thermal boundary layers • For larger Ra flow usually not steady state 2D convection experiments, isoviscous case Ra = 107 H = 0 '(T)= 1 / = 1 h hlm hum Courtesy of Allen McNamara Temperature dependent viscosity Ra = 107 H = 0 '(T) = 1000 / = 1 h hlm hum Courtesy of Allen McNamara Thermal convection in the mantle Ra = 107 H = 0 '(T) = 1000 / = 50 h hlm hum Courtesy of Allen McNamara Increase of internal heating Ra = 2.4e5 Heating mode from (Davies, 1999) • bottom/internal heating • passive/active upwellings (MOR?) • time dependence • bottom/top boundary layer independent (plumes vs. plates) Convection with η(T) (1) • large aspect ratios • large viscosity variations in top 200 km • asymmetry between up- & downwellings from Bercovici et al., (1996) Convection with η(T) (2) Stagnant lid regime on Earth? missing rheology! from Bercovici et al. (1996).

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