Mantle Dynamics and Convective Mixing

Transcription

1 Mantle Dynamics and Convective Mixing ( Chemical Geodynamics ) Henri Samuel

2 Mantle Dynamics and Convective Mixing ( Dynamical Geochemistry ) Henri Samuel

3 General Motivation [Ballentine et al., 2002]

4 [Ballentine et al., 2002] General Motivation The Earth s mantle is heterogeneous

5 [Ballentine et al., 2002] General Motivation The Earth s mantle is heterogeneous? [Tarney et al., 1980]

6 [Ballentine et al., 2002] General Motivation The Earth s mantle is heterogeneous? [Tarney et al., 1980]

7 [Ballentine et al., 2002] General Motivation The Earth s mantle is heterogeneous? [Tarney et al., 1980]

8 General Motivation [Ballentine et al., 2002] Convective motions homogenization The Earth s mantle is heterogeneous? [Tarney et al., 1980]

9 General Motivation [Ballentine et al., 2002] Convective motions homogenization The Earth s mantle is heterogeneous? [Tarney et al., 1980]

10 General Motivation [Ballentine et al., 2002] Convective motions homogenization The Earth s mantle is heterogeneous? [Tarney et al., 1980] Survival time of mantle heterogeneities? How efficient is convective mixing?

11 Preliminary Considerations

12 Convective mixing in the Earth Scale dependent: Scale of heterogeneity Convective system: whole mantle vs. magma chambers L~10 6 m t~ yr L~ m t~ yr

13 Convective mixing in the Earth Scale dependent: Scale of heterogeneity Convective system: whole mantle vs. magma chambers 1. Convective mixing in the (solid) Earth s mantle! 2. Convective stirring in magma chambers L~10 6 m t~ yr L~ m t~ yr

14 Mixing & Stirring: Homogenisation BUT Mixing Stirring Mixing vs. Stirring

15 Mixing vs. Stirring Mixing & Stirring: Homogenisation Example: Steady Vortex flow BUT Mixing Stirring (x, z) = 1 sin( x) 2 sin( z) 2

16 Mixing vs. Stirring Mixing & Stirring: Homogenisation BUT Mixing Stirring Example: Steady Vortex flow (x, z) = 1 sin( x) 2 sin( z) 2 Stirring: Advection only

17 Mixing vs. Stirring Mixing & Stirring: Homogenisation BUT Mixing Stirring Example: Steady Vortex flow (x, z) = 1 sin( x) 2 sin( z) 2 Stirring: Advection only Diffusion only

18 Mixing vs. Stirring Mixing & Stirring: Homogenisation BUT Mixing Stirring Example: Steady Vortex flow (x, z) = 1 sin( x) 2 sin( z) 2 Stirring: Advection only Diffusion only Mixing: Stirring + Diffusion

19 Mixing vs. Stirring Mechanical stirring & diffusion required for efficient & complete mixing Mixing & Stirring: Homogenisation BUT Mixing Stirring Example: Steady Vortex flow (x, z) = 1 sin( x) 2 sin( z) 2 Stirring: Advection only Diffusion only Mixing: Stirring + Diffusion

20 Requirements for efficient stirring

21 End-member flows Given U and δ0 = δ(t=0), what is δ(t)? δ

22 End-member flows Given U and δ0 = δ(t=0), what is δ(t)? Simple shear flow Regular stirring δ d dt = Linear evolution Weak/slow stirring

23 End-member flows Given U and δ0 = δ(t=0), what is δ(t)? Simple shear flow Regular stirring δ Pure shear flow Chaotic/turbulent stirring Hyperbolic point d dt = Linear evolution Weak/slow stirring d dt = Exponential evolution Strong/fast stirring

24 Time-dependent vs. steady flow Steady flow Time-dependent flow

25 Time-dependent vs. steady flow Hyperbolic point Steady flow Elliptic point Time-dependent flow

26 Time-dependent vs. steady flow Hyperbolic point Steady flow Elliptic point Time-dependent flow

27 Time-dependent vs. steady flow Hyperbolic point Steady flow Elliptic point Time-dependent flow

28 Time-dependent vs. steady flow Hyperbolic point Steady flow Elliptic point Closed trajectories = Weak stirring Time-dependent flow

29 Time-dependent vs. steady flow Hyperbolic point Steady flow Elliptic point Closed trajectories = Weak stirring Time-dependent flow

30 Time-dependent vs. steady flow Hyperbolic point Steady flow Elliptic point Closed trajectories = Weak stirring Time-dependent flow

31 Time-dependent vs. steady flow Hyperbolic point Steady flow Elliptic point Closed trajectories = Weak stirring Time-dependent flow Crossed trajectories = Efficient stirring

32 Time-dependent flow generates crossings between fluid trajectories Efficient stirring Time-dependent vs. steady flow Hyperbolic point Steady flow Elliptic point Closed trajectories = Weak stirring Time-dependent flow Crossed trajectories = Efficient stirring

33 Requirements for U to produce efficient (turbulent) stirring Presence of hyperbolic points Time dependence

34 Requirements for U to produce efficient (turbulent) stirring Presence of hyperbolic points Time dependence a. Stretching

35 Requirements for U to produce efficient (turbulent) stirring Presence of hyperbolic points Time dependence a. Stretching b. Folding

36 Requirements for U to produce efficient (turbulent) stirring Presence of hyperbolic points Time dependence Repeated action of stretching & folding ( horse shoe map )

37 Requirements for U to produce efficient (turbulent) stirring Presence of hyperbolic points Time dependence Repeated action of stretching & folding ( horse shoe map ) a. Stretching

38 Requirements for U to produce efficient (turbulent) stirring Presence of hyperbolic points Time dependence Repeated action of stretching & folding ( horse shoe map ) a. Stretching b. Folding

39 Requirements for U to produce efficient (turbulent) stirring Presence of hyperbolic points Time dependence Repeated action of stretching & folding ( horse shoe map ) a. Stretching b. Folding (a) & (b)= = hors

40 Requirements for U to produce efficient (turbulent) stirring Presence of hyperbolic points Time dependence Repeated action of stretching & folding ( horse shoe map ) a. Stretching b. Folding (a) & (b)= = hors 2 X

41 Requirements for U to produce efficient (turbulent) stirring Presence of hyperbolic points Time dependence Repeated action of stretching & folding ( horse shoe map ) a. Stretching b. Folding... (a) & (b)= = hors 2 X 3 X

42 Requirements for U to produce efficient (turbulent) stirring Presence of hyperbolic points Time dependence Repeated action of stretching & folding ( horse shoe map ) a. Stretching b. Folding... (a) & (b)= = hors 2 X 3 X What influences U affects stirring efficiency

43 Mantle dynamics: U? Hot material is lighter & rises up Mantle dynamics ~ upwellings & downwellings Ra = g T H3 apple Buoyancy Viscous + thermal diffusivity eat loss/gain via diffusion Conservation equations Mass.U = 0 Momentum p +.( )+Ra T z =0 Cold material is heavier and sinks down Energy DT Dt = 2 T

44 Mantle dynamics & Rayleigh-Bénard convection Ra = g T H3 apple Buoyancy Viscous + thermal diffusivity Present day Earth s mantle parameters: Ra >> 1 Convection: deformation of mantle material mixing

45 Mantle dynamics & Rayleigh-Bénard convection Ra = g T H3 apple Buoyancy Viscous + thermal diffusivity Present day Earth s mantle parameters: Ra >> 1 Convection: deformation of mantle material mixing

46 Stirring efficiency & (mantle) dynamics Stirring efficiency ~ mantle dynamics Mantle dynamics ~ Rayleigh-Bénard convection + (many) complications: Internal heating Phase changes Complex rheology Active compositional heterogeneities Plate tectonics Continental lids...

47 Stirring efficiency & (mantle) dynamics Stirring efficiency ~ mantle dynamics Mantle dynamics ~ Rayleigh-Bénard convection + (many) complications: Internal heating Phase changes Complex rheology Active compositional heterogeneities Plate tectonics Continental lids... The influence of each contribution on mixing processes should be studied separately

48 Key questions How to create geochemical reservoirs in a convecting mantle? How to preserve reservoirs in a vigorously convecting mantle? (Partial) answers/examples: How to sample geochemical reservoirs?

49 Key questions How to create geochemical reservoirs in a convecting mantle? How to preserve reservoirs in a vigorously convecting mantle? (Partial) answers/examples: How to sample geochemical reservoirs? I. Origin of chemical heterogeneities MORB/OIB dichotomy?

50 Key questions How to create geochemical reservoirs in a convecting mantle? How to preserve reservoirs in a vigorously convecting mantle? (Partial) answers/examples: How to sample geochemical reservoirs? I. Origin of chemical heterogeneities MORB/OIB dichotomy? II. Origin of isotopic variability among MORB?

51 I. Origin of chemical heterogeneities MORB/ OIB dichotomy?

52 Origin of chemical heterogeneities MORB/OIB dichotomy? MORB He/4He x Ra OIB OIB (3He/4He)/Ra x Ra [Farley & Neroda, 1998]

53 Origin of chemical heterogeneities MORB/OIB dichotomy? 1. Primordial? Satisfies rare gas constraints Low 3 He/ 4 He High 3 He/ 4 He [Allègre et al., 1986]

54 Origin of chemical heterogeneities MORB/OIB dichotomy? 1. Primordial? Satisfies rare gas constraints 2. Recycled? Satisfies HIMU constraint [Christensen & Hofmann, 1994] [Coltice & Ricard, 1999]...

55 Origin of chemical heterogeneities MORB/OIB dichotomy? 1. Primordial? Satisfies rare gas constraints 2. Recycled? Satisfies HIMU constraint Both? Satisfies 1 & 2 Focus on He isotopes Degassed Mantle: Low 3 He/ 4 He Undegassed Mantle: High 3 He/ 4 He

56 Origin of chemical heterogeneities MORB/OIB dichotomy? 1. Primordial? Satisfies rare gas constraints 2. Recycled? Satisfies HIMU constraint Partial melting & degassing at plumes and ridges Equilibrium melting Determined by the flow Creation & recycling of oceanic crust and lithosphere Both? Satisfies 1 & 2 Focus on He isotopes Degassed Mantle: Low 3 He/ 4 He Undegassed Mantle: High 3 He/ 4 He Plume Ridge!"#$%&"'".>,)'/01'2*1'31' 4 5#1' 6 5#7 ".>,)'''''''''''''' '@A'B<'!"#$%&"'(&)*+,-*#.#'/01'2*1'31' 4 5#1' 6 5#7 (&)*+,-*#.#'''''''''''''''''' C sol i = C 0 i D sol i melt + F (1 Di sol melt )?#;$,,#:'<$%)(#'/01'2*1'31' 4 5#1' 6 5#7 :#; $,,#:'''''' ρ = ' ' 8++.(9':#;$,,#:'<$%)(#'/01'2*1'31' 4 5#1' 6 5#7 -++.(9':#;$,,#:''' ρ = ρ χ C melt i = C sol i D sol i melt

57 Convection: driven by density differences = T + 0 Thermal Chemical? Ra=10 6, B=0.5 Buoyancy Number: B = T B stabilizes convection More complex than purely thermal convection Even small β (i.e., 1%) greatly affects the dynamics

58 Convection: driven by density differences = T + 0 Thermal Chemical? Ra=10 6, B=0.5 Buoyancy Number: B = T B stabilizes convection More complex than purely thermal convection Even small β (i.e., 1%) greatly affects the dynamics

59 Stirring efficiency measured with Finite-time Lyapunov exponents σ- σ- σ+ σ+ t t=0 ± = 1 ln t ± Finite Time Lyapunov Exponents Efficient stirring Weak stirring [Farnetani & Samuel, EPSL, 2003]

60 Helium ratios Observed MORB Predicted MORB He/4He x Ra OIB OIB (3He/4He)/Ra x Ra [Farley & Neroda, 1998] Good agreement with observations OIB [Samuel & Farnetani, EPSL, 2003] OIB: Various components Large spectrum of He Ratios MORB: Homogeneous Narrow spectrum of He Ratios

61 Predicted Helium ratios Predicted 100 R/Ra=26 MORB Proportion (%) ~ 83% Oceanic lithosphere 0 ~ 8% ~ 8% Oceanic Undegassed crust material { OIB [Samuel & Farnetani, 2003] OIB: Various components Large spectrum of He Ratios [Samuel & Farnetani, EPSL, 2003] MORB: Homogeneous Narrow spectrum of He Ratios

62 Conclusions I Compositional density contrasts prevents homogenisation of primordial reservoirs over geological times Thermochemical plumes sampling the denser and undegassed reservoir and accumulated recycled lithosphere and crust create a broad spectrum of helium isotopes at OIBs! The remaining degassed mantle is more efficiently mixed and more homogeneous than OIB sources, leading to a narrower spectrum of helium isotopes for MORBs

63 II. Understanding Isotopic Variability at Mid-ocean Ridges

64 MORB variability & mantle dynamics [Graham, 2002] Pacific N Indian 20 N Atlantic S Atlantic He/ 4 He (R/R A )

65 MORB variability & mantle dynamics ( 3 He/ 4 He) variability σ(( 3 He/ 4 He)/R a ) Interpretation: Weak stirring From [Georgen et al., 2003] [Graham, 2002] Interpretation: Efficient stirring Ridge spreading rate (cm/yr) General trend: isotopic variability ~ 1/(spreading rate)

66 MORB variability & mantle dynamics ( 3 He/ 4 He) variability σ(( 3 He/ 4 He)/R a ) Interpretation: Weak stirring From [Georgen et al., 2003] [Graham, 2002] Interpretation: Efficient stirring Ridge spreading rate (cm/yr) General trend: isotopic variability ~ 1/(spreading rate) Standard interpretation: Isotopic variability = local mantle heterogeneity due to plate motion

67 MORB variability & mantle dynamics ( 3 He/ 4 He) variability σ(( 3 He/ 4 He)/R a ) Interpretation: Weak stirring From [Georgen et al., 2003] [Graham, 2002] SWIR Interpretation: Efficient stirring Ridge spreading rate (cm/yr) General trend: isotopic variability ~ 1/(spreading rate) Standard interpretation: Isotopic variability = local mantle heterogeneity due to plate motion

68 MORB variability & mantle dynamics ( 3 He/ 4 He) variability σ(( 3 He/ 4 He)/R a ) Interpretation: Weak stirring From [Georgen et al., 2003] [Graham, 2002] SWIR Interpretation: Efficient stirring Ridge spreading rate (cm/yr) General trend: isotopic variability ~ 1/(spreading rate) Standard interpretation: Isotopic variability = local mantle heterogeneity due to plate motion SWIR data incompatible with standard assumption

69 MORB variability & mantle dynamics ( 3 He/ 4 He) variability σ(( 3 He/ 4 He)/R a ) Interpretation: Weak stirring From [Georgen et al., 2003] [Graham, 2002] SWIR Interpretation: Efficient stirring Ridge spreading rate (cm/yr) General trend: isotopic variability ~ 1/(spreading rate) Standard interpretation: Isotopic variability = local mantle heterogeneity due to plate motion SWIR data incompatible with standard assumption Mantle dynamics= large-scale flow (plates) + small-scale convection

70 MORB variability & mantle dynamics ( 3 He/ 4 He) variability σ(( 3 He/ 4 He)/R a ) Interpretation: Weak stirring From [Georgen et al., 2003] [Graham, 2002] SWIR Interpretation: Efficient stirring Ridge spreading rate (cm/yr) General trend: isotopic variability ~ 1/(spreading rate) Standard interpretation: Isotopic variability = local mantle heterogeneity due to plate motion SWIR data incompatible with standard assumption Mantle dynamics= large-scale flow (plates) + small-scale convection Plate motion = Large-scale convection

71 MORB variability & mantle dynamics ( 3 He/ 4 He) variability σ(( 3 He/ 4 He)/R a ) Interpretation: Weak stirring From [Georgen et al., 2003] [Graham, 2002] SWIR Interpretation: Efficient stirring Ridge spreading rate (cm/yr) General trend: isotopic variability ~ 1/(spreading rate) Standard interpretation: Isotopic variability = local mantle heterogeneity due to plate motion SWIR data incompatible with standard assumption Mantle dynamics= large-scale flow (plates) + small-scale convection [Parsons & McKenzie, 1978]

72 MORB variability & mantle dynamics ( 3 He/ 4 He) variability σ(( 3 He/ 4 He)/R a ) Interpretation: Weak stirring From [Georgen et al., 2003] [Graham, 2002] SWIR Interpretation: Efficient stirring Ridge spreading rate (cm/yr) General trend: isotopic variability ~ 1/(spreading rate) Standard interpretation: Isotopic variability = local mantle heterogeneity due to plate motion SWIR data incompatible with standard assumption Mantle dynamics= large-scale flow (plates) + small-scale convection [Parsons & McKenzie, 1978] Small-scale convection Predicted by theory, observed experimentally Explains the observed flattening of the lithosphere at old ages [Stein & Stein 1992]

73 MORB variability & mantle dynamics ( 3 He/ 4 He) variability σ(( 3 He/ 4 He)/R a ) Interpretation: Weak stirring From [Georgen et al., 2003] [Graham, 2002] SWIR Interpretation: Efficient stirring Ridge spreading rate (cm/yr) General trend: isotopic variability ~ 1/(spreading rate) Standard interpretation: Isotopic variability = local mantle heterogeneity due to plate motion Mantle dynamics= large-scale flow (plates) + small-scale convection [Parsons & McKenzie, 1978] Small-scale convection Predicted by theory, observed experimentally Explains the observed flattening of the lithosphere at old ages [Stein & Stein 1992] SWIR data incompatible with standard assumption Effect of small-scale convection on stirring efficiency?

74 Governing equations Mass Energy.U =0 D t T = Momentum Stream function formulation + Finite Volume discretisation: StreamV [Samuel, 2009] Model Setup 2 T p +. + Ra T z= 0 half-spreading rate: Vx=Pe Imposed ridge motion 2H/L = [2-6] L Governing Parameters g TL 3 Rayleigh #: Ra 0 = 0 = Stirring efficiency measured with Finite-time Lyapunov exponents [Farnetani & Samuel, 2003] Rheology: = 0 (z)[ ln( )T ] Pe = Vx surface = Aspect ratio 2H = 2L-6L, γ= σ - σ + t=0 ± = 1 t ln ± σ - σ + t

75 Results: Temperature field Pe = 100 (~ 0.2 cm/yr) 2H

76 Results: Temperature field Pe = 100 (~ 0.2 cm/yr) 2H

77 Results: Temperature field Pe = 100 (~ 0.2 cm/yr) 2H Motion= Plate (large scale) motion + Small-Scale Convection

78 Results: Temperature field Pe = 100 (~ 0.2 cm/yr) dssc donset 2H Motion= Plate (large scale) motion + Small-Scale Convection Small Scale Convection (SSC) initiates within tonset! Drifting SSC onset distance donset = Vsurface x tonset Distance where SSC is fully developed dssc = H - donset

79 Results: Temperature field Pe = 100 (~ 0.2 cm/yr) dssc donset 2H Motion= Plate (large scale) motion + Small-Scale Convection Small Scale Convection (SSC) initiates within tonset! Drifting SSC onset distance donset = Vsurface x tonset Distance where SSC is fully developed dssc = H - donset Faster spreading rate = weaker SSC

80 Stirring efficiency vs. plate velocity (Vsurface~ 0.2 cm/yr) (Vsurface~ 1 cm/yr) (Vsurface~ 2.2 cm/yr) (Vsurface~ 9.6 cm/yr) Pe Proportion (%) Proportion (%) Proportion (%) Proportion (%) Finite Time Lyapunov Exponent, λ Finite Time Lyapunov exponent Pe=100 Pe= Finite Time Lyapunov Exponent, λ + Pe= Finite Time Lyapunov Exponent, λ + Pe=8000 Weak stirring Strong stirring Finite Time Lyapunov Exponent, λ +

81 Stirring efficiency vs. plate velocity (Vsurface~ 0.2 cm/yr) (Vsurface~ 1 cm/yr) (Vsurface~ 2.2 cm/yr) (Vsurface~ 9.6 cm/yr) Pe Proportion (%) Proportion (%) Proportion (%) Proportion (%) Finite Time Lyapunov Exponent, λ Finite Time Lyapunov exponent Pe=100 Pe= Finite Time Lyapunov Exponent, λ + Pe= Finite Time Lyapunov Exponent, λ + Pe=8000 Weak stirring Strong stirring Finite Time Lyapunov Exponent, λ +

82 Stirring efficiency vs. plate velocity (Vsurface~ 0.2 cm/yr) (Vsurface~ 1 cm/yr) (Vsurface~ 2.2 cm/yr) (Vsurface~ 9.6 cm/yr) Pe Proportion (%) Proportion (%) Proportion (%) Proportion (%) Finite Time Lyapunov Exponent, λ Finite Time Lyapunov exponent Pe=100 Pe= Finite Time Lyapunov Exponent, λ + Pe= Finite Time Lyapunov Exponent, λ + Pe=8000 Weak stirring Strong stirring Finite Time Lyapunov Exponent, λ +

83 Stirring efficiency vs. plate velocity (Vsurface~ 0.2 cm/yr) (Vsurface~ 1 cm/yr) (Vsurface~ 2.2 cm/yr) (Vsurface~ 9.6 cm/yr) Pe Proportion (%) Proportion (%) Proportion (%) Proportion (%) Finite Time Lyapunov Exponent, λ Finite Time Lyapunov exponent Pe=100 Pe= Finite Time Lyapunov Exponent, λ + Pe= Finite Time Lyapunov Exponent, λ + Pe=8000 Weak stirring Strong stirring No simple Finite relationship Time Lyapunov Exponent, between λ + velocity & stirring efficiency!

84 Stirring efficiency vs. spreading rate Pe=100 V=Pe dssc H Pe=1000 dssc Pe=2000 dssc dssc ~H - Pe tssc

85 Stirring efficiency vs. spreading rate 3000 Average (RMS): V RMS Dimensionless velocity SSC No SSC Dimensionless half spreading rate, Pe

86 Stirring efficiency vs. spreading rate Dimensionless velocity SSC Average (RMS): V RMS No SSC Dimensionless half spreading rate, Pe Finite Time Lyapunov Exponent SSC 0 Average (RMS): λ + RMS No SSC Dimensionless half spreading rate, Pe

87 Stirring efficiency vs. spreading rate Dimensionless velocity SSC Average (RMS): V RMS No SSC Dimensionless half spreading rate, Pe Finite Time Lyapunov Exponent SSC 0 Average (RMS): λ + RMS No SSC Dimensionless half spreading rate, Pe Two regimes: Pe < Pec SSC + large-scale flow Pe > Pec No SSC : pure large-scale flow

88 Stirring efficiency vs. spreading rate Dimensionless velocity SSC Average Average (RMS): V(RMS): V RMS Temporal variations: V RMS ± σ(v RMS )x5 No SSC Dimensionless half spreading rate, Pe Finite Time Lyapunov Exponent SSC 0 Average (RMS): λ + RMS No SSC Dimensionless half spreading rate, Pe Two regimes: Pe < Pec SSC + large-scale flow Pe > Pec No SSC : pure large-scale flow

89 Stirring efficiency vs. spreading rate Dimensionless velocity SSC Average Average (RMS): V(RMS): V RMS Temporal variations: V RMS ± σ(v RMS )x5 No SSC Dimensionless half spreading rate, Pe Finite Time Lyapunov Exponent Average Average (RMS): (RMS): λ + RMS λ + RMS Spatial variations: λ + RMS ± σ(λ+ ) SSC No SSC Dimensionless half spreading rate, Pe Two regimes: Pe < Pec SSC + large-scale flow Pe > Pec No SSC : pure large-scale flow

90 Stirring efficiency vs. spreading rate Dimensionless velocity SSC Average Average (RMS): V(RMS): V RMS Temporal variations: V RMS ± σ(v RMS )x5 No SSC Dimensionless half spreading rate, Pe Finite Time Lyapunov Exponent Average Average (RMS): (RMS): λ + RMS λ + RMS Spatial variations: λ + RMS ± σ(λ+ ) SSC No SSC Dimensionless half spreading rate, Pe Two regimes: Pe < Pec SSC + large-scale flow Pe > Pec No SSC : pure large-scale flow SSC Strong time dependence of the flow Homogeneous stirring efficiency No SSC Quasi-steady flow Heterogeneous stirring efficiency

91 Influence of rheology Dimensionless half spreading rate, Pe RMS FTLE, λ + RMS Constant rheology Stress dependent rheology Temperature dependent rheology H= Half spreading rate (cm/yr)

92 RMS FTLE, λ + RMS Influence of rheology Dimensionless half spreading rate, Pe Constant rheology Stress dependent rheology Temperature dependent rheology H= Half spreading rate (cm/yr) V-shaped curves hold for various rheologies

93 Summary 1500 SSC Homogeneous stirring efficiency No SSC: large scale flow only Heterogeneous stirring efficiency λ + RMS Pe

94 Summary 1500 SSC Homogeneous stirring efficiency No SSC: large scale flow only Heterogeneous stirring efficiency λ + RMS H=1 H=2 H= Pe

95 Summary 1500 SSC Homogeneous stirring efficiency No SSC: large scale flow only Heterogeneous stirring efficiency λ + RMS H=1 H=2 H= Pe V-shaped curves hold for various rheologies & domain aspect ratios

96 Summary 1500 SSC Homogeneous stirring efficiency No SSC: large scale flow only Heterogeneous stirring efficiency λ + RMS H=1 H=2 H= Pe V-shaped curves hold for various rheologies & domain aspect ratios Pe c = 500 exp[ln(2) H]

97 Analytical mixing model 1. Flow = quasi-linear superposition of Large Scale (LS) and SSC motions: = LS (1 )+ SSC 2. Lagrangian strain rate velocity: ape LS SSC bra2/3 e (Pe,H) = exp ( 3.8 P e/p e c ) φ H= Stirring is chaotic about the ridge axis: d dt = = 1 ln Mixing time: 0 f Lyapunov exponent (a) Pe (b) H=2 Analytical model Numerical experiments Pe

98 80 N Predicted vs. observed isotopic variability (a) World s seafloor age 40 N MAR Age (Myr) CIR 120 EPR SAR S PAR CR SWIR SEIR S 160 W 120 W 80 W 40 W 0 40 E 80 E 120 E 160 E 0 [Samuel & King, 2014]

99 Predicted vs. observed isotopic variability 10 1 σ[( 3 He/ 4 He)/R atm ] 10 0 SWIR MAR SAR CIR CR PAR SEIR DATA EPR Half spreading rate (cm/yr) [Samuel & King, 2014]

100 Predicted vs. observed isotopic variability 10 1 σ[( 3 He/ 4 He)/R atm ] 10 0 SWIR MAR SAR CIR CR PAR SEIR DATA EPR Half spreading rate (cm/yr) Mixing time in Myr Pe SWIR MAR SAR CIR PAR,CR Mixing = Large-scale flow only SEIR EPR Half spreading rate (cm/yr) [Samuel & King, 2014]

101 Predicted vs. observed isotopic variability 10 1 σ[( 3 He/ 4 He)/R atm ] 10 0 SWIR MAR SAR CIR CR PAR SEIR DATA EPR Half spreading rate (cm/yr) Mixing time in Myr Pe SWIR SWIR MAR MAR SAR SAR CIR PAR,CR CIR PAR,CR Mixing=Large-scale flow only SEIR SEIR Mixing = large-scale flow + SSC EPR EPR Half spreading rate (cm/yr) [Samuel & King, 2014]

102 Predicted vs. observed isotopic variability 10 1 σ[( 3 He/ 4 He)/R atm ] Mixing time in Myr SWIR Pe SWIR SWIR MAR MAR MAR SAR SAR SAR CIR CR PAR CIR PAR,CR CIR PAR,CR SEIR DATA Half spreading rate (cm/yr) Mixing=Large-scale flow only SEIR SEIR Mixing = large-scale flow + SSC Half spreading rate (cm/yr) EPR EPR EPR!!! Standard assumption leads to monotonous mixing time-spreading rate relationship Accounting for SSC matches data better: efficient stirring for both slow and fast spreading rates [Samuel & King, 2014]

103 σ[( 3 He/ 4 He)/R atm ] Mixing time in Myr Predicted vs. observed isotopic variability Missing link: mixing time Pe to chemical variability! SWIR SWIR SWIR MAR MAR MAR SAR SAR SAR CIR CR PAR Half spreading rate (cm/yr) CIR PAR,CR CIR PAR,CR SEIR DATA Mixing=Large-scale flow only SEIR SEIR Mixing = large-scale flow + SSC EPR EPR EPR!!! Standard assumption leads to monotonous mixing time-spreading rate relationship Accounting for SSC matches data better: efficient stirring for both slow and fast spreading rates Half spreading rate (cm/yr) [Samuel & King, 2014]

104 Linking mixing time to chemical variability 4. Normal distribution for sampling age of mante material at ridge: " # sampling 1 (t = p sampling ) 2 exp 2 sampling Standard deviation, σ sampling with σsampling and τsampling =functions of (Φ,H,Pe,Raeff) 5. Chaotic mixing var =σ 2 = exp (-t/τ) 6. Mantle variability trend not affected by melting processes Weighted variance over [0-te] var = Z te t=0 sampling exp [ t / ] dt Z te t=0 sampling dt H=1 e Elapsed time t e =100 Myrs Elapsed time t e =200 Myrs Elapsed time t e =500 Myrs Pe

105 Predicted vs. observed isotopic variability 10 1 σ[( 3 He/ 4 He)/R atm ] 10 0 SWIR MAR SAR CIR CR PAR SEIR DATA EPR Half spreading rate (cm/yr) [Samuel & King, 2014]

106 Predicted vs. observed isotopic variability SAR 10 1 (b) CR SEIR σ[( 3 He/ 4 He)/R atm ] 10 0 MAR SWIR MAR SWIR SAR CIR CIR PAR CR SEIR PAR Data: ( 3 He/ 4 He)/R atm DATA EPR EPR 10 1 Predicted standard deviation, σ Dimensionless half spreading rate, Pe Half spreading rate (cm/yr) (d) SWIR SWIR MAR SAR CIR PAR,CR MAR After t e =200 Myr of evolution SAR CIR PAR,CR SEIR SEIR Large Scale flow accounted for only Small Scale Convection + Large Scale flow Half spreading rate (cm/yr) EPR EPR [Samuel & King, 2014]

107 Predicted vs. observed isotopic variability SAR 10 1 (b) CR SEIR σ[( 3 He/ 4 He)/R atm ] Predicted standard deviation, σ SWIR SWIR MAR MAR SAR CIR CR PAR SEIR DATA CIR PAR Data: ( 3 He/ 4 He)/R atm Half spreading rate (cm/yr) (d) Dimensionless half spreading rate, Pe SWIR SWIR MAR SAR CIR PAR,CR MAR After t e =200 Myr of evolution SAR CIR PAR,CR SEIR SEIR Large Scale flow accounted for only Small Scale Convection + Large Scale flow Half spreading rate (cm/yr) EPR EPR EPR EPR!!! Standard assumption incompatible with dynamics Accounting for SSC matches data better [Samuel & King, 2014]

108 Sampling reservoirs: Further Implications

109 Further implications: MORB vs. OIB variability Half-spreading rate ~ 0.1 cm/yr Moderate ridge spreading rate (SSC) Homogeneous stirring efficiency MORB variability = OIB variability Half-spreading rate ~ 8 cm/yr Fast ridge spreading rate (No SSC) Heterogeneous stirring efficiency MORB variability < OIB variability Weak stirring Strong stirring

110 Further implications: MORB vs. OIB variability Slow-spreading mid-ocean ridge Oceanic plate Mantle Homogeneous lavas erupted at ridge Less-vigorous plate-scale mantle convection Ridge axis Seamounts and ocean islands Vigorous small-scale mantle convection Moderate ridge spreading rate (SSC) Homogeneous stirring efficiency MORB variability = OIB variability Weak Mixing Strong Fast-spreading mid-ocean ridge Oceanic plate Homogeneous lavas erupted at ridge Mantle Ridge axis Heterogeneous lavas erupted at seamounts and ocean islands Seamounts and ocean islands Fast ridge spreading rate (No SSC) Heterogeneous stirring efficiency MORB variability < OIB variability Domains of poorly mixed mantle Vigorous plate-scale mantle convection Less-vigorous small-scale mantle convection [Graham, 2014]

111 Plume Sampling Mechanism Fast-spreading mid-ocean ridge Homogeneous lavas erupted at ridge Ridge axis Heterogeneous lavas erupted at seamounts and ocean islands Seamounts and ocean islands Oceanic plate Mantle Domains of poorly mixed mantle Vigorous plate-scale mantle convection Less-vigorous small-scale mantle convection [Graham, 2014]

112 Plume Sampling Mechanism Fast-spreading mid-ocean ridge Homogeneous lavas erupted at ridge Ridge axis Heterogeneous lavas erupted at seamounts and ocean islands Seamounts and ocean islands Oceanic plate Mantle Domains of poorly mixed mantle Vigorous plate-scale mantle convection Less-vigorous small-scale mantle convection [Graham, 2014]

113 Plume Sampling Mechanism Fast-spreading mid-ocean ridge Homogeneous lavas erupted at ridge Ridge axis Heterogeneous lavas erupted at seamounts and ocean islands Seamounts and ocean islands Oceanic plate Mantle Domains of poorly mixed mantle Vigorous plate-scale mantle convection Job done? Less-vigorous small-scale mantle convection [Graham, 2014]

114 Plume Sampling Mechanism Fast-spreading mid-ocean ridge Homogeneous lavas erupted at ridge Ridge axis Heterogeneous lavas erupted at seamounts and ocean islands Seamounts and ocean islands Oceanic plate Mantle Domains of poorly mixed mantle Vigorous plate-scale mantle convection Job done? Less-vigorous small-scale mantle convection [Graham, 2014]

115 Plume Sampling Mechanism C.G. Farnetani, A.W. Hofmann / Earth and Fast-spreading mid-ocean ridge Homogeneous lavas erupted at ridge Ridge axis Heterogeneous lavas erupted at seamounts and ocean islands Seamounts and ocean islands Oceanic plate Mantle Domains of poorly mixed mantle Vigorous plate-scale mantle convection Job done? Less-vigorous small-scale mantle convection [Graham, 2014] [Farnetani & Hofmann, 2009]

116 Plume Sampling Mechanism C.G. Farnetani, A.W. Hofmann / Earth and Fast-spreading mid-ocean ridge Homogeneous lavas erupted at ridge Ridge axis Heterogeneous lavas erupted at seamounts and ocean islands Seamounts and ocean islands Oceanic plate Mantle Domains of poorly mixed mantle Vigorous plate-scale mantle convection Job done? Less-vigorous small-scale mantle convection [Graham, 2014] [Farnetani & Hofmann, 2009] Unlikely: plume sample material located in thermal boundary layers

117 Possible Plume Sampling Mechanisms Fast-spreading mid-ocean ridge Homogeneous lavas erupted at ridge Ridge axis Heterogeneous lavas erupted at seamounts and ocean islands Seamounts and ocean islands Oceanic plate Mantle Domains of poorly mixed mantle Slow-spreading mid-ocean ridge Vigorous plate-scale mantle convection Homogeneous lavas erupted at ridge [Graham, 2014] Ridge axis Phase change Seamounts and ocean islands Less-vigorous small-scale mantle convection Oceanic plate Mantle Less-vigorous plate-scale mantle convection Vigorous small-scale mantle convection Mixing Weak Strong [Ballmer et al., 2013]

118 Possible Plume Sampling Mechanisms II: compositional heterogeneity Tank experiments [Kumagai et al., 2007] Increasing B Numerical experiments [Samuel & Bercovici, 2006]

119 Possible Plume Sampling Mechanisms II: compositional heterogeneity Tank experiments [Kumagai et al., 2007] Increasing B Numerical experiments [Samuel & Bercovici, 2006]

120 Possible Plume Sampling Mechanisms II: compositional heterogeneity Tank experiments [Kumagai et al., 2007] Increasing B Numerical experiments [Samuel & Bercovici, 2006]

121 Possible Plume Sampling Mechanisms II: compositional heterogeneity Tank experiments [Kumagai et al., 2007] Increasing B Numerical experiments [Samuel & Bercovici, 2006]

122 Possible Plume Sampling Mechanisms II: compositional heterogeneity Tank experiments [Kumagai et al., 2007] Increasing B Numerical experiments [Samuel & Bercovici, 2006]

123 Possible Plume Sampling Mechanisms II: compositional heterogeneity Tank experiments [Kumagai et al., 2007] Sampling can occur through oscillatory diapirism (moderate B) or via Increasing the separation B of thermal and compositional components (larger B) Numerical experiments [Samuel & Bercovici, 2006]

124 A dynamically inconsistent scenario: the Plum pudding mantle model Fig. 1. The blob model of convection (cartoon). Note the following features: Convection is in the whole mantle mode with varying morphology of slab penetration through the 670-km-transition zone. The blobs reside mainly in the cores of the convective cells and represent the primitive reservoir. Surrounding material and especially the upper mantle region are depleted and degassed by melting at the ridges and earlier continent-formation. Blobs are sampled by rising plumes that entrain material and lead to a heterogeneous OIB isotope source. [Becker et al.,1999]

125 A dynamically inconsistent scenario: the Plum pudding mantle model Fig. 1. The blob model of convection (cartoon). Note the following features: Convection is in the whole mantle mode with varying morphology of slab penetration through the 670-km-transition zone. The blobs reside mainly in the cores of the convective cells and represent the primitive reservoir. Surrounding material and especially the upper mantle region are depleted and degassed by melting at the ridges and earlier continent-formation. Blobs are sampled by rising plumes that entrain material and lead to a heterogeneous OIB isotope source. [Becker et al.,1999]

126 A dynamically inconsistent scenario: the Plum pudding mantle model Fig. 1. The blob model of convection (cartoon). Note the following features: Convection is in the whole mantle mode with varying morphology of slab penetration through the 670-km-transition zone. The blobs reside mainly in the cores of the convective cells and represent the primitive reservoir. Surrounding material and especially the upper mantle region are depleted and degassed by melting at the ridges and earlier continent-formation. Blobs are sampled by rising plumes that entrain material and lead to a heterogeneous OIB isotope source. [Becker et al.,1999] [Manga, 2010]

127 A dynamically inconsistent scenario: the Plum pudding mantle model Fig. 1. The blob model of convection (cartoon). Note the following features: Convection is in the whole mantle mode with varying morphology of slab penetration through the 670-km-transition zone. The blobs reside mainly in the cores of the convective cells and represent the primitive reservoir. Surrounding material and especially the upper mantle region are depleted and degassed by melting at the ridges and earlier continent-formation. Blobs are sampled by rising plumes that entrain material and lead to a heterogeneous OIB isotope source. Cartoon:! Plumes sample primitive blobs! Ridges sample depleted mantle only [Becker et al.,1999] [Manga, 2010]

128 A dynamically inconsistent scenario: the Plum pudding mantle model Fig. 1. The blob model of convection (cartoon). Note the following features: Convection is in the whole mantle mode with varying morphology of slab penetration through the 670-km-transition zone. The blobs reside mainly in the cores of the convective cells and represent the primitive reservoir. Surrounding material and especially the upper mantle region are depleted and degassed by melting at the ridges and earlier continent-formation. Blobs are sampled by rising plumes that entrain material and lead to a heterogeneous OIB isotope source. Cartoon:! Plumes sample primitive blobs! Ridges sample depleted mantle only Dynamics:! Plumes do not sample primitive blobs! Ridges sample depleted mantle and primitive blobs [Becker et al.,1999] [Manga, 2010]

129 Conclusions II Num. Experiments: spreading rate has a considerable & subtle influence Two regimes depending on spreading rate & domain aspect ratio: a. Moderate spreading rate : average stirring efficiency decrease with spreading rate but homogeneous stirring efficiency identical MORB and OIB variabilities b. Fast spreading rate : average stirring efficiency increases with spreading rate but heterogeneous stirring efficiency develops significant differences between MORB and OIB without the need of additional geochemical reservoir.! Stirring efficiency in MORB source region may seem completely uncorrelated with ridge spreading rate!! The interpretation of geochemical data must consider these relationships

130 General conclusions! The creation, the survival and the sampling of mantle geochemical reservoirs is directly linked multi-scale mantle convective dynamics!! Dynamical considerations provide strong constraints for the interpretation of geochemical data!! Much remains to be done to understand the influence of the complexities in Earth s mantle dynamics on convective mixing at different scales (e.g., plate tectonics vs. magma chambers)!! Attempts to interpret geochemical data must be tested dynamically