Dynamics in the Earth s core Philippe Cardin, ISTerre, Université Grenoble Alpes et CNRS Doctoral training on internal Earth, Barcelonnette, oct 2016
Sources of motions inside the core Core cooling and thermal convection in the outer core. Inner core growth and chemical convection in the outer core. Precession and nutations of the Earth mantle. 1-2
Influence of the Coriolis force When an object is moving in a rotating frame, two inertial forces apply to explain its trajectory. The Coriolis force applies only in particle in motion and tends to deviate the particle towrs the right in the northen hemisphere. 1-3
Carousel Exercice W A B In the absolute frame In the rotating frame A throw a ball to B The trajectory is a straight line The ball deviate to the right and changes its velocity What do see the child in the carousel? 1-4
Trajectory Core dynamics, P. 1-5
Rotation and viscosity The viscous forces/ Coriolis forces is the definition of the dimensionless Ekman number. For the Core, As in the oceans and atmospheres, the rotation is very important for the dynamics. 1-6
Navier and Stokes equation In the rotating frame, it writes: Using R as length scale, W -1 as time scale, we have: 1-7
Geostrophic balance If the flow is quasi stationnary (T> W -1 ), of small amplitude (u<< WR) and E <<1. Flow is parallel to isobare lines Proudman-Taylor theorem The flow is invariant along the axis of rotation. Magntic dipole // rotation axis? 1-8
Taylor column The column form in a few periods of rotation Inertial waves of propagation 1-9
Inertial waves (inviscid) Inertial wave equation 1-10
Inertial wave dispersion relation Plane wave Transverse wave 1-11
Time to built a geostrophic column h k d 1-12
Excitation and normal modes 13 Kelley et al Core
The Ekman layer If there is a wall, the boundary condition on the velocity will generate a boundary layer where the viscous term comes back into the balance. It is the Ekman layer. Geostrophic flow Ekman layer wall 1-14
Horizontal flow in the layer Let s suppose a geostrophic flow u g along x. We get to solve: And we found: The Ekman spiral 1-15
Ekman spiral 1-16
The Ekman pumping The incompressibility gives: Hence When z>>d, More generally (Greenspan 1968) MOVIE 1-17
Precession of the Earth Core dynamics, P. 1-18
Hypothesis: The fluid in the precessing cavity rotates as a solid. (Poincaré, 1910). Question: Where is the axis of rotation of the fluid? W p = [-10,0] rpm w c = [50,300] rpm a = [0,45] a = 0.125 m a = 0.120 m Dimensionless numbers: E= n/w c a 2 = 10-5-10-6 W = W p /w c Experimental set-up h = 1/25 19
Axis of the core <> Axis of the mantle w c w r f W r p 1-20
Poistion of the axis w r c w r f W r p Résonnance non linéaire 21
Detection of a jump of the axis of rotation 22
Theory Torque Balance: Precession torque, G i Pressure torque, G p Viscous torque, G v (A,B,C) = w f C A B 23
One finds the spin-up condition : G p.w f = G i.w f = 0. which means no differential rotation along w f : k c.w f = w f 2 The spin over torque: We get the non linear system to solve in A,B,C: 24
Comparison between experiment and theory 25
Geostrophic cylinders? 1-26
Open questions Resonances in Earth history (precession ans nutations) Did Venus go through its precession resonance? Moon dynamo? W p = -4.94 rpm Open questions: Turbulence? W p = -3 rpm W p = -7 rpm Elliptical instabilities? Amplitude of the geostrophic velocity? w c = 300 Core rpm, dynamics, a = 25 P. Cardin 27
Thermal convection in rotating system Dynamics equations + heat equation z k, thermal diffusivity a, thermal expansion g, gravity 2L r 0 y T 0 T 1 d x 1-28
Dimensionless equations Scales: d, d 2 /k, DT 1-29
Basic state Cartesian coordonate for simplification Linear approach 1-30
Geostrophic balance Asymptotic approach: Zero order Streamfunction : 1-31
First order : quasi-geostrophic equation z-component of the vorticity equation z - integration 1-32
Vertical component of the flow and QG model Ekman pumping : Slope effect : 1-33
Marginal stability analysis If b=0, b=0, stationary instability If very weak rotation Rayleigh Bénard convection 1-34
Rotating flat cylinder if We get the critical parameters : 1-35
Rotating cylinder with depth varying height The imaginary part leads to: The instability is oscillatory with a pulsation linear in b. It is a thermal Rossby wave (Busse, 1970). 1-36
Rossby waves Let s consider a inviscid fluid with no archimedean forces in a sloping annulus Quasi 2D, stream function z x Beta <0 (sphere), prograde Small scales, slow 1-37
Thermal convection in a rotating sphere Busse Dormy Plaut Aubert Busse 70 1-38
3D numerical simulation : onset E = 10-4 E = 10-5 E = 10-6 Dormy, 1998 1-39
Ekman pumping At the onset 2 times critical Dormy 1998 1-40
Numerical quasi geostrophic model Z-invariance of the motion Z-integration of the equations Solving in the equatorial plane Presence of the non linear terms 1-41
Comparison 2D/3D Aubert et al, 2003 1-42
Benchmark (GJI 2014) 1-43
Internal heating in full sphere Guervilly, 2010 1-44
Very small Ekman number 1-45
Calcul 3D : Earth simulator 1-46
Experience of convection in a rotating sphere Gravity is the centrifugal acceleration Temperature gradient is reversed Flakes and fluoreceine E = 10-6 Pr =7 1-47
Columnar flow Side view with a sheet of light 1-48
Turbulent in the equator Top view 1-49
Experimental set-up 10-4 > E > 10-6 1-50
Doppler velocimetry Plexiglass sphere (water +pollen) Cupper sphere (Gallium+ Zirconium borure) Doppler measurement f=4mz 1-51
Time/space diagram of the velocity water gallium Aubert, 2001 5-52
Scaling law for convective velocity 1-53
Scaling law for the convective velocity 1-54
Full sphere calculation 1-55
Zonal flows Movie 1-56
Zonal flows 1-57
New experiment ZoRo (Sylvie Su PhD) 1-58
Surveys of parameter 1-59
Sub criticality of thermal convection 1-60
3D calculations 1-61
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Surveys of parameter 1-63
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Weak branche 1-65
Strong branche 1-66
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Strong branches 1-68
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Surveys of parameter 1-73
Time evolution of u for different Ra Quasi périodicity? growth rate? 1-74
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Movie linear 1-76
Movie linéaire normé 1-77
Movie lineaire/log 1-78
Solitary wave? (Soward 77) 1-79
Cycles in phase space Exponential growth 1-80
Kuo criteria Zonal flow instability : Rossby waves. Kuo 1949 Schaeffer & Cardin, 2005 Guervilly & Cardin, 2010 Core dynamics, P. 1-81
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summary 1-84