On the Generation of Core Dynamo Action

Transcription

1 On the Generation of Core Dynamo Action CIDER Meeting, KITP 7/7/16 Jonathan Aurnou UCLA Earth & Space Sciences

2 Beyond record players? Treat core dynamics as a rotating magnetoconvection problem Theoretical insight Reduction in problem complexity (?) Cheng et al. GJI 15

3 Movie: J. Cheng

4 Movie: Daphné Lemasquerier

5 Movie: Daphné Lemasquerier

6 Image: Glatzmaier & Olson, SciAm 05 The Geodynamo Generates a magnetic fields and then continually regenerate that field Magnetohydrodynamic (MHD) process Converts kinetic energy of flowing conductor into magnetic energy

7 Successful MHD Modeling + u ru +2 u = 1 o rp + T g + r 2 u + 1 o J B = r (u B)+ r2 B + u rt = appler2 T + S Current Density Continuity Ampere s Law: J = 1 µ o r B r u =0 r B =0

8 Successful MHD Modeling + u ru +2 u = 1 o rp + T g + r 2 u + 1 o J B = r (u B)+ r2 B + u rt = appler2 T + S Current Density Continuity Ampere s Law: J = 1 µ o r B r u =0 r B =0

9 Lorentz Force on a Particle i m i d~v i dt = q i~v i ~ B Moving charged particle pushed off to its right R gr = m iv i q i B

10 Lorentz Force in a Continuum m i d~v i dt = q i~v i ~ B Using the current density J, where n is the charge density ~J = nq < ~v i >= I A Gives: d~u dt V = nq~v d V ~ B

11 Lorentz Force in a Continuum m i d~v i dt = q i~v i ~ B Using the current density J, where n is the charge density ~J = nq < ~v i >= I A Gives: d~u dt = ~ J ~ B NB: u is now the material velocity!

12 Lorentz Force in a Continuum An example maybe? MHD Pump Opposite of a dynamo: J and B drive u 0= ~ J ~ B + µr 2 ~u

13 MHD Pump Top View Leads Cu walls J J x B Cross- Section Austin Chadwick, Lorraine Esturas, Alex Kerelsky 0.05 M CuSO 4 solution

14 MHD Pump r = 0 not to scale H

15 MHD Pump r = 0 not to scale J = I A = I 2 rh / 1 r H H r

16 MHD Pump r = 0 not to scale J = I A = I 2 rh / 1 r H Thin layer: H << rinner u (z = 0) = 0 0= ~ J ~ B + µr 2 ~u IBH 1 0= IB 2 rh + µd2 u dz 2 u (H) = 4 µ r

17 MHD Pump u (H) = IBH 4 µ 1 r Austin Chadwick, Lorraine Esturas, Alex Kerelsky r i ' r o /2! u i ' 2 u o C i ' C o /2! t i ' t o /4

18 MHD Pump Opposite of a dynamo J x B drives flow, u, in an electrically conducting fluid Now to dynamos where u generates (J, B)

19 Dynamo Essentials + u ru +2 u = 1 o rp + T g + r 2 u + 1 o J B = r (u B)+ r2 B + u rt = appler2 T + S Current Density Continuity Ampere s Law: J = 1 µ o r B r u =0 r B =0

20 The Induction ~ = r2 ~ B + r ~u ~ B Compare to the (scalar) heat equation dt dt = appler2 T +

21 The Induction ~ = r2 ~ B! L2 Compare to the (scalar) heat equation dt dt = appler2 T! L2 apple

22 The Induction ~ = r2 ~ B! L2 Magnetic diffusivity ~ 1 m 2 /s in core fluids Magnetic diffusion time ~ 15 kyr or so Geologically rapid decay

23 The Induction ~ = r2 ~ B + r ~u ~ B Compare to the (scalar) heat equation dt dt = appler2 T +

24 The Induction ~ = r2 ~ B + r ~u ~ B For a self-sustaining dynamo, the inductive source term must be able to exceed the diffusion term

25 The Induction ~ B Lab vs. = r2 ~ B + r ~u ~ B Dynamos? For a self-sustaining dynamo, the inductive source term must be able to exceed the diffusion term Magnetic Reynolds number: Rm = UL & 40 for dynamo action

26 The Induction ~ = r2 ~ B + r ~u ~ B For a self-sustaining dynamo, the inductive source term must be able to exceed the diffusion term Magnetic Reynolds number: Rm = UL NB: magnetic field lines are carried along with the fluid at large Rm

27 The Omega-Effect Shearing out of magnetic field orthogonal to its direction (axisymmetric) Poloidal Magnetic Field Ω Inductive stretching term: Zonal Velocity (u ˆ φ ) (B oˆ s ) = B oˆ s u ˆ φ Zonal Velocity = B o u s φ ˆ

28 The Omega-Effect Shearing out of magnetic field orthogonal to its direction (axisymmetric) Inductive stretching term: (u ˆ φ ) (B oˆ s ) = B oˆ s u ˆ φ Poloidal Magnetic Field Ω = B o u s φ ˆ Toroidal Magnetic Field

29 The Omega-Effect Poloidal Toroidal Omega generates axisymmetric toriodal field from poloidal field But nothing regenerates initial poloidal field citation?

30 The Alpha-Effect Helical twisting of magnetic field lines (Non-axisymmetric!!!) Can generate J parallel to toroidal B Ensemble parallel J generates toloidal B Now Omega-effect can re-generate toroidal field DYNAMO! Poloidal Magnetic Field Toroidal Magnetic Field J Ω

31 The Alpha-Effect Helical twisting of magnetic field lines (Non-axisymmetric!!!) Can generate J parallel to toroidal B Ensemble parallel J generates toloidal B Now Omega-effect can re-generate toroidal field DYNAMO! Roberts 2008

32 Alpha-Omega Dynamo Poloidal Self-sustaining process! Toroidal Poloidal citation?

33 Observations Models Jackson, Nature 2003 Br CMB Soderlund et al. EPSL 2012 Br CMB Tangent Cylinder Lmax ~ 13

34 Observations Models Jackson, Nature 2003 Br CMB Soderlund et al. EPSL 2012 z-vorticity Lmax ~ 13

35 Observations Models Jackson, Nature 2003 Br CMB Soderlund Christensen, et Enc. al. EPSL Solid 2012 Earth Geophys z-vorticity Ω g Lmax ~ 13

36 Models Christensen, Enc. Solid Earth Geophys Ω z-vorticity g Aubert et al. GJI 2008

37 Observations Models Jackson, Nature 2003 Br CMB Christensen, Enc. Solid Earth Geophys z-vorticity Ω g Lmax ~ 13 Present paradigm: helical columns extrapolate to planetary cores; continually regenerate field

38 Summary Dynamos require: Energy source Organized flow Conducting fluid Rm = UL Large length scales Two stage regeneration process Non-axisymmetric flow in at least one stage Paradigm: Large-scale helical columns

39 Sheyko s Turbulent Dynamo Vr Slice (E = 3e-7, Pm=0.05); [Rm = 3e3, Re ~ 5e4, Ro ~ 0.02] Andrey Sheyko (2014)

40 Sheyko s Turbulent Dynamo Br Slice (E = 3e-7, Pm=0.05); [Rm = 3e3, Re ~ 5e4, Ro ~ 0.02] Andrey Sheyko (2014)

41 Sheyko s Turbulent Dynamo Br CMB (E = 3e-7, Pm=0.05); [Rm = 3e3, Re ~ 5e4, Ro ~ 0.02] Andrey Sheyko (2014)

42 Sheyko s Turbulent Dynamo Lmax = 12 Br CMB Jackson, Nature 2003 Br CMB (E = 3e-7, Pm=0.05); [Rm = 3e3, Re ~ 5e4, Ro ~ 0.02] Andrey Sheyko (2014) Lmax ~ 13

43 Sheyko s Turbulent Dynamo Extreme model: magnetic flux patches no longer directly related to columns (scale separation)

44 Length Scale Questions ' Ro 1/2 coreh Linear HD onset scale: In a turbulent HD core, ` E 1/3 H ` Ro 1/2 H

45 Length Scale Questions ' Ro 1/2 coreh Linear HD onset scale: In a turbulent HD core, ` E 1/3 H ` Ro 1/2 H

46 MHD Experiments Magnet Rotating magnetoconvection in liquid metal Strong magnetic field can balance Coriolis Different behaviors predicted

47 Image: Adolfo Ribeiro Top View: Axial Velocity

48 Lab MHD Experiments In liquid metals, fast oscillatory columns and slow wall modes Wall modes related to low latitude waves in observations Missing in present dynamo models

49 Summary II 2000 s model: large, helical columns associated with B-flux patches State-of-the-art: helical flow, but flux patches no longer directly related to columns Many open questions e.g., Dynamo action driven by rotating turbulence in liquid metals (Featherstone CIG/ALCF) Paradigm shift? Or waiting for Godot?

50 Thanks Thomas Gastine (IPGP)