Part 1 : solar dynamo models [Paul] Part 2 : Fluctuations and intermittency [Dario] Part 3 : From dynamo to interplanetary magnetic field [Paul]

Transcription

1 Dynamo tutorial Part 1 : solar dynamo models [Paul] Part 2 : Fluctuations and intermittency [Dario] Part 3 : From dynamo to interplanetary magnetic field [Paul] ISSI Dynamo tutorial 1 1

2 Dynamo tutorial Part 1 : solar dynamo models The solar cycle magnetic field Global 3D MHD simulations 2D axisymmetric solar dynamo models Mean-field electrodynamics and models Babcock-Leighton models Models based on HD/MHD instabilities ISSI Dynamo tutorial 1 2

3 The solar magnetic cycle ISSI Dynamo tutorial 1 3

4 Maxwell s equations ISSI Dynamo tutorial 1 4

5 From Maxwell to MHD (1) Step 1: Drop displacement current to revert to the original form of Ampère s Law: Step 2: Write down Ohm s Law in a frame co-moving with the fluid: Step 3: Non-relativistic transformation back to the laboratory frame of reference: ISSI Dynamo tutorial 1 5

6 From Maxwell to MHD (2) Step 4: Combine with Ampère s Law to express the electric field as: Step 5: Substitute into Faraday s Law to get the justly famous magnetohydrodynamical induction equation: where we defined the magnetic diffusivity as: ISSI Dynamo tutorial 1 6

7 The MHD equations ISSI Dynamo tutorial 1 7

8 Global 3D MHD simulations ISSI Dynamo tutorial 1 8

9 Simulation framework Simulate anelastic convection in thick rotating fluid shell, convectively unstable in upper two thirds, using EULAG-MHD in «ideal» mode. This type of ILES simulation is often refered to as «minimally diffusive»; This allows to reach a maximally turbulent state on a given mesh; A flow or magnetic structure can develop gradients reaching the mesh scale and remain nonlinearly stable; Simulations can be run in strongly turbulent regimes over very long times. ISSI Dynamo tutorial 1 9

10 Kinetic and magnetic energies (120 s.d.=10 yr) ISSI Dynamo tutorial 1 10

11 Magnetic cycles (1) QuickTime et un décompresseur sont requis pour visionner cette image. Large-scale organisation of the magnetic field takes place primarily at and immediately below the base of the convecting fluid layers ISSI Dynamo tutorial 1 11

12 Zonally-averaged Magnetic B phi at r/r cycles =0.718 (1) Zonally-averaged B phi at -58 o latitude ISSI Dynamo tutorial 1 12

13 Successes and problems KiloGauss-strength large-scale magnetic fields, antisymmetric about equator and undergoing regular polarity reversals on decadal timescales. Cycle period four times too long, and strong fields concentrated at mid-latitudes, rather than low latitudes. Internal magnetic field dominated by toroidal component and strongly concentrated immediately beneath core-envelope interface. Well-defined dipole moment, well-aligned with rotation axis, but oscillating in phase with internal toroidal component. Reasonably solar-like internal differential rotation, and solar-like cyclic torsional oscillations (correct amplitude and phasing). On long timescales, tendency for hemispheric decoupling, and/or transitions to non-axisymmetric oscillatory modes. ISSI Dynamo tutorial 1 13

14 REALITY CHECK The numerical treatment of unresolved scales influences a lot the global dynamo behavior! ISSI Dynamo tutorial 1 14

15 Axisymmetric+kinematic formulation of solar dynamo models ISSI Dynamo tutorial 1 15

16 Model setup Solve MHD induction equation in spherical polar coordinates for large-scale (~R), axisymmetric magnetic field in a sphere of electrically conducting fluid: Evolving under the influence of a steady, axisymmetric large-scale flow: Match solutions to potential field in r > R. ISSI Dynamo tutorial 1 16

17 Kinematic axisymmetric dynamo ISSI Dynamo tutorial 1 17

18 Differential rotation Slow Fast ISSI Dynamo tutorial 1 18

19 Shearing by axisymmetric differential rotation ISSI Dynamo tutorial 1 19

20 Kinematic axisymmetric dynamo ISSI Dynamo tutorial 1 20

21 Meridional circulation ISSI Dynamo tutorial 1 21

22 REALITY CHECK Many independent global MHD simulations suggest that magnetic backreaction on large-scale flows is an important (perhaps the dominant) amplitude-limiting mechanism ISSI Dynamo tutorial 1 22

23 Kinematic axisymmetric dynamo ISSI Dynamo tutorial 1 23

24 Kinematic axisymmetric dynamos ISSI Dynamo tutorial 1 24

25 Poloidal source terms 1. Turbulent alpha-effect 2. Active region decay (Babcock-Leighton mechanism) 3. Helical hydrodynamical instabilities 4. Magnetohydrodynamical instabilities (flux tubes, Spruit-Tayler) ISSI Dynamo tutorial 1 25

26 REALITY CHECK We are forcing fundamentally nonaxisymmetric processes into an axisymmetric model formulation ISSI Dynamo tutorial 1 26

27 Mean-field electrodynamics and dynamo models ISSI Dynamo tutorial 1 27

28 The basic idea [ Parker, E.N., ApJ, 122, 293 (1955) ] Cyclonic convective updraft/downdrafts acting on a pre-existing toroidal magnetic field will twist the fieldlines into poloidal planes (in the high Rm regime) The collective effect of many such events is the production of an electrical current flowing parallel to the background toroidal magnetic field; such a current system contributes to the production of a poloidal magnetic component ISSI Dynamo tutorial 1 28

29 Scale separation Separate flow and magnetic field into large-scale, «laminar» component, and a small-scale, «turbulent» component: Assume now that a good separation of scales exists between these two components, so that it becomes possible to define an averaging operator: such that: This is NOT a linearization! No assumption is made at this stage with regards to the relative magnitudes of flow and field components ISSI Dynamo tutorial 1 29

30 The turbulent EMF (1) Substitute into MHD induction equation and apply averaging operator: with : TURBULENT ELECTROMOTIVE FORCE! Now subtract averaged induction equation from original induction equation to obtain evolution equation for b : with : Still no approximation!! ISSI Dynamo tutorial 1 30

31 The turbulent EMF (2) Now, the whole point of the mean-field approach is NOT to have to deal explicitly with the small scales; since the PDE for b is linear, with the term acting as a source; therefore there must exit a linear relationship between b and B, and also between B and ; We develop the mean emf as Where the various tensorial coefficients can be a function of, of the statistical properties of u, on the magnetic diffusivity, but NOT of. Specifying these closure relationships is the crux of the mean-field approach ISSI Dynamo tutorial 1 31

32 The alpha-effect (1) Consider the first term in our EMF development: If u is an isotropic random field, there can be no preferred direction in space, and the alpha-tensor must also be isotropic: This leads to: The mean turbulent EMF is parallel to the mean magnetic field! This is called the «alpha-effect» ISSI Dynamo tutorial 1 32

33 The alpha-effect (2) Computing the alpha-tensor requires a knowledge of the statistical properties of the turbulent flow, more precisely the cross-correlation between velocity components; under the assumption that b << B, if the turbulence is only mildly anisotropic and inhomogeneous, the so-called Second-Order Correlation Approximation leads to where is the correlation time for the turbulence. The alpha-effect is proportional to the fluid helicity! If the mild-anisotropy is provided by rotation, and the inhomogeneity by stratification, then we have ISSI Dynamo tutorial 1 33

34 Turbulent diffusivity Turn now to the second term in our EMF development: In cases where u is isotropic, we have, and thus: The mathematical form of this expression suggests that can be interpreted as a turbulent diffusivity of the large-scale field. for homogeneous, isotropic turbulence with correlation time, it can be shown that This result is expected to hold also in mildly anisotropic, mildly inhomogeneous turbulence. In general, ISSI Dynamo tutorial 1 34

35 REALITY CHECK All commonly-used formulations for the alpha and beta-tensors apply to physical regimes that are far removed from solar interior conditions ISSI Dynamo tutorial 1 35

36 Scalings and dynamo numbers Length scale: solar/stellar radius: Time scale: turbulent diffusion time: 0 Three dimensionless groupings have materialized: ISSI Dynamo tutorial 1 36

37 The mean-field zoo The alpha-effect is the source of both poloidal and toroidal magnetic components; works without a large-scale flow! planetary dynamos are believed to be of this kind. Differential rotation shear is the source of the toroidal component; the alpha-effect is the source of only the poloidal component. the solar dynamo is believed to be of this kind. Both the alpha-effect and differential rotation shear contribute to toroidal field production; stellar dynamos could be of this kind if differential rotation is weak, and/or if dynamo action takes place in a very thin layer. ISSI Dynamo tutorial 1 37

38 Linear alpha-omega solutions (1) Solve the axisymmetric kinematic mean-field alpha-omega dynamo equation in a differentially rotating sphere of electrically conducting fluid, embedded in vacuum; in spherical polar coordinates: Choice of alpha: ISSI Dynamo tutorial 1 38

39 Linear alpha-omega solutions (2) The growth rate, frequency, and eigenmode morphology are completely determined by the product of the two dynamo numbers ISSI Dynamo tutorial 1 39

40 Linear alpha-omega solutions (3) QuickTime et un décompresseur codec YUV420 sont requis pour visionner cette image. QuickTime et un décompresseur codec YUV420 sont requis pour visionner cette image. Positive alpha-effect Negative alpha-effect ISSI Dynamo tutorial 1 40

41 Linear alpha-omega solutions (4) Time-latitude «butterfly» diagram [ ] Equivalent in axisymmetric numerical model: constant-r cut at r/r=0.7, versus latitude (vertical) and time (horizontal) ISSI Dynamo tutorial 1 41

42 Linear alpha-omega solutions (5) QuickTime et un décompresseur codec YUV420 sont requis pour visionner cette image. QuickTime et un décompresseur codec YUV420 sont requis pour visionner cette image. ISSI Dynamo tutorial 1 42

43 Nonlinear models: alpha-quenching (1) We expect that the Lorentz force should oppose the cyclonic motions giving rise to the alpha-effect; We also expect this to become important when the magnetic energy becomes comparable to the kinetic energy of the turbulent fluid motions, i.e.: This motivates the following ad hoc expression for «alpha-quenching»: ISSI Dynamo tutorial 1 43

44 Nonlinear models: alpha-quenching (2) ISSI Dynamo tutorial 1 44

45 Nonlinear models: alpha-quenching (3) The magnetic diffusivity is the primary determinant of the cycle period ISSI Dynamo tutorial 1 45

46 Nonlinear models: alpha-quenching (4) Magnetic fields concentrated at too high latitude; Try instead a latitudinal dependency for alpha: QuickTime et un décompresseur codec YUV420 sont requis pour visionner cette image. QuickTime et un décompresseur codec YUV420 sont requis pour visionner cette image. ISSI Dynamo tutorial 1 46

47 Nonlinear models: alpha-quenching (5) ISSI Dynamo tutorial 1 47

48 REALITY CHECK Alpha-quenching is a ridiculously oversimplistic representation of the complex nonlinear interactions between a flow and a magnetic field ISSI Dynamo tutorial 1 48

49 Alpha-Omega dynamos with meridional circulation (1) QuickTime et un décompresseur codec YUV420 sont requis pour visionner cette image. Equatorward propagation of the deep toroidal field is now due to advection by the meridional flow, not «dynamo waves» effect. ISSI Dynamo tutorial 1 49

50 Models with meridional circulation (2) ISSI Dynamo tutorial 1 50

51 Models with meridional circulation (3) ISSI Dynamo tutorial 1 51

52 REALITY CHECK Both observations and numerical simulations indicate that the solar meridional flow is spatiotemporally far more complex than the steady, single-cell pattern used in the vast majority of models ISSI Dynamo tutorial 1 52

53 Models based on the Babcock-Leighton mechanism ISSI Dynamo tutorial 1 53

54 Sunspot as as emerging toroidal flux ropes Hale et al. 1919, ApJ, 49, 153 Tilt Parker 1955, ApJ, 121, 491 Latitude ISSI Dynamo tutorial 1 54

55 1952 «magnetograph» ISSI Dynamo tutorial 1 55

56 2001 magnetogram ISSI Dynamo tutorial 1 56

57 Active region decay (1) Peak polar cap flux: ~10^14 Wb Synoptic magnetogram courtesy D. Hathaway, NASA/MSFC [ ] Toroidal flux emerging in active regions in one cycle: ~10^17 Wb ISSI Dynamo tutorial 1 57

58 (2b) ISSI Dynamo tutorial 1 58

59 Active region decay (3) Zonal means ISSI Dynamo tutorial 1 59

60 REALITY CHECK Is the Babcock-Leighton effect an essential part of the dynamo loop, or a mere surface side effect of sunspot decay? ISSI Dynamo tutorial 1 60

61 Babcock-Leighton dynamo model (1) ISSI Dynamo tutorial 1 61

62 Kinematic axisymmetric dynamo ISSI Dynamo tutorial 1 62

63 Babcock-Leighton dynamo model (2) A Babcock-Leighton source term for the axisymmetric dynamo equations: Peaking at mid-latitudes Non-local in B Concentrated in surface layers The source term operates only in a finite range of toroidal field strengths. ISSI Dynamo tutorial 1 63

64 Babcock-Leighton dynamo model (3) QuickTime et un décompresseur codec YUV420 sont requis pour visionner cette image. The turnover time of the meridional flow is the primary determinant of the cycle period ISSI Dynamo tutorial 1 64

65 Babcock-Leighton dynamo model (4) ISSI Dynamo tutorial 1 65

66 REALITY CHECK Can the solar polar field really be kinematically subducted by the meridional flow, like assumed by all Babcock-Leighton models? ISSI Dynamo tutorial 1 66

67 Hinode ISSI Dynamo tutorial 1 67

68 Babcock-Leighton versus alpha-effect There are serious potential problems with the operation of the alpha-effect at high field strength; not so with the B-L mechanism The B-L mechanism operates only in a finite range of field strength; potentially problematic in the presence of large cycle amplitude fluctuations. Both models can produce tolerably solar-like toroidal field butterfly diagrams, and yield the proper phase relationship between surface poloidal and deep toroial components (with circulation included in the mean-field model) The alpha-effect (or something analogous) appears unavoidable in stratified, rotating turbulence. A decadal period arises «naturally» in B-L models; in mean-field models, it requires tuning the value of the turbulent magnetic diffusivity ISSI Dynamo tutorial 1 68

69 Models based on HD and/or MHD instabilities ISSI Dynamo tutorial 1 69

70 Buoyant rise of toroidal flux ropes [ Caligari et al. 1995, ApJ, 441, 886 ] Destabilization and buoyant rise of thin toroidal magnetic flux tubes stored immediately below the core-envelope interface (overshoot) Non-axisymmetric modes of low order (m=1 or m=2) are most easily destabilized; Conservation of angular momentum generates a flow along the axis of the rising loop; The Coriolis force acting on the flow in the legs of the loop imparts a twist that shows up as an E-W tilt upon emergence through the surface. ISSI Dynamo tutorial 1 70

71 Instability of toroidal flux tubes stored in the tachocline Toroidal flux ropes stored In the overshoot region of the tachocline are sensitive to a buoyantly-driven undulatory instability with long growth rate; these «helical waves» provide a mean EMF parallel to the tube axis, I.e., a form of alpha-effect. Diagram courtesy A. Ferriz Mas Dynamo regions Emergence region ISSI Dynamo tutorial 1 71

72 REALITY CHECK We currently do not have a «concensus dynamo model» for the basic solar cycle ISSI Dynamo tutorial 1 72

73 Coming up after recess: Dynamo tutorial Part 2 : Fluctuations and intermittency ISSI Dynamo tutorial 1 73