Nonlinear Effects in Mean Field Dynamo Theory

1. Nonlinear Effects in Mean Field Dynamo Theory David Hughes Department of Applied Mathematics University of Leeds Chicago, October 2003

2. Magnetogram X-ray emission from solar corona Temporal variation of sunspots Chicago, October 2003

3. Starting point is the magnetic induction equation of MHD: where B is the magnetic field, u is the fluid velocity and η is the magnetic diffusivity (assumed constant for simplicity). Assume scale separation between large- and small-scale field and flow: where B and U vary on some large length scale L, and u and b vary on a much smaller scale l. where averages are taken over some intermediate scale l « a « L. Kinematic Mean Field Theory Chicago, October 2003

4. For simplicity, ignore large-scale flow, for the moment. Induction equation for mean field: where mean emf is This equation is exact, but is only useful if we can relate to Chicago, October 2003

5. Consider the induction equation for the fluctuating field: where (and hence between and and Under this assumption, the relation between ) is linear and homogeneous. Traditional approach is to assume that the fluctuating field is driven solely by the large-scale magnetic field. i.e. in the absence of B0 the fluctuating field decays. i.e. No small-scale dynamo Chicago, October 2003

6. Postulate an expansion of the form: where αij and βijk are pseudo-tensors. Simplest case is that of isotropic turbulence, for which αij = αδij and βijk = βεijk. Then mean induction equation becomes: α: regenerative term, responsible for large-scale dynamo action. Since is a polar vector whereas Bis an axial vector then α can be non-zero only for turbulence lacking reflexional symmetry (i.e. possessing handedness). β: turbulent diffusivity. Chicago, October 2003

7. Mean field dynamo theory is very user friendly. e.g. butterfly diagrams for dipolar and quadrupolar fields: (Tobias 1996) Mean Field Theory – Applications For example, Cowling’s theorem does not apply to the mean induction equation – allows axisymmetric solutions. With a judicial choice of α and β (and differential rotation ω) it is possible to reproduce a whole range of observed astrophysical magnetic fields. Chicago, October 2003

8. Crucial questions • What is the role of the Lorentz force on the transport • coefficients α and β? • How weak must the large-scale field be in order for it to be • dynamically insignificant? Dependence on Rm? • 3. What happens when the fluctuating field may exist of its • own accord, independent of the mean field? • What is the spectrum of the magnetic field generated? Is the • magnetic energy dominated by the small scale field? Chicago, October 2003

9. Two-dimensional MHD turbulence Field co-planar with flow. Field of zero mean guaranteed to decay. Can address Q1 and Q2, for β. In two dimensions and the induction equation becomes: Averaging, assuming incompressibility and u.n = 0 and either A = 0 or nA = 0 on the boundaries, gives Question of interest is: What is the rate of decay? Kinematic turbulent diffusivity given by ηt = Ul. Kinematic rate of decay of large-scale field of scale L is: Follows that: i.e. strong small-scale fields generated from a (very) weak large-scale field. Chicago, October 2003

10. Dynamic effects of magnetic field significant once the total magnetic energy is comparable to the kinetic energy. Leads to the following estimate for decay time (Vainshtein & Cattaneo): where M2 = U2/VA2, the Alfvénic Mach number based on the large-scale field. Diffusion suppressed for very weak large-scale fields, M2 < Rm. Physical interpretation: Strong (equipartition strength) fields on small-scales prevent the shredding of the field to the diffusive length scale. The field imbues the flow with a “memory”, which inhibits the separation of neighbouring trajectories. cf. the Lagrangian representation Chicago, October 2003

11. Magnetic Energy time Randomly-forced flow: periodic boundary conditions. (Wilkinson 2003) Chicago, October 2003

12. We may argue that so that ifηT is suppressed in three-dimensions, then so is α. α can be computed through the measurement of the e.m.f. for an applied uniform field. Consider the following two numerical experiments. Three-dimensional Fields and Flows In three dimensions we again expect strong small-scale fields. Lagrangian (perfectly conducting) representation of α is: (Moffatt 1974) Chicago, October 2003

13. Forced three-dimensional turbulence where F is a deterministic, helical forcing term. In the absence of a field the forcing drives the flow α is calculated by imposing a uniform field of strength B0. We then determine the dependence of α on B0 and the magnetic Reynolds number Rm. Chicago, October 2003

14. Imposed vertical field with B02 = 10-2, Rm = 100. Chicago, October 2003

15. Components of e.m.f. versus time. Chicago, October 2003

16. α versus B02 (Cattaneo & Hughes 1996) Suggestive of the formula: for γ = O(1). α versus Rm (C, H & Thelen 2002) Chicago, October 2003

17. Boussinesq convection. Taylor number, Ta = 4Ω2d4/ν2 = 5 x 105, Prandtl number Pr = ν/κ = 1, Magnetic Prandtl number Pm = ν/η = 5. Critical Rayleigh number = 59 008. Anti-symmetric helicity distribution anti-symmetric α-effect. Rotating turbulent convection T0 Ω g T0 + ΔT Chicago, October 2003

18. Ra = 150 000 Weak imposed field in x-direction. Temperature on a horizontal slice close to the upper boundary. Chicago, October 2003

19. Ra = 75 000. No dynamo at this Rayleigh number – but still an α-effect. Mean field of unit magnitude imposed in x-direction. Chicago, October 2003

20. emf versus time – well-defined α-effect. Chicago, October 2003

21. Ra = 140,000 Convergence of Exand Ey but not Ez. Chicago, October 2003

22. Ra = 106 Box size: 10 x 10 x 1 Temperature. No imposed field. Chicago, October 2003

23. Bx Chicago, October 2003

24. Objections to strong α-suppression From Ohm’s law, we can derive the exact result: Under certain assumptions one can derive the expression for strong suppression from the term (Gruzinov & Diamond). term? What about the Magnetic helicity governed by: For periodic boundary conditions, divergence terms vanish. Then, for stationary turbulence Can the surface flux terms act in such a manner as to dominate the expression for α? Maybe ……… Chicago, October 2003

25. Conclusions • Even the kinematic “eigenfunction” has very little power in the large-scale field. • α-effect suppressed for very weak fields. • 3. It is far from clear whether boundary conditions will change this result – or, indeed, • in which direction any change will be. • 4. β-effect suppressed for two-dimensional turbulence. No definitive result for • three-dimensional flows. • Some evidence of adjustment to a more significant-large scale field, but on an • Ohmic timescale. • 6. So how are strong astrophysical fields generated? • (i) Velocity shear probably essential. • (ii) Spatial separation of α-effect and region of strong shear (Parker’s interface model). Chicago, October 2003