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

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

Chicago, October 2003 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 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 relateto

Chicago, October 2003 where Consider the induction equation for the fluctuating field: Traditional approach is to assume that the fluctuating field is driven solely by the large-scale magnetic field. i.e. in the absence of B 0 the fluctuating field decays. i.e. No small-scale dynamo Under this assumption, the relation between and (and hence between and ) is linear and homogeneous.

Chicago, October 2003 Postulate an expansion of the form: where α ij and β ijk are pseudo-tensors. Simplest case is that of isotropic turbulence, for which α ij = αδ ij and β ijk = βε ijk. Then mean induction equation becomes: α : regenerative term, responsible for large-scale dynamo action. Since is a polar vector whereas B is 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 Mean Field Theory – Applications Mean field dynamo theory is very user friendly. For example, Cowlings 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. e.g. butterfly diagrams for dipolar and quadrupolar fields: (Tobias 1996)

Chicago, October 2003 Crucial questions 1.What is the role of the Lorentz force on the transport coefficients α and β? 2.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? 4.What is the spectrum of the magnetic field generated? Is the magnetic energy dominated by the small scale field?

Chicago, October 2003 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 n A = 0 on the boundaries, gives Question of interest is: What is the rate of decay? Kinematic turbulent diffusivity given by η t = 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 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 M 2 = U 2 /V A 2, the Alfvénic Mach number based on the large-scale field. Diffusion suppressed for very weak large-scale fields, M 2 < 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 Randomly-forced flow: periodic boundary conditions. Magnetic Energy time (Wilkinson 2003)

Chicago, October 2003 Three-dimensional Fields and Flows In three dimensions we again expect strong small-scale fields. Lagrangian (perfectly conducting) representation of α is: We may argue that so that if η T 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. (Moffatt 1974)

Chicago, October 2003 Forced three-dimensional turbulence where F is a deterministic, helical forcing term. α is calculated by imposing a uniform field of strength B 0. We then determine the dependence of α on B 0 and the magnetic Reynolds number Rm. In the absence of a field the forcing drives the flow

Chicago, October 2003 Imposed vertical field with B 0 2 = 10 -2, Rm = 100.

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

Chicago, October 2003 α versus B 0 2 (Cattaneo & Hughes 1996) α versus Rm (C, H & Thelen 2002) Suggestive of the formula: for γ = O(1).

Chicago, October 2003 Rotating turbulent convection g T 0 + ΔT T0T0 Ω Boussinesq convection. Taylor number, Ta = 4Ω 2 d 4 /ν 2 = 5 x 10 5, Prandtl number Pr = ν/κ = 1, Magnetic Prandtl number Pm = ν/η = 5. Critical Rayleigh number = Anti-symmetric helicity distribution anti-symmetric α-effect.

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

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

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

Chicago, October 2003 Ra = 140,000 Convergence of E x and E y but not E z.

Chicago, October 2003 Ra = 10 6 Box size: 10 x 10 x 1 Temperature. No imposed field.

Chicago, October 2003 BxBx

Objections to strong α-suppression From Ohms law, we can derive the exact result: Under certain assumptions one can derive the expression for strong suppression from the term (Gruzinov & Diamond). What about the term? 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 Conclusions 1.Even the kinematic eigenfunction has very little power in the large-scale field. 2.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. 5.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 (Parkers interface model).