Chicago, October 2003 David Hughes Department of Applied Mathematics University of Leeds Nonlinear...

Chicago, October 2003

David Hughes

Department of Applied Mathematics

University of Leeds

Nonlinear Effects in Mean Field Dynamo Theory


Magnetogram

X-ray emission from solar corona

Temporal variation of sunspots


Starting point is the magnetic induction equation of MHD:

,)( 2BBuB

t

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 fieldand flow:

,, 00 uUUbBB

where B and U vary on some large length scale L, and u and bvary on a much smaller scale l.

,, 00 UUBB

where averages are taken over some intermediate scale l « a « L.

Kinematic Mean Field Theory


For simplicity, ignore large-scale flow, for the moment.

,020 B

B

E

t

. buE

Induction equation for mean field:

where mean emf is

This equation is exact, but is only useful if we can relate E to .0B


,)( 20 bGBu

b

t

. bubuGwhere

Consider the induction equation for the fluctuating field:

Traditional approach is to assume that the fluctuating field is driven solely bythe large-scale magnetic field.

i.e. in the absence of B0 the fluctuating field decays.

i.e. No small-scale dynamo

Under this assumption, the relation between b and 0B (and hence between

Eand 0B ) is linear and homogeneous.


Postulate an expansion of the form:

k

jijkjiji x

BB 0

0 E

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:

.)() 02

00 BB

B

(

t

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.

E


Mean Field Theory – Applications

Mean field dynamo theory is very user friendly.

.)()) 02

0000 BBBU

B

((

t

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.

e.g. butterfly diagrams for dipolar and quadrupolar fields:

(Tobias 1996)


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?


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 ,zB A and the induction equation becomes:

.. 2 AAt

A

u

Averaging, assuming incompressibility and u.n = 0 and either A = 0 or nA = 0 on the boundaries, gives

.2 22 BAt

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: .2

TT

L

Follows that: .22 BRmB

i.e. strong small-scale fields generated from a (very) weak large-scale field.


Dynamic effects of magnetic field significant once the total magnetic energyis comparable to the kinetic energy.Leads to the following estimate for decay time (Vainshtein & Cattaneo):

,1

112

2

MRm

LT

where M2 = U2/VA2, the Alfvénic Mach number based on the large-scale field.

Diffusion suppressed for very weak large-scale fields, M2 < Rm.

.3

1 2 dt

dT

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


Randomly-forced flow: periodic boundary conditions.

MagneticEnergy

time

(Wilkinson 2003)


Three-dimensional Fields and Flows

In three dimensions we again expect strong small-scale fields.

Lagrangian (perfectly conducting) representation of α is: .dt

d

We may argue that ,3

||2

lldt

d T

so that if ηT 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)


Forced three-dimensional turbulence

.0.,)(

,0.,.

2

2

BBBuB

uFuBjuuu

t

pt

where F is a deterministic, helical forcing term.

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.

))sin(sin())cos(cos(2

3);,,( tytxxy u

In the absence of a field the forcing drives the flow


Imposed vertical field with B02 = 10-2, Rm = 100.


Components of e.m.f.versus time.


α versus B02

(Cattaneo & Hughes 1996)

α versus Rm(C, H & Thelen 2002)

Suggestive of the formula:

20

0

1 BRm

for γ = O(1).


Rotating turbulent convection

g

T0 + ΔT

T0Ω

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.


Ra = 150 000

Weak imposed field in x-direction.

Temperature on a horizontal slice close to the upper boundary.


Ra = 75 000. No dynamo at this Rayleigh number – but still an α-effect.

Mean field of unit magnitude imposed in x-direction.


emf versus time – well-defined α-effect.


Ra = 140,000

Convergence of Ex and Ey but not Ez.


Ra = 106 Box size: 10 x 10 x 1Temperature.

No imposed field.


Bx


Objections to strong α-suppression

From Ohm’s law, we can derive the exact result:

...1

020 bebjB

E.B

Under certain assumptions one can derive the expression for strong suppression from the bj. term (Gruzinov & Diamond).

What about the be. term?

Magnetic helicity governed by:

.).().(.2. eabbeba t

For periodic boundary conditions, divergence terms vanish. Then, for stationary turbulence .0. be

Can the surface flux terms act in such a manner as to dominate the expression for α?

Maybe ………


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 (Parker’s interface model).

Chicago, October 2003 David Hughes Department of Applied Mathematics University of Leeds Nonlinear...

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