Transcript Transport Effects in MHD Turbulence

Nonlinear Effects in Mean Field
Dynamo Theory
David Hughes
Department of Applied Mathematics
University of Leeds
Chicago, October 2003
Magnetogram
X-ray emission
from solar corona
Temporal variation of sunspots
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Kinematic Mean Field Theory
Starting point is the magnetic induction equation of MHD:
B
   (u  B)  2B,
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 field
and flow:
B  B0  b, U  U0  u,
where B and U vary on some large length scale L, and u and b
vary on a much smaller scale l.
 B  B0 ,  U  U0 ,
where averages are taken over some intermediate scale l « a « L.
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For simplicity, ignore large-scale flow, for the moment.
Induction equation for mean field:
B0
   E  2B0 ,
t
where mean emf is
E   u  b.
This equation is exact, but is only useful if we can relate
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E
to
B0 .
Consider the induction equation for the fluctuating field:
b
   (u  B0 )    G  2b,
t
where G  u  b   u  b.
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
Under this assumption, the relation between b and B0 (and hence between
E and B0) is linear and homogeneous.
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Postulate an expansion of the form:
Ei  ij B0 j  ijk
B0 j
xk

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:
B0
   (B0 )  (   )2B0 .
t
α: regenerative term, responsible for large-scale dynamo action.
Since E 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.
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Mean Field Theory – Applications
Mean field dynamo theory is very user friendly.
B0
   ( U0  B0 )    (B0 )  (   )2B0 .
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:
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(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?
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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 B    Az, and the induction equation becomes:
A
 u.A  2 A.
t
Averaging, assuming incompressibility and u.n = 0 and either A = 0 or
nA = 0 on the boundaries, gives
 t  A2   2  B 2 .
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:
T 
L2
T
.
 B2   Rm  B2 .
i.e. strong small-scale fields generated from a (very) weak large-scale field.
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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):
L2  1
1 
T  
 2 ,
  Rm M  1 
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
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1d 2
T 
 .
3 dt
Magnetic
Energy
time
Randomly-forced flow: periodic boundary conditions.
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(Wilkinson 2003)
Three-dimensional Fields and Flows
In three dimensions we again expect strong small-scale fields.
Lagrangian (perfectly conducting) representation of α is:
 
d
 .   
dt
(Moffatt 1974)
d  2  3T

,
We may argue that |  |
dt l
l
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.
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Forced three-dimensional turbulence
u
 u.u  p  j  B  2u  F,
t
B
   (u  B)  2B,
t
.u  0,
.B  0.
where F is a deterministic, helical forcing term.
In the absence of a field the forcing drives the flow
u  ( y , x , );

3
cos( x  cos(t ))  sin( y  sin( t )) 
2
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.
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Imposed vertical field with B02 = 10-2, Rm = 100.
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Components of e.m.f.
versus time.
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α versus B02
(Cattaneo & Hughes 1996)
Suggestive of the formula:

0

1  Rm B
2
0
for γ = O(1).
α versus Rm
(C, H & Thelen 2002)
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Rotating turbulent convection
T0
Ω
g
T0 + ΔT
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
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anti-symmetric α-effect.
Ra = 150 000
Weak imposed field in x-direction.
Temperature on a horizontal slice close to
the upper boundary.
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Ra = 75 000. No dynamo at this Rayleigh number – but still an α-effect.
Mean field of unit magnitude imposed in x-direction.
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emf versus time – well-defined α-effect.
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Ra = 140,000
Convergence of Ex and Ey but not Ez.
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Ra = 106
Box size: 10 x 10 x 1
Temperature.
No imposed field.
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Bx
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Objections to strong α-suppression
From Ohm’s law, we can derive the exact result:
B02  E.B0  
1

 j.b   e.b.
Under certain assumptions one can derive the expression for strong suppression from
the  j.b term (Gruzinov & Diamond).
What about the  e.b term?
Magnetic helicity governed by:
t a.b  2e.b  .(b )  .(a  e).
For periodic boundary conditions, divergence terms vanish. Then, for stationary
turbulence  e.b  0.
Can the surface flux terms act in such a manner as to dominate the expression for α?
Maybe ………
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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).
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