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
Temporal variation of sunspots Magnetogram X-ray emission from solar corona Chicago, October 2003
Kinematic Mean Field Theory
Starting point is the magnetic induction equation of MHD:
B
t
(
u
B
) 2
B
, 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
B
0
b
,
U
U
0
u
,
where
B
and
U
vary on some large length scale
L
, and
u
and
b
vary on a much smaller scale
l.
B
B 0
,
U
U 0
, 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:
B
0
t
E 2
B
0 , where mean emf is E
u
b
.
This equation is exact, but is only useful if we can relate E to
B
0 .
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Consider the induction equation for the fluctuating field:
b
t
(
u
B
0 )
G
2
b
, 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
B
0
the fluctuating field decays.
i.e. No small-scale dynamo
Under this assumption, the relation between
b
E and
B
0 ) is linear and homogeneous.
and
B
0 (and hence between Chicago, October 2003
Postulate an expansion of the form: E
i
ij B
0
j
ijk
B
0
x k j
where α
ij
and β
ijk
are
pseudo-
tensors.
Simplest case is that of isotropic turbulence, for which α
ij
Then mean induction equation becomes: = αδ
ij
and β
ijk =
βε
ijk.
B
0
t
(
B
0 ) ( ) 2
B
0 .
α
: regenerative term, responsible for large-scale dynamo action. E
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.
B
0
t
(
U
0
B
0 ) (
B
0 ) ( ) 2
B
0 .
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) 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?
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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
A
t A
z
, and the induction equation becomes:
u
.
A
2
A
.
Averaging, assuming incompressibility and
u.n =
0
n A
= 0 on the boundaries, gives and either
A = 0
or
t
A
2 2
B
2 .
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:
T
L
2
T
.
Follows that:
B
2
Rm
B
2 .
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):
T
L
2 1
Rm
M
2 1 1 , 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
T
1 3
d dt
2 .
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Magnetic Energy Randomly-forced flow: periodic boundary conditions.
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:
d dt
.
(Moffatt 1974) We may argue that | |
d dt
l
2 3
T l
, 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.
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Forced three-dimensional turbulence
t
B
u
t
u
.
u
(
u
p
B
)
j
B
2
B
, 2
u
F
, .
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 2 cos(
x
cos(
t
)) sin(
y
sin(
t
)) α is calculated by imposing a uniform field of strength
B
0.
We then determine the dependence of α on
B
0 Reynolds number
Rm.
and the magnetic Chicago, October 2003
Imposed vertical field with
B
0 2 = 10 -2 ,
Rm
= 100.
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Components of e.m.f.
versus time.
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Suggestive of the formula: 1 0
Rm
B
0 2 for γ = O(1).
α versus
Rm
(C, H & Thelen 2002) Chicago, October 2003 α versus
B
0 2 (Cattaneo & Hughes 1996)
Rotating turbulent convection
T 0
Ω g
T 0 + ΔT 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 = 59 008.
Anti-symmetric helicity distribution anti-symmetric α-effect.
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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 E
x
and E
y
but not E
z .
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Ra =
10 6 Box size: 10 x 10 x 1 Temperature.
No imposed field.
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B x
Objections to strong α-suppression
From Ohm’s law, we can derive the
exact
result:
B
0 2 E.
B
0 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 ……… 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 (Parker’s interface model).
Chicago, October 2003