Transcript 20040909103511001-148717

Different Limiting Mechanisms for
Nonlinear Dynamos
Robert Cameron
Max-Planck-Institut für Sonnensystemforschung
D-37191 Katlenburg-Lindau, Germany
David Galloway
School of Mathematics and Statistics
University of Sydney, NSW 2006, Australia
Outline
1.
2.
Need for strong-field dynamos
Problems with filamentary dynamos (eg
ABC)
3. Ways of escape
4. The Archontis dynamo
5.
A modified ABC strong-field dynamo
6. Conclusions
1. Need for strong-field dynamos
•
80s issue: can there be fast dynamos that
grow on the turnover timescale of the flow,
rather than the diffusion timescale?
Essentially a kinematic question. Answer is
yes, providing underlying flow is chaotic
(Klapper and Young, 1995, plus several
numerical examples).
•
Fast dynamos are astrophysically necessary
because for many objects with observed
magnetic fields, the laminar diffusion time is
longer than the age of the Universe. (If, that
is, laminar diffusion is relevant!)
• 90s question: can a dynamo generate fields as
significant as those observed, once the back
reaction of the Lorentz force limits its growth?
• Considerable pessimism (Vainshtein and Cattaneo,
1992; Gruzinov and Diamond, 1994): total
Magnetic Energy expected to be much less than
total Kinetic Energy at high Rm/Re.
• Difficulties are particularly acute for mean field
dynamos: proponents have been fighting to
overcome them (see other talks).
• This talk exhibits strong field dynamos for the
non-mean-field case; these turn out to have ME  KE
2. Problems with ABC forcing
Nonlinear dynamos driven by prescribed
force field F(r, [t]), in 2π -periodic geometry
Governing equations:
u
1
1
 u  (  u)  ( P /   u 2 )  F 
(  B)  B    2u
t
2
0 
u
=   (u  B)    2 B
t
 u B0
F   1 ( A sin z  C cos y, B sin x  A cos z , C sin y  B cos x);
where we will use scaled units such that  =1/Re,  =1/Rm.
Here is the kind of thing that happens.
This is for Re=5, Rm=400, at two times
where the flow has become statistically
steady. The field is filamentary. The total
magnetic and kinetic energies are are of
the same order.
But…this is at low Re (chosen so underlying
ABC flow wants to be stable). Does similar
behaviour exist at high Re? Answer: no
(numerically shown by Galanti, Pouquet &
Sulem 1992).
Scaling argument (Galloway 2003): assume upper
bound for ohmic dissipation of magnetic field is
viscous dissipation in the absence of any field.
Then
Total ME 1

Total KE Re
(filaments of thickness Rm -1/2 )
1/ 2
Total ME
1  
 1/ 2  
Total KE Re   
(filaments of order 1 radius with Rm-1/2 edges)
This is bad news. Same conclusion reached by Brummell,
Cattaneo & Tobias (2001), for time-dependent ABC forcing.
3. Ways of Escape
This last limiting mechanism is one that clearly
doesn’t work! Real flows (and even real
numerical experiments) are turbulent. They
have far more viscous dissipation than the
laminar value. The KE is severely
overestimated. We need a good theory of MHD
turbulence before we can cure this.
A crude fix: go with Schatzmann, who told us to
take Re=100 in astrophysics (typical value for
instability to the next scale down). Then, at a
price, the difficulty might go away…
A better fix: do all dynamos really have to be
filamentary? Socratic dialogue between the current
authors took place, finally yielding the answer no.
The Lorentz force can balance the applied force either
directly or via the nonlinear u  (  u) term. The latter
can be much larger than the forcing, but does no work.
An example of the second possibility appeared already
in Archontis (2000; PhD, www.astro.ku.dk /~bill).
We have concocted our own example of the first.
Both of these dynamos are laminar and have velocity
and magnetic fields everywhere approximately equal to
one another, so that the ME/KE ratio is 1.
4. A version of the Archontis
dynamo
Archontis took a forcing F proportional to
(sin z,sin x,sin y ) . This was designed to produce a
velocity of the same form; there is numerical
evidence suggesting that such a flow is a fast
kinematic dynamo (Galloway and Proctor, 1992). In
fact such a flow does result, but by a very circuitous
route involving the generation of a magnetic field.
This is illustrated in the following diagram:
Fsines  usines  usines  ( usines )
Time-dependent
non-sines flow
(properly forced
sines flow is unstable
for Re > 8)
Time-dependent
non-sines flow with
non-sines B
Eventual evolution to
u B  usines
Some differences:
Archontis used a time-dependent forcing amplitude
designed to control the total kinetic energy to be
constant, whereas we set our forcing to a constant
level.
Archontis dealt with a compressible fluid, whereas
ours is incompressible (hopefully to give a better
chance of analytic progress).
A similar dynamo results in all cases. The first
impression is
u=B=u
as Rm  
sines
that the dynamo is asymptoting to
But in fact this is not the case. Although this state is formally a
solution to the diffusionless version of the problem (as indeed is
any state with u=B), the limit is singular. In the diffusive case there
are small additional corrections to the sines-flow solution, and these
persist at the level of a few percent for all Rm. An attempt to derive
a high Rm perturbation expansion makes clear why these terms
have to be there.
Most significant Fourier modes of u-B
…and of u+B
Mode Structure in Elsasser Variables
First 8 modes of U-B, x-component
First 8 modes of U+B, x-component
sin y cos z
2.024Rm-1
sin z
0.958
sin x sin 2y sin z
-1.024Rm-1
sin x sin 2y sin z
-0.066
sin 2x sin y sin 2z
-0.728Rm-1
cos x sin 2y cos 2z -0.055
cos 2x cos 2y sin z -0.56 Rm-1
cos x sin 2y
-0.05
cos x sin 2y
-0.496Rm-1
sin 2x sin y sin 2z
-0.04
sin 2x cos 2z
0.352Rm-1
cos 2x sin y cos z
0.032
cos 2x sin 3z
-0.328Rm-1
cos 2y sin z
-0.03
sin 2x sin 3y sin 2z
0.32 Rm-1
cos 2x sin z
0.024
Evolution of KE and ME starting from
small seed field
Tubes around heteroclinic orbits
Isosurface of |u-B| (0.75 of max)
Results for Re=Rm=200
5. A Modified ABC dynamo
Idea: try and make a dynamo with u close to the 1:1:1
ABC flow (which is Beltrami with u  (  u) = 0 ), and B
parallel to it. Select “target field” BT=α(r)uABC where r is
the least distance of a point’s KAM surface from the
chaotic region (thus α is constant on KAM surfaces, cf.
Arnold). Typically, take α to be logistic in shape with a
near zero value in the chaotic regions and a maximum
furthest away from them. Then force the dynamo with
F=Fext+Fν where Fext=BT(  BT) and Fν is the same
viscosity-based forcing as earlier. The Lorentz force of the
target field is thus balanced by an artificially supplied
external force---dynamos to order!
Target field
Computational results
Rm=Re=100
Evolution of energies with time
6. Conclusions
●
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Strong field dynamos with comparable total
magnetic and kinetic energies are possible even at
high kinetic and magnetic Reynolds numbers.
The examples we have found are almost laminar
and steady and relate to a known class of solutions
with u=B (though our final example merely has u
parallel to B). It is interesting that these solutions
seem to be attractors.
In all cases the evolution onto these solutions is
slow, and may indeed take a large number of
diffusion times due to nonlinear switching between
different states (cf intermittency).
In astrophysics, this last fact may be problematic
but there the flows seem likely to be turbulent