Different Limiting Mechanisms for Nonlinear Dynamos

Robert CameronMax-Planck-Institut für Sonnensystemforschung

D-37191 Katlenburg-Lindau, Germany

David GallowaySchool of Mathematics and Statistics

University of Sydney, NSW 2006, Australia

Outline

1. Need for strong-field dynamos

2. 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:

2 2

0

2

1 1) ( / ) ( )

2

( )

0

( sin cos , sin cos , sin cos );

where we will use scaled units such that =1/Re, =1/Rm.

P ut

t

A z C y B x A z C y B x

1

uu ( u F B B u

u= u B B

u B

F

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

-1/2

1/ 2

-1/21/ 2

Total ME 1 (filaments of thickness Rm )

Total KE Re

Total ME 1 (filaments of order 1 radius with Rm edges)

Total KE Re

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 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.

( ) u u

4. A version of the Archontis dynamo

Archontis took a forcing F proportional to

. 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:

(sin ,sin ,sin )z x y

sines sines sines sines( ) F u u u

No B

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 sinesu

B

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

that the dynamo is asymptoting to sines as Rm u = B = u

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

sin y cos z 2.024Rm-1

sin x sin 2y sin z -1.024Rm-1

sin 2x sin y sin 2z -0.728Rm-1

cos 2x cos 2y sin z -0.56 Rm-1

cos x sin 2y -0.496Rm-1

sin 2x cos 2z 0.352Rm-1

cos 2x sin 3z -0.328Rm-1

sin 2x sin 3y sin 2z 0.32 Rm-1

First 8 modes of U-B, x-component

sin z 0.958

sin x sin 2y sin z -0.066

cos x sin 2y cos 2z -0.055

cos x sin 2y -0.05

sin 2x sin y sin 2z -0.04

cos 2x sin y cos z 0.032

cos 2y sin z -0.03

cos 2x sin z 0.024

First 8 modes of U+B, x-component

Mode Structure in Elsasser Variables

Isosurface of |u-B| (0.75 of max)

Tubes around heteroclinic orbits

Evolution of KE and ME starting from small seed field

Results for Re=Rm=200

5. A Modified ABC dynamoIdea: try and make a dynamo with u close to the 1:1:1 ABC flow (which is Beltrami with ), 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!

( 0 u u) =

( )

Target field Computational results

Evolution of energies with time

Rm=Re=100

6. Conclusions● 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 anyway.