Convection, Magnetoconvection, and Dynamo Theory Cargese, 20-25 September, 2010

Happy Birthday,

Mike!

Some Topological Aspects of Dynamo Theory

Keith Moffatt Trinity College, Cambridge

Helicity H =<u.ω> is an integral invariant of the Euler equations, representing the degree of linkage of vortex lines (which are frozen in the fluid). H is a pseudo-scalar!

For linked vortex tubes carrying fluxes Ф_1, Ф_2, H = 2n Ф_1 Ф_2 where n is the Gauss linking number of the two tubes.

For a single knotted vortex tube carrying flux Ф, H = Ф^2( Wr + Tw), where Wr is the writhe of the centreline of the tube, and Tw is the twist of the field within the tube (a combination of torsion and ‘internal twist’).

These results provide a natural bridge between continuum fluid mechanics and rigorous results of differential geometry/topology.

Origin of Magnetic Fields in Astrophysics (Turbulent "Dynamo" Mechanisms) Vaĭnshteĭn, S.I. & Zel'dovich, Ya B. (1972) Soviet Physics Uspekhi, 15, 159-172 .

This was the paper in which the ‘stretch-twist-fold’ mechanism for the ‘fast dynamo’ was introduced, i.e. a mechanism for the intensification of a magnetic field that appears to operate without any need to invoke ‘ohmic’ diffusive effects. A magnetic field is doubled in intensity, and its energy increased four-fold with each iteration of this cycle.

But there is a twist ----

Ya.B.Zel’dovich 1914-1987

Stretch, twist, fold: the fast dynamo By Stephen Childress, Andrew D. Gilbert Springer 1995

HKM & Proctor (1985) Topological constraints associated with fast dynamo action. J. Fluid Mech. 154, 493–507.

Bajer and M 1990 (JFM 212, 337-363): Steady Stokes flows in a sphere with chaotic streamlines

But can these flows act as dynamo? We still don’t know.

Stretch- Twist- Fold

Unfold- Untwist-Relax!

Many problems in Fluid Mechanics are topological in character:

In Euler flow of an ideal fluid, vortex lines are transported with the fluid, so the topology of the vorticity field is conserved.

Similarly in ideal MHD, magnetic field lines are frozen in the fluid.

In either case, finite diffusivity (viscous or ohmic respectively) leads to reconnection processes, with associated change of topology. This can have

dramatic consequences:

Magnetic Field Topology in Solar Flare Activity

The magnetic field emerging from below the sun’s surface evolves in response to movement of the ‘footpoints’ on the photosphere. Tangential field discontinuities

form and field lines reconnect, with change of topology. Strong Joule heating in these regions underlies observed solar flare activity.

NASA image of a magnetic flux tube erupting from the solar surface

Filament destabilization observed by TRACE in Active Region 9957 on 27 May 2002.

This filament started to rise shortly before 18 UT; the X-ray flux (recorded by the GOES satellite) rose in step, reaching a maximum around 18:10 UT, just as the twisting filament

reached a maximum elevation of about 80,000 km. The magnetic field subsequently remained steady, while the plasma within the field cooled and flowed back down to the

solar surface along the distorted magnetic flux tubes.

I want to discuss an interesting fluid dynamical problem in which a sudden topological transition occurs, under slow change of the controlling parameters.

t = 0.1 0.25 0.4

t = 0.6 0.8 0.9

x(s) = (x,y,z) = [-t cos s + (1-t) cos 2s, -t sin s + (1- t) sin 2s, -2t (1- t ) sin s]/l(t), 0 ≤ s ≤ 2π, Parametric equations of an untwisting wire (Maggioni & Ricca 2006, Proc Roy Soc., 462,3151)

Now dip this wire (at t = 0.1) in soap solution to make a one-sided Möbius strip soap film, and unfold: What happens?

Courant, R. 1940 Soap film experiments with minimal surfaces. Amer. Math. Mon. 47, 167–174 .

In this informal article, Courant mentions that a one-sided soap-film in the form of a Möbius strip is easily formed, and that it can jump to a two-sided film if the wire boundary is suitably distorted .

We decided to repeat this experiment ...

A distinguished precedent:

Richard Courant (1888 – 1972), founder of the Courant Institute in New York

(a) b = 0.1 (b) b = 0.3 (c) b = bc = 0.6627

The two-wire problem dates back to Euler and Lagrange:

Euler, L. 1744 Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematic isoperimetrici latissimo sensu accepti. Opera Omnia I, Vol. XXIV, C. Caratheodory (ed) (1952).

Lagrange, J.L. 1762 Essai d’une nouvelle méthode pour determiner les maxima et les minima des formules intégrals indefinites. Miscellanea Taurinensia 2, 173-195.

Recent experiments: Robinson, N.D. & Steen, P.H. 2001 Observations of singularity formation during the capillary collapse and bubble pinch-off of a soap film bridge. J. Coll. Int. Sci. 241, 448-458

Nitsche, M. & Steen, P.H. 2004 Numerical simulations of inviscid capillary pinchoff. J. Comp. Phys. 200, 299-324

b=0.6, two solutions, one stable,one unstable

Area of catenoid

b=0.7, no minimum-area solution

Soap film with ‘connecting disc-film’ on the plane z = 0, when the rings are at maximum separation for a minimum area solution.

Films connect at an angle of 120°

We get a clue from the family of ruled surfaces:

where −1 ≤ µ ≤ 1. The centreline µ = 1 is a circle of radius 1 − t in the plane z = 0.

t = 0.1 0.25 0.4

t = 0.6 0.8 0.9

Topology changes (the hole disappears) somewhere between 0.6 and 0.8, actually at 2/3

Topological transition of a soap film Möbius strip.

As the wire frame is gradually distorted, the initial one-sided film (a) transforms to a two-sided solution (f).

Frames from a high-speed movie in (b)-(e) at intervals of 5.4 ms show the collapse process leading to a collision with the frame.

The twisted Plateau border far from the collapse time (h) is shown schematically in (g) as a red line wrapping around the wire (gray). Just after the collapse (i) the border is twisted around the frame in the

opposite sense (!) as shown in (j).

The caustic in (i) is a consequence of the twisted Plateau border. Scale bar in (a) is 2 cm.

Joint work with with Ray Goldstein, Adriana Pesci and Renzo Ricca

Movie

Expanded view of the twisted fold singularity just after collapse; the twist of the Plateau border round the wire can be seen here.

The resulting residual twist in the surface shows up through the amplified optical effects.

This pattern can persist for some minutes; but eventually disappears through a process of viscous diffusion.

Dynamics of the topological transition (over a period of about 0.05s, and resolved to 0.2ms)

Throat diameter D(t), averaged over multiple runs; tp = ‘pinch time’.

Red lines indicate power-law behaviour with exponents of 1/3 (far from the singularity) (viscous control), and 2/3 closer to the pinch time tp (air inertia control)

Local analysis of the ruled surface shows that a twisted fold forms at the boundary at t=2/3, and propagates towards the centre for t > 2/3;

The surface is also self-intersecting in a neighbourhood of this fold.

The real soap film cannot sustain this structure!

t = 0.63 t = 2/3 t = 0.7

Projection of a portion of the surface (−1 < s < 1, −1 < µ < − 0.9) on the z-plane for values of t in the neighbourhood of the critical value tc = 2/3.

The hole disappears at the critical value, when the cusp in the boundary curve is apparent.

At the instant t = 2/3, the wire has the form of a twisted cubic; and at this instant, ‘writhe plus torsion’ jumps to ‘internal twist’ (as with the twisted belt). HKM & Ricca, Proc Roy Soc 1992

(a) Examples of numerically obtained minimal surfaces (gray) and ruled surfaces (copper) as a function of the parameter t of the model. Black and red lines indicate the minimal energy for the two classes of

surfaces. Blue line is the energy of the ruled surface.

(b) (c) The geometry in the unstable region very close to the singularity.

(d) The universal ‘twisted saddle’ geometry from the ruled-surface model.

Energy and geometry of surfaces.