Magnetic structures created by localised velocity shear and

magnetic buoyancy

Nic BrummellKelly Cline

Fausto Cattaneo

Nic Brummell (303) 492-8962JILA, University of Colorado

brummell@solarz.colorado.edu

Large-scale dynamo: Intuitive picture

toroidal poloidal

toroidal poloidal

Here, we will examine the -effect and the role of magnetic buoyancy

Philosophy: Examine nonlinear versions of concepts with as few assumptions as possible

The role of magnetic buoyancy

Dual roles of magnetic buoyancy in the large-scale dynamo:

Limiter:

• Magnetic buoyancy limits the growth of the magnetic field by removing flux from the region of dynamo amplification

• Magnetic buoyancy instabilities then control the dynamo amplitude

• BUT magnetic buoyancy does not actively contribute to the amplification process

Driver:

• If the poloidal field regeneration is associated with rising and twisting structures, then magnetic buoyancy is the very mechanism that drives the dynamo.

First case – dynamo operates IN SPITE of magnetic buoyancy

Second case – dynamo operates BECAUSE of magnetic buoyancy

Examine via solutions of fully nonlinear MHD equations

Mimic some properties of the tachocline : • Use a convectively stable layer

• Force* a velocity shear in both the vertical (z) and one horizontal (y) direction.

e.g. U(y,z) = f(z) cos(2 y/ym)

where f(z) is a polynomial function chosen to confine the shear to a particular layer between zu and zl (and to be sufficiently continuous)

• Shear flow is hydrodynamically stable

Then add an initial magnetic field:

B0 = (0, By , 0) with By = 1

Model: Localised velocity shear

By +

* Add term in the equations that induces desired flow in absence of magnetic effects

Basic “–effect’’ mechanism

• Start from zero velocity initial conditions (technical reasons: avoids long transients)

• As velocity builds up in shear regions, it stretches the transverse (poloidal) field out into streamwise (toroidal) field.

• Fieldlines become close together where the velocity shear in y is strongest (y=/2, 3/2) creating structure in the magnetic field.

• Mathematically, strong Bx is created from weak By by the inductance of the shear:

t Bx ~ BydyU0

Induction of strong toroidal field by shear

Evolution

What happens next depends upon the parameters!

Low Rm: Non-static equilibrium

A very boring movie!

Low Rm: Non-static equilibrium• A balance between stretching production and diffusive removal of Bx is achieved.

• However, advection plays a role: buoyancy-driven roll cells make the equilibrium non-static.

• Magnetic field is dynamically active

Increasing Rm: Magnetic buoyancy instabilityA more interesting movie!

• Instability

• Cyclic activity

• Two out-of-phase sequences of identical but oppositely-directed magnetic structures

• Instability driven by interaction of induced poloidal flow with background shear

Higher Rm: Chaotic behaviour

For higher Rm, structures are formed and rise but in a chaotic manner:

• Size, strength and geometry vary

• Cyclic but irregular position of eruption and therefore polarities of rising structures

Larger Re: Secondary K-H instability

Rotation of

a snapshot

Instability mechanism:

• Initial field purely poloidal

• Poloidal field sheared -> toroidal

• Toroidal field creates magnetic buoyancy

• Magnetic buoyancy induces roll-like poloidal flows

• These steepen the shear

• If shear is steepened enough, becomes K-H unstable

Dynamo

Hmmm…

A possible dynamo mechanism?

Model: Localised velocity shear 2

Sawtooth profile

By -

By +

Other configurations used too:

Velocity shear:

Early work: U(y,z) = f(z) cos(2 y/ym)

Dynamo work: U(y,z) = f(z) [sawtooth(y)]

Magnetic field:

B0=(0,By,0)

Early work: By = 1

Dynamo work:

+1 (z>0.5)

- 1 (z<0.5){By =

Weak initial field: Non-static quasi-equilibrium

• System eventually decays due to diffusion between the By = +/- parts (hence quasi-equilibrium)

Stronger initial field: A dynamo!

A dynamo!

• Strong magnetic field maintained!

• Strong toroidal field is generated in a cyclic manner

• Polarity of the strong field reverses

Bx - ve Bx + ve

cf. By(t=0)=1!

A dynamo! Longer time …

• Diffusion time ~ 300 time units

• => even more convincing is a dynamo

• Remarkably, also shows periods of reduced activity!

Dynamo conclusions

A new class of dynamo mechanisms (as far as we know)

A dynamo driven solely by the action of shear and magnetic buoyancy

Fully self-consistent

No Coriolis forces required to twist toroidal into poloidal

Intrinsically nonlinear … cannot quantify in terms of an “-effect” (and if you do attempt to, get meaningless result).

What is a flux tube? Examine magnetic fieldlines

These structures appear to fit with our natural ideas of a magnetic flux tube:

• compact, cylindrical, isolated

• clear inside/out

To be true, need magnetic flux surfaces. Do they really exist?

We will examine the nature of magnetic fieldlines in the three general states found:

• equilibrium

• primary instability

• secondary instability

We take a 3-D snapshot of the magnetic fields, pick a starting point and integrate along the magnetic field lines.

Fieldlines in equilibrium state:x-invariant, y-mirror symmetric

Recurrence maps of 15 fieldlines stacked vertically in XY- and YZ-planes.

Invariant sets are isolated points – hits same points over and over again.

Fieldlines map out only a line (degenerate surfaces)

Projection of 1 fieldline onto XY-plane (i.e. viewed from above)

Projection of 15 fieldlines stacked vertically onto YZ-plane (i.e. viewed from the end)

Fieldlines – primary instability:Break y-mirror symmetry

Recurrence maps of 15 fieldlines stacked vertically in XY- and YZ-planes.

Points of return migrate due to loss of symmetry. Invariant set is a line.

Fieldlines map out a PLANE, i.e. FLUX SURFACES.

Projection of 1 fieldline onto XY-plane (i.e. viewed from above)

Projection of 15 fieldlines stacked vertically onto YZ-plane (i.e. viewed from the end)

Fieldlines – primary instability:

Recurrence maps of 15 fieldlines (stacked vertically) in YZ-planes.

Surfaces remain as surfaces throughout.

Time sequence:

Contours in YZ-plane

Fieldlines – secondary instability:Break x-translational symmetry too

Recurrence maps of 15 fieldlines (stacked vertically) in YZ-planes.

Fieldlines fill volume during the 3D stages.Invariant set is some complicated object!

Time sequence:

Fieldlines – secondary instability:

Recurrence map (YZ-plane)

• single instance in time

• 3D KH kinked structure

• 5 returns

• initial positions inside “structure”

Fieldlines do NOT remain within structure.

Neighbouring fieldlines diverge rapidly (chaotic?)

Fieldlines – secondary instability:

“Lyapunov” map (YZ-plane)

• single instance in time

• 3D KH kinked structure

• Points within 3D structure show large “Lyapunov” exponents

• Trajectories diverge rapidly

• Chaotic!

Comments, thoughts, conclusions(?)Three types of fieldline topology found:

• Very symmetric: Fieldlines lie on surfaces but individual lines do not cover the surface

• No y mirror symmetry: Fieldlines lie on surfaces and individual lines do cover the surface

• No x translational symmetry: Fieldlines occupy some complex topological object!

Flux surfaces only exist where there is a great deal of symmetry

Structures are not necessarily encased in flux surfaces

There is no easily defined inside/outside (therefore cannot define writhe, twist etc uniquely)

Fluid is free to flow in and out (leak out) of the structure

Despite the fact that this is not our idealised picture, this may actually HELP in many problematic circumstances, e.g. axisymmetric rise of a flux tube.

Questions:

• Fieldlines ever lie CLOSE to flux surfaces?

• Can reconnection isolate entities? (only if remains symmetric?)

• Or do we re-think magnetic structures?