A dynamo without many of the usual

ingredients!A public service announcement

Nic BrummellKelly Cline

Fausto Cattaneo

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

brummell@solarz.colorado.edu

Large-scale dynamo theory

We seem to strive very hard to build a large-scale dymamo out of the “usual suspects ingredients”:

effects, rotation, differential rotation, turbulent diffusivities, turbulent transport …

Everytime we look nonlinearly, our intuitive ideas come up against obstacles:

Turbulent Re stresses are complex

effects and turbulent diffusivities are quenched

etc

Does a dynamo NEED turbulence to work, or can it work IN SPITE of turbulence?

A dynamo! The movie …

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!

WTF!

WHAT THE HECK IS THIS THING?!

Answer:

• A dynamo driven entirely by magnetic buoyancy

• That does requires NEITHER rotation NOR turbulence!

• But is intrinsically nonlinear and non-kinematic

Large-scale dynamo: intuitive picture

Model configuration: Localised velocity shear

Sawtooth profile

By -

By +

Configurations used:

Build one strong structure

Have field that will diffuse

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

• Velocity ramps up

• Magnetic field By stretched into Bx by velocity shear.

• Strong “tube”-like magnetic structure forms in region of strongest shear.

Weak initial field: Non-static quasi-equilibrium

• Magnetic structures created by shear.

• Density drops due to contribution from magnetic pressure.

• Density drives a roll-like flow up through the centre of the structure.

• Balance achieved between creation of magnetic field by induction due to the shear and resistive diffusion and advection by magnetic buoyancy driven flow.

• Flows => non-static

Bx (shaded, +ve dark) (By,Bz) arrows

Density perturbation (shaded, +ve dark) (v,w) arrows

Weak initial field: Non-static quasi-equilibrium

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

Stronger initial field: K-H instability• Increasing initial field strength increases the poloidal flow strength induced by the toroidal magnetic structure.

• At t~40, an instability occurs

• Instability is of Kelvin-Helmholtz type: sinusoidal variations in velocity components associated with shears in vertical and horizontal.

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

• Shear then becomes K-H unstable

Hydrodynamic instability but magnetically induced (=> non-kinematic)

K-H

Stronger initial field: K-H instability

• Effect of instability is to kink geometry of structures.

• Note this is NOT a magnetic kink instability!

• K-H modes advect/wrap magnetic field into helical shape

Stronger initial field: poloidal field generation

• K-H flows create two poloidal loops -- CCW above, CW below – out of STRONG toroidal field

• => STRONG poloidal field created

• Stronger poloidal field => stronger toroidal field

B components

K-H

Feedback loop created for dynamo! So … let it run …

A dynamo! The movie …

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!

A dynamo!

Mechanisms (complicated!):

Dynamo:

• Two poloidal loops created, upper one opposing original field

• Sign of By reversed between loops

• Weaker toroidal field created which rises

• K-H acts on this to create poloidal loop in upper region with original direction

• Combines with lower loop (diffusion) to start process again.

Reversal:

• Strongest structure created

• Dredges in toroidal field from sides to switch polarity

Inactivity periods:

• Failed polarity reversal

An even wackier dynamo!

Things to note: There is a minimum initial magnetic field required to trigger the K-H instability and therefore the dynamo i.e. the mechanism is NOT KINEMATIC.

The dynamo saturates in equipartition with the shear energy source

Higher Rm (e.g. Rm ~ 2000 cf earlier Rm ~ 1000, varying the magnetic Prandtl number):

dynamo behaves irregularly – irregular production of structures, polarity no longer so obvious

Work in progress to determine large Rm behaviour: does it turn off (no more reconnection)?

Lower Rm (e.g. Rm ~ 500 cf earlier Rm ~ 1000, varying the magnetic Prandtl number):

Diffuses away UNLESS raise initial field strength significantly

Then can trigger K-H => dynamo, but different

NO RISE! Statistically steady travelling wave K-H rather than intermittent K-H

An even wackier dynamo!

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

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 the role of turbulence? This is VERY LAMINAR! Does hydrodynamic turbulence enhance or decrease the dynamo effeciency? Enhanced diffusion helps reconnection processes? OR loss of coherence kills dynamo? Add noise to the dynamo simulations … ( work in progress )

Obvious questions:What determines the strength of the emerging structures?

tshear-buoyant >> tequilib => structures have characteristics of equilibrium

tshear-buoyant ~ tequilib => structures have characteristics set by instability

tshear-buoyant depends on stratification (poloidal flows)

tequilib does not (depends on balance of stretching and tension)

Buoyancy forces set upper limit on strength of structures by setting maximum time for shear amplification mechanism to act

What are the writhe and twist of 3D structures (observational signatures)?

= components of the magnetic helicity, invariant in the limit of zero resistive diffusion when integrated over a volume (flux tube) surrounded by unmagnetized material.

• Writhe and twist could be defined by thresholding, but would be ambiguous.

• Integral would not be invariant, due to fieldlines entering and exiting volume.

Leads to question: are our structures isolated or encased in flux surfaces?

See Geoff’s

talk next!

Question: What is a flux tube?

The observation of intense, intermittent ,“isolated”, “frozen-in” elements of magnetic field on the sun has led to the notion of a MAGNETIC FLUX TUBE:

Do such flux surfaces really exist?

Important question because can lead to very different models of evolution.

e.g. Do not need non-axisymmetric rise of annulus or drainage down tubes to remove mass if the tube is not defined by a closed surface.

Usefulness of the flux tube concept hinges on the existence of flux surfaces (although may hold up even if magnetic field lies close to surfaces).

So …

• compact, typically cylindrical, region of magnetic field

• really isolated – magnetic field inside, none outside

• divided by magnetic flux surface

• flux surfaces are material surfaces (in an ideal fluid)

• fluid inside stays inside, fluid outside stays outside, unless leaves through “ends”

Examine magnetic fieldlines

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:

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

Points of return are commensurate – hits same points over and over again; periodicity of lines is same as box.

Fieldlines map out only a line

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 XY- and YZ-planes.

Points of return migrate in X and Y but not Z

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.

Planes remain as planes throughout.

Time sequence:

Contours in YZ-plane

Fieldlines – secondary instability:

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

Fieldlines fill volume during the 3D stages.

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, depending on degree of symmetry present:

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

• Fieldlines lie on surfaces and individual lines do cover the surface

• Fieldlines are volume filling (chaotic)

Structures are not necessarily encased in flux surfaces

There is no easily defined inside and outside

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.

What are the dynamics of leaky structures?