A dynamo without many of the usual ingredients! A public service announcement

1. A dynamo without many of the usual ingredients! A public service announcement Nic Brummell Kelly Cline Fausto Cattaneo Nic Brummell (303) 492-8962 JILA, University of Colorado brummell@solarz.colorado.edu

2. Large-scale dynamo theory We seem to strive very hard to build a large-scale dymamo out of the “usual suspects ingredients”: • a-effects, rotation, differential rotation, turbulent diffusivities, turbulent transport … Everytime we look nonlinearly, our intuitive ideas come up against obstacles: • Turbulent Re stresses are complex • a-effects and turbulent diffusivities are quenched • etc Does a dynamo NEED turbulence to work, or can it work IN SPITE of turbulence?

3. A dynamo! The movie …

4. Bx - ve Bx + ve cf. By(t=0)=1! A dynamo! Strong magnetic field maintained! Strong toroidal field is generated in a cyclic manner Polarity of the strong field reverses

5. A dynamo! Longer time … Diffusion time ~ 300 time units => even more convincing is a dynamo Remarkably, also shows periods of reduced activity!

6. 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

7. Large-scale dynamo: intuitive picture

8. By - By + Model configuration: Localised velocity shear Configurations used: Build one strong structure Have field that will diffuse Velocity shear: Early work:U(y,z) = f(z) cos(2 p 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) Sawtooth profile { By =

9. 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.

10. Weak initial field: Non-static quasi-equilibrium Bx (shaded, +ve dark) (By,Bz) arrows Density perturbation (shaded, +ve dark) (v,w) arrows 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

11. Weak initial field: Non-static quasi-equilibrium System eventually decays due to diffusion between the By = +/- parts (hence quasi-equilibrium)

12. 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 K-H Hydrodynamic instability but magnetically induced (=> non-kinematic)

13. 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

14. K-H 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 Feedback loop created for dynamo! So … let it run …

15. A dynamo! The movie …

16. Bx - ve Bx + ve cf. By(t=0)=1! A dynamo! Strong magnetic field maintained! Strong toroidal field is generated in a cyclic manner Polarity of the strong field reverses

17. A dynamo! Longer time … Diffusion time ~ 300 time units => even more convincing is a dynamo Remarkably, also shows periods of reduced activity!

18. 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

19. 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

20. An even wackier dynamo!

21. 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 BECAUSEof magnetic buoyancy

22. 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 “a-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  )

23. 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 See Geoff’s talk next! 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?

26. 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”

27. 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.

28. 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 15 fieldlines stacked vertically onto YZ-plane (i.e. viewed from the end) Projection of 1 fieldline onto XY-plane (i.e. viewed from above)

29. Fieldlines – primary instability: Recurrence maps of 15 fieldlines stacked vertically in XY- and YZ-planes. Points of returnmigratein X and Y but not Z Fieldlines map out aPLANE, i.e. FLUX SURFACES. Projection of 15 fieldlines stacked vertically onto YZ-plane (i.e. viewed from the end) Projection of 1 fieldline onto XY-plane (i.e. viewed from above)

30. Fieldlines – primary instability: Time sequence: Recurrence maps of 15 fieldlines (stacked vertically) in YZ-planes. Planes remain as planes throughout. Contours in YZ-plane

31. Fieldlines – secondary instability: Recurrence maps of 15 fieldlines (stacked vertically) in YZ-planes. Fieldlines fill volume during the 3D stages. Time sequence:

32. 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?)

33. 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!

34. 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 actuallyHELP in many problematic circumstances, e.g. axisymmetric rise of a flux tube. What are the dynamics of leaky structures?