1. Magnetic structures created by localised velocity shear and magnetic buoyancy Nic Brummell Kelly Cline Fausto Cattaneo Nic Brummell (303) 492-8962 JILA, University of Colorado brummell@solarz.colorado.edu

2. 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

3. 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

4. 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 p y/y m ) where f(z) is a polynomial function chosen to confine the shear to a particular layer between z u and z l (and to be sufficiently continuous) Shear flow is hydrodynamically stable Then add an initial magnetic field: B 0 = (0, B y, 0) with B y = 1 Model: Localised velocity shear B y + * Add term in the equations that induces desired flow in absence of magnetic effects

5. 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 B x is created from weak B y by the inductance of the shear:  t B x ~ B y d y U 0

6. Induction of strong toroidal field by shear

7. Evolution What happens next depends upon the parameters!

8. Low Rm: Non-static equilibrium A very boring movie!

9. Low Rm: Non-static equilibrium A balance between stretching production and diffusive removal of B x is achieved. However, advection plays a role: buoyancy-driven roll cells make the equilibrium non-static. Magnetic field is dynamically active

10. Increasing Rm: Magnetic buoyancy instability A 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

11. 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

12. 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

13. Dynamo Hmmm… A possible dynamo mechanism?

14. Model: Localised velocity shear 2 Sawtooth profile B y - B y + Other configurations used too: Velocity shear: Early work: U(y,z) = f(z) cos(2 p y/y m ) Dynamo work: U(y,z) = f(z) [sawtooth(y)] Magnetic field: B 0 =(0,B y,0) Early work: B y = 1 Dynamo work: +1 (z>0.5) - 1 (z<0.5) { B y =

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

16. Stronger initial field: A dynamo!

17. A dynamo! Strong magnetic field maintained! Strong toroidal field is generated in a cyclic manner Polarity of the strong field reverses B x - veB x + ve cf. B y (t=0)=1!

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

19. 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).

20. 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.

21. 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)

22. 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)

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

24. 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:

25. 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?)

26. 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!

27. 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?