1. Magnetohydro- dynamic instability in stars Jonathan Braithwaite MPA Garching

2. Summary of talk ● Instability in various magnetic fields: – Poloidal fields – Toroidal fields ● Differential-rotation-driven dynamo ● Conclusions

3. Equilibrium and stability ● An arbitrary magnetic field is not in equilibrium, as the Lorentz and pressure forces will not generally be balanced. It will change on an Alfv é n time-scale. ● Certain equilibrium field configurations have been shown to be unstable in non-rotating stars, including: – All purely poloidal fields 1, – All purely toroidal fields 2. 1. Wright 1973, Markey & Tayler 1973. 2. Tayler 1973.

4. Instability of a poloidal magnetic field A purely poloidal field is unstable, as one half of the star can rotate with respect to the other, and the magnetic energy outside the star goes down. A poloidal field of uniform direction inside the star with a potential (zero current) field in the atmosphere

5. Form of instability in poloidal field

6. Plan ● It has been shown analytically that any purely poloidal field in a non-rotating star is unstable ● In a rotating star, analytic results inconclusive ● Use numerical MHD to follow evolution of a poloidal magnetic field in a rotating star ● But first, look at non-rotating star

7. The model ● Star modelled as self- gravitating ball of plasma, (a polytrope of index n=3,) embedded in a hot, non-conductive atmosphere. ● Fitted into cubic computational box of side 4.5R *.

8. Initial magnetic field Field strength: ratio of thermal to magnetic energy ~ 400. Still in `weak' regime, but stronger than in most real stars to speed things up computationally Poloidal, of form inside the star, potential (zero current) field outside star

9. Numerical code ● Adapted from a code by Nordlund & Galsgaard (1995). ● Grid-based, finite-difference MHD code. ● Cartesian coordinates: – No singularities or special points, – Simpler, and therefore faster, – Some wastage at corners of computational box.

10. Simulation of decay of poloidal field (not rotating) Frames at t = 0, 4.4, 5.0 and 5.7 Alfv é n crossing times Red & blue show regions of +ve and -ve B r

11. Rotating star ● Solid-body rotation has no significant effect on stability unless at least comparable to A (Frieman & Rotenberg 1960) ● If >> A, is the field stabilised? ● Examine numerically: – Transform to rotating frame by adding Coriolis force – Do not add centrifugal force, which is an unnecessary complication ● Look first at aligned and then at oblique rotators

12. Maps of B r on stellar surface white = +ve B r black = -ve B r Left: = 0 Right: = 0.5 crit = 10 A

13. Azimuthal Fourier modes of v z above: amplitudes of m=5,9,13,17,21 below: growth rates left: = 0 right: = 0.5 crit = 10 A

14. Rotation only slows loss of magnetic energy Total magnetic energy against time, for five values of / crit : 0, 0.125, 0.25, 0.5 and 1.

15. The oblique rotator The field is still unstable when the rotation and magnetic axes are not aligned Total magnetic energy against time, for three values of the angle  between the rotation and magnetic axes: 0º, 45º and 90º.

16. Different initial magnetic fields Below: uniform field inside star, potential (zero current) outside Above: a field given by ABAB typ e

17. Different initial magnetic fields Magnetic energy against time, for simulations using fields of types A and B, both rotating ( / crit = 0.5) and non-rotating. Clearly these fields are unstable too. Conclusion: no evidence has been found for stable poloidal fields in rotating stars This result contradicts a recent study (Geppert & Rheinhardt 2006) which finds that rotation can stabilise a poloidal field. However, that study used a non-compressible fluid.

18. Instability of toroidal field A magnetic field exerts a positive pressure perpendicular to its direction. In a toroidal field, this creates a situation like that in a compressed spinal column. Consequently: any purely toroidal field is subject to instability on or near the magnetic axis. This is called the Tayler instability

19. Tayler instability modes

20. Simulation of toroidal-field instability ● Instead of modelling the whole star, model part of the star around the magnetic axis ● B = B( ,z) e  and B ∝  at small  ● Addition of gravity and rotation, and magnetic and thermal diffusion possible ● Field is initially in equilibrium; small perturbation is added at t=0 to the velocity field

21. Surfaces of constant initial radial coordinate  ; frames separated by roughly an Alfv é n time- scale

22. Unstable modes Amplitude (in displacement field) against time Only the m=1 mode is unstable

23. Quantitative results ● In absence of gravity, rotation and diffusion, growth rate is proportional to field strength and equal to ~0.9 A where A = v  /  ● Now investigate the effects of rotation, gravity (stable stratification) and diffusion...

24. Unstable wavelengths ● We expect instability at the following (vertical) wavenumbers: ● A > n 2 > N 2 / A 2 r 2 ● Magnetic diffusion (i.e. finite conductivity) provides a maximum unstable wavenumber ● Run code with various values of magnetic diffusivity  and measure growth rate at a particular wavenumber ● A /  = n 2 if  1.6 x 10 -3 ● Result not quite as expected, the instability is not entirely suppressed, perhaps to do with zero thermal conductivity Expected cutoff Growth rate of the m=1, n=20 mode against magnetic diffusivity

25. Gravity ● Instability does work against gravity, which is highest at long vertical wavelengths. Stratification produces therefore a minimum unstable wavenumber ● Run code with various values of gravitational acceleration g and measure growth rate at a particular wavenumber ● Expected cutoff at N = n  A r ● Result as expected

26. Effect of rotation on instability ● Rotation parallel to magnetic axis ● We expect stability in the regime  ≫   ● Code run with   = 0.64, and various values of  ● Instability is suppressed if  >  

27. Rotation and diffusion ● Rotation should suppress instability only in the adiabatic case. ● This can be understood by means of an analogy: in a non-rotating system, a ball on the top of a hill is in unstable equilibrium. In a rotating system, and in the absence if energy dissipation, if the ball is perturbed it will perform epicycles and eventually come back to the top of the hill. But if there is any friction, the ball will not have enough energy to come back to the top. ● Preliminary results from simulations do confirm this. Therefore, it looks likely that any stable field in a rotating star must be of the mixed poloidal-toroidal form.

28. Tayler instability and dynamo effect ● Dynamo loop: – Seed field is wound up by differential rotation, creating a predominantly toroidal field – Toroidal field decays via Tayler instability, creating new poloidal field – New poloidal field is wound up by differential rotation – [Differs from some other dynamos, e.g. `convective' dynamo in the Sun, in that no small-scale velocity field needs to be imposed] ● Differential rotation will eventually disappear unless it is continuously replenished ● This can be investigated by means of numerical MHD

29. The model Gas in box made to rotate by applying a force Begin with a weak seed field

30. Results Log energy density in toroidal (solid line) and poloidal (dashed line) components, against time At first, the toroidal component grows from its zero initial value until it is stronger than the poloidal field. Then the poloidal field strength starts to increase. Both increase until saturation is reached. A self-sustaining field has been created. With  0 = 0 and g = 0 Field reversals?

31. Field reversals Contour plots of azimuthally-averaged components B z and B  (shown with shading and lines respectively) at nine points in time. Gradually +ve B z and B  are replaced by -ve. This process is also observed in many other types of dynamo, e.g. Earth’s mantle

32. Check: is the torque really magnetic? Magnetic and kinetic torques exerted on gas are given by: The magnetic torque is much greater than the kinetic (except during field reversals) Vertically-averaged magnetic (solid line) and kinetic (dotted line) torques plotted against time

33. With rotation:  0  0 Log magnetic energy against time, for four values of  0 /  s : 0, 0.1, 0.3 and 1 (solid, dotted, dashed and dot-dashed lines). We would expect fast rotation to prevent dynamo action, as it slows the growth of the Tayler instability. Here, this does happen - above some rotation speed, the dynamo is suppressed.

34. With gravity: g  0 Log magnetic energy against time, in toroidal and poloidal components (solid and dotted lines) The addition of gravity should stabilise the longer vertical wavelengths. Here, it does not prevent the dynamo reaching saturation, but does appear to change its properties. This is not yet well understood.

35. Some applications of dynamo process ● Could be important in stellar evolution – rotation and magnetism are perhaps the biggest unsolved problem in stellar evolution ● Uniform rotation of solar core (e.g. Schou et al. 1998) ● Core-envelope coupling in AGB stars and supernova progenitors: rotation rates of white dwarfs and neutron stars (e.g. Heger et al. 2005, Yoon & Langer 2005) ● GRBs: are hypermassive neutron stars formed? If so, how long do they last?

36. Differential Rotation in the Sun: Observations Schou et al. 1998 Charbonneau et al. 1999

37. Summary of results ● No evidence of a stable poloidal magnetic field, even in a rotating star. ● Many properties of Tayler instability in toroidal fields confirmed, and some holes in our theoretical understanding identified. ● Existence of differential-rotation-driven dynamo confirmed. However, detail still very unclear. More work needed!