1. A generalization of the Taylor-Green vortex to MHD: ideal and dissipative dynamics Annick Pouquet Alex Alexakis*, Marc-Etienne Brachet*, Ed Lee, Pablo Mininni^ & Duane Rosenberg * ENS, Paris ^ Universidad de Buenos Aires Cambridge, October 31st, 2008 pouquet@ucar.edupouquet@ucar.edu

2. OUTLINE Magnetic fields in the Universe The MHD equations and some of their properties Numerical simulations in the ideal case Dissipation and structures Energy transfer Conclusion

3. Magnetic fields in astrophysics The generation of magnetic fields occurs in media for which the viscosity and the magnetic diffusivity  are vastly different, and the kinetic and magnetic Reynolds numbers R v and R M are huge. B [Gauss] T [days] P M = /  RVRV RMRM Earth/ liquid metals 1.9110 -6 10 9 10 2 Jupiter5.30.4110 -6 10 12 10 6 Sun10 4 2710 -7 10 15 10 8 Disks10 -2 0.1 10 1110 Galaxy10 -6 7·10 10 1000 ++10 6 10 9

4. Many parameters and dynamical regimes Many scales, eddies and waves interacting * The Sun, and other stars * The Earth, and other planets - including extra-solar planets The solar-terrestrial interactions, the magnetospheres, …

5. Predictions of the next solar cycle, due (or not) to the effect of long-term memory in the system (Wang and Sheeley, 2006) How strong will be the next solar cycle?

6. Surface (1 bar) radial magnetic fields for Jupiter, Saturne & Earth versus Uranus & Neptune (16-degree truncation, Sabine Stanley, 2006) Axially dipolar Quadrupole ~ dipole

7. Brunhes Jamarillo Matuyama Olduvai Reversal of the Earth’s magnetic field over the last 2Myrs (Valet, Nature, 2005) Temporal assymmetry and chaos in reversal processes

8. Taylor-Green turbulent flow at Cadarache Numerical dynamo at a magnetic Prandtl number P M =/=1 (Nore et al., PoP, 4, 1997) and P M ~ 0.01 (Ponty et al., PRL, 2005). I n liquid sodium, P M ~ 10 -6 : does it matter?   R H=2R Bourgoin et al PoF 14 (‘02), 16 (‘04)… Experimental dynamo in 2007 R ~800, U rms ~1, ~80cm

9. The MHD equations P is the pressure, j = ∇ × B is the current, F is an external force, ν is the viscosity, η the resistivity, v the velocity and B the induction (in Alfvén velocity units); incompressibility is assumed, and .B = 0. ______ Lorentz force

10. The MHD invariants ( =  =0) * Energy: E T =1/2 (direct cascade to small scales, including in 2D) * Cross helicity: H C = (direct cascade) And: * 3D: Magnetic helicity: H M = with B=  x A (Woltjer, mid ‘50s) E A * 2D: E A = (+) [A: magnetic potential] Both H M and E A undergo an inverse cascade (evidence: statistical mechanics, closure models and numerical simulations)

11. Elsässer z ± = v ± b The Elsässer variables z ± = v ± b  t z + + z -.  z + = -  P (ideal case) ______ No self interactions [(+,+) or (-,-)] Alfvén waves: ± = 0 or v = ± b Alfvén waves: z ± = 0 or v = ± b Ideal invariants: ± 2 > / 2 = / 2 = E T ± H c E ± = / 2 = / 2 = E T ± H c

12. Numerical set-up Periodic boundary conditions, pseudo-spectral code, de-aliased with the 2/3 rule Direct numerical simulations from 64 3 to 1536 3 grid points, and to an equivalent 2048 3 with imposed symmetries No imposed uniform magnetic field (B 0 =0) V and B in equipartition at t=0 (E V =E M ) Decay runs (no external forcing), and =  Taylor-Green flow (experimental configuration) Or ABC flow + random noise at small scale or 3D Orszag-Tang vortex (neutral X-point configuration)

13. A Taylor-Green flow for MHD v(x, y, z ) = v 0 [(sin x cos y cos z )e x (cos x sin y cos z )e y, 0] Taylor & Green, 1937; M.E. Brachet, C. R. Acad. Sci. Paris 311, 775 (1990) And, for example, b x = b 0 cos(x) sin(y) sin(z) b y = b 0 sin(x) cos(y) sin(z) b z = −2b 0 sin(x) sin(y) cos(z) Lee et al., ArXiv 0802.1550, Phys. Rev. E, to appear * Current j =  b contained within what can be called the impermeable (insulating) box [0, π] 3 * Mirror and rotational symmetries allow for computing in the box [0, π/2] 3 : sufficient to recover the whole (V,B) fields

14. Two current sheets in near collision Ideal case =  =0 2048 3 TG symmetric

15. Fit to spectra: E(k,t)=C(t)k -n(t) exp[-2  (t)k]  (t) ~ exp[-t/  ] n(t) resolution limit on a given grid 2048 3 TG symmetric ideal run Rate of production of small scales  (t) Spectral inertial index n(t)

16. Fit: E(k,t)=C(t)k -n(t) exp[-2  (t)k]  (t) ~ exp[-t/  ] n(t) resolution limit 2048 3 TG symmetric ideal run Rate of production of small scales And spectral inertial index

17. Fit: E(k,t)=C(t)k -n(t) exp[-2  (t)k]  (t) ~ exp[-t/  ] n(t) resolution limit 2048 3 TG symmetric ideal run Rate of production of small scales And spectral inertial index

18. Fit to spectra: E(k,t)=C(t)k -n(t) exp[-2  (t)k]  (t) ~ exp[-t/  ] n(t) resolution limit on a given grid 2048 3 TG symmetric ideal run Rate of production of small scales  (t) Spectral inertial index n(t) Spectra appear shallower than in the Euler case

19. 1) Time-step halved twice 2) RK2 and RK4 temporal scheme 3) Energy spectrum at t=2.5 512 3 T-G MHD symmetric ideal run (diamonds) versus 512 3 Full DNS (solid line) How realistic is this break-point in time evolution of 

20. E(k,t)= C(t)k -n(t) exp[-2  (t)k] k max =N/3

21. E(k,t)=C(t)k -n(t) exp[-2  (t)k] k max =N/3

22. 2048 3 TG symmetric ideal run, v 0 = b 0 = 1 Maximum current J max =f(t) Exponential phase followed by (steep) power law (see insert)

23. Two current sheets in near collision 2048 3 TG, symmetric ideal run

24. A magnetic quasi rotational discontinuity behind the acceleration of small scales Strong B outside (purple) Weak B between the two current sheets B-line every 2 pixels Rotational discontinuity, as observed in the solar wind (Whang et al., JGR 1998, …)?

25. A magnetic quasi rotational discontinuity behind the acceleration of small scales Strong B outside (purple) Weak B between the two current sheets B-line each 2 pixels 1

26. A magnetic quasi rotational discontinuity behind the acceleration of small scales Strong B outside (purple) Weak B between the two current sheets B-line each 2 pixels 12

27. A magnetic quasi rotational discontinuity behind the acceleration of small scales Strong B outside (purple) Weak B between the two current sheets B-line each 2 pixels 123

28. Some conclusions for the ideal case in MHD * Need for higher resolution and longer times with more accuracy * Can we start from the preceding resolution run at say k max /x? * Could we use a filter (instead of dealiasing 2/3 rule) (hyperviscosity?)? * What about other Taylor-Green MHD configurations? (in progress) * What about other flows (e.g., Kerr et al., …; MHD-Kida flow, … ? * What is a good candidate for an eventual blow-up in MHD? Is a rotational discontinuity a possibility? * Effect of v-B correlation growth (weakening of nonlinear interactions)?

29. The dissipative case

30.  2 + J 2 J 2 = f(t)  2  *k max = f(t) Dissipative case Taylor-Green flow in MHD Equivalent 2048 3 grid

31. The energy dissipation rate  T decreases at large Reynolds number * The decay of total energy is slow: t -0.3 Energy dissipation rate in MHD for several R V = R M, first TG flow Low R v High R v

32. Low R v High R v (2048 3 equiv. grid) A different Taylor Green flow in MHD, again with imposed symmetries The energy dissipation rate  T is ~ constant at large Reynolds number 2D-MHD: Biskamp et al., 1989, Politano et al., 1989

33. Scaling with Reynolds number of energy dissipation in MHD

34. 2048 3 TG Symmetric dissipative run

35. 512 3 TG - Different symmetric dissipative run

36. MHD dissipative ABC+noise decay simulation on 1536 3 grid points Visualization freeware: VAPOR http://www.cisl.ucar.edu/hss/dasg/software/vapor Zoom on individual current structures: folding and rolling-up Mininni et al., PRL 97, 244503 (2006) Magnetic field lines in brown At small scale, long correlation length along the local mean magnetic field (k // ~ 0)

37. Hasegawa et al., Nature (2004); Phan et al., Nature (2006), … Recent observations (and computations as well) of Kelvin-Helmoltz roll-up of current sheets

38. Current and vorticity are strongly correlated in the rolled-up sheet J2J2 22 1536 3 dissipative run, early time VAPOR freeware, cisl.ucar.edu/hss/dasg/software/vapor

39. V and B are aligned in the rolled-up sheet, but not equal (B 2 ~2V 2 ): Alfvén vortices? (Petviashvili & Pokhotolov, 1992. Solar Wind: Alexandrova et al., JGR 2006) J2J2 cos(V, B) Early time (end of ideal phase)

40. Rate of energy transfer in MHD 1024 3 runs, either T-G or ABC forcing (Alexakis, Mininni & AP; Phys. Rev. E 72, 0463-01 and 0463-02, 2005) R ~ 800 Advection terms

41. Rate of energy transfer in MHD 1024 3 runs, either T-G or ABC forcing (Alexakis, Mininni & AP; Phys. Rev. E 72, 0463-01 and 0463-02, 2005) R ~ 800 Advection terms All scales contribute to energy transfer through the Lorentz force This plateau seems to be absent in decay runs (Debliquy et al., PoP 12, 2005)

42. Second conclusion: need for more numerical resolution and ideas Temporal evolution of maximum of current and vorticity and of logarithmic decrement points to a lack of evidence for singularity in these flows as yet Constant energy dissipation as a function of Reynolds number Piling, folding & rolling-up of current & vorticity sheets Energy transfer and non-local interactions in Fourier space Energy spectra and anisotropy Strong intermitency in MHD Role of strong imposed uniform field? Role of magnetic helicity? Of v-B correlations? (Both, invariants)

43. Thank you for your attention!