A generalization of the Taylor-Green vortex to MHD: ideal and dissipative dynamics

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.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 Rv and RM are huge.

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

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

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

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

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

9. The MHD equations • P is the pressure, j = ∇ × B is the current, F is an external force, ν is the viscosity, ηthe 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: ET=1/2< v2 + B2 > (direct cascade to small scales, including in 2D) * Cross helicity: HC= < v.B > (direct cascade) And: * 3D: Magnetic helicity: HM=< A.B > with B=  x A (Woltjer, mid ‘50s) * 2D: EA= < A2 > (+)[A: magnetic potential] Both HM and EAundergo an inverse cascade (evidence: statistical mechanics, closure models and numerical simulations)

11. The Elsässer variablesz± = v ± b tz+ + z- .z+= - P (ideal case) ______ No self interactions [(+,+) or (-,-)] Alfvén waves: z± = 0 or v = ± b Ideal invariants: E±= < z±2 > / 2 = < v2 + B2 ± 2 v.B > / 2 = ET ± Hc

12. Numerical set-up • Periodic boundary conditions, pseudo-spectral code, de-aliased with the 2/3 rule • Direct numerical simulations from 643 to 15363gridpoints, and to an equivalent 20483 with imposed symmetries • No imposed uniform magnetic field (B0=0) • V and B in equipartition at t=0 (EV=EM) • 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 ) = v0[(sin x cos y cos z )ex(cos x sin y cos z )ey, 0] Taylor & Green, 1937; M.E. Brachet, C. R. Acad. Sci. Paris 311, 775 (1990) And, for example, bx = b0cos(x) sin(y) sin(z) by = b0sin(x) cos(y) sin(z) bz = −2b0sin(x) sin(y) cos(z) Lee et al., ArXiv 0802.1550, Phys. Rev. E, to appear * Currentj = b contained within what canbe 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 20483 TG symmetric

15. 20483 TG symmetric ideal run Rate of production of small scales (t) Spectral inertial index n(t) (t) ~ exp[-t/] resolution limit on a given grid Fit to spectra: E(k,t)=C(t)k-n(t)exp[-2(t)k] n(t)

16. 20483 TG symmetric ideal run Rate of production of small scales And spectral inertial index (t) ~ exp[-t/] resolution limit Fit: E(k,t)=C(t)k-n(t)exp[-2(t)k] n(t)

17. 20483 TG symmetric ideal run Rate of production of small scales And spectral inertial index (t) ~ exp[-t/] resolution limit Fit: E(k,t)=C(t)k-n(t)exp[-2(t)k] n(t)

18. 20483 TG symmetric ideal run Rate of production of small scales (t) Spectral inertial index n(t) (t) ~ exp[-t/] resolution limit on a given grid Fit to spectra: E(k,t)=C(t)k-n(t)exp[-2(t)k] n(t) Spectra appear shallower than in the Euler case

19. How realistic is this break-point in time evolution of 1) Time-step halved twice 2) RK2 and RK4 temporal scheme 3) Energy spectrum at t=2.5 5123 T-G MHD symmetric ideal run (diamonds) versus 5123 Full DNS (solid line)

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

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

22. 20483 TG symmetric ideal run, v0=b0= 1 • Maximum current Jmax=f(t) Exponential phase followed by (steep) power law (see insert)

23. Two current sheets in near collision20483 TG, symmetric ideal run

24. A magnetic quasi rotational discontinuity behind the acceleration of small scales • StrongB outside (purple) • WeakB 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 1 2

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 1 2 3

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 kmax/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 +J2 J2 = f(t) 2 *kmax = f(t) Dissipative case Taylor-Green flow in MHD Equivalent 20483 grid

31. Energy dissipation rate in MHD for several RV = RM,first TG flow Low Rv The energy dissipation rate Tdecreases at large Reynolds number * The decay of total energy is slow: t-0.3 High Rv

32. Low Rv High Rv (20483 equiv. grid) A different Taylor Green flow in MHD, again with imposed symmetries The energy dissipation rate Tis ~ 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. 20483 TG Symmetricdissipativerun

35. 5123 TG-Differentsymmetricdissipativerun

36. MHD dissipative ABC+noise decay simulation on 15363gridpoints Visualization freeware: VAPORhttp://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. Recent observations (and computations as well) of Kelvin-Helmoltz roll-up of current sheets Hasegawa et al., Nature (2004); Phan et al., Nature (2006), …

38. Current and vorticity are strongly correlatedin the rolled-up sheet 2 J2 15363dissipative 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 (B2 ~2V2): Alfvén vortices?(Petviashvili & Pokhotolov, 1992. Solar Wind: Alexandrova et al., JGR 2006) J2 cos(V, B) Early time (end of ideal phase)

40. Rate of energy transfer in MHD10243 runs, either T-G or ABC forcing(Alexakis, Mininni & AP; Phys. Rev. E 72, 0463-01 and 0463-02, 2005) Advection terms R~ 800

41. Rate of energy transfer in MHD10243 runs, either T-G or ABC forcing(Alexakis, Mininni & AP; Phys. Rev. E 72, 0463-01 and 0463-02, 2005) Advection terms R~ 800 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!