1 / 43

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

This paper explores the ideal and dissipative dynamics of the generalization of the Taylor-Green vortex to magnetohydrodynamics (MHD). It covers magnetic fields in the universe, MHD equations and properties, numerical simulations, dissipation and structures, energy transfer, and concludes with a discussion of magnetic fields in astrophysics.

Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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!

More Related