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

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

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 Jupiter Sun Disks Galaxy ·

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, …

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?

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

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

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 ~ : does it matter?   R H=2R Bourgoin et al PoF 14 (‘02), 16 (‘04)… Experimental dynamo in 2007 R ~800, U rms ~1, ~80cm

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

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)

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

Numerical set-up Periodic boundary conditions, pseudo-spectral code, de-aliased with the 2/3 rule Direct numerical simulations from 64 3 to grid points, and to an equivalent 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)

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 , 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

Two current sheets in near collision Ideal case =  = TG symmetric

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 TG symmetric ideal run Rate of production of small scales  (t) Spectral inertial index n(t)

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

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

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 TG symmetric ideal run Rate of production of small scales  (t) Spectral inertial index n(t) Spectra appear shallower than in the Euler case

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

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

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

Two current sheets in near collision TG, symmetric ideal run

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, …)?

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

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

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

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)?

The dissipative case

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

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

Low R v High R v ( 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

Scaling with Reynolds number of energy dissipation in MHD

TG Symmetric dissipative run

512 3 TG - Different symmetric dissipative run

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

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

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

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

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

Rate of energy transfer in MHD runs, either T-G or ABC forcing (Alexakis, Mininni & AP; Phys. Rev. E 72, and , 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)

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)

Thank you for your attention!