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

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 [email protected]

mailto:[email protected]

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 Rv and RM are huge.

B [Gauss]

T [days]

PM=/ RV RM

Earth/ liquid metals

1.9 1 10-6 109 102

Jupiter 5.3 0.41 10-6 1012 106

Sun 104 27 10-7 1015 108

Disks 10-2 0.1 0.1 1011 1010

Galaxy 10-6 7·1010 1000 ++ 106 109

€

RV =LU

ν

RM =LU

η

PM =RMRV

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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?

QuickTime™ and a decompressor


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 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?

R

H=2R

Bourgoin et al PoF 14 (‘02), 16 (‘04)…

Experimental dynamo in 2007

R ~800, Urms~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: 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: EEAA= < A2 > (+) [A: magnetic potential]

Both HM and EA undergo an inverse cascade (evidence: statistical mechanics, closure models and numerical simulations)

The ElsässerElsässer variables zz± ± = = vv ± ± bb

t z+ + z- .z+ = - P (ideal case)

______ No self interactions [(+,+) or (-,-)]

Alfvén waves: Alfvén waves: z±± = 0 or = 0 or vv = ± = ± bb

Ideal invariants:

E± = < z±± 22 > / 2 = < v > / 2 = < v22 + B + B22 ± 2 ± 2 v.Bv.B > / 2 = E > / 2 = ETT ± H ± Hcc

Numerical set-up

• Periodic boundary conditions, pseudo-spectral code, de-aliased with the 2/3 rule

• Direct numerical simulations from 643 to 15363 grid

points, 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)

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 = b0 cos(x) sin(y) sin(z) by = b0 sin(x) cos(y) sin(z)bz = −2b0sin(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

Two current sheets in near collision

• Ideal case==0

20483 TGsymmetric

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

20483 TGsymmetric 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


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


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


Rate of production of small scales (t)

Spectral inertial index n(t)

Spectra appear shallower than in the Euler case

1) Time-step halved twice2) RK2 and RK4 temporal

scheme

3) Energy spectrum at t=2.5 5123 T-G MHDsymmetric ideal run

(diamonds)versus5123 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]kmax=N/3

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

20483 TG symmetric ideal run, v0 = b0 = 1

• Maximum current

Jmax=f(t)

Exponential phase followed by (steep) power law

(see insert)

Two current sheets in near collision20483 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, …)?



• Weak B between


B-line each 2 pixels

1



• Weak B between



1 2



• Weak B between



1 2 3

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)?

The dissipative case

2 + J2

J2 = f(t)

2

•

*kmax = f(t)

Dissipative caseTaylor-Green flow in MHDEquivalent 20483 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 RV = RM, first TG flow

Low Rv

High Rv

Low Rv High Rv (20483 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

20483 TG Symmetric dissipative run

5123 TG - Different symmetric dissipative run



MHD dissipative ABC+noise decay simulation on 15363 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)



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

J2 2

15363 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 (B2 ~2V2): Alfvén vortices?

(Petviashvili & Pokhotolov, 1992. Solar Wind: Alexandrova et al., JGR 2006)

J2 cos(V, B)Early time (end of ideal phase)

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)

R~ 800

Advection terms

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)

R~ 800

Advection terms

All scales contribute to energy transfer through the Lorentz force

This plateau seems to be absent in decay runs (This plateau seems to be absent in decay runs (Debliquy et al., PoP Debliquy et al., PoP 12,12, 2005 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!

Documents

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