1. A Few Issues in MHD Turbulence
Alex Alexakis*, Bill Matthaeus%, Pablo Mininni^ , Duane Rosenberg
and Annick Pouquet
NCAR / CISL / TNT
* Observatoire de Nice
% Bartol, U. Delaware
^ Universidad de Buenos Aires
Warwick, September 17, 2007

2. A Few Issues in MHD Turbulence: The strong and the weak

3. * Introduction * Conclusion
Some examples of MHD turbulence in astro, geophysics & the lab.
Equations, invariants, exact laws, phenomenologies
* Direct Numerical Simulations, mostly for the decay case
Is there any measurable difference with fluid turbulence?
What are the features of an MHD flow, both spatially and spectrally?
Discussion
Accessing high Reynolds numbers for better scaling laws
Can modeling of MHD flows help understand their properties?
An example : The generation of magnetic fields at low PM
Can adaptive mesh refinement help unravel characteristic features of MHD?
* Conclusion

4. 1 parsec ~ 3.26 light years ~ 3 1013 km
Observations of galactic magnetic fields (after Brandenburg & Subramanian, 2005)

5. Cyclical reversal of the solar magnetic field over the last 130 years
Prediction of the next solar cycle, because of long-term memory in the system (Dikpati, 2007)

6. HINODE / Sunrise
November 2006

8. Surface (1 bar) radial magnetic fields for Jupiter, Saturne & Earth, Uranus & Neptune (16-degree truncation, Sabine Stanley, 2006)
Axially dipolar
Pablo, duane compare location and intensity of jmax
Quadrupole ~ dipole

9. Brunhes Jamarillo Matuyama Olduvai
Reversal of the Earth’s magnetic field over the last 2Myrs
(Valet, Nature, 2005)
Temporal assymmetry of reversal process
Brunhes Jamarillo Matuyama Olduvai

10. Experimental dynamo within a constrained flow: Riga
Gailitis et al., PRL 84 (2000)
Karlsruhe, with a Roberts flow (see e.g. Magnetohydrodynamics 38, 2002, special issue)

11. Taylor-Green turbulent flow at Cadarache
Bourgoin et al PoF 14 (‘02), 16 (‘04)… W R
H=2R
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

12.   
Small-scale
ITER (Cadarache)

13. The MHD equations [1]
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

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

15. The Elsässer variables z± = v ± b
t z+ + z- .z+ = - P + + ∆ z+ + - ∆ z- + F±
with  ± = [ ± ]/2 and F± = FV ± FM
Invariants:
E± = < z ± 2 >/2 = < v2 + B2 ± 2 v.B >/2

16. Parameters in MHD * Magnetic Prandtl number: PM = RM / RV = ν / η
RV = Urms L0 / ν >> 1
Magnetic Reynolds number RM = Urms L0 / η
* Magnetic Prandtl number: PM = RM / RV = ν / η
PM is high in the interstellar medium.
PM is low in the solar convection zone, in the liquid core of the Earth, in liquid metals and in laboratory experiments.
Energy ratio EM/EV
Uniform magnetic field B0
Amount of magnetic helicity HM

17. <  z-/+L i  zi± 2 > = - [ 4/d ] ± r
Two scaling laws in MHD [1] à la Yaglom (1949), and Antonia et al. (1997)
F(r) = F(x+r) - F(x) : structure function for field F ;
longitudinal component FL(r) = F . r/ |r|
<  z-/+L i  zi± 2 > = - [ 4/d ] ± r
in dimension d, with z± = v ± b, ± = - dtE± = T ± C,
omitting dissipation and forcing and assuming stationarity
[Politano and AP, 1998, Geophys. Res. Lett. 25 (see also Phys. Rev. E 57)]
Note: <  z+  z+  z- > ~ < ( v +  B)2 ( v -  B) > ~ +
A lack of equipartition between kinetic and magnetic energy is observed
in many instances (e.g., in the solar wind)

18. Two scaling laws in MHD [2]
In terms of velocity and magnetic field, these two scaling laws become:
< vLvi2 >+ <vLbi2 > -2 < bLvibi > = - (4/d) Tr
- <bLbi2 > - < bLvi2 >+2 < vLvibi > = - (4/d) c r
with T = - dtET and c = - dtHc
Strong V regime, or strong B regime or Alfvénic (v~B) regime (Ting et al 1986, …);
in the latter case, v-B correlations play a dynamical role
(Politano+AP,GRL 25, 1998; see also Boldyrev, 2006)
Where such laws apply, the energy input T can be measured, e.g. in the solar wind

19. Phenomenologies for MHD turbulence
If MHD is like fluids Kolmogorov spectrum EK41(k) ~ k-5/3 Or
Slowing-down of energy transfer to small scales because of Alfvén waves propagation along a (quasi)-uniform field B0: EIK(k) ~ (T B0)1/2 k-3/2
(Iroshnikov - Kraichnan (IK), mid ‘60s)
transfer ~ NL * [NL/A] , or 3-wave interactions but still with isotropy.
Eddy turn-over time NL~ l/ul and wave (Alfvén) time A ~ l/B0
And
Weak turbulence theory for MHD (Galtier et al PoP 2000): anisotropy develops and the exact spectrum is: EWT(k) = Cw k-2 f(k//)
Note: WT is IK -compatible when isotropy (k// ~ k ) is assumed: NL~ l/ul and A~l// /B0
Or k-5/3 (Goldreich Sridhar, APJ 1995) ? Or k-3/2 (Nakayama, ApJ 1999; Boldyrev, PRL 2006) ?

20. Spectra of three-dimensional MHD turbulence
EK41(k) ~ k-5/3 as observed in the Solar Wind (SW) and in DNS (2D & 3D)
Jokipii, mid 70s, Matthaeus et al, mid 80s, …
EIK(k) ~ k-3/2 as observed in SW, in DNS (2D & 3D), and in closure models
Müller & Grappin 2005; Podesta et al. 2007; Mason et al. 2007; Yoshida 2007
EWT(k) ~ k as may have been observed in the Jovian magnetosphere, and
recently in a DNS, Mininni & AP arXiv: v1, astro-ph (see later)
Is there a lack of universality in MHD turbulence (Boldyrev; Schekochihin)? If so, what are the parameters that govern the (plausible) classes of universality? The presence of a strong guiding uniform magnetic field?
* Can one have different spectra at different scales in a flow?
* Is there it a lack of resolving power (instruments, computers)?
* Is an energy spectrum the wrong way to analyze / understand MHD?

21. Recent results using direct numerical simulations and models of MHD

22. Numerical set-up
Periodic boundary conditions, pseudo-spectral code, de-aliased with the 2/3 rule
From 643 to grid points
No imposed uniform magnetic field
Either decay runs or forcing
ABC flow (Beltrami) + random noise at small scale, with V and B in equipartition (EV=EM), or Orszag-Tang (X-point configuration), or Taylor-Green flow

23. Temporal evolution of maximum of current and vorticity
Energy transfer and non-local interactions in Fourier
space
Temporal evolution of maximum of current and
vorticity
Energy dissipation and scaling laws
Piling, folding & rolling-up of current and vorticity sheets
Alignment of v and B fields in small-scale structures
Energy spectra and anisotropy
Intermittency
Beyond Moore’s law

24. Energy Transfer
Let uK(x) be the velocity field with wave numbers in the range
K < |k| < K+1 Q K
K+1
Q+1
^ Isotropy
^ Sharp filters
Fourier space

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

26. Rate of energy transfer Tub(Q,K) from u to b for different K shells
Scaling for NS is like R^0.5
The magnetic field at a given scale receives energy in equal amounts from the velocity field from all larger scales (but more from the forcing scale).
The nonlocal transfer represents ~25% of the total energy transfer at this Reynolds number (the scaling of this ratio with Reynolds number is not yet determined).

27. Jmax for the Orszag-Tang flow Mininni et al
Jmax for the Orszag-Tang flow Mininni et al., PRL 97, (2006) (res.: 1283 to 10243)
Growth of the current maximum: linear phase, followed by an algebraic tn regime with n ~ 3

28. Jmax for a random flow, resolutions up to 15363 grid points (RV from 690 to 10100)
OT flow
Growth of the current maximum: linear phase followed by an algebraic tn regime with n ~ 3

29. AMR in incompressible 2D - MHD turbulence at R~1000
Adaptive Mesh Refinement using spectral elements of different orders
Accuracy matters when looking at Max norms, here the current, although there are no noticeable differences on the L2 norms (for both the energy and enstrophy)
Rosenberg et al., New J. Phys. 2007

30. Energy dissipation rate in MHD for several RV OT- vortex
Low Rv
High Rv
Orszag-Tang simulations at different Reynolds numbers (X factor of 10)
Is the energy dissipation rate T constant in MHD at large Reynolds number (Mininni + AP, arXiv: v1, astro-ph), as presumably it is in 2D-MHD in the reconnection phase?
There is evidence of constant  in the hydro case (Kaneda et al., 2003)

31. Scaling with Reynolds number of times for max. current & dissipation
Time Tmax(1) at which
global maximum of dissipation
is reached in (ABC+ random) flow
and
Time Tmax(2) at which the current
reaches its first maximum
Both scale as Rv0.08

32. Scaling with Reynolds number of dissipation

33. MHD decay simulation @ NCAR on 15363 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)

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

35. Current and vorticity are strongly correlated in the rolled-up sheet
Current J2
Vorticity 2
15363 run, early time

36. V and B are aligned in the rolled-up sheet, but not equal (B2 ~2V2): Alfvén vortices? (Petviashvili & Pokhotolov, Solar Wind: Alexandrova et al., JGR 2006)
combination of Bessel functions
Current J2
cos(V, B)
15363 run, early time

37. Hc : Velocity - magnetic field correlation
PdFs of cos(v,B):
Flow with weak normalized total cross helicity Hc
Flow with strong Hc
Matthaeus et al. arXiv.org/abs/

38. Velocity - magnetic field correlation [3]
Local map in 2D
of v & B alignment:
|cos (v,B)| > 0.7 (black/white)
(otherwise, grey regions).
even though the global normalized correlation coefficient is ~ 10-4
Quenching of nonlinear terms in MHD
(Meneguzzi et al., J. Comp. Phys.
123, 32, 1996)
similar to the Beltramisation (v // ) of fluids

39. Vorticity =xv & Relative helicity intensity h=cos(v,)
Local v- alignment (Beltramization). Tsinober & Levich, Phys. Lett. (1983); Moffatt, J. Fluid Mech. (1985); Farge, Pellegrino, & Schneider, PRL (2001), Holm & Kerr PRL (2002).
--> no mirror symmetry, together with weak nonlinearities in the small scales
Blue h> 0.95 Red h<-0.95

40. Strong relative magnetic helicity (~  1): change of topology across sheet
Current J2
cos(A, B) , with B=xA
15363 run, early time

41. Current at peak of dissipation
Zoom
Global view

42. MHD scaling at peak of dissipation [1]
Energy spectra compensated by k3/2
Solid: ET
Dash: EM
Dot: EV
Insert: energy flux
Dash-dot: k5/3-compensated
10% smaller for V for the 2 lengths
LintM ~ 3.1
M ~ 0.4

43. MHD scaling at peak of dissipation [2]
Anomalous exponents of structure functions for Elsässer variables, with isotropy assumed (similar results for v and B)
Note 4 ~ 0.98  0.01,
i.e. far from fluids and with more intermittency

44. MHD decay run at peak of dissipation [3]
Isotropy ratio
RS = S2b / S2b//
Isotropy obtains in the large scales, whereas anisotropy develops at smaller scales
RS is proportional to the so-called Shebalin angles
LintM ~ 3.1, M ~ 0.4

45. Structure functions at peak of dissipation [4]
(Mininni + AP, arXiv: v1, astro-ph)
LintM ~ 3.1 , M ~ 0.4
L1/2 -compensation of S2b
 and // structure fns.
Flat at large scales,
with equipartition of the  and // components, hence
E(k) ~ k-3/2 ,
isotropic IK (1965) spectrum
Solid: perpendicular
Dash: parallel
Insert: l 2/3-compensated

46. Structure functions at peak of dissipation [5]
(Mininni + AP, arXiv: v1, astro-ph)
LintM ~ 3.1 , M ~ 0.4
Structure function S2 ,
with 3 ranges:
L2 (regular) at small scale
L at intermediate scale, as for weak turbulence: Ek~k-2.
Is it the signature of weak MHD turbulence?
L1/2 at largest scales (Ek~ k-3/2)
Insert: anisotropy ratio
Solid: perpendicular
Dash: parallel

47. Solid: perpendicular Dash: parallel
Solid: parallel, Dash: perpendicular, LintM ~ 3.1, M ~ 0.4
Solid: perpendicular
Dash: parallel

48. Kolmogorov-compensated Energy Spectra: k5/3 E(k)
Navier-Stokes, ABC forcing
Small Kolmogorov k-5/3 law
(flat part of the spectrum)
K41 scaling increases in range, as the Reynolds number increases
Bottleneck at dissipation scale
Solid: 20483, Rv= 104, R~ 1200
Dash: 10243, Rv=4000
Kolmogorov k-5/3 law
Linear resolution: X 2
Cost: X 16
Mininni et al., submitted

49. Evidence of weak MHD turbulence
with a k-2 spectrum
in Reduced MHD computations (Dmitruk et al., PoP 10, 2003)
in the Jovian magnetosphere (Saur et al., Astron. Astrophys. 386, 2002)

50. Extreme events at high Rv unraveled by high-resolution runs (grid from 483 to 15363 points)

51. Discussion
Can we derive a dynamic model for the Kelvin-Helmoltz rolling-up of current and vorticity sheets?
Are Alfvén vortices, as observed for example in the magnetosphere, present in MHD at high Reynolds number?
What is the physical origin of the tn ~ 3 evolution of the current and vorticity maxima in MHD (time-dependent velocity shear?)?
Is another scaling range possible at scales smaller than where the weak turbulence spectrum is observed?
What is the effect of the non-locality of nonlinear energy transfer observed in MHD on the flow dynamics, e.g. on the dynamo problem (generation of magnetic fields)?
Quantification of anisotropy in MHD, including in the absence of a large-scale magnetic field? How much // vs. perp. transfer is there?
Large-scale coherent forcing versus random forcing, i.e. universality?

52. Thank you for your attention!