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A Few Issues in MHD Turbulence and how to cope with them using computers Annick Pouquet
Alex Alexakis!, Julien Baerenzung&, Marc-Etienne Brachet!,
Jonathan Pietarila-Graham&&, Aimé Fournier, Darryl Holm@, Giorgio Krstulovic*,
Ed Lee#, Bill Matthaeus%, Pablo Mininni^, Jean-François Pinton!!,
Hélène Politano*, Yannick Ponty* and Duane Rosenberg
! and !! ENS, Paris and Lyon
& MPI, Postdam
&& LANL
@ Imperial College & LANL
* Observatoire de Nice
# U. Leuwen
% Bartol, U. Delaware
^ Universidad de Buenos Aires
Les Houches, February 28, 2011
[email protected]
•  Introduction
•  Some examples of MHD turbulence in astrophysics,
geophysics and the laboratory
•  Invariants, cascades, exact laws, …
•  Features of an MHD flow, both spectrally and spatially
* Breaking of universality in MHD turbulence?
* Need for a paradigm shift in MHD turbulence?
•  How can various physical and numerical modeling
techniques help in MHD turbulence?
* Conclusion
(Geophysical & astrophysical) (MHD) Turbulence •  Mul9-­‐scale interac9ons -­‐ non-­‐linear phenomena •  Lack of predictability -­‐ spa.o-­‐temporal chao.c behavior •  Extreme events -­‐ non-­‐Gaussian sta.s.cs •  Abnormal transport proper9es (momentum, chemical tracers, …) -­‐ self-­‐similar (power) law dependencies or mul.-­‐
fractal, eddy viscosity, eddy noise, … Extreme events as a func9on of Reynolds number (and thus, of resolu9on) for 3D MHD
•  Probability Distribu9on Func9on of current density (with Jrms ~ 1): purple (483 grid points), 963, 1923, 3843, 7683 & 15363 The flow becomes more non-­‐Gaussian, with large wings Extrema from ~ 12 to 130 (strong but rare events) I
I
F nyc &fl & i Gcu r
&
o
Bourgoin et al PoF 14 (‘02), 16 (‘04)…
R
H=2R
Rλ ~800, Urms~1, λ~80cm
Numerical dynamo at a magnetic Prandtl number PM=nu/
eta=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 a turbulent flow in 2007
Reversal of the Earth’s
magnetic field over the last 2 million years
(Virtual Axial Dipole Moment, from
sediment cores)
Valet et al., Nature 2005
Temporal assymmetry and chaos in
reversal processes for the Earth
polarities
* Galaxies, the Sun, and other stars
* The Earth, and other planets
- including extra-solar planets?
•  The solar-terrestrial interactions,
magnetospheres, …
Many parameters and
dynamical regimes
Many scales, eddies,
structures and waves
in interaction
How strong will be the next solar cycle?
••  Predictions
of the next
solar cycle,
due to effect
(or not) of
long-term
memory in
the system (Wang and Sheeley, 2006)
Observations of galactic magnetic fields (after Brandenburg & Subramanian, 2005)
The MHD equa9ons High Reynolds number, to the detriment of all other concerns
•  P is the pressure, j = ∇ × B is the current, F is an external force, ν is the viscosity, η the resis9vity, v the velocity and B the induc9on (in Alfvén velocity units); incompressibility is assumed (div. v =0), and div.B = 0. o
pi T& pi rt c &ν = η W*
Turbulent spectra and the cascade scenario Energy
injection
Towards
dissipation
FGM data in the magnetosheath
18/02/2002
Blue: perp
Red : parallel
After Sahraoui, 2005
Parameters in MHD turbulence •  RV = Urms L0 / ν >> 1 •  Magnetic Reynolds number RM = Urms L0 / η
* Magne9c Prandtl number: PM = RM / RV = ν/η
PM is high in the interstellar medium. PM is low in the solar convec9on zone, in the liquid core of the Earth, in liquid metals and in laboratory experiments And PM= 1 in most numerical experiments un.l recently … •  Energy ratio EM/EV or time-scale ratio TNL/TA
with TNL= l/ul & TA=l/B0
•  (Quasi-) Uniform magnetic field B0 •  Amount of magnetic helicity HM and of cross helicity HC
•  Boundaries, geometry, cosmic rays, rotation, stratification, …
Magne9c fields in astrophysics •  The genera9on of magne9c fields occurs in media for which the viscosity ν and the magne9c diffusivity η are vastly different, and the kine9c and magne9c Reynolds numbers Rv and RM are large to huge. B
T
[Gauss] [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
LU
RV =
ν
LU
RM =
η
RM
PM =
RV
In most numerical computations up until recently, PM = 1
The energy spectrum of a turbulent flow
Assume simple self-similarity: E(k) ~ k-m
Determine m by assuming that E(k) depends only on
the energy injection rate e (an input parameter), and
the wavenumber k itself. With a dimensional
evaluation of transfer time of energy to small scales
l/ul :

E(k) = CK ε2/3 k-5/3 : Kolmogorov (1941) or K41
Eddy turn-over time TL ~ l/ul ~ k2/3
+ anisotropic variant using kE(k) ~ ul2
The energy spectrum of a turbulent flow
Assume simple self-similarity: E(k) ~ k-m
Determine m by assuming that E(k) depends only on
the energy injection rate e (an input parameter), and
the wavenumber k itself. With a dimensional
evaluation of transfer time of energy to small scales
l/ul :

E(k) = CK ε2/3 k-5/3 : Kolmogorov (1941) or K41
Eddy turn-over time TL ~ l/ul ~ k2/3
using kE(k) ~ ul2
Other characteristic times (e.g., waves, shear, …)?  Other spectra?
o i c s i c yc ep t c &
rD& Dy i
r ycu c && y) c i
u l u c i &en r&i t &rc ts y yt y t
D t c yT i
u T t a&c ae ) c i yc i e ql Dt p=flDi pc &s ( y * qi c i flyc y prn
c pi r & ) c i t ru i u p yn t a& r t y t=F
r& i t &.
; ×
1 é
u pro
ζζ
h.+&( f epc . bhpt Gc tbrr ( eci ethc.hm) g
) cG.p&hf .b ~ r7Gr
f 5 r f pTb ~ r7 P
5 .hti peahf & . e i p p 5 T8 e 1LPtT
B pethc.hme f .e pc
a10 fld
M7= a
E
P
( K
z
n c i &en
t
Di ) c i c
ps ac t Di pc &s
s e i ) ( y *
Bigot et al. PRE 2008
i &en & i t &
r I qb= ro T yc prn ( y u pro u T i Ds
ζ : h: ζ +g
^ Isotropy ^ Sharp filters
Slide from Alex Alexakis, 2006+
&t pi ro &i e
Fourier space
Energy transfer in MHD Large Velocity
Scales
Small Velocity
Scales
Large Magnetic
Scales
Small Magnetic
Scales
Slide from Alex Alexakis, 2006+
Rate of energy transfer in MHD 10243 runs, either T-­‐G or ABC forcing (Alexakis, Mininni & AP; Phys. Rev. E 72, 0463-01 and 0463-02, 2005)
Advection terms
Rλ~ 800
All scales contribute to energy transfer through the Lorentz force r c
i &en r&i t &pi
MP=: + .Gpt8 eci . & h.
5 r w aet8 epeppeA
h. epl
3Phys. Rev. E 72, 0463-01 and 0463-02, 2005)
Advection terms
Rλ~ 800
All scales contribute to energy transfer through the Lorentz force Solar Wind Voyager 10 data k-5/3
Mabhaeus and Goldstein, 1983 Observations of non-Kolmogorov behavior
B
Wind and ACE data
Podesta, 2010 (in preparation)
V
Extreme events in ac9ve regions on the Sun
•  Scaling exponents of structure func9ons δF(r) = F(x+r) - F(x) < δF[r]q > ~ rζq from solar magnetograms of ac9ve regions Abramenko 2002 (review, 2007) br& s
T i rt pi ro
tc y& c &c i
ypi e bac i i rt c
tr&D rD& Di ) c i t
δF(r) = F(x+r) - F(x)
ζ δF[r]q > ~ rEl
• 
&c s tc y& s e i rc e&s t
. pah =PP= 5. f e ( 8=PP/ T
c y& c &c i
& 5 Grr& eta .el i cp tt8
wc. rc. f ehr c l epl 8
r hG epe . MééNT
c y &s bps Ds pt
qtypeoryn=s c & pi r &s pH i r
.ecta) c r9
=PP/
6
Extreme events in direct numerical simula9ons (DNS) of MHD turbulence
•  Scaling exponents of structure func9ons δF(r) = F(x+r) - F(x)
< δF[r]p > ~ rζp 5123 DNS with varying B0: •  As B0 increases, so does the intermilency/curvature Müller & Biskamp, PRE 67 (2003) Extreme events for supersonic super-Alfvénic MHD turbulence
Scaling exponents of structure functions
δF(r) = F(x+r) - F(x)
< δF[r]p > ~ rζp
Ms=9.5 (bottom) to Ms=0.4 (top)
Padoan et al. PRL 2004
2D MHD
A F
&p c [ g] / / ac pi rt
thr c
epeb r G.. pc ti
mh. r f hrGbhp
Loureiro et al., 2009
cg
2D MHD
Turbulent reconnecAon Grids of up to 163842 points, in a turbulent environment with ini9al condi9ons at intermediate scale Sta.s.cs at both O-­‐points and X-­‐points in the magne.c poten.al in 1/3 X 1/3 of the box Servidio et al. PoP 2010; also Biskamp, 1989; Politano et al., 1989
2D MHD
Reconnec9on data, up to 163842 points Turbulent environment Auto-­‐correla9on func9on C of magne9c field Histogram of thickness δ and length λ of current sheets Servidio et al. PoP 2010
ReconnecAon in 3D MHD turbulence, grid of 15363 points Algorithm which selects connected regions, showing the con.nuity in the ``ver.cal’’ of events seen in 2D cuts One such event is selected in red Uritsky et al. 2010
Reconnec9on, 3D MHD, 15363 points One tenth of the current structures shown (top), and zoom on two such structures (bolom) Several sta9s9cs are computed: Scaling laws are sought for the spatial clusters so identified,
at a fixed time (i.e., not SOC yet, à la Lu & Hamilton)
Uritsky et al. 2010
Decomposi9on of structures into coherent and incoherent components in 3D MHD turbulence (5123 points) using wavelets Energy spectra
Energy fluxes
 The coherent part made up of structures (current and vorticity sheets)
captures the turbulent dynamics, as for fluids
Yoshimatsu et al., PoP 2009
Structures Alexakis et al., NJP 2007
Two exact scaling laws in MHD Politano+AP,GRL 25, 1998 In terms of velocity and magnetic field, two scaling laws for MHD for
δF(r) = F(x+r) - F(x) : structure function for field F ;
longitudinal component δFL(r) = dF . r/ |r|
< δvLδvi2 >+ <δvLδbi2 > - 2 < δbLδviδbi > = - (4/D) εTr
- <δbLδbi2 > - < δbLδvi2 >+2 < δvLδviδbi > = - (4/D) εc r
with the dissipation rates of total energy and cross-correlation:
εT = - dt(EV + EM) and
D is the space dimension
εc = - dt<v.b>
u
i
A
hrec phB 8
uA
=N8MééW
THE ROLE OF VELOCITY-­‐MAGNETIC FIELD CORRELATIONS: exact laws Dimensionally, < δb2(l) δv(l) > ~ l , … (+ other similar terms); does it imply δv ~ l 1/3 as for Kolmogorov (1941) fluid turbulence?  Role of v-­‐b correla9ons •  Boldyrev, 2006: what if < δb2(l) δv(l) θ(l) > ~ l , so e.g., δv(l) ~ δb(l) ~ l1/4 , θ(l)~ l1/4 where θ(l) is the angle (degree of alignment) of V & B This scaling leads to E(kperp) ~ kperp-­‐3/2 (role of anisotropy) It implies a varia9on of V-­‐B alignment with scales [as measured by θ (l)] Variation of
V,b angle θ
with scale
compensated by l1/4
and in inset,
for different
imposed
uniform
magnetic fields
Mason et al. 2006
Phenomenologies for MHD turbulence at low correla9on •  MHD could be like fluids Kolmogorov spectrum EK41(k) ~ ε2/3 k-­‐5/3 Or •  Slowing-­‐down of energy transfer to small scales because of Alfvén waves propaga.on along a (quasi)-­‐uniform field B0: EIK(k) ~ [εT B0]1/2 k-­‐3/2 (Iroshnikov -­‐ Kraichnan (IK), mid ‘60s) Ttransfer ~ TNL * [TNL/TA] , or 3-­‐wave interac.ons but s.ll with isotropy. Eddy turn-­‐over .me TNL~ l/ul and wave (Alfvén) .me TA ~ l/B0 Or •  Weak turbulence theory for MHD (Gal.er et al PoP 2000): anisotropy develops and the exact spectrum is: EWT(k) = Cw kperp-­‐2 f(k//) Note: WT is IK -­‐compa.ble when isotropy (k// ~ kperp ) is assumed: TNL~ lperp/ul and TA~l// /B0 Or kperp-­‐5/3 (Goldreich Sridhar, APJ 1995) ? Or kperp-­‐3/2 (Nakayama 1999; Boldyrev 2006, Yoshida 2007) ? 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, Mabhaeus 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) ~ kperp-­‐2 as may have been observed in the Jovian magnetosphere, and recently in DNS, Mininni & AP PRL 99, 254502; Lee et al., arXiv 0802xxx •  Is there a lack of universality in MHD turbulence? •  If so, what are the parameters that govern the (plausible) classes of universality? The presence of a strong guiding (quasi)-­‐uniform magne.c 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 right/wrong way to analyze and understand MHD? * Can the amount of correla9on between v and b change the spectra? Some recent results using direct numerical simula9ons GHOST: Geophysical High Order Suite for Turbulence
Pablo Mininni, principal developer, Duane Rosenberg, software engineer
•  2D, 3D, Navier-Stokes, rotation, MHD, Hall MHD
•  Ideal or dissipative (including hyper-diffusivity & friction)
•  Passive tracer(s)
•  Several modeling algorithms available
•  Pseudo-spectral
•  Hybrid MPI/OpenMP parallelization up to ~ 30,000 processors in
production run
Mininni et al. 2010, arxiv:1003.4322
•  Code available for the community, and data as well
Just ask: mininni@,
duaner@,
pouquet@
ucar.edu
Slide after Rich Loft, TOY Workshop (NCAR), 2008
Numerical set-­‐up •  Periodic boundary condi9ons, de-­‐aliased with the 2/3 rule •  From 643 to 15363 grid points, and to an ``equivalent’’ 20483 run (code: Brachet, mid 80’s and ~ 2009 for MHD) •  No imposed uniform magne9c field (B0=0) •  Decay runs (no external forcing, F=0), and ν= η (PM=1) •  V and B in equipar99on at t=0 (EV=EM) •  ABC flow (Beltrami) + random noise at small scale at t=0 or Taylor-­‐Green flow (cf. experimental configura9on) Energy dissipa9on rate in MHD for several RV OT-­‐ vortex Low Rv
High Rv
Orszag-­‐Tang simula9ons in 3D at different Reynolds numbers (~100 to ~1000) •  Is the energy dissipa9on rate εT constant in MHD at large Reynolds number (Mininni + AP, Phys. Rev. Leb., 99, 254502), as presumably it is in 2D-­‐MHD in the reconnec9on phase (Mabhaeus & Lamkin, 1986; Biskamp, 1989; Politano et al. 1989)? * There is evidence of constant ε in the hydrodynamic case (Kaneda et al., 2003) Energy dissipa9on rate ε for several Rλ in MHD OT-­‐ vortex Low Rv
High Rv
Orszag-­‐Tang simula9ons at different Reynolds numbers (~100 to ~1000) •  Is the energy dissipa9on rate εT constant in MHD at large Reynolds number (Mininni + AP, Phys. Rev. Leb., 99, 254502), as presumably it is in 2D-­‐MHD in the reconnec9on phase? * There is evidence of constant ε in the hydrodynamic case (Kaneda et al., 2003) MHD 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)
movie at stacks.iop.org/NJP/10/125007/mmedia
Visualization freeware: VAPOR http://www.cisl.ucar.edu/hss/dasg/software/vapor
Current roll-up in sheets
J2
At small scale, long correlation length along the local mean magnetic field (k// ~ 0)
15363 dissipative run
Convoluted sheets at later times
Recent observations of Kelvin-Helmoltz roll-up of current sheets
Hasegawa et al., Nature (2004); Phan et al., Nature (2006), …
Current and vor9city are strongly correlated in the rolled-­‐up sheet J2
15363 dissipative run, early time
ω2
VAPOR freeware, cisl.ucar.edu/hss/dasg/software/vapor
Growth of velocity -­‐ magne9c field correla9on Matthaeus et al. , PRL 100, 085003
c &) prn , g
i s
0
y) T o yp prn pi r i tprn oWhtqs3, =
c y Tfl, y pei s i r q yr&s p9 ) c i = Tsinober & Levich, Phys. Lett. (1983); Moffatt, J. Fluid Mech.
(1985); Farge, Pellegrino, & Schneider, PRL (2001), Holm & Kerr PRL (2002).
Velocity – magne9c field alignment PdFs of cos(v,B): •  Flow with strong normalized total cross helicity Hc •  Flow with weak Hc: the pdf peaks at both ends Mabhaeus et al. , PRL 100 (2008) Contours of r2(x)=v.B/[v2+b2]: local plages of maximal correla9ons (r2=0.5) except in the central current sheet of the Orszag-­‐Tang vortex (for which globally, r2=0.25) (Meneguzzi et al., JCP 123, 32 (1996) Contours and PdFs of cos(v,B)
(weak global correlation 10-4):
Matthaeus et al. , PRL 100, 085003
c && y r
Gc u t pi z
Correlation coefficient, 2D cut
Weak global correlation, strong B0, decay case
Bigot et al., 2008
Current & vorticity
Energy spectra in the presence of v-­‐B correla9ons
Elsässer z±=v±b E+(k) ~ k-­‐p E-­‐(k) ~ k-­‐m with here m + p = 3 2D MHD direct numerical simula9ons, 5122 points Pouquet et al., 1986 Similar and analy9c result for correlated MHD flows in weak turbulence Elsässer z±=v±b E+(k) ~ k-­‐m+ E-­‐(k) ~ k-­‐m-­‐ m+ + m-­‐ = 3 (2D MHD direct numerical simula9ons, AP et al., 1986) Weak MHD turbulence, m + p = 4 Varia.on of the ``Kolmogorov’’ constant with spectral index (and thus with degree of correla.on) (Gal.er et al., 2000) c && y r
Gc u t3z
Weak vB correlation, strong B0
Whatever the global v.B correlation, the ± spectra both tend to
kperp-3/2
(RMHD, forced)
Strong correlation, strong B0, different Reynolds numbers
Perez & Boldyrev, 2010
c && y r
Gc u t z
Whatever the global v.B correlation, the ± spectra both tend to
kperp-5/3
(MHD, forced)
Strong correlation, different B0
Bereshnyak & Lazarian, 2009, 2010
Is there a lack of universality in MHD turbulence, even for a given and weak VB correla9on? No universality!
•  Reduced MHD eqs. + boundary •  Various 9me ra9o TA/TNL •  Various spectral indices Dmitruk et al. 2003 No universality!
• 
Total energy spectrum: Kolmogorov law or Iroshnikov-­‐Kraichnan law • 
In some cases (above), kine.c (dash) & magne.c (dots) spectra have different indices Müller & Grappin 2005 Is there a lack of universality in MHD turbulence? •  Decay case, no uniform magne9c field B0, same ini9al velocity (Taylor-­‐Green or TG flow) in the large scale: F=0, B0 =0 •  Three different ini9al magne9c fields (I, A & C) at k0 with at t=0: –  EV=EM –  HM=0 –  Normalized HC < 4% The code enforces the 4-­‐fold symmetries of the TG flow and its generaliza9ons to MHD, to reach equivalent resolu9ons of 20483 grid points, using a pseudo-­‐spectral method Three different ini9al condi9ons for B, same V •  Decay case, no uniform B0, same ini9al velocity (Taylor-­‐Green or TG flow) in the large scale •  Three different ini9al B fields (I, A & C flows) at k0, and with at t=0, EV=EM, HM=0, normalized HC < 4%, using a code that enforces the 4-­‐fold symmetries of the TG-­‐MHD flow I-
C-
A-
t
• 
c
i c & pi e tns s r&p t
u c &Di t3ffg/ z e&p t3c i
i c & pi e ro
ci
e i &p at D c flta r&y c
R
nyc &fl & i tns s r&p t3
PdF of cos(V,B)
Pouquet et al. GAFD 104, 2010
Is a code enforcing symmetries OK? •  Two runs, 5123 grids, one enforcing the Taylor-­‐Green symmetries, one a generic pseudo-­‐spectral code Pouquet et al. GAFD 104, 2010
k5/3
B0 = 0, F = 0 ET(k)
C
k-2
Arrows:
Magnetic Taylor scales
A
Energy spectra I
Symmetric MHD Taylor-Green 20483 equivalent res. Rλ ~ 1300, Lee et al. 2010
3 different initial magnetic fields, same initial velocity field
k5/3 ET(k)
k-2
IK
K41
Lack of universality in MHD turbulence, B0 = 0, F=0 WT
•  Do these spectral indices persist in time for a given flow?
•  Will they be observed as well in the statistically steady state?
Symmetric MHD Taylor-Green 20483 equivalent res. Rlam ~ 1300, Lee et al. 2010
3 different initial magnetic fields, same initial velocity field
k5/3 ET(k)
k-2
IK
K41
Lack of universality in MHD turbulence B0 = 0, F=0 WT
•  Do these spectral indices persist in time for a given flow?
•  Will they be observed as well in the statistically steady state?
•  Does the difference in indices persist at higher Reynolds number?
•  Is there, in the IK case, a follow-up steeper (k-2 weak turb.) spectrum?
k5/3 ET(k)
Ratio of time-scales
B0 = 0, F=0 TNL / TA = f(k)
Energy spectra Arrows:
Magnetic Taylor scales
Symmetric MHD Taylor-Green 20483 equivalent res. Rlam ~ 1300, Lee et al. 2010
Tp i
c u h
rD& Dy i
u pro ha &a fl/
ta r&Ds
•  i & D
c s aDr) c i t 5 c r98 h I E8=PP+T
• 
i ro c Tpi
s ei rc tao &
ec.Ga
5 G.
c r98 tc.hp9 tc.hmi ) t9( /~8
=PP=T
Parametric study of MHD turbulence, for various PM from 0.01 to 10, and both for decay (tc is max. of dissipation) And for forced flows
(Sahoo et al., 2010)
EV & EM ~ kx e-ak in the dissipative range Pm=10
PdF of alignment
for Pm from
0.01 to 10
u, b
Pm=0.01
u, w
b, w
w, j
Sahoo et al.,
2010
u, j
b, j
PdF of vorticity and current for Pm from 0.01 to 10 for decay (top) and forced(bottom) flows
(Sahoo et al., 2010)
Measures of Intermittency for various PM
Left:
ζp/ ζ3 = f(p) (V & B)
Right:
hyper-flatness
(V & B) Sahoo et al., 2010
Q
R
QR plane = f(PM) [most points are at (0,0)]
Pm from 0.01 to 10, decay-top / forced-down
Q>>0: vortex
and Q<<0: strain
R>0: vortex stretching
Solid line: discriminant D=0
Sahoo et al., 2010
Numerical modeling Direct Numerical Simulations
(DNS)
versus
Large Eddy Simulations
(LES)
* Resolve all scales
vs.
* Model (many) small scales
Slide from Comte, Cargese Summer school on turbulence, July 2007
3D space
&
Spectral space
Numerical modeling Slide from Julien Baerenzung, 2008 Lagrangian-­‐averaged (or alpha) Model for Navier-­‐Stokes and MHD (LAMHD): the velocity & induc.on are smoothed on lengths αV & αM, but not their sources (vor.city & current) 
Equations preserve invariants (in modified - filtered L2  H1 form)
McIntyre (mid ‘70s), Holm (2002), Marsden, Titi, …, Montgomery & AP (2002)
Sorriso-Valvo
et al., PoP 9
(2002)
s ac &y Tc yD) c i c s e i ) pttpa) c i i
i
yy) c i bac i i r F c s a& ptc i u pro
. i K=[d-dF]/2 (see Sorriso-Valvo et al.
PoP, 2002)
c r98
-­‐ K8=PPN
Large-­‐Eddy Simula9on (LES) •  Add to the momentum equa9on a turbulent viscosity νt(k,t), à la Chollet-­‐Lesieur, with no modifica9on to the induc9on equa9on: with Kc a cut-off wave-number
The first numerical dynamo within a turbulent flow
at a magnetic Prandtl number below PM ~ 0.25
down to 0.02 (Ponty et al., PRL 94, 164502, 2005).
Turbulent dynamo at PM ~ 0.002 on the Roberts flow (Mininni, 2006).
Turbulent dynamo at PM ~ 10-6 , using second-order EDQNM closure (Léorat et al., 1980)
Critical
magnetic
Reynolds
number
RMc for
dynamo
action
LES based on spectral closure Using the formulation of the EDQNM (Eddy Damped
Quasi-Normal Markovian) closure for MHD
turbulence, à la Chollet Lesieur and:
^ including all the eddy transport coefficients, for both the viscosity and resistivity, ^ including eddy noise, ^ allowing for energy spectra that may not follow a Kolmogorov law (through a high-k fit of the resolved flow)
Baerenzung et al., PRE 78, 2008
Taylor-Green flow
Energy spectrum
difference (DNS – LES)
for two different
formulations of LES
based on two-point
closure EDQNM
Noticeable improvement
in the small-scale
spectrum
(Baerenzung et al., 2008)
RCM
Red is LES MHD 1/Pm
h e mht
t) ε***z l DpTy i r3
c.) i .
v
. / ] **
. i c r98tG ev
V & M Taylor scales & alpha filter wavenumbers
k0.3
B0 = 0, F=0
Variation with wavenumber of timescale ratio compatible with IK or K41
ε***z l DpTy i r3
. i c r98ch
B0 = 0, F=0
Variation with wavenumber of
total energy spectrum
k3/2ET(k)
and
magnetic to kinetic energy ratio
tG ev
f .) thhp
v
. / ] **
B0 = 0, F=0
Variation with wavenumber of
total energy spectrum and
magnetic to kinetic ratio of energies
and of time-scales
ε***z l DpTy i r3
. i c r98ch
tG ev
f .) thhp
v
. / ] **
Slide from Sebastien Galtier, 2009?, data from Bigot et al. 2008
Another way to go to higher Reynolds numbers …
Can we go beyond Moore’s law? Doubling of speed of processors every 18 months  doubling of resolu9on for DNS in 3D every 6 years …  Develop models of turbulent flows (Large Eddy Simula9ons, closures, Lagrangian-­‐averaged, …)  Improve numerical techniques  Be pa9ent •  Is Adap9ve Mesh Refinement (AMR) a solu9on? •  If so, how do we adapt? How much accuracy do we need? The need for Adap9ve Mesh Refinement Examples of AMR Parallel flux tubes in 3D, ideal run
With effective resolution up to 40963
Hairpin vortex, Euler case
Grauer et al. PRL 80 (1998)
Grauer Marliani PRL 84 (2000)
2D -­‐MHD OT vortex with AMR •  Magne9c X-­‐point configura9on in 2D •  Error in temporal deriva9ve of total energy (compared to dissipa9on) is ~ 10-­‐3 (computed every 10 .me steps) •  Error in div.v is ~ 10-­‐5 (controlled by code parameter) Rosenberg et al., New J. Phys. 2007 AMR in incompressible 2D -­‐ MHD turbulence at R~1000 •  AMR using spectral elements of different orders, P •  No no9ceable differences when using the L2 norms (energy and its dissipa9on) •  But accuracy maXers when looking at Max norms, here the current Rosenberg et al., New J. Phys., 2007 o
pi T& pi rt c &ν = η W*
Inverse or direct
cascades?
3D statistical
equilibria (ν=η =0)
of a truncated
system of modes
(phase space)
Relaxation to
Gibbs ensemble with 3 invariants
Frisch et al. JFM 1975
> 0
Inverse or direct
cascades?
3D statistical
equilibria (ν=η =0)
of a truncated
system of modes
(phase space)
> 0
Relaxation to
Gibbs ensemble with 3 invariants
Frisch et al. JFM 1975
EV(k) = EM(k) - c2 | HC(k)/γ |
Equipartition of energy or of helicity?
•  Energy equipartition obtains only for
^ k∞
^ or zero magnetic helicity
^ or maximal cross correlation HC
Otherwise, EM > EV in statistical equilibrium
•  Helicity equipartition
never obtains,
except for maximal HC
Frisch et al. JFM 1975
Inverse or direct
cascades?
3D statistical
equilibria (ν=η =0)
`Partial condensation’ of cross-helicity HC,
due to condensation of magnetic helicity
(Stribling & Matthaeus, 1990, 1991)
> 0
Magne9c helicity HM=<a.b> •  In decaying MHD, inverse transfer to large scales HM≠ 0
EM(k) at different times,
Christensson et al. 2001
Magne9c helicity HM=<a.b> •  In decaying MHD, inverse transfer to large scales HM≠ 0
EM(k) at different times,
Christensson et al. 2001
Magne9c helicity HM=<a.b> •  In decaying MHD, inverse transfer to large scales HM≠ 0
EM(k) at different times,
HM≠ 0
HM= 0
Christensson et al. 2001
And what about the cross helicity Hc=<v.b> ? The direct cascade of cross helicity <v.b> to small sales versus the inverse cascade of magne9c helicity to large scales? •  Inverse transfer in decaying MHD, with HC≠ 0 , HM≠ 0 Spectral transfer of EM & EV
Spectral transfer of HM & HC,
all following each other?
EM
323 data, …, AP & Patterson, JFM 1978
The ideal non-dissipative case with turbulent behavior
Can the ideal interac9ons of a truncated ensemble of modes behave as in the dissipa9ve case, with Kolmogorov-­‐like spectra? YES: Fluids (Euler eq.) in three dimensions (D=3), Cichowlas et al., 2005 YES: Ideal MHD for D=2, Krstulovic et al. arXiv:1101.1078 In both cases, the thermalized part of the spectrum at small scale (say kD-­‐1) produces an effec9ve eddy viscosity and eddy resis9vity ac9ng on the large-­‐scale spectra, which are found to follow the expected (forced/dissipa9ve) spectra Ideal MHD in 2D EM(t) & EV(t),
HC~ 0
EM /EV
<J2 > /<ω2> Up to 40962 data, …, Krstulovic et al. arXiv:1101.1078 Ideal MHD in 2D Spectra k3/2EM(k) & k3/2EV(k)
EM(t) & EV(t),
HC~ 0
High k, E(k)~ k
Low k: EM > EV
EM /EV
<J2 > /<w2> Up to 40962 data, …, Krstulovic et al. arXiv:1101.1078 Ideal MHD, current
Frisch et al., 1983; …, Krstulovic et al. arXiv:1101.1078 Ideal MHD, current B0=0, later times
Frisch et al., 1983; Krstulovic et al. arXiv:1101.1078 Some issues in need of more invesAgaAons •  Temporal evolution of current and vorticity, and the problem of
existence or not of singularities in the non-dissipative case
•  Limit R ∞
Energy dissipation & fast reconnection
Energy transfer & non-local interactions in Fourier space
Scaling laws (energy & higher order moments) and universality classes
Observations of weak turbulence in MHD?
Global large-scale isotropy, small-scale anisotropy?
Partial Alfvénization of energies and helicities to govern the dynamics?
•  Structures, piling, folding & rolling-up of current and vorticity sheets
•  Dynamo, small/large magnetic Prandtl number, …
•  Large-scale and small-scale properties of magnetic helicity and cross-helicity,
direct or inverse cascade, and validity of dimensional analysis?
Thank you for your alen9on! [email protected]
The Taylor-­‐Green flow ⎡ sin( k0 x) cos(k0 y ) cos(k0 z ) ⎤
F = ⎢⎢− cos(k0 x) sin( k0 y ) cos(k0 z )⎥⎥
⎥⎦
⎢⎣
0
z
•  A (globally) non-helical forcing
Taylor & Green, Proc. Roy. Soc. A 151, 421, 1935.
* The resulting flow shares similarities with several experiments (ENS
Paris & Lyon, U. Maryland, Cornell & MPI), and with the Cadarache
dynamo experiment (at PM ~ 10-6)
Marié et al., Magnetohydrodynamics, 38, 163, 2002.
* TG gives a dynamo at PM = 1 (Nore et al., Phys. Plasmas, 4, 1, 1997),
down to PM = 0.02 (Ponty et al., PRL 94, 164502, 2005), and now in the lab.
(Monchaux et al., PRL 98, 2007)