A Few Issues in fluid and MHD Turbulence

Alex Alexakis*, Aimé Fournier, Jonathan Pietarila-Graham&, Darryl Holm@, Bill Matthaeus%, Pablo Mininni^ and Duane Rosenberg

NCAR / CISL / TNT

* Observatoire de Nice & MPI, Lindau

@ Imperial College & LANL % Bartol, U. Delaware

^ Universidad de Buenos Aires

Boulder, October 15, 2007 pouquet@ucar.edu

• Theme of the year on Geophysical Turbulence (GT) Sponsored by GT Program (IMAGe)

• A Workshop: Theory and Modeling for GT, 27-29 February 2008

• B Workshop: Petascale computing for GT, 5-7 May 2008

• C Workshop: New sensors, new observations for GT, 28-30 May 2008

• A,B,C School, July 14 to August 1, 2008

Keith.Julien@colorado.edu, pouquet@ucar.edu, rothney@ucar.edu

Related activities:• This workshop

• Nonlinear processes in atmospheric chemistry, 5 lectures, ~ November ‘07

* Introduction• Some examples of turbulence

• Equations and phenomenologieies

• What are the features of a turbulent flow, both spatially and spectrally?

• Is there any measurable difference between fluid & MHD turbulence?

** 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• Combine all approaches …

Observations of galactic magnetic fields (after Brandenburg & Subramanian, 2005)

• Prediction of the next solar cycle, because of long-term memory in the system (Dikpati, 2007)

Cyclical reversal of the solar magnetic field over the last 130 years

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Brunhes Jamarillo Matuyama Olduvai

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Reversal of the Earth’s magnetic field over the last 2Myrs (Valet, Nature, 2005)

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Temporal assymmetry of reversal processes

Surface (1 bar) radial magnetic fields for

Jupiter, Saturne & Earth, Uranus & Neptune

(16-degree truncation, Sabine Stanley, 2006)

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Axially dipolar

Quadrupole ~ dipole

Experimental dynamo within a constrained flow: Riga

Gailitis et al., PRL 84 (2000)

Karlsruhe, with a Roberts flow (see e.g. special issue of Magnetohydrodynamics 38, 2002)

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 TG dynamo obtained in 2007

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.

Parameters in MHD: how many? How relevant?

• 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 = <A.B>

• Boundaries, geometry, rotation, stratification, anisotropy, forcing mechanisms, coupling to compressibility, cosmic rays, supernovae, …

Phenomenologies for MHD turbulence

• If MHD is like fluids Kolmogorov spectrum EK41(k) = CK(T)2/3 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)=CIK (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) ?

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-2 as may have been observed in the Jovian magnetosphere, and

recently in a DNS, Mininni & AP arXiv:0707.3620v1, astro-ph (see later)

• Is there a lack of universality in MHD turbulence (Boldyrev; Pouquet et al., 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 given flow? * Is there it a lack of resolving power (instruments, computers)? * Is an energy spectrum the wrong way to analyze / understand MHD?

Recent results using direct numerical simulations and models of MHD

Numerical set-up

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

• From 643 to 15363 grid points (6 WE, 500 proc., NCAR)

• No imposed uniform magnetic field• Either decay runs (or forcing at lower resolutions)

• 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 (experimental dynamo configuration)

Speeding on Lemieux and BigBen

Left: Speed--up for the code on Lemieux (HP-Digital) and Bigben (XT3) at PSC. Right: Time in seconds to compute one time step (Navier-Stokes).Grids of 1283 points [squares], 2563 [triangles] (left), with also (right) 5123 [diamonds], 10243 [*], and 20483 [+][sustained 3Terafl., ~ 30%] [MPI + FFTW2&3].Dotted lines: Lemieux. Dashed lines: Bigben.Slopes of dashed lines indicate ideal scaling. Similar scaling obtains on IBM (P4, P5; ~ 8%).

NSF: 12000NSF: 1200033 NS up to t=1 in 40 hours with 1 Petaflops NS up to t=1 in 40 hours with 1 Petaflops [2D domain decomposition …][2D domain decomposition …]

• Energy transfer & non-local interactions of Fourier modes• Evolution of maximum of current and vorticity• Total energy dissipation

• Piling, folding & rolling-up of current and vorticity sheets• Alignment of v and B fields in small-scale structures: weakening of nonlinear interactions

• Energy spectra and anisotropy• Intermittency and extreme events

Scaling laws and structures

Rate of energy transfer Tub(Q,K) from u to b for different K shells

The magnetic field at a given scale receives energy in equal The magnetic field at a given scale receives energy in equal amounts from the velocity field from all larger scales amounts from the velocity field from all larger scales (but more (but more from the forcing scale).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).

K= 10

K= 20

K= 30

Orszag-Tang (=) simulations at different Reynolds numbers (X 10)• Is the energy dissipation rate T constant in MHD at large Reynolds number

(Mininni + AP, arXiv:0707.3620v1, 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)

Energy dissipation rate in MHD for several RV

OT- vortexLow Rv High Rv

Scaling with Reynolds number of dissipation: Are we there yet?

• But: Other scaling laws harder to get, e.g. the Kolmogorov constant

MHD decay simulation @ NCAR on 15363 grid pointsVisualization 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?)

Current and vorticity are strongly correlated in the rolled-up sheets

Current J2 Vorticity 2

15363 run, early time

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)

Current J2 cos(V, B)15363 run, early time

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 in 2D; Matthaeus et al,

2007, in 3D)

similar to the Beltramisation (v // ) of fluids

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

Blu

e h

> 0

.95

Red

h

<-0

.95

Blu

e h

> 0

.95

Red

h

<-0

.95

MHD scaling at peak of dissipation [1]

Total energy spectra compensated by k3/2 (IK, ‘65)

Solid: ET

Dash: EM

Dot: EV

Insert: energy flux

Dash-dot: k5/3-compensated

LintM ~ 3.1

M ~ 0.4

MHD scaling at peak of

dissipation [2]• Anomalous exponents

of structure functions for Elsässer variables, with isotropy assumed (similar results for v and B): intermittency and extreme events

Note 4 ~ 0.98 0.01, i.e. far from fluids and with

more intermittency

MHD decay run at peak of dissipation [3]

Isotropy ratio

RS = S2b / S2

b//

Isotropy obtains in the large scales, whereas anisotropy develops at smaller scales

RS is proportional to the so-called Shebalin angles Lint

M ~ 3.1, M ~ 0.4

Structure functions at peak of dissipation [5]

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: perpendicularDash: parallel

(Mininni + AP, arXiv:0707.3620v1, astro-ph)

LintM ~ 3.1 , M ~ 0.4

Solid: parallel, Dash: perpendicular, LintM ~ 3.1, M ~ 0.4

Solid: perpendicularDash: parallel

Anisotropic energy spectrum compensated for the weak turbulence case

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

Linear resolution: X 2Cost: X 16

Kolmogorov k-5/3 law

Mininni et al., submitted

Kaneda et al. (2003), 40963 run, R~ 1300

The dynamo problem of generation of magnetic field

at small magnetic Prandtl number

• Is a turbulent dynamo possible at all?

• Is the magnetic field present at small scales?

Small magnetic Prandtl number

• PM << 1: it is 10-6 in liquid metals

Resolve two dissipative ranges, the inertial range and the energy containing range

And

Run at a magnetic Reynolds number RM larger than some critical value

(RM governs the importance of stretching of magnetic field lines over Joule dissipation)

Resort to modeling of small scales

Lagrangian-averaged (or alpha) Model for Navier-Stokes and MHD (LAMHD):the velocity & induction are smoothed on lengths αV & αM, but not their sources (vorticity & current)

Equations preserve invariants (in modified - filtered L2 --> H1 form)McIntyre (mid ‘70s), Holm (2002), Marsden, Titi, …, Montgomery & AP (2002)

-->

Lagrangian-averaged NS & MHD Non-dissipative Model Equations

• ∂v/∂t + us · v = −vj u j s − ∇ ∇ ∇

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Cancellation exponent and magnetic dissipation: comparison with LAMHD

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=[d-dF]/2 (Sorriso-Valvo et al. PoP, 2002)

Comparison of DNS and Lagrangian model

• RM = 41, Rv=820,

PM = 0.05 dynamo

• Solid line: DNS• - - - : LAMHD• Linear scale in inset

Comparable growth rate and saturation level of Direct Numerical Simulation and model

Phys. Fluids 17; Phys Rev E 71 (2005)

Large-Eddy Simulation (LES)• Add to the momentum equation a turbulent viscosity νt(k,t) (à la Chollet-Lesieur) (no modification to the induction

equation

(study of similar LES for MHD in progress, Baerenzung et al.)

with Kc a cut-off wave-number

The first numerical dynamo within a turbulent flowat 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 for --> dynamo action

Can we go beyond Moore’s law?

Doubling of speed of processors every 18 months--> doubling of resolution for DNS in 3D every 6 years …

Develop models of turbulent flows (Large Eddy Simulations, closures, Lagrangian-averaged, …)

Improve numerical techniquesBe patient

• Is Adaptive Mesh Refinement (AMR) a solution?

• If so, how do we adapt? How much accuracy do we need?

Another way to go to higher Reynolds numbers …

The need for Adaptive Mesh Refinement

AMR on 2D Navier-Stokes

Duane Rosenberg et al, JCP 2006; Aimé Fournier et al., 2008

• Decay for long times (incompressible)

• Formation of dipolar vortex structures

• Gain in the number of degrees of freedom (~ 4) with adaptive mesh refinement (AMR), compared to an equivalent pseudo-spectral code (periodic boundary conditions)

(but ….)

2D -MHD OT vortex with AMR

• Error in temporal derivative of total energy (compared to dissipation)

is ~ 10-3

(computed every 10 time steps)

• Error in .v is ~ 10-5 (controlled by code parameter)

Duane Rosenberg et al, JCP 2006

AMR in 2D - MHD turbulence

• Magnetic X-point configuration in 2D

• Temporal variation of:• Dissipation• Jmax

• Degrees of freedom normalized by the number of

modes in a pseudo-spectral code at the same Rv,

~33%

Refinement and coarsening criteria …

Duane Rosenberg et al, New J. Phys. 2007

AMR in 2D - MHD turbulence

• 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

Discussion (1)

• Questions of universality in turbulent flows, wrt forcing, wrt parameters, geometry, boundaries, …• Succession of dynamical regimes for a given problem, can we resolve them all?• Exploration of physical space: rotation, stratification, compressibility, supernovae, convection, …• Choice of grids?• Ensemble computations?

• One run at the largest Reynolds number and parametric investigations at less demanding values of relevant parameters

• Numerical codes developed for the community?• Community discussions on the choice of parameters for a few very large runs?

Discussion (2)

• Parallel codes: issue of total number of operations per grid point

• Fastest code to reach ``T=1’’ at given accuracy

• Combining large simulations, modeling, adaptive mesh refinement, data assimilation?

• Essential (energetic) diagnostics (analysis and graphics) while we run, as well as post-processing

• Development of tools (analysis and graphics)

• Access to data

Possible IPAM 4-month meeting on petascale computing for geophysics, in 2009 (Bjorn Stevens et al.)

Thank you for your attention!