A Few Issues in fluid and MHD Turbulence
A Few Issues in fluid and MHD Turbulence 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 phenomenologies ie • 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)
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)
Reversal of the Earth’s magnetic field over the last 2 Myrs (Valet, Nature, 2005) Temporal assymmetry of reversal processes Brunhes Jamarillo Matuyama Olduvai
Surface (1 bar) radial magnetic fields for Jupiter, Saturne & Earth, Uranus & Neptune (16 -degree truncation, Sabine Stanley, 2006) 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 Bourgoin et al Po. F 14 (‘ 02), 16 (‘ 04)… W R H=2 R W Numerical dynamo at a magnetic Prandtl number PM= / =1 (Nore et al. , Po. P, 4, 1997) and PM ~ 0. 01 (Ponty et al. , PRL, 2005). In liquid sodium, PM ~ 10 -6 : does it matter? 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 L 0 / ν >> 1 • Magnetic Reynolds number RM = Urms L 0 / η * Magnetic Prandtl number: P M = 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 B 0 • 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 EK 41(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 B 0: EIK(k)=CIK ( T B 0)1/2 k-3/2 (Iroshnikov - Kraichnan (IK), mid ‘ 60 s) transfer ~ NL * [ NL/ A] Eddy turn-over time , or 3 -wave interactions but still with isotropy. NL~ l/ul and wave (Alfvén) time A ~ l/B 0 And • Weak turbulence theory for MHD (Galtier et al Po. P 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// /B 0 Or k -5/3 (Goldreich Sridhar, APJ 1995) ? Or k -3/2 (Nakayama, Ap. J 1999; Boldyrev, PRL 2006) ?
Spectra of three-dimensional MHD turbulence • EK 41(k) ~ k-5/3 as observed in the Solar Wind (SW) and in DNS (2 D & 3 D) Jokipii, mid 70 s, Matthaeus et al, mid 80 s, … • EIK(k) ~ k-3/2 as observed in SW, in DNS (2 D & 3 D), and in closure models • EWT(k) ~ k -2 as may have been observed in the Jovian magnetosphere, and recently in a DNS, Mininni & AP ar. Xiv: 0707. 3620 v 1, astro-ph (see later) Müller & Grappin 2005; Podesta et al. 2007; Mason et al. 2007; Yoshida 2007 • 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 Big. Ben Left: Speed--up for the code on Lemieux (HP-Digital) and Bigben (XT 3) 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 3 Terafl. , ~ 30%] [MPI + FFTW 2&3]. Dotted lines: Lemieux. Dashed lines: Bigben. Slopes of dashed lines indicate ideal scaling. Similar scaling obtains on IBM (P 4, P 5; ~ 8%). NSF: 120003 NS up to t=1 in 40 hours with 1 Petaflops [2 D domain decomposition …]
Scaling laws and structures • 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
Rate of energy transfer Tub(Q, K) from u to b for different K shells K= 10 K= 20 K= 30 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).
Energy dissipation rate in MHD for several RV OT- vortex Low Rv High Rv Orszag-Tang ( = ) simulations at different Reynolds numbers (X 10) • Is the energy dissipation rate T constant in MHD at large Reynolds number (Mininni + AP, ar. Xiv: 0707. 3620 v 1, astro-ph), as presumably it is in 2 D-MHD in the reconnection phase? There is evidence of constant in the hydro case (Kaneda et al. , 2003)
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 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? )
Current and vorticity are strongly correlated in the rolled-up sheets Current J 2 15363 run, early time Vorticity 2
V and B are aligned in the rolled-up sheet, but not equal (B 2 ~2 V 2): Alfvén vortices? (Petviashvili & Pokhotolov, 1992. Current J 2 15363 run, early time Solar Wind: Alexandrova et al. , JGR 2006) cos(V, B)
Velocity - magnetic field correlation [3] • Local map in 2 D 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 2 D; Matthaeus et al, 2007, in 3 D) 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). Blue h> 0. 95 Red h<-0. 95 --> no mirror symmetry, together with weak nonlinearities in the small scales
MHD scaling at peak of dissipation [1] Total energy spectra compensated by k 3/2 (IK, ‘ 65) Solid: ET Dash: EM Dot: EV Insert: energy flux Lint. M ~ 3. 1 M ~ 0. 4 Dash-dot: k 5/3 compensated
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 = S 2 b / S 2 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] (Mininni + AP, ar. Xiv: 0707. 3620 v 1, astro-ph) Lint. M ~ 3. 1 , M ~ 0. 4 Structure function S 2 , with 3 ranges: L 2 (regular) at small scale L at intermediate scale, as for weak turbulence: Ek~k -2. Is it the signature of weak MHD turbulence? L 1/2 at largest scales (Ek~ k-3/2) Solid: perpendicular Dash: parallel
Solid: parallel, Dash: perpendicular, Lint. M ~ 3. 1, M ~ 0. 4 Solid: perpendicular Dash: parallel Anisotropic energy spectrum compensated for the weak turbulence case
Kolmogorov-compensated Energy Spectra: k 5/3 E(k) Navier-Stokes, ABC forcing k-5/3 Small Kolmogorov law (flat part of the spectrum) Kolmogorov k-5/3 law K 41 scaling increases in range, as the Reynolds number increases • Bottleneck at dissipation scale Solid: 20483, Rv= 104, R ~ 1200 Dash: 10243, Rv=4000 Mininni et al. , submitted Kaneda et al. (2003), 40963 run, R ~ 1300 Linear resolution: X 2 Cost: X 16
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 L 2 --> H 1 form) Mc. Intyre (mid ‘ 70 s), Holm (2002), Marsden, Titi, …, Montgomery & AP (2002)
Lagrangian-averaged NS & MHD Non-dissipative Model Equations • ∂v/∂t + us · ∇v = −vj ∇u j s − ∇P P* + j × Bs, • ∂Bs/∂t + us · ∇Bs = Bs · ∇us • The above equations have invariants that differ in their formulation from those of the primitive equations: the filtering prevents the small scales from developing. • For example, kinetic energy invariant EV = <v 2>/2 --> EV, α = < v 2 + α 2ω2 >/2
Cancellation exponent and magnetic dissipation: comparison with LAMHD =[d-d. F]/2 (Sorriso-Valvo et al. Po. P, 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 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 for --> dynamo action
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 resolution for DNS in 3 D every 6 years … à Develop models of turbulent flows (Large Eddy Simulations, closures, Lagrangian-averaged, …) à Improve numerical techniques à Be patient • Is Adaptive Mesh Refinement (AMR) a solution? • If so, how do we adapt? How much accuracy do we need?
The need for Adaptive Mesh Refinement
AMR on 2 D 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 …. )
2 D -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 2 D - MHD turbulence • Magnetic X-point configuration in 2 D • 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 2 D - 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 L 2 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!
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