Post on 17-Dec-2015
Modeling of turbulence using filtering,and the absence of ``bottleneck’’ in MHD
Annick Pouquet
Jonathan Pietarila-Graham& , Darryl Holm@, Pablo Mininni^
and David Montgomery!
& MPI, Lindau @ Imperial College
! Dartmouth College
^ Universidad de Buenos AiresCambridge, October 2008 pouquet@ucar.edu
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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, …
Extreme events in active regions on the Sun
• Scaling exponents of structure functions for magnetic fields in solar active regions
(differences versus distance r, and assuming self-similarity)
Abramenko, review (2007)
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Surface (1 bar) radial magnetic fields for
Jupiter, Saturne & Earth versus Uranus & Neptune
(16-degree truncation, Sabine Stanley, 2006)
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Axially dipolar
Quadrupole ~ dipole
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
The MHD equationsMulti-scale interactions, high R runs
• 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
Parameters in MHD
• 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
And PM=1 in most numerical experiments until recently …
• Energy ratio EM/EV or time-scale ratio NL/A with
NL= l/ul and A=l/B0
• (Quasi-) Uniform magnetic field B0 • Amount of correlations <v.B> or of magnetic helicity <A.B>
• Boundaries, geometry, cosmic rays, rotation, stratification, …
Small magnetic Prandtl number
• PM << 1: ~ 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
• Equations for the alpha model in fluids and MHD
* Some results comparing to DNS
•The various small-scale spectra arising for fluids
• The MHD case
• Some other tests both in 2D and in 3D
* An example : The generation of magnetic fields at low magnetic Prandtl number and the contrast between two models
* Conclusion
Numerical modeling
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Slide from Comte, Cargese Summer school on turbulence, July 2007
Direct Numerical Simulations(DNS)versusLarge Eddy Simulations(LES)
Resolve all scalesvs. Model (many) small scales
1D space & Spectral space
)
• Probability
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Higher grid resolutions, higher Reynolds numbers, more multi-scale interactions: study the 2D case
(in MHD, energy cascades to small scales, and it models anisotropy …)
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 model for Navier-Stokes & MHD Non-dissipative case
• ∂v/∂t + us · v = −vj u j s − ∇ ∇ ∇
Lagrangian-averaged NS & MHD dissipative equations
• ∂v/∂t + us · v = −vj u j s − ∇ ∇ ∇
B ~ k2 Bs --> hyperdiffusive term
Navier-Stokes: vortex filaments
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Alpha model
DNS
MHD: magnetic energy structures at 50% threshold (nonlinear phase of a PM=1 dynamo regime)
Alpha model, 643
DNS, 2563 grid
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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)
3D Navier-Stokes: intermittency
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Chen et al., 1999; Kerr, 2002 Pietarila-Graham et al., PoF 20, 035107 2008
DNS: X
Largest filter length& smaller cost:more intermitency
Third-order scaling law for fluids(4/5th law) stemming from energy conservation
• v is the rough velocity and
us is the smooth velocity,
is the filter length and
is the energy transfer rate
• A priori, two scaling ranges:– For small , Kolmogorov law (at high Reynolds number)
– For large , us3 ~ r3, we have an advection by a smooth field,
or Eu ~ k-3, hence E ~ Euv ~ k-1
r ~ < v us2 + [2 / r2 ] us
3 >
Third-order scaling law stemming from energy conservation
• v is the rough velocity and
us is the smooth velocity, is the filter length and is the energy transfer rate
• Two ranges:– For small , Kolmogorov law
– For large , us3 ~ r3, we have
an advection by a smooth field,
or Eu ~ k-3, hence E ~ Euv ~ k-1
But we observe rather ~ k+1
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Solid line: model, for large (k =3)
r ~ < v us2 + [2 / r2 ] us
3 >
Why?
k+1
k
Regions with /< > ~ 0
Black: u//3(r=2/10) < 0.01
Filling factor ff of regions with very low energy transfer
ff~ 0.26 for DNS
DNSrun
~ 10-4
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Regions with /< > ~ 0
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3D Run with large (2 /10)
Black: u//3(r=2/10) < 0.01
Filling factor ff of regions with very low energy transfer (at scales smaller than ):
ff~ 0.67 for LA-NS Versus ff~ 0.26 for DNS
DNSrun
~ 10-4
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3D Run with large (2 /10)
Black: u//3(r=2/10) < 0.01
DNSrun
~ 10-4
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3D Run with large (2 /10)
Black: u//3(r=2/10) < 0.01
DNSrun
~ 10-4
``rigid bodies’’ (no stretching):
us(k) = v(k) / [ 1 + 2 k2]
and take limit oflarge : the flow is advectedby a uniform field U (no degrees of freedom)
us=constantv ~ k2 us for large usv ~ k2 ~ k E(k)
E(k) ~ k+1
``rigid bodies’’:
us(k) = v(k) / [ 1 + 2 k2]
and take limit oflarge : the flow is advectedby a uniform field Us (no degrees of freedom)
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3D Run with large (2 /10)
Black: u//3(l=2/10) < 0.01
DNSrun
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Solid line: model, for large (k =3)Dash line: same model withoutregions of negligible transfer
Kinetic Energy Spectra in MHD
Solid: DNS, 15363
Dash: LAMHD, 5123
Dot: Navier-Stokes , 5123
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k
Energy Fluxes
Solid/dash: LAMHD (Elsässer variables)
Dots: alpha-fluid
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Circulation conservation is broken by Lorentz force
Magnetic Energy Spectra
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Solid: DNS, 15363 gridDash: LAMHD, 5123
k
Energy transfer in MHD is more non local than for fluids
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Transfer of kinetic energy to magnetic energyfrom mode Q (x axis)
to mode K =10 (top panel)
K =20
K =30
Alexakis et al., PRE 72, 046301
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Sorriso-Valvo et al., P. of Plas. 9 (2002)
Currentsheets in 2D MHD
DNS
Comparison in 2D with LAMHD: cancellation exponent (thick lines)
& magnetic dissipation (thin lines) Graham et al., PRE 72, 045301 r (2005)
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Solid: DNS
2D - MHD, forced
Kinetic (top) and
magnetic (bottom) energies
and
squared mag. potential growth: DNS vs. LAMHD
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Inverse cascade of <A2>
associated with a negative eddy
resistivityassociated with a lack of equipartition in the
small scales
turb~ EkV - Ek
M < 0
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DNS
Rädler;AP, mid ‘80s
Dynamo regime at PM=1: the growth of magnetic energy at the
expense of kinetic energy : all three runs display similar
temporal evolutions and energy spectra
DNS at 2563 grid (solid line) and α runs ( 1283 or 643 grids, (dash or dot)
Beltrami ABC flow
at k0=3
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
Beyond testing …
Solid: DNS, 15363, R ~ 1100Dash: LAMHD, 2563
Dot: DNS, 2563
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Temporal evolution of total energy (top),kinetic (bottom) andmagnetic energies
Temporal evolution of total enstrophy j2 +2
Solid: DNS, 15363
Dash: LAMHD, 2563
Dot: DNS, 2563
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Magnetic energy spectra compensated by k3/2
Solid: DNS, 15363
Dash: LAMHD, 2563
Dot: DNS, 2563
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Summary of results
• For large , for fluids, the model has large portions of the flow with low energy transfer (67% vs. 26% for DNS)
• This results in an enhancement of spectra at small scales, akin to a bottleneck
• This phenomenon is absent in MHD, perhaps because of nonlocal interactions
• The -model in MHD allows a sizable savings over DNS (X6 in resolution for second-order correlations)
• Applications: low-PM (experiments, Earth) and high PM (interstellar medium) dynamos, MHD turbulence spectra, parametric studies (e.g., effect of resolution on high-order statistics, energy spectra, anisotropy, role of velocity-magnetic field correlations, role of magnetic helicity, …)
• There are other models in MHD, …
Conclusions
• Deal with peta and exa-scale computers: parallelism! But keep the absolute time of computation and usage of memory at their lowest, and watch for accuracy.
Collaborations on large projects (shared codes, shared data, …)
• Be creative:– Tricks, as symmetric flows– Models (many …)– Adaptive Mesh Refinement, keeping accuracy
– Combine and contrast all approaches!
Conclusions
• Deal with peta and exa-scale computers: parallelism! But keep the absolute time of computation and usage of memory at their lowest, and watch for accuracy.
Collaborations on large projects (shared codes, shared data, …) GHOST: Geophysical High-Order Suite for Turbulence
• Be creative:– Tricks, as symmetric flows– Models (many …)– Adaptive Mesh Refinement, keeping accuracy
– Combine and contrast all approaches!
Pietarila-Graham et al., PRE 76, 056310 (2007); PoF 20, 035107 (2008); and arxiv:0806.2054
Thank you for your attention!
Scientific framework
• Understanding the processes by which energy is distributed and dissipated down to kinetic scales, and the role of nonlinear interactions and MHD turbulence, e.g. in the Sun and for space weather
• Understanding Cluster observations in preparation for a new remote sensing NASA mission (MMS: Magnetospheric Multi-Scale)
• Modeling of turbulent flows with magnetic fields in three dimensions, taking into account long-range interactions between eddies and waves, and the geometrical shape of small-scale eddies
Computational challenges• Pseudo-spectral 3D-MHD code parallelized using MPI, periodic boundary
conditions & 2/3 de-aliasing rule, Runge-Kutta temporal scheme of various orders, runs for ~ 10 turnover times at the highest Reynolds number possible in order to obtain multi-scale interactions.
• Parallel FFT with a 2D domain decomposition in real and Fourier space with linear scaling up to thousands of processors.
• Planned pencil distribution to scale to a larger number of processors. • MHD computation on a grid of 20483 points up to the peak of
dissipation will take ~ 22 days on 2000 single core IBM POWER5 processors with a 1.9-GHz clock cycle, using ~230 s/ time step
• A 40963 MHD grid, needed in order to resolve inertial interactions between scales, will require much more and represents a substantial computing challenge• And add kinetic effects …
Some questions• Are Alfvén vortices, as observed e.g. in the magnetosphere, present in MHD at high Reynolds number, and what are their properties?
• Is another scaling range possible at scales smaller than where the weak turbulence spectrum is observed (non-uniformity of theory)?
• How to quantify anisotropy in MHD, including in the absence of a large-scale magnetic field? How much // vs. perp. transfer is there?
• Universality, e.g. does a large-scale coherent forcing versus a random forcing influence the outcome?
• And how can one travel through parameter space, at high Reynolds number, thus at high 3D resolution?
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
with Kc a cut-off wave-number
Taylor-Green flow
Energy spectrum difference for two different formulations of LES based on two-point closure EDQNM
Noticeable improvement in the small-scale spectrum
(Baerenzung et al., 2008)
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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 RM
c for dynamo action