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
Cambridge, October 2008
Aires
pouquet@ucar.edu
* The Sun, and other stars
* The Earth, and other planets including extra-solar planets
• The solar-terrestrial interactions,
the magnetospheres, …
Many parameters and
dynamical regimes
Many scales, eddies
and waves interacting
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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)
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Abramenko, review (2007)
Surface (1 bar) radial magnetic fields for
Jupiter, Saturne & Earth versus Uranus & Neptune
(16-degree truncation, Sabine Stanley, 2006)
Axially dipolar
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Quadrupole ~ dipole
Taylor-Green turbulent flow at Cadarache
Bourgoin et al PoF 14 (‘02), 16 (‘04)…
W
R
H=2R
W
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
ITER (Cadarache)
Small-scale
The MHD equations
Multi-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
Direct Numerical Simulations
(DNS)
versus
Large Eddy Simulations
(LES)
1D space
&
Spectral space
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Resolve all scales
vs.
Model (many) small scales
Slide from Comte, Cargese Summer school on turbulence, July 2007
)
• 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 − ∇ 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 = <v2>/2 for NS:
Evα = < v2 + α2ω2 >/2
MHD: ETα, Hcα and HMα are invariant (+ Alfven theorem)
Lagrangian-averaged NS & MHD
dissipative equations
• ∂v/∂t + us · ∇v = −vj ∇u j s − ∇ P* + j × Bs + ∇2 v
• ∂Bs/∂t + us · ∇Bs = Bs · ∇us +  ∇2 B
B ~ k2 Bs --> hyperdiffusive term
Navier-Stokes: vortex filaments
DNS
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Alpha model
MHD: magnetic energy structures at 50% threshold
(nonlinear phase of a PM=1 dynamo regime)
DNS, 2563 grid
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Alpha model, 643
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)
3D Navier-Stokes: intermittency
DNS: X
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Chen et al., 1999; Kerr, 2002
Largest
filter length
& smaller cost:
more intermitency
Pietarila-Graham et al., PoF 20, 035107 2008
Third-order scaling law for fluids
(4/5th law) stemming from energy conservation
 r ~ < v us2 + [2 / r2 ] us3 >
• 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
Third-order scaling law
stemming from energy conservation
 r ~ < v us2 + [2 / r2 ] us3 >
• 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
k
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– 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
Why?
Solid line:  model, for large  (k =3)
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
 ~ 10-4
DNS
run
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Regions with  /< > ~ 0
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
3D Run
with
large 
(2 /10)
 ~ 10-4
DNS
run
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Black:
u//3(r=2/10) < 0.01
3D Run
with
large 
(2 /10)
 ~ 10-4
DNS
run
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Black:
u//3(r=2/10) < 0.01
3D Run
with
large 
(2 /10)
 ~ 10-4
``rigid bodies’’ (no stretching):
us(k) = v(k) / [ 1 + 2 k2]
DNS
run
and take limit of
large : the flow is advected
by a uniform field U (no degrees of freedom)
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``rigid bodies’’:
us(k) = v(k) / [ 1 + 2 k2]
us=constant
v ~ k2 us for large 
usv ~ k2 ~ k E(k)
E(k) ~ k+1
and take limit of
large : the flow is advected
by a uniform field Us (no degrees of freedom)
Black:
u//3(l=2/10) < 0.01
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3D Run
with
large 
(2 /10)
 ~ 10-4
Solid line:  model, for large  (k =3)
Dash line: same  model without
regions of negligible transfer
DNS
run
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Kinetic Energy Spectra in MHD
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k
Solid: DNS, 15363
Dash: LAMHD, 5123
Dot: Navier-Stokes , 5123
Energy Fluxes
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Solid/dash: LAMHD (Elsässer variables)
Dots: alpha-fluid
Circulation conservation is broken by Lorentz force
Magnetic Energy Spectra
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k
Solid: DNS, 15363 grid
Dash: LAMHD, 5123
Energy transfer in MHD is
more non local than for fluids
Transfer of kinetic energy to magnetic energy
from mode Q (x axis)
to mode K =10 (top panel)
K =20
K =30
Alexakis et al., PRE 72, 046301
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Current
sheets in
2D MHD
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DNS
Sorriso-Valvo et
al., P. of Plas. 9
(2002)
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
resistivity
associated with a lack
of equipartition in the
small scales
turb~ EkV - EkM < 0
Rädler;
AP, mid ‘80s
DNS
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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 …
Temporal evolution
of total energy (top),
kinetic (bottom) and
magnetic energies
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Solid: DNS, 15363, R ~ 1100
Dash: LAMHD, 2563
Dot: DNS, 2563
Temporal evolution of total enstrophy j2 +2
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Solid: DNS, 15363
Dash: LAMHD, 2563
Dot: DNS, 2563
Magnetic energy spectra
compensated by k3/2
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Solid: DNS, 15363
Dash: LAMHD, 2563
Dot: DNS, 2563
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 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