A few issues in turbulence and how to cope with them using computers

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A few issues in turbulence and how to cope with
them using computers
Annick Pouquet,
NCAR Alex Alexakis!, Julien Baerenzung,
Marc-Etienne Brachet!, Jonathan
Pietarila-Graham, Aimé Fournier, Darryl Holm_at_,
Giorgio Krstulovic, Ed Lee, Bill Matthaeus,
Pablo Mininni, Jean-François Pinton!!, Hélène
Politano, Yannick Ponty, Duane Rosenberg,
Amrik Sen Josh Stawarz
! !!
ENS, Paris and Lyon
MPI, Postdam
LANL
_at_ Imperial College
LANL
Observatoire de Nice
U. Leuwen
Bartol, U. Delaware,
and
Universidad de Buenos Aires

CU Boulder, July 2011 , pouquet_at_ucar.edu
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GENERAL OUTLINE for LECTURES
Physical complexity of flows on Earth and
beyond Vorticity and helicity dynamics
Kinematics of tensors and methodology
Exact laws, structures and different
energy spectra in MHD?
Complexity of phenomenology beyond
Kolmogorov Weak turbulence and beyond,
towards strong turbulence with closures
II Some results for MHD and for
rotation II - Modeling why and how II - The
Lagrangian averaging model, for MHD and perhaps
for fluids II - Adaptive mesh refinement with
spectral accuracy II - Application to the dynamo
problem at low magnetic Prandtl number
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• Energy dissipation (Celani)
• Eulerian velocity
• Acoustic intensity (vorticity fluctuations)

From Baudet, Cargèse Summer school
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Observations of galactic magnetic fields (after
Brandenburg Subramanian, 2005)
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Hurricane Francis from Space
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• The Sun, and other stars
• The Earth, and other planets -
• including extra-solar planets
• The solar-terrestrial interactions
• (space weather), the magnetospheres,

Many parameters and dynamical regimes Many
scales, eddies and waves interacting Cluster ?
MMS,
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Weather, climate and all that

• In order to progress, one needs
• A deeper understanding of underlying
  fundamental
• processes (minimalist approach)
• An added complexity in models
• (maximalist approach Physics, Chemistry,
  Biology, Socio-economics, )
• An adequate resolution, both in space and
  time, both observationally, experimentally and
  numerically (expensive approach)

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Seemingly simple questions

• By how much is the sea-level going to rise by,
  say, year 2030?
• What does it take to control the global
  temperature so that it be increasing by
• at most two degrees? Inverse
  problem

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Surface-Atmosphere Interactions

• Influences of atmospheric stability, orography,
  and plants all matter on ecosystem exchanges

From Peter Sullivan, NCAR
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Surface-Atmosphere Interactions

• Mathematical model Trees viewed as fractal
• ? ensuing evaluation of transport (drag)
  coefficients through Random Numerical Simulations
  (RNS)

Meneveau et al. (JHU)
Other possible model corrugation?
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Seamless predictions across scales, from hourly
to decadal
Greenland ice cover
By how much is the sea level going to rise?
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Seamless predictions across scales, from hourly
to decadal
chemistry, biology, economy, policies, society,

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Slide after Mark Rast, TOY Workshop (NCAR)
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Modeled SST, West coast
Observed SST, East coast
Sea Surface Temperatures (SST)
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Slide after Krueger, TOY Workshop (NCAR)
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Slide after Ian Foster (Argonne), National Energy
Modeling system
One modeling example of societal complexity
wiring diagrams
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Models are advancing to tackle different issues
Continental Scale ?
Focus of modelers
Different Scales (space/time) Different Issues
Problems Different Stakeholders Different
Decisions
Human/watershed Scale ?
Slide after Roy Rasmussen, NCAR
Center for Hydrometeorology and Remote Sensing,
University of California, Irvine
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Complex interactions

• Rotation, stratification (waves eddies),
  radiation,
• Compressibility, moisture,
• Chemistry, biology, hydrology
• Math algorithms
• Boundaries, geometry,
• Socio-politico-economical processes

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Multi-scale Interactions

• Large ? small, or climate ? weather
• Small ? large, or weather ? climate
• orography / bathymetry ? circulation
  pattern and pluviosity ? cloud cover ? climate
• Butterfly effect
• Eddy-viscosity and anomalous transport
  coefficients
• Beating of 2 small frequencies (eddy-noise)
• Inverse cascades (Statistical mech. with 2 or
  more temperatures)
• The Andes, the Rockies, the Alps, the Pacific
  coast
• Memory effect (decadal time scales, El Niño, )

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Vorticity dynamics

• Vorticity equation ?t? curl (v x ?) ? ? ?
  curl F
• Also
• Dt ? ?t ? v. grad ? ?. Grad v
  ? ? ? curl F
• advection stretching
  by velocity gradients dissipation forcing
• Model w ?
• Dt w w . grad v, grad v O(1) exponential
  growth of vorticity at early times
• But
• You can also view w grad v so
• Dt w w2 explosive growth
  w(t) (t-t)-1
• What is really happening?
• What can get us out of this explosive growth?

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Vorticity dynamics

• Vorticity equation ?t? curl (v x ?) ? ? ?
  curl F
• Also
• Dt ? ?t ? v. grad ? ?. Grad v
  ? ? ? curl F
• advection stretching
  by velocity gradients dissipation forcing
• Model w ?
• Dt w w . grad v, grad v O(1) exponential
  growth of vorticity at early times
• But
• You can also view w grad v so
• Dt w w2 explosive growth
  w(t) (t-t)-1
• What is really happening? What can get us out of
  this explosive growth?
• What is the role of the geometry of structures?

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Vorticity dynamics and dissipation

• DE/DT e -2 ? lt ?2gt
• ? ? 0, ? ? infinity?, e O(1) U3/L
  with U1, L1
• E(k)k-5/3 ? O (k) k2-5/3 k1/3 the
  vorticity peaks at the dissipation scale
• Could there be other expressions for e?

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Vorticity dynamics and dissipation

• DE/DT e -2 ? lt ?2gt
• ? ? 0, ? ? infinity?, e O(1) U3/L
  with U1, L1
• Could there be other expressions for e?
• Should we introduce another time-scale, such as a
  wave period? For example, for rotating flows
• e U3/L Ro U4/L2O with RoU/LO the
  Rossby number
• Roltlt1 at high rotation less nonlinear transfer
  because of linear waves

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Energy conservation ?

• Energy equation
• Dt ltv2gt/2 e
• - ? lt?2gt
• What happens to the energy conservation when
• ? ? 0 does lt?2gt ? infinity ?
• There are indications that the energy dissipation
  rate is finite
• e Urms3 / L0
• i.e. e O(1) for Urms 1 and L0 1

After Sreenivasan (1998) and Ishihara and
Kaneda (2002)
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x
After Sreenivasan (1998) and Kaneda (Cargèse,
2007) X Kaneda et al. (2002, 2009), R? 1500
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Nth-order scaling exponents of structure
functions (velocity and temperature differences)
n/3
Slide from Lanotte, Cargèse Summer school
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Invariants of the Euler equations

• Invariants in the absence of dissipation
  forcing (?0F)
• Kinetic energy EV ltv2gt/2 , together
  with
• In three dimensions kinetic helicity HV ltv.
  ? gt (mid 60s, Moreau Moffatt after Woltjer
  for MHD, mid 50s)

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Invariants of the Euler equations

• Invariants in the absence of dissipation
  forcing (?0F)
• Kinetic energy EV ltv2gt/2 , together
  with
• In three dimensions kinetic helicity HV ltv.
  ? gt (mid 60s, Moreau Moffatt after Woltjer
  for MHD, mid 50s)
• Statistical equilibria no indication of inverse
  cascade (Kraichnan, 1973)
• In two dimensions lt ?2 gt, the enstrophy
• (and more, )

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?
Helicity dynamics H is a pseudo (axial) scalar
u
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?
Helicity dynamics H is a pseudo (axial) scalar
u
ltui(k)uj(-k)gt UE(k) Pij(k)
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?
Helicity dynamics H is a pseudo (axial) scalar
u
ltui(k)uj(-k)gt UE(k) Pij(k) eijlkl UH(k)
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?
Helicity dynamics H is a pseudo (axial) scalar
u
ltui(k)uj(-k)gt UE(k) Pij(k) eijlkl UH(k)
L
R
For particles, helicity is S.P where S is spin
vector and P is momentum, versus chirality for
fluids and knots (Kelvin, 1873, 1904). L,R
important differences thalidomide, aspartame,
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?
Helicity dynamics H is a pseudo (axial) scalar
u
ltui(k)uj(-k)gt UE(k) Pij(k) eijlkl UH(k)
Two defining functions UE UH, or E(k) and
H(k) ? A priori two different scaling laws
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?
Helicity dynamics H is a pseudo (axial) scalar
u
ltui(k)uj(-k)gt UE(k) Pij(k) eijlkl UH(k)

• Two defining functions UE UH,
• or E(k) and H(k)
• A priori two different scaling laws
• And more if anisotropic (polarization)

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?
Helicity dynamics H is a pseudo (axial) scalar
u
A wild knot
L
R
For particles, helicity is S.P where S is spin
vector and P is momentum, versus chirality for
fluids and knots (Kelvin, 1873, 1904). L,R
important differences thalidomide, aspartame,
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Shear helicityin the atmosphere
Helicity in tropical cyclones versus
shear Molinari Vollaro, 2010
Helicity spectrum in the Planetary Boundary
Layer K41
Koprov, 2005
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Helicity in Hurricane Andrew, Xu Wu, 2003
Strong helicity where magnetic field is active,
Komm et al., 2003 Role of small-scale helicity
in the dynamo process (Parker, 1950s, )
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Helicity dynamics
hrcos(v, ?), non-helical TG flow
Blue, hrgt0.95 Red, hrlt-.95

• Zoom on 3D NS flow
• Perspective volume rendering of relative helicity
  i.e. the degree of alignment between velocity and
  vorticity for the Taylor-Green (TG) vortex,
  non-helical globally

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Helicity dynamics
hrcos(v, ?), non-helical TG flow
Blue, hrgt0.95 Red, hrlt-.95

• Evolution equation for the local helicity density
  (Matthaeus et al., PRL 2008)
• ?t(v. ?) v. grad(v. ?)
• ?.grad(v2/2 - P) ?? (v. ?)
• forcing
  
• v. ? (x) can grow locally
• on a fast (nonlinear) time-scale

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Helicity dynamics
hrcos(v, ?), non-helical TG flow
Blue, hrgt0.95 Red, hrlt-.95

• Evolution equation for the local helicity density
  (Matthaeus et al., PRL 2008)
• ?t(v. ?) v. grad(v. ?)
• ?.grad(v2/2 - P) ?? (v. ?)
• forcing
  
• v. ? (x) can grow locally
• on a fast (nonlinear) time-scale

The several ways to a 0 solution
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• Vorticity ?curl v
  Relative helicity intensity hcos(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 local mirror symmetry, and weak
  nonlinearities in the small scales

Blue, hgt 0.95 Red, hlt-0.95
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Exact laws in turbulence

• With 5 hypotheses
• Homogeneity
• Isotropy
• Incompressibility
• Stationarity
• High Reynolds number
• Kolmogorov 1941,

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Exercise The 12th law for 1D Burgers
equation xxr uu(x)




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The magnetohydrodynamics (MHD) equations

• P is the pressure, B is the induction (in Alfvén
  velocity units), j ? B is the current, ? is
  the resistivity, and div B 0 (not an
  assumption).

______ Lorentz force
Elsässer z v B ? ... What is different
from the Navier-Stokes eqs.?
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Invariants of the MHD equations

• Invariants (no dissipation, no forcing, ?0?F)
• Total energy ET ltv2 b2gt/2 , together
  with
• In three dimensions cross-helicity HC ltv.bgt,
  and magnetic helicity HM ltA.b gt/2
  (Woltjer, mid 50s)
• In two dimensions HM 0 lt A2 gt, the square
  magnetic potential, is invariant (and more, ),
  with b curl A
• Does it matter for the MHD equations when ??0, ?
  ?0?
• Does conservation hold for vanishing viscosity
  resistivity?

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Two coupled exact laws in MHD 1PolitanoAP,GRL
25, 1998

• In terms of velocity magnetic field, two
  invariants (ltv2b2gt/2 ltv.bgt ), and two
  scaling laws for MHD for
• dF(r) F(xr) - F(x) structure function for
  field F
• longitudinal
  component dFL(r) dF . r/ r
• lt dvLdvi2 gt ltdvLdbi2 gt -2 lt dbLdvidbi gt -
  (4/D) eTr
• - ltdbLdbi2 gt - lt dbLdvi2 gt2 lt dvLdvidbi gt -
  (4/D) ec r
• with eT - dt(EV EM) and ec - dtltv.bgt
  D is the space dimension

• 
  
  

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Two coupled scaling laws in MHD
2PolitanoAP,GRL 25, 1998

• In terms of velocity magnetic field, two
  invariants (ltv2b2gt/2 ltv.bgt ), and two
  scaling laws for MHD for
• dF(r) F(xr) - F(x) structure function for
  field F
• longitudinal
  component dFL(r) dF . r/ r
• lt dvLdvi2 gt ltdvLdbi2 gt -2 lt dbLdvidbi gt -
  (4/D) eTr
• - ltdbLdbi2 gt - lt dbLdvi2 gt2 lt dvLdvidbi gt -
  (4/D) ec r
• with eT - dt(EV EM) and ec - dtltv.bgt
  D is the space dimension

• 
  
  

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Two coupled scaling laws in MHD
2PolitanoAP,GRL 25, 1998

• In terms of velocity magnetic field, two
  invariants (ltv2b2gt/2 ltv.bgt ), and two
  scaling laws for MHD for
• dF(r) F(xr) - F(x) structure function for
  field F
• longitudinal
  component dFL(r) dF . r/ r
• lt dvLdvi2 gt ltdvLdbi2 gt -2 lt dbLdvidbi gt -
  (4/D) eTr
• - ltdbLdbi2 gt - lt dbLdvi2 gt2 lt dvLdvidbi gt -
  (4/D) ec r
• with eT - dt(EV EM) and ec - dtltv.bgt
  D is the space dimension
• Strong V regime, or strong B regime or Alfvénic
  (vB) regime (Ting et al 86) in the latter
  case, v-B correlations play a dynamical role
• Where such laws apply, the energy input can be
  measured, e.g. in the solar wind

• 
  
  

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THE ROLE OF VELOCITY-MAGNETIC FIELD CORRELATIONS

• Exact law (Politano AP,1998) dimensionally, lt
  db2(l) dv(l) gt l ,
• does it imply dv l 1/3 as for
  fluid turbulence?
• No, because of v-b correlations
• Boldyrev, 2006 what if
• lt db2(l) dv(l) ?(l) gt l , so e.g., dv(l)
  db(l) l1/4 , d?(l) l1/4
• where ? is the angle (degree of
  alignment) of V B
• This scaling is compatible with IK, E(k)
  k-3/2 (but role of anisotropy)
• It implies a variation of V-B alignment with
  scales (observed numerically)

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Several time scales in a turbulent flow

• Local turn-over time TNL l/ul k-2/3 for
  a Kolmogorov spectrum
• Integral time Tint Lint/Urms k0
• Global advection time TE l/Urms k-1
• Diffusive time Tdiss l2/? k-2
• T(l) li and E(l)
  lf(i)
• One can define other time scales, e.g.,
  associated with waves TW
• ? Different regimes and different spectral
  indices for energy spectra?
• E.g., rotation, stratification,
  compressibility, magnetic fields,

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The GReC experiment

• Flow Reynolds Numbers and Scales
• Reynolds Numbers Re 8.105 to 109
• Integral Scale L 30 cm
• Taylor Reynolds R? 1300 to 6000
• Taylor Scale ? 1.7 mm to 0.4 mm
• Kolmogorov Scale 20 µm to 0.4 µm
• Time series up to 109 samples w/ sampling _at_ 1.25
  MHz

Slide from Baudet, Cargèse Summer school
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2nd order experimental data
1962
Slide from Baudet, Cargèse Summer school
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Compensated energy spectra ?
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Numerical data

• Kaneda Ishihara, 2006
• Red 40963 grid for Navier-Stokes turbulence

Exercise What is the criterium to choose the
Reynolds number at a given grid resolution?
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K41-Compensated spectra
Kaneda Ishihara 2006
Kolmogorov constant
Slow convergence

Slide from Kaneda, Cargèse Summer school
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Compensated energy spectrum, with bottleneck
Is equipartition stemming from statistical
mechanics equilibrium (?0) responsible for the
bottleneck effect (Frisch et al., 2008)? Result
from a 2-pt closure of turbulence (Bos et al.)
Slow convergence

Slide from Cambon, Cargèse Summer school
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VAPOR freeware (John Clyne Alan Norton,
CISL)

• What happens to a flow when the Reynolds
  number is increased?
• Navier-Stokes grids with N3 points, Taylor-Green
  flow
• 643 2563
• 10243 20483 grids
• Pablo Mininni
• (NCAR, and Pittsburgh for the highest
  resolution run)

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VAPOR freeware (John Clyne Alan Norton,
CISL)

• What happens to a flow when the Reynolds
  number is increased?
• Navier-Stokes grids with N3 points, Taylor-Green
  flow
• 643 2563
• 10243 20483 grids
• Pablo Mininni
• (NCAR, and Pittsburgh for the highest
  resolution run)

Answer more structures, more scales in
interaction, more complexity?
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• SEMI-ZOOM
• Vortex filament complex with particle paths in
  red
• Navier-Stokes
• R? 1200
• 20483 grid points

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Zoom on vortex tube with helical field lines
Navier-Stokes equations in three dimensions R?
1000
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The classical picture of turbulence
F
ul,l
ul/2,l/2
ul/4, l/4
ul/8, l/8
? dE/dt energy dissipation rate E kE(k)
(locality) and t l/ul (turn-over time), So
? ul3/l
and E(k) CK ?2/3 k -5/3
v
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The classical picture of turbulence
F
ul,l
ul/2,l/2
ul/4, l/4
ul/8, l/8

• ? dE/dt , EkE(k) and tl/ul, ? ul3/l
  E(k) ?2/3 k -5/3
• Can we ignore long-range interactions
• e.g. climate at large scale weather
  at small scale
• Can we ignore the aspect ratio the structure
  of eddies?

v
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Some history on transfer and locality in fluid
turbulence

• Theory Kolmogorov (1941) Kraichnan (1971, 1976)
• Direct Numerical Simulations
• Importance of non local interactions
• Domaradzki PRL (1987), Phys. Fluids (1988),
  Kerr (1990)
• - Local energy transfer with non local
  interactions
• Domaradzki Rogallo (1990), Yeung
  Brasseur (1991), Ohkitani (1992)
• Models and different measures of locality
• Herring (70s), Zhou (1993, 1996)
• Locality in wavelet space Kishida (1999)
• Bounds on locality Eyink (1994, 2005)
• Nonlocality for NS and MHD fluids, including Hall
  MHD

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Scaling with Reynolds number of the ratio of
nonlocal to local energy flux at the Taylor scale
20483
Scaling laws barely appear for the largest
resolution run A self-similar Kolmogorov-like
behavior becomes apparent only for sufficiently
large Reynolds numbers (Navier-Stokes run)
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Intermittency higher-order data
p2
3
9
Slide from Baudet, Cargèse Summer school
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Extreme events as a function of resolution, and
thus of Reynolds number (MHD example)

• Probability Distribution Function of current
  density (with Jrms 1)
• purple (483 grid points), 963, 1923, 3843,
  7683 15363 grids
• The flow becomes
• progressively more
• non-Gaussian,
• with large wings
• Extrema from 12 to 130 in units of Jrms strong
  but rare events, or intermittency

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NS intermittency Anomalous exponents (using
Extended Self Similarity, or ESS) for the TG
ABC flows there are small but measurable
differences between them

T2(K,Q)
1.9 10-5
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Conclusions for Navier-Stokes

• Slow return to homogeneity and isotropy at small
  scales
• Kolmogorov spectrum but departures from
  self-similarity at higher order
• Intermittency discrepancies viewed on PDFs and
  anomalous exponents between different flows
  between active / non-active regions of a given
  flow is there a weak break of universality?
• Influence of forcing scale, as seen on enstrophy
  as measured on energy transfer, leading to
  quantifiable degree of interactions between
  disparate scales
• Models such as RDT, Lagrangian model
  leading to advection by smooth flow may increase
  in value
• Scaling of such properties with Reynolds number?
• What is the role of helicity (2nd invariant of
  the Euler equations)?

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Thank you for your attention!