Vlasov Simulations of the IonAcoustic Instability

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Vlasov Simulations of the Ion-Acoustic Instability
Panagiota Petkaki and Mervyn Freeman British
Antarctic Survey (BAS) Cambridge, UK
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Collaborators

• Mervyn Freeman (BAS)
• Clare Watt (University of Alberta, Canada)
• Richard Horne (BAS)
• Tobias Kirk (University of Cambridge)

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Reconnection and Geospace

• Geospace is the only environment in which
  reconnection can be observed both
• In-situ (locally) by spacecraft
• Remotely from ground (globally)
• Reconnection between interplanetary magnetic
  field and geomagnetic field at magnetopause
• Drives plasma convection cycle involving
  reconnection in the magnetotail.

Earth
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Evidence for reconnection in Earths magnetosphere

• Dependence of many phenomena on Interplanetary
  Magnetic Field orientation relative to Earths
  magnetic dipole
• convection strength and pattern
• auroral activity
• Mixing of solar and terrestrial plasmas
• energy dispersion and cut-offs
• In-situ observation of X- and O-type magnetic
  reconnection structures
• magnetic nulls
• magnetic islands (plasmoids)
• bi-directional jets

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Change in Electron inertia from wave-particle
interactions

• Reconnection at MHD scale requires violation of
  frozen-in field condition.
• Kinetic-scale wave turbulence can scatter
  particles to generate anomalous resistivity.
• Change in electron momentum pe contributes to
  electron inertial term Davidson and Gladd, 1975
  with effective resistivity given by
• Broad band waves seen in crossing of reconnecting
  current sheet Bale et al., Geophys. Res. Lett.,
  2002.

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Anomalous Resistivity due to Ion-Acoustic Waves

• Resistivity from Wave-Particle interactions is
  important in collisionless plasmas (Watt et al.,
  GRL, 2002)
• We have studied resistivity from Current Driven
  Ion-Acoustic Waves (CDIAW)
• Used 1D Electrostatic Vlasov Simulations
• Realistic plasma conditions i.e. TeTi
  Maxwellian and Lorentzian distribution function
  (Petkaki et al., JGR, 2003,2006)
• Found substantial resistivity at quasi-linear
  saturation
• How does anomalous resistivity depend on MHD
  variables (n, T,J)?

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Outline of Seminar

• Why Ion-Acoustic waves
• Vlasov Simulation description of the Ion-Acoustic
  Resistivity - Maxwellian Plasma
• Non-linear Evolution of Ion-Acoustic Instability
• Cluster Observations of wave activity

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Why study Ion-Acoustic (IA) Waves?

• Previous analytical estimates and simulations of
  the resistivity due to current-driven
  ion-acoustic waves have concentrated on the
  regime where electron temperature far exceeds ion
  temperature. Not always the case in space
  plasmas.
• A Maxwellian plasma with similar electron and ion
  temperatures, needs a large current to excite
  unstable ion-acoustic waves.
• Ion-acoustic waves are measured in many regions
  of space plasma, and in laboratory plasma
  experiments indicates the need to study them in
  more detail for a range of plasma parameters.

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Evolution of Vlasov Simulation
Finite Difference Equations

• 1-D and electrostatic with periodic boundary
  conditions.
• Plasma species ? modelled with f?(z, v, t) on
  discrete grid
• The B 0 in the current sheet, but curl B
  ?0c2J.
• Second-Order Splitting Upwind Method (Petkaki,
  2005)

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Vlasov Simulation Initial Conditions

• CDIAW- drifting electron and ion distributions
• Apply white noise Electric field
• f? close to zero at the edges
• Maxwellian
• Drift Velocity - Vde 1.2 x ?
• (? (2T/m)1/2 )
• Mi25 me
• Ti1 eV, Te 2 eV
• nine 7 x 106 /m3
• Nz 642, Nve 891, Nvi 289

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• Maxwellian Run
• Evolution from linear to quasi-linear saturation
  to nonlinear
• Distribution function changes
• Plateau formation at linear resonance
• Ion distribution tail

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Time-Sequence of Full Electron Distribution
Function

• Top figure Anomalous resistivity
• Lower figure Electron DF

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Ion-Acoustic Resistivity Post-Quasilinear
Saturation

• Resistivity from Wave-Particle interactions is
  important in Collisionless plasmas
• We have studied resistivity from Current Driven
  Ion-Acoustic Waves using Vlasov Simulations
• Realistic plasma conditions i.e. TeTi
  Maxwellian and Lorentzian distribution function
• Found Substantial resistivity at quasi-linear
  saturation (saturation of fastest growing mode)
• What happens after quasi-linear saturation
• We investigate the non-linear evolution of the
  ion-acoustic instability and its resulting
  anomalous resistivity by examining the properties
  of two statistical ensembles of Vlasov
  simulations.
• Resistivity after saturation also important
• Behaviour of resistivity highly variable
• Ensemble of simulation runs probability
  distribution of resistivity values, study its
  evolution in time
• Evolution of each individual simulation in the
  nonlinear regime is very sensitive to initial
  noise field
• Require Statistical Approach
• 104 ensemble run on High Performance Computing
  Edinburgh and 10 ensemble run on local cluster.

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Superposition of the time evolution of 104 Vlasov
Simulations
Mean of the IA resistivity (?) 3?
? 75 35
? 188 105
Mean of the IA wave energy 3?
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Skewness and kurtosis of PD of resistivity values
PD of resistivity values at Quasilinear phase
skewness 0, kurtosis 3for a Gaussian DF
PD of resistivity values in Nonlinear phase
PD of resistivity values after Quasilinear phase
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Real Mass Vlasov Simulation Initial Conditions

• CDIAW- drifting electron and ion distributions
• Apply white noise Electric field
• f? close to zero at the edges
• Maxwellian
• Drift Velocity - Vde 1.2 x ?
• (? (2T/m)1/2 )
• Mi1836.15 me
• Ti1 eV, Te 2 eV
• nine 7 x 106 /m3
• Nz 529, Nve 3729, Nvi 307

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Real Mass Ratio Simulations

• Maxwellian Run
• Evolution from linear to quasi-linear saturation
  to nonlinear
• Distribution function changes
• Plateau formation at linear resonance

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Real Mass Ratio Simulations

• Superposition of Ensemble of 10 Vlasov
  Simulations of the IA Instability
• Mean of the IA anomalous resistivity 1 (green)
  and 3 standard deviations (red)
• Mean of the IA wave energy 1 (green) and 3
  (red) standard deviation

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Time Evolution of Electron Distribution Function
Real Mass Ratio Simulations
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Time Evolution of Electron Distribution Function
Real Mass Ratio Simulations
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Time Evolution of Ion Distribution Function
Real Mass Ratio Simulations
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Electron and Ion Bounce Frequencies

• Calculate Electron and Ion bounce frequencies
  using
• Compare with Fluctuations in Anomalous Resistivity

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Discussion

• Ensemble of 104 Vlasov Simulations with reduced
  mass ratio of the current driven ion-acoustic
  instability with identical initial conditions
  except for the initial phase of noise field
• Ensemble of 10 Vlasov Simulations with real mass
  ratio of the current driven ion-acoustic
  instability as before
• Variations of the resistivity value observed in
  the quasilinear and nonlinear phase
• Timescale of variations consistent with electron
  and proton bounce motion in reduced mass ratio
  Vlasov simulations.
• Timescale of variations consistent with electron
  bounce motion in reduced mass ratio Vlasov
  simulations
• The probability distribution of resistivity
  values Gaussian in Linear, Quasilinear,
  Non-linear phase
• A well-bounded uncertainty can be placed on any
  single estimate of resistivity, e.g., at
  quasi-linear saturation
• Estimation at quasi-linear saturation provides
  underestimation of Resistivity
• May affect likehood of magnetic reconnection and
  current sheet structure

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References

• Watt C.E.J., Horne R. Freeman M., Geoph. Res.
  Lett., 29, 10.1029/2001GL013451, 2002
• Petkaki P., Watt C.E.J., Horne R., Freeman M.,
  JGR, 108, A12, 1442, 10.1029/2003JA010092, 2003
• Petkaki P., Freeman M., Kirk T., Watt C.E.J.,
  Horne R., JGR,111, 10.1029/2004JA010793, 2006
• Petkaki P., Freeman M., ApJ, to be submitted,
  2007

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CLUSTER observations of electromagnetic waves in
a reconnection diffusion region in the Earths
magnetotail current sheet.

• Panagiota Petkaki1, Mervyn Freeman1, Andrew
  Walsh1,2,
• 1 British Antarctic Survey, Cambridge, UK
• 2 Mullard Space Science Lab., Dorking, UK 

Acknowledgements A. Buckley, E. Lucek, C. Owen,
A. Fazakerley, G. Abel, and R. Horne for
discussions. N. Cornilleau-Wehrlin, M.
Maksimovic, L. Mirioni (STAFF), E. Lucek (FGM),
H. Reme, I. Dandouras (CIS-CODIF), M. Andre, A.
Vaivads (EFW) and the Cluster Active Archive for
providing the data and associated support.
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Motivation

• Investigate the relationship of wave activity to
  a possible magnetic reconnection region
• Lower-Hybrid waves (Huba et al. 1977, Bale et
  al., 2002), Ion-acoustic waves (Galeev and
  Sagdeev, 1984, Scarf et al., 1984), Whistler
  waves (Deng and Matsumoto, 2001)
• Recent research in ion-acoustic anomalous
  resistivity (Watt et al., GRL, 2002, Petkaki et
  al., JGR, 2003, 2006)
• Cluster measurements of a magnetotail event on
  the 11th October 2001, 300-400 UT. Mean
  position in GSE X -15.65 Re, Y 10.9 Re, Z
  1.93 Re
• We used all four spacecraft, but Cluster 1 is
  shown here.

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(No Transcript)
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Cluster Tail event 11-Oct-2001 300-400 UT
Mean position in GSE X -15.65 Re Y 10.9
Re Z 1.93 Re
Minimum separation 1752km Maximum separation
1996km. 1500 Km separation in Z
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Observations
magnetic
electric

• Cluster move from northern lobe ( Bx) to
  southern lobe (- Bx) over whole interval, making
  several current sheet crossings.
• Flows reverse from tailward (-Vp?x) and duskward
  (Vp?y) to earthward ( Vp?x) and dawnward (-
  Vp?y) suggesting reconnection site moves over
  spacecraft (4).
• Strong wave activity is seen how is related to
  reconnecting current sheet structure.

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X-Z Plane
1
3
4
Tail
2
5
2

• Schematic of data interpretation
• SC1 in central plasma sheet (plasma ? ? 0.5)
• SC1 Bx gt 0 in 1 and 3 to Bx lt 0 in regime 5
• Transient partial crossing of the current sheet
  center in 2
• Several transient partial or complete crossings
  in regime 4 repeated magnetic reversals of the
  Bx component.

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5
1
3
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• SC1 Alfvén speed
• Compared (panel e)
• with the Alfvén speed from the magnetic field
  and density measurements.
• The flow is Alfvénic when B is small
• These times correspond to when the spacecraft is
  in the current sheet (low Alfvén speed).
• Provides further support for reconnection
  structure.

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• Magnitude of magnetic field as proxy for position
  in the current sheet
• Re-ordering STAFF magnetic and electric field
  spectra by B (i.e., finding average spectrum
  for all occasions when B 1, 2, 3 nT, etc.)
• Wave power reduced in centre of current sheet and
  maximizes in the lobes.
• Re-ordered spectra are broadband and relatively
  unstructured.
• Fce (Hz) 28 B (nT) (white line)
• Peak at a few tens of Hz in the magnetic field
  wave spectrum.
• Indicates a turbulent cascade extending to
  frequencies outside those measured by the STAFF
  instrument.

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• The electric energy density integrated over the
  STAFF frequency range is a minimum of 5.9 x 10-22
  J m-3 at B lt 1 nT and a maximum of 6.9 x 10-17
  J m-3 at B 19 nT (within a broad plateau
  between 15 and 21nT).
• The magnetic energy density is a minimum of 3.1 x
  10-18 J m-3 at B lt 1 nT and a maximum
  of 7.4 x 10-16 J m-3 at B 13 nT (within a
  broad plateau between 10 and 20 nT).
• Similar spectra and levels of wave activity are
  seen on the other spacecraft.

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Extended Wave Spectrum 032857 to 032917 UT
(1737-1757 seconds after 0300 UT)

• The magnetic field was stable (10 ? B ? 12 nT)
  in a location within the current sheet.
• EFW and FGM range of frequencies is from 0.04 Hz
  to 12Hz
• Overlaps with the STAFF spectra (8Hz to 4096 Hz)
• FFT EFW and FGM data to produce the power spectra
  at low frequencies
• Time averaged the STAFF electric and magnetic
  field spectra over the time interval to yield the
  spectra at high frequencies.

Magnetic Field
Electric Field
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Extended Wave Spectrum 032857 to 032917 UT
(1737-1757 seconds after 0300 UT)

• Overplotted are several linear wave frequencies
• The wave activity is electromagnetic in the range
  of 1Hz to 1kHz
• B spectrum comprises
• a broadband power-law component from 0.1 to
  1000Hz with exponent aB -2.4 (black solid line)
  
• a narrowband component peaking close to fpp
  (proton plasma f) 95Hz.
• E spectra comprises
• a broadband power-law component from 1 to 1000Hz
  with exponent aE -1.4 (black dashed line)
• a superposed narrowband enhancement peaking close
  to fpp
• an additional narrowband component at 400Hz,
  just above fce (electron cyclotron f)

aB -2.4
aE -1.4
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Extended Wave Spectrum 032857 to 032917 UT
Are they Lower Hybrid Waves?

• No enhancement of wave power at LH frequency
• From
• 
  
• Approximate B spectrum by a power law aB
  -2.36 (solid black line)
• Linear LH dispersion relation for finite plasma
  beta and Te lt Ti Davidson et al.1977
• Predict the electric field spectrum (solid purple
  line).
• Electric energy density 7 orders of magnitude
  lower than observed.
• Spectral slope is -1 (dashed purple line)
  different to aE -1.4 of the observed spectrum
• Observed spectrum is unlikely to be due to LH
  waves

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Extended Wave Spectrum 032857 to 032917 UT
Are they Whistler Waves?

• Evidence for whistler waves associated with fast
  plasma flows Deng and Matsumoto, 2001
• Whistler frequency for high beta plasma below
  fce.
• Narrowband peak in 20-140Hz, similar fpp -
  frequency of obliquely propagating whistler waves
  in the small wavelength limit
• Broadband spectral for whistler waves in a high
  beta plasma ? k2 and hence
• Consistent with aB - aE 1
• Solid green curve - linear whistler dispersion
  relation for high plasma beta and propagation
  parallel to the B (Biskamp 2000).
• Energy density an order low - the spectral slope
  of -1.4 consistent with observed (dot-dash
  green line).

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Summary

• Wave activity from 0.04 to 4000 Hz measured by
  EFW, FGM and STAFF on Cluster, possible
  reconnection event.
• Plasma flows of order of the local Alfvén speed
  reversed from tailward to earthward.
• Strong broadband electric and magnetic wave
  activity.
• Ordered the observed wave spectrum by the
  position within the current using the magnitude
  of the magnetic field.
• Electric and magnetic wave power decreased at all
  frequencies when the magnetic field strength
  approached zero.
• No evidence of Lower Hybrid waves
• Evidence of Whistler waves
• The wave environment of a reconnecting current
  sheet is likely characterized by non-linear
  whistler wave turbulence whose power maximises in
  the high-speed outflow jets close to the X-line
  and towards the edge of the current sheet.

Petkaki, Freeman and Walsh, Geophys. Res. Lett.,
33, L16105, doi10.1029/2006GL027066, 2006
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• Please come to- Natural Complexity Theory and
  Data
• in dialogue, Clare College, Cambridge, 13-17th
  August 2007
• www.antarctica.ac.uk/Meetings/2007/complexity2007/
  

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Natural Complexity Programme at BAS

• New cross-disciplinary programme
• Smallest BAS programme-7 scientists
• But strong links to other BAS scientists and data
  
• Visiting scientist programme to foster
  collaboration and knowledge transfer

www.antarctica.ac.uk/Meetings/2007/complexity2007/
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courtesy Rudolf Treumann
Few effective variables
Many independent variables
Complex systems are those with many strongly
interdependent variables. This excludes systems
with only a few effective variables, the kind we
meet in elementary dynamics. It also excludes
systems with many independent variables we learn
how to deal with them in elementary statistical
mechanics. Complexity appears where coupling is
important, but doesn't freeze out most degrees of
freedom Shalizi, Physics Today, 58(2), 65, Feb.
2005
courtesy Nick Watkins
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But why is the British Antarctic Survey
interested in complexity ?
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Complex Magnetosphere
Motivation Evidence for fractal behaviour in
magnetosphere e.g. Lui et al GRL, 2000
Uritsky et al JGR, 2002
courtesy Nick Watkins
solar wind
Solar wind
Magnetosphere
Freeman Watkins Science, 2002)
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How does solar wind drive the
magnetosphere ?

• Mass, momentum and energy are transferred into
  magnetosphere via magnetic reconnection at solar
  wind - magnetosphere interface.
• Plasma circulates from day to night over poles
  and from night to day around flanks.

courtesy Mervyn Freeman
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The Magnetospheric Substorm-a global scale energy
release event

BANG!

• Convection cycle is unsteady.
• Irregular, large-scale releases of energy in
  magnetotail called substorms (c.f. earthquakes).
• Intense magnetic field-aligned currents
  accelerate particles to cause aurora.

courtesy Mervyn Freeman