Anomalous resistivity and the non-linear evolution of the ion-acoustic instability
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Anomalous resistivity and the non-linear evolution of the ion-acoustic instability
Panagiota Petkaki
British Antarctic Survey, Cambridge
Magnetic Reconnection Theory
Isaac Newton Institute
Cambridge, UK
Tobias Kirk (Uni. Cambridge), Mervyn Freeman (BAS), Clare Watt (Uni. Alberta), Richard Horne (BAS)
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Change in Electron inertia from wave-particle
interactions
...1
2
*
+∂∂
⎟⎠
⎞⎜⎝
⎛==tJ
JJE
nemeη
• 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].
• The Measured Electric Field is more than 100 times the analytically estimated due to Lower Hybrid Drift Instability
tJ
Jtp
p peo
e
epeo ∂∂
=∂∂
−= 22
11ωεωε
η
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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. Te~Ti’
Maxwellian and Lorentzian distribution function (Petkaki et al., JGR, 2003)
– Found substantial resistivity at quasi-linear saturation
• What happens after quasi-linear saturation
• Study resistivity from the nonlinear evolution of CDIAW
• We investigate the non-linear evolution of the ion-acoustic instability and its resulting anomalous resistivity by examining the properties of a statistical ensemble of Vlasov simulations.
ηω ε
=W
n k TE
o B e pe
1
0
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Evolution of Vlasov Simulation One-dimensional and electrostatic with periodic boundary conditions.
• Plasma species modelled with f(z, v, t) on discrete grid
• f evolves according to Vlasov eq. E evolves according to Ampère’s Law
• In-pairs method
• The B = 0 in the current sheet, but curl B = 0c2J.
• MacCormack method
• Resistivity
• Grid - Nz = 642, Nve = 891, Nvi = 289
zz v
fE
m
q
z
fv
t
f
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛−
∂∂
−=∂∂
xtBeJc
t
E×∇+−=
∂∂ 2
0
∑∫∞
∞−
=
vvfqJ d
t
p
pe
epe ∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
11
02 εω
η
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Vlasov Simulation Initial Conditions• CDIAW- drifting electron and ion
distributions – Natural Modes in Unmagnetised Plasmas driven unstable in no magnetic field and in uniform magnetic field Centre of Current Sheet - driven unstable by current
• Apply white noise Electric field
• f close to zero at the edges• Maxwellian • Drift Velocity - Vde = 1.2 x (2T/m)1/2 • Mi=25 me, Ti=1 eV, Te = 2 eV• ni=ne = 7 x 106 /m3
( )2/1
30
11
2
sin)0,(
⎟⎟⎠
⎞⎜⎜⎝
⎛=
+= ∑=
De
eBtf
n
N
ntf
TkE
zkEzE
λε
ϕ
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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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Time-Sequence of Full Ion Distribution Function
• Top figure : Anomalous resistivity
• Lower figure : Ion DF
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Ion-Acoustic Resistivity Post-Quasilinear Saturation
• Resistivity at saturation of fastest growing mode• 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 the nonlinear regime is very sensitive to
initial noise field– Require Statistical Approach
• 104 ensemble run on High Performance Computing (HPCx) Edinburgh (1280 IBM POWER4 processors)
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Superposition of the time evolution of ion-acoustic anomalous resistivity of 104 Vlasov Simulations
Superposition of the time evolution of ion-acoustic
wave energy of 104 Vlasov Simulations
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Mean of the ion – acoustic anomalous resistivity (η) ± 3
Mean of the ion-acoustic Wave Energy ± 3
220ωpet (blue) η= 75 35
250ωpet (yellow) η= 188 105
300ωpet , η= 115 204
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PD of resistivity values at Quasilinear phase
PD of resistivity values in the Linear phase
Approximately Gaussian?
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PD of resistivity values after Quasilinear phase
PD of resistivity values in Nonlinear phase
Distribution in Nonlinear regime Gaussian?
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Histogram of Anomalous resistivity values
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Skewness and kurtosis of probability
distribution of resistivity values
skewness = 0kurtosis = 3
for a Gaussian
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Discussion• Ensemble of 104 Vlasov Simulations of the current driven
ion-acoustic instability with identical initial conditions except for the initial phase of noise field
• Variations of the resistivity value in the quasilinear and nonlinear phase
• The probability distribution of resistivity values Gaussian in Linear, Quasilinear, Non-linear phase
• A well-bounded uncertainty on any single estimate of resistivity.
• Estimation of resistivity at quasi-linear saturation is an underestimate.
• May affect likehood of magnetic reconnection and current sheet structure
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ReferenceReferencess
1. Petkaki P., Watt C.E.J., Horne R., Freeman M., 108, A12, 1442, 10.1029/2003JA010092, JGR, 2003
2. Watt C.E.J., Horne R. Freeman M., Geoph. Res. Lett., 29, 10.1029/2001GL013451, 2002
3. Petkaki P., Kirk T., Watt C.E.J., Horne R., Freeman M., in preparation
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Conclusions
• Ion-Acoustic Resistivity can be high enough to break MHD frozen-in condition
• Form of the distribution function of ions and electrons is important
• Gaussian statistics describes variation in ion-acoustic resistivity values
• Estimation of ion-acoustic resistivity can be used as input by other type of simulations
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Superposition of the time evolution
of ion-acoustic anomalous
resistivity of 3 Vlasov
Simulations
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Linear Dispersion Relation
Dispersion Relation from Vlasov Simulation
642 k modes
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Finite Difference Equations
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Grid of Vlasov SimulationGrid of Vlasov Simulation Significant feature of the Code : Number of grid points to reflect
expected growing wavenumbers - ranges of resonant velocities
• Spatial Grid : Nz=Lz/Δz
• Largest Wavelength (Lz)
• Δz is 1/12 or 1/14 of smallest wavelength
• Velocity Grid Nv{e,i} =2 X (vcut/Δv{e,i}) +1
• vcut > than the highest phase velocity
• Vcut,e = 6 + drift velocity or 12 + drift velocity
• Vcut,i = 10 or 10 maximum phase velocity
• Time resolution
• Courant number
• One velocity grid cell per timestep
vcut
zt
Δ≤Δ
maxE
v
q
mt
Δ≤Δ
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Electron DF
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Ion DF
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= 2Te/Ti = 1.0Mi/Me = 25
Vde = 1.2 x e
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2/122/3
⎟⎠
⎞⎜⎝
⎛ −=
mTkB
Critical Electron Drift
Velocity Normalized
to
Mi=1836me
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Compare Anomalous Resistivity from Three Simulations
• S1 - Maxwellian - Vde = 1.35 x ( = (2T/m)1/2 ) Nz=547, Nve=1893, Nvi=227• S2 - Lorentzian - Vde = 1.35 x
( = [(2 -3)/2]1/2 (2T/m)1/2 ) Nz=593, Nve=2667, Nvi=213• S3 - Lorentzian - Vde = 2.0 x ( = [(2 -3)/2]1/2 (2T/m)1/2 ) Nz=625, Nve=2777, Nvi=215• Mi=25 me • Ti=Te = 1 eV• ni=ne = 7 x 106 /m3
• Equal velocity grid resolution =2
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Effect of the reduced mass ratio on the stability curves.
The Maxwellian case is plotted as = 80 for illustration purposes.
Curves are plotted forTe / Ti = 1.
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The reconnecting universe
• Most of the universe is a plasma.
• Most plasmas generate magnetic fields.
Cusp-shaped soft X-ray structure on the northeast limb of the Sun observed by the soft X-ray telescope on the Yohkoh spacecraft. Reconnection above the cusp structure may drive a coronal mass ejection and eruptive flare.
• Magnetic reconnection Magnetic reconnection is a universal is a universal phenomenonphenomenon– Sun and other starsSun and other stars– Solar and stellar windsSolar and stellar winds– CometsComets– Accretion disksAccretion disks– Planetary Planetary
magnetospheresmagnetospheres– GeospaceGeospace
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z
x
Hall MHD reconnection
• Physics
– Hall effect separates ion and electron length scales.
– Whistler waves important (not Alfven waves)
• Consequences
– fast reconnection
– insensitive to mechanism which breaks frozen-in
• Evidence– Generates quadrupolar out-
of-plane magnetic field.– Observed in geospace
[Ueno et al., J. Geophys. Res., 2003]
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SOC Reconnection?
• Distributions of areas and durations of auroral bright spots are power law (scale-free) from kinetic to system scales [Uritsky et al., JGR, 2002; Borelov and Uritsky, private communication]
• Could this be associated with multi-scale reconnection in the magnetotail?
• Self-organisation of reconnection to critical state (SOC) [e.g., Chang, Phys. Plasmas, 1999]
• cf SOC in the solar corona [Lu, Phys. Rev. Lett., 1995]
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Previous analytical work
• Analytical estimates of the resistivity due to ion-acoustic waves:– Sagdeev [1967]:
– Labelle and Treumann [1988]:
• Both estimates assume Te » Ti which is not the case for most space plasma regions of interest (e.g. magnetopause).
01.0 where2
≈= εω
ωη
i
e
s
de
ope
pi
TT
cv
eB
E
ope Tnk
W
εωη 1
=
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Ion-Acoustic Waves in Space Plasmas
• Ionosphere, Solar Wind, Earth’s Magnetosphere• Ion-Acoustic Waves – Natural Modes in Unmagnetised
Plasmas– driven unstable in no magnetic field and in uniform
magnetic field – Not affected by the magnetic field orientation (under
certain conditions)• Centre of Current Sheet - driven unstable by current• Source of diffusion in Reconnection Region• Current-driven Ion-Acoustic Waves – finite drift between
electrons and ions
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Reconnection and Geospace
Earth
• Geospace is the only Geospace is the only natural environment in natural environment in which reconnection can be which reconnection can be observed bothobserved both– in-situ (locally) by in-situ (locally) by
spacecraftspacecraft– remotely from ground remotely from ground
(globally)(globally)• Reconnection between
interplanetary magnetic field and geomagnetic field at magnetopause.
• Drives plasma convection Drives plasma convection cycle involving cycle involving reconnection in the reconnection in the magnetotail.magnetotail.
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Anomalous Resistivity due to Ion-Acoustic Waves
• 1-D electrostatic Vlasov simulation of resistivity due to ion-acoustic waves.
• Resistivity is 1000 times greater than Labelle and Treumann [1988] theoretical (quasi-linear) estimate (depending on realistic mass ratio)
– must take into account the changes in form of the distribution function.
• Consistent with observations in reconnection layer [Bale et al., Geophys. Res. Lett., 2002]
• Resistivity in non-Maxwellian and non-linear regimes.
ηω ε
=W
n k TE
o B e pe
1
0
[Watt et al., Geophys. Res. Lett., 2002]
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Reconnection in Collisionless Plasmas
• Magnetosphere• Magnetopause• Magnetotail• Solar Wind• Solar Corona• Stellar Accretion Disks• Planetary Magnetospheres• Pulsar Magnetospheres
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Δη
Δη ηi - ηi+1
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1. Calculated ion-acoustic anomalous resistivity for space plasmas conditions, for low Te/Ti 4, Lorentzian DF.
2. A Lorentzian DF enables significant anomalous resistivity for conditions where none would result for a Maxwellian DF.
3. At wave saturation, the anomalous resistivity for a Lorentzian DF can be an order of magnitude higher than that for a Maxwellian DF, even when the drift velocity and current density for the Maxwellian case are larger.
4. The anomalous resistivity resulting from ion acoustic waves in a Lorentzian plasma is strongly dependent on the electron drift velocity, and can vary by a factor of 100 for a 1.5 increase in the electron drift velocity.
5. Anomalous resistivity seen in 1-D simulation
6. Resistivity I) Corona = 0.1 m, II) Magnetosphere = 0.001 m
Important Conclusions on The Ion-Acoustic Important Conclusions on The Ion-Acoustic ResistivityResistivity