Transcript Analytical estimates of the resistivity due to ion

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)
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

p e
1 J

2
2
o  pe
p e t
o  pe
J t
1
• Broad band waves seen in crossing
of reconnecting
sheet [Bale et
E*  mcurrent
e  1 J

 Res.2 Lett.,
 ...

al., Geophys.
2002].
J  ne  J t
• The Measured Electric Field is more
than 100 times the analytically
estimated due to Lower Hybrid Drift
Instability
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 ionacoustic instability and its resulting anomalous
resistivity by examining the properties of a
statistical ensemble of Vlasov simulations.

WE
1
no k B Te  pe  0
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
E
   0c 2J    Bext
t
f
f  q  f
E
 v z
 
t
z  m  v z
• In-pairs method

J    q vf dv


• The B = 0 in the current sheet, but curl B = 0c2J.
• MacCormack method
• Resistivity
 1  1 pe
   2 
  pe 0  pe t
• Grid - Nz = 642, Nve = 891, Nvi = 289
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
N
E1 ( z ,0)   Etf sin k n z   
n 1
1/ 2
 2k BTe 

Etf  
3 
  0 De 
•
•
•
•
•
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
Maxwellian Run
• Evolution from
linear to quasilinear saturation to
nonlinear
• Distribution
function changes
• Plateau formation
at linear resonance
• Ion distribution tail
Time-Sequence of Full Electron
Distribution Function
• Top figure :
Anomalous resistivity
• Lower figure :
Electron DF
Time-Sequence of Full Ion Distribution
Function
• Top figure :
Anomalous
resistivity
• Lower figure : Ion
DF
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)
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
Mean of the ion – acoustic
anomalous resistivity () ± 3
220pet (blue) = 75 35
250pet (yellow) = 188 105
300pet , = 115 204
Mean of the ion-acoustic
Wave Energy ± 3
PD of resistivity values in the
Linear phase
Approximately Gaussian?
PD of resistivity values at
Quasilinear phase
PD of resistivity values
after Quasilinear phase
PD of resistivity values in
Nonlinear phase
Distribution in Nonlinear regime Gaussian?
Histogram of Anomalous resistivity values
Skewness and
kurtosis of
probability
distribution of
resistivity values
skewness = 0
kurtosis = 3
for a Gaussian
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
References
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
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 ionacoustic resistivity values
• Estimation of ion-acoustic resistivity can be used
as input by other type of simulations
Superposition of
the time evolution
of ion-acoustic
anomalous
resistivity of 3
Vlasov
Simulations
Linear Dispersion
Relation
Dispersion Relation
from Vlasov
Simulation
642 k modes
Finite
Difference
Equations
Grid 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
•
z
t 
vcut
One velocity grid cell per timestep
m v
t 
q Emax
Electron DF
Ion DF
k=2
Te/Ti = 1.0
Mi/Me = 25
Vde = 1.2 x e
Critical
Electron Drift
Velocity
Normalized
to
 k  3 / 2 2 k BT 
 

m 
 k
1/ 2
Mi=1836me
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 k-3)/2k]1/2 (2T/m)1/2 )
Nz=593, Nve=2667, Nvi=213
S3 - Lorentzian - Vde = 2.0 x 
( = [(2 k-3)/2k]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
k=2
Effect of the reduced
mass ratio on the
stability curves.
The Maxwellian case
is plotted as k = 80 for
illustration purposes.
Curves are plotted for
Te / Ti = 1.
The reconnecting universe
• Most of the universe is a
plasma.
• Most plasmas generate
magnetic fields.
• Magnetic reconnection is a
universal phenomenon
–
–
–
–
–
–
Sun and other stars
Solar and stellar winds
Comets
Accretion disks
Planetary magnetospheres
Geospace
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.
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 outof-plane magnetic field.
– Observed in geospace
[Ueno et al., J. Geophys.
Res., 2003]
z
x
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]
Previous analytical work
• Analytical estimates of the resistivity due to ion-acoustic
waves:
– Sagdeev [1967]:
pi v de Te
 2
where   0.01
 pe o cs Ti
– Labelle and Treumann [1988]:
1
WE

 pe o nkBTe
• Both estimates assume Te » Ti which is not the case for most
space plasma regions of interest (e.g. magnetopause).
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
Reconnection and Geospace
• Geospace is the only
natural 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
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]

WE
1
no k B Te  pe  0
• Resistivity in non-Maxwellian and nonlinear regimes.
[Watt et al., Geophys. Res. Lett., 2002]
Reconnection in Collisionless Plasmas
•
•
•
•
•
•
•
•
Magnetosphere
Magnetopause
Magnetotail
Solar Wind
Solar Corona
Stellar Accretion Disks
Planetary Magnetospheres
Pulsar Magnetospheres

•   i - i+1
Important Conclusions on The Ion-Acoustic
Resistivity
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