Characterizing Fluid and Kinetic Instabilities using Field-Particle Correlations on Single-Point Time Series

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by  Kristopher G Klein

Characterizing Fluid and Kinetic Instabilities using Field-Particle Correlations on Single-Point Time
Series
Kristopher G. Klein1,a)
Department of Climate and Space Sciences and Engineering, University of Michigan, Ann Arbor,
MI 48109, USA
Arecentlyproposedtechniquecorrelatingelectricfieldsandparticlevelocitydistributionsisapplied
7 tosingle-pointtime seriesextractedfromlinearlyunstable,electrostaticnumericalsimulations. The
1
form of the correlation, which measures the transfer of phase-space energy density between the
0
electric field andplasma distributions and had previouslybeen applied to damped electrostatic sys-
2
tems, is modified to include the effects of drifting equilibrium distributions of the type that drive

n
counter-streaming and bump-on-tail instabilities. By using single-point time series, the correlation
a
is ideal for diagnosing dynamics in systems where access to integrated quantities, such as energy, is
J
observationally infeasible. The velocity-space structure of the field-particle correlation is shown to
3
characterize the underlying physical mechanisms driving unstable systems. The use of this correla-
1
tion in simple systems will assist in its eventual application to turbulent, magnetized plasmas, with

] the ultimate goal of characterizing the nature of mechanisms that damp turbulent fluctuations in
h
the solar wind.
p
-
m
I. INTRODUCTION fast flows, the time evolution is essentially frozen
s and the measurement traces out the spatial struc-
a
l A significant goal of plasma physics research is tureoftheturbulence;areviewoftheapplicationof
p the characterization of mass, momentum, and en- Taylor’s Hypothesis to solar wind observations can
s. ergytransportinawidevarietyofcomplexsystems. be found in Klein et al 2014.4
c
In particular, the question of what mechanisms me- Toaddressthesecondcomplicationofinaccessible
i
s diate the transfer ofenergybetween turbulentfields spatiallyintegratedquantities,onemayconsiderthe
y
and distributions of plasma particles, leading to the dynamics of the phase-space energy density rather
h
p eventual damping and dissipation of turbulence, is than the total energy. A technique has been re-
[ open. One system which displays turbulent behav- cently proposedto measure the local-in-phase-space

ior is the solar wind, a hot, diffuse emanation from energy transfer between fields and plasma distribu-
1
the Sun’s surface that fills the heliosphere. While tions using single-pointtime series of simple plasma
v
lacking the precise control over conditions afforded systems.5,6 Bycorrelatingtheproductoftheelectric
7
8 in a laboratory setting, the large volume of in situ field and velocity derivative of the particle distribu-
6 measurements of the solar wind over the last half tion measured at a single point in space, the veloc-
3 centuryhasledtotheaccrualofobservationsoftur- ity structure of the transferof energybetween fields
0 bulence with a wide variety of plasma parameters. and particles is obtained. By averaging this cor-
.
1 Such measurements have provenuseful to the study relationovera selectedtime interval, the oscillatory
0 of phenomena in magnetized turbulence.1 energytransferbetweenthefieldsandparticlesisre-
7 A limitation of these in situ observations is that moved,leavingonlythesecularenergytransfer. The
1
they are taken at a single-point in space at a given mechanisms responsible for this energy transfer can
:
v time.2 This raises at least two significant compli- be identified by the velocity space structure of the
i cations; one must disentangle the dynamics associ- field-particle correlation. Initial work applied this
X
ated with spatial and temporal variation, and track correlation to systems that damp via the Landau
ar the evolutionofspatiallyintegratedquantities,such resonance.7 Here, we considerthe transferof energy
as the energy content of a field or distribution of in linearly unstable systems, and show that field-
charged particles, given only single-point measure- particlecorrelationsareabletoidentifythepresence
ments. The first of these complications is addressed ofbothfluidandkineticinstabilitiesinsuchsystems.
by invoking Taylor’s Hypothesis,3 the conjecture The unstable systems under consideration are re-
that for single-point measurements of sufficiently viewed in Section II. In Section III, we present the
nonlinearnumericalcodeemployedinthiswork,VP,
aswellasthethreesimulationsunderconsideration.
In Section IV, the field-particle correlation is pre-
a)Electronicmail: [email protected] sented and applied to the three simulations. Anal-
1
ysis and discussion related to the correlations are
TABLE I. Electron Bulk Parameters
foundinSectionV.Applicationoffield-particlecor-
relations to simple, homogeneous, linearly unstable
ne1/ni vd1/vte ne2/ni vd2/vte
systemsenablestheconstructionofsignaturesofba-
Case 1 0.5 0.75 0.5 −0.75
sicenergytransfermechanisms. Combinedwithsig-
Case 2 0.5 1.75 0.5 −1.75
natures for all relevant energy transfer mechanisms,
such correlations may be usefully employed to diag- Case 3 0.9 0.00 0.1 4.25
nose the behavior of more complex systems.
populations take the Maxwellian form
II. LINEARLY UNSTABLE SYSTEMS
n (v v )2
sj dsj
F (v)= exp − − (3)
The 1D1Velectrostatic systems of interestin this s0 √2π " 2vt2s #
work are governedby the Vlasov and Poissonequa-
tions where vdsj is the population’s drift velocity. The
linear normal mode behavior for such equilibria is
∂fs ∂fs qs ∂φ∂fs governedby solutions of the dispersion relation
+v =0, (1)
∂t ∂x − m ∂x ∂v
s
2
q n T
D(ω,k)=k2λ2 + s sj e [1+ξ Z (ξ )]
and De q n T sj 0 sj
j (cid:18) e(cid:19) i s
X
∂2φ ∞ (4)
∂x2 =−4π qs −∞dvfs (2) whereDisafunctionofwavenumberkandcomplex
s Z frequency (ω,γ), q is the species charge, Z is the
X s 0
plasma dispersion function11 with argument ξ =
sj
which evolve the distribution of species s, f (x,v,t)
s
(ω/ω kλ ) T m /2T m v /v , with the
and the electric field E(x,t) = ∂φ(x,t)/∂x. Re- pe De e s s e − dsj ts
cent work has applied field-part−icle correlations to sumtakenove(cid:16)rpthethreeplasm(cid:17)apopulationsj. The
electrostaticstablesystemsasameansofextracting electron plasma frequency ω 4π n q2/m
pe ≡ j ej e
the velocity dependent signature of Landau damp-
ing from single-point time series.5,6 Here, we con- andDebyelengthλDe = Te/4πqjnePjq2,bothde-
siderelectrostaticsystemsunstabletoeitherfluidor fined using the total elecqtron density, normalize the
P
kinetic instabilities, and characterize the signature time andlength scales ofour system. Values for the
of the associated growth using field-particle corre- normalizeddensity of electron populationj, n /n ,
ej i
lations. For fluid instabilities, the behavior of the and bulk velocity, normalized by the electron ther-
plasma depends on the bulk parameters of the sys- malvelocityv = T /m ,aregiveninTableI.For
te e e
tem and the energy transfer is not organized by all three cases, the electron populations have equal
p
characteristic velocities such as the resonant wave temperatures;theionsaresinglyionized,andinitial-
phase speed; energy transfer for kinetic instabilities izedwith T =T , m =100m , andv =0. Case1
i e i e di
dependsonsuchcharacteristicvelocities,astheelec- isstableto theeffects ofthecounter-streamingelec-
trostatic field acts to exchange particles of higher trons,whiletheincreasein v forcase2yieldsthe
dej
| |
and lower kinetic energies near the field’s resonant classic counter-streaming instability. Case 3 is un-
velocity. For both types ofinstabilities, one ormore stable to the bump-on-tail instability. Cases 2 and
of the distributions lose energy, while the fields and 3 serve as examples of fluid and kinetic instabilities
other distributions gain energy, resulting in an in- respectively.
verse damping. General discussion of plasma insta- In Fig 1, solutions to Eqn 4 for the three cases
bilities, as well as particular treatments of the elec- are presented. In panel a, the complex frequency
tron drift instabilities of interest in this work, can solutions (ω,γ)/ω are given for fixed kλ = 0.2.
pe De
be found in many plasma textbooks.8–10 For case 1 (black circles) all modes are shown to be
To highlight the distinct velocity space structure damped. For case 2 (red triangles), the increase in
of energy transfer in unstable systems, we consider v leaves the frequency and damping rate of the
dej
| |
threecases,eachwithdistinctsetsofparametersfor Langmuir modes, the modes with ω ω , largely
pe
| |≈
one population of ions and two populations of elec- unaffected. Thepairofleastdampedacousticmodes
trons. The equilibrium distributions for the three from case 1 are now both unstable, having ω = 0
2
Normal Mode Solutions: kλ =0.2 Langmuir Acoustic
De 0.3
(b) 1
Case 1 (a) e 0.2 10
0.2 Case 2 ωp 0.1 ωpe100
Case 3 γ/ 0 ω/10-1 (d) (e)
-0.1 -2
e 0.1 10
p -0.1 0 0.1
ω 0.3 0.1 1 0.1 1
γ/ (c) 100
0 pe 0 pe10-1
ω ω 10-2 (f)
γ/-0.3 γ||/10-3 γ < 0 γ > 0 (g)
-0.1 -0.6 10-4
-2 -1 0 1 2 0 1 2 0.1 1 0.1 1
ω/ω ω/ω kλ kλ
pe pe De De
FIG. 1. Normal mode properties for the three cases examined in this work. The complex frequency solutions to
Eqn4forkλDe=0.2andtheparametersofcases1,2,and3(black,red,andblue)areshowninpanela. Parametric
paths for normal mode solutions from |v |=0 (v =0) to |v |=1.75 (v =4.25) are plotted in panel b (c),
dej de2 dej de2
indicatingtheconnectionbetweenstableandunstableacousticmodes. Frequencydispersioncurvesω(kλDe)/ωpe for
the least damped and/or fastest growing Langmuir and acoustic modes are illustrated in panels d and e. Damping
rates γ(kλDe)/ωpe for the same modes are plotted in panels f and g, with solid (dashed) lines indicating growth
(damping). Frequency estimates extracted from linear and small amplitude nonlinear VP simulations are shown as
points in panels d-g.
and two distinct growthrates, γ >0. The paramet- a third-order Adams-Bashforth scheme in time. As
ric path from stability (v = 0.0, open triangles) a test of this extension, we compare numerical solu-
dej
| |
toinstability(v =1.75,filledtriangles)forthese tions of Eqn. 4 for the three cases described in sec-
dej
| |
twomodesisillustratedasafunctionofcomplexfre- tion II to frequencies and damping rates extracted
quency in panel b. For case 3, the ω < 0 Langmuir fromtimetracesoftheelectrostaticfieldenergyfrom
modeandtheleastdampedacousticmodesareneg- linearandsmall-amplitudenonlinearVPsimulations
ligibly affected by the bump distribution. By para- for a rangeofwavevectors,givenas points in Fig.1.
metric variation of v from 0.0 (open diamonds Agreement between the dispersion relation and VP
de2
in panel c) to 4.25 (filled diamonds), it is observed arecloseaslongasthe mode ofinterestisnotheav-
that the growing mode for the bump-on-tail insta- ily damped and unstable modes supported by the
bilityarisesfromanacousticmodewhichisstrongly system do not grow too quickly with respect to the
damped in the stable regime (solid line), while the damped modes.
ω > 0 Langmuir mode (dashed line) becomes heav- For the evaluation of the field-particle correla-
ily damped. The dispersion relations for the least tion, we perform three nonlinear simulations cor-
dampedand/orfastestgrowingLangmuir(acoustic) responding to the three cases from section II. For
modes as a function of kλ for the three cases are these simulations, we add a sinusoidal perturba-
De
presented in panels d and f (e and g) illustrating tion to the ion’s equilibrium distribution of the
the wavelengths for which linear instabilities arise form 0.1F (x,v)sin(k x) with k λ = 0.2. 256
i0 0 0 de
for cases 2 and 3 (solid lines in panel g). (128) points in velocity (coo¨rdinate) space are re-
solved over the interval v/v [ 8,8] (x/λ
ts De
∈ − ∈
[ 5π,5π]). Each simulation is run for longer than
t−=40ω−1 and the total energy in the system
III. NUMERICAL SIMULATIONS pe
E2 m v2
s
W = dx + dx dv f (5)
To evaluate the time evolution of these systems, total 8π 2 s
we have extended the Vlasov-Poisson solver VP6 to Z Xs Z Z
allow for the inclusion of an arbitrary number of is conserved to better than a few tenths of a per-
drifting plasma populations. VP solves the nonlin- cent. Changes in the electrostatic energy W =
φ
ear Vlasov-Poison system using second-order finite dxE2/8π as well as energy in the three plasma
differencing for spatial and velocity derivatives and populations, W = dx dvm v2f /2, from their
j j j
R
R R
3
Run 1 Run 2 Run 3
0.4 2 8
(a) (b) (c)
0.2 1 4
0
Wj
0 0 0
W-j
-0.2 -1 We1 -4 Wφ
W W
e2 i
-0.4 -2 -8
0 10 20 30 40 0 10 20 30 40 0 10 20 30 40
tω tω tω
pe pe pe
FIG.2. Thechangeinenergy,Wj(t)−Wj(t=0),forthetwopopulationsofelectrons,theions,andtheelectrostatic
field (black, red, blue, and grey lines), for thethree nonlinear simulations, normalized by Te.
initial values are shown in Fig. 2, with energy nor- x10-2 ∫dtCE(-qv2∂vδf/2,E) ∫dtCE(-qv2∂vf/2,E)
malizedbyTe. Inrun1,mostoftheenergydamped 11..55 (a) 20 (b) 20
from the electric field is equally partitioned be- 11 15 15
00..55
tween the two electron populations, with little en-
00 10pe 10pe
ergytransferredtotheions. Forrun2,aninstability ω ω
--00..55 t t
is clearly triggered, with significant losses of energy --11 5 5
from the electrons and both Wφ and Wi growing --11..55 0 0
from their initial values. A more virulent instabil- -4 -2 0 2 4 -4 -2 0 2 4
v/v v/v
te te
ity is triggered in run 3, with the bump electron
population losing a significant fraction of its energy FIG. 3. The accumulated change in the electron phase-
to the core electron population, with little energy space energy density for damped, counter-propagating
transferred to the ions. Langmuirwavescalculated usingtheperturbedelectron
distributioninpanela,Eqn.2fromKlein&Howes20165
and the total electron distribution in panel b, Eqn. 6.
The resonant velocities for the system are indicated by
IV. FIELD-PARTICLE CORRELATIONS
dashed grey lines.
Whiletrackingthechangeinenergyissufficientto
identifyinstabilitiesinsystemswithcompleteknowl-
space between the electric field and velocity distri-
edge of spatial and velocity structure, we seek the
bution. Byaveragingoveraselectedtimeintervalτ,
signature of unstable behavior given limited, single-
oscillatory energy transfer between E and f is re-
s
pointmeasurementsofasystemofthetypeavailable
moved, leaving only the secular energy transfer. To
to spacecraft in the solar wind. The application of
tracktheaccumulatedchangeofthephase-spaceen-
field-particle correlations to such measurements al-
ergy density, we integrate the correlation over time,
lows for a local observation of secular energy trans- t ′ ′
defining ∆w (x,v,t,N) dtC (x,v,t,N).
fer. s ≡ 0 E
We define the field-particle correlation for a dis- UnlikeEqn.2inKlein&RHowes2016,5 weinclude
crete set of measurements of fs(x,v,t) and E(x,t) thefulldistributionfunctioninourdefinitionofCE,
with timestep dt, taken at a single point x=x0 as ratherthenonlytheperturbedcomponentδfs. Pre-
vious studies had focused on particular cases where
1 i+N q v2∂f (x ,v,t ) the equalibria Fs0 were even with respect to v = 0,
C (x ,v,t ,N)= s s 0 j E(x ,t ). ensuring that they would not contribute to net en-
E 0 i 0 j
−N 2 ∂v
j=i ergy transfer. For the cases under consideration
X
(6) in this work, the equilibrium electron distributions
The correlation averages the field-particle interac- have odd components and therefore may contribute
tion term in the Vlasov equation, the third term in toaseculartransferofenergybetweenthefieldsand
Eqn. 1, over a time interval of length τ = Ndt. As distributions. To show that the two forms of the
theballisticterm,thesecondinEqn.1,doesnotlead correlation obtain similar results when F is even,
s0
tonetenergytransfer,6theproductinC represents we apply both correlations to single-point field and
E
the energy density transferredat one point in phase distribution data from a Landau damped counter-
4
propagating Langmuir wave simulation, case 1 in V. VELOCITY-SPACE STRUCTURE OF
Klein & Howes 2016, and plot ∆w at x = 0 with ENERGY TRANSFER
e
τω = 6.28 in Fig. 3. We see that both correla-
pe
tions produce qualitatively similar structure in the
Withanappropriateintervalτ selected,wecalcu-
accumulated phase-space energy density, especially
late the velocity dependent field-particle correlation
in regards to production of a plateau surrounding
for the three simulations. By retaining the velocity
theresonantvelocitiesofthesystem, v =2.86v ,
| res| te dependence, we are able to address the question of
which serves as the key velocity-space signature of
wheretheenergytransferoccursinphasespace,and
Landau damping.
use the structure of this energy transfer to charac-
With our correlation defined, we next select an terize the nature of the underlying instability.
appropriate correlation interval τ for the three sim- For the stable, counter-streaming electron case,
ulations. Byaveragingoveraparticularinterval,we run1,thephase-spaceenergytransferobtainedfrom
removethe transferofenergybetweenthe fieldsand the field-particle correlation is a fairly regular func-
particles which oscillates with frequency of the or- tion of velocity. At a given point in co¨ordinate
derω 2π/τ,leavingthesecular,ornon-oscillatory, space, for example x = 0λ shown in the first
∼ De
component. PlottedinFig.4arevelocityintegrated row of Fig. 5, one of the electron populations gains
correlations, dvCE(x,v,t,τ), between E and the energy from, while the other population loses en-
three plasma populations for a range of correlation ergy to, the electric field. The energy transfer to
R
lengths τωpe [0,30] at a single spatial location the ions from the electric field is an odd function
∈
x=0. of velocity, meaning that when the correlationis in-
tegrated over v, there is no net energy transfer to
Thereissignificantoscillatorytransferforsmallτ
the ions. There is no evidence for the dependence
correlations for run 1, panels a-c of Fig. 4. This os-
ofthe phase-spaceenergytransferondriftvelocities
cillatorytransferisreducedforlongerintervals,with
(black vertical dashed lines) or the resonant veloci-
nearlyalloftheoscillationsremovedforτω =5.64,
pe
ties of either the Langmuir (green lines) or acoustic
a correlation length corresponding to the period of
(magenta) modes for any of the three populations.
Langmuir waves supported by the system with fre-
quency ω = 1.11ω . This τ leaves the velocity in- Increasingthespeedofthecounter-streamingelec-
pe
tegrated correlations for all three distribution func- trons for run 2, shown in panels e-h of Fig. 5, the
tionsnearlymonotonic,whileslightlylongercorrela- structure of the energy transfer is altered. The sign
tions reintroduce some oscillatory behavior. As the of the field-particle correlation changes at vdej due
Langmuirwaveistheleastdamped,finite frequency toachangeinsignof∂fej/∂v. Unlikeforcase1,this
linear mode supported by this system, correlating change in sign occurs for electrons which exchange
over its period is physically justified. For run 2, significant energy with the electric field. For the
panels d-f, correlatingoverthe interval τω =5.15, ions, a small, even component in the field-particle
pe
which corresponds to the Langmuir wave frequency correlation arises, yielding a net transfer of energy
ω = 1.21ω , removes significant oscillatory energy from the fields to the ions, as seen in panel h. This
pe
transfer. Averaging over the time scale associated transferofenergytotheions,whichdoesnotdepend
with the least damped modes, rather than the un- on either the Langmuir or acoustic resonant veloci-
stable modes, is motivated by the fact that the un- ties, serves as a phase-space signature for the fluid
stablemodesofthis systemhavezerofrequency;see instability that arises for this system.
Fig 1. For run 3, we see that there does not ex- For the bump-on-tail instability, case 3 shown in
ist a single value of τ for which the oscillatory en- panels i-l of Fig. 5, the velocity-space structure of
ergytransferiscompletelyremoved. Thisduetothe the field-particle correlation is significantly differ-
factthatthesystemsupportsbothaweaklydamped ent. As we have correlated over the unstable acous-
Langmuirwaveaswellasafinite-frequencyunstable tic period, the oscillatorystructure from the weakly
acoustic mode. Correlating over the Langmuir pe- damped Langmuir mode is evident. We also see in
riod retains some of the acoustic oscillations, while thebumpelectronpopulationthattheenergytrans-
correlating over the acoustic period retains some of ferchangessignacrosstheacousticresonanceandat
the Langmuir oscillations. We choose τω = 7.11, later time across the Langmuir resonance. This res-
pe
corresponding to the period of the growing acoustic onantstructureisthesignatureofthetransferofen-
mode with ω =0.88ω and acknowledge that some ergy from the bump population to the core electron
pe
oscillatory contributions from Langmuir waves will populationasmediatedbytheelectricfield. Wenote
persist. that the resonant structure is maintained for other
5
∫dvCE(fe1) (a) ∫dvCE(fe2) (b) ∫dvCE(fi) (c) τωpe
0.04 0.04 0.04
30
0.02 0.02 0.02
1
n 0 0 0
u 25
R -0.02 -0.02 -0.02
-0.04 -0.04 -0.04
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
0.3 0.3 0.3 20
(d) (e) (f)
0.15 0.15 0.15
2
n 0 0 0 15
u
R -0.15 -0.15 -0.15
-0.3 -0.3 -0.3
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 10
0.04 0.04 0.04
(g) (h) (i)
0.02 0.02 0.02
3
n 0 0 0 5
u
R -0.02 -0.02 -0.02
-0.04 -0.04 -0.04 0
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
tω tω tω
pe pe pe
FIG.4. Thevelocityintegratedfield-particlecorrelationCE(fs)atx=0forarangeofcorrelationintervalsτ ∈[0,30],
differentiatedincolor. Eachrowrepresentsthethreesimulationsunderconsideration, whilethecolumnspresentthe
correlation with fe1, fe2 and fi. The black line highlights theτ selected for Fig 5.
choices of τ, including an interval corresponding to the electric field regardless of spatial position. For
theLangmuirwaveperiod,notshown. Thevelocity- the unstable counter-streamingelectrons,run2, the
integrated correlation confirms that the core popu- same pattern of shifting sign and amplitude for the
lation has a net gain of energy, and as expected for energy transfer to and from the electron holds. The
thisinstability,weseethattheelectronsthatreceive ions gain energy from the electric field regardless of
the energyarenearthe resonantvelocities. This ex- which electron population is gaining or losing en-
plicitdependence ofthe phase-spaceenergytransfer ergy. The amplitude of the ion energy gain changes
onresonantcouplingbetweenthefieldsandparticles withthe amplitude ofthe electronfield-particlecor-
serves as a distinct signature of kinetic instabilities relation, going to zero at the nodes and having its
when compared to the fluid instability in run 2. largest value at the anti-nodes. This phase space
structure serves as a distinct signature of growing
The field-particle correlations presented in Fig. 5
fluid instabilities. For the bump-on-tail instability,
werecalculatedatasinglepointinco¨ordinatespace.
run 3, there is no regular spatial variation in the
An obvious question arises as to how the correla-
transfer of energy between the beam and core elec-
tion and the associated phase-space energy transfer
tron populations; the core gains energy at the ex-
change as a function of position within the simu-
penseofthe beam,withthe energydensityaccumu-
lation. To asses this question, we calculate C at
E lating at nearly the same rate regardless of spatial
five other points within the simulation and present
position, as expected for a kinetic instability.
the velocity integrated accumulated change of the
phase-space energy density, dv∆w , in Fig. 6. For
s
thestablecase,run1,weseethattheenergytransfer
R VI. CONCLUSION
toorfromthetwoelectronpopulationschangessign
and amplitude as a function of the position in the
simulation, with the transfer passing through zero Field-particle correlations of the form defined in
atthetwonodesoftheinitialstandingwavepattern Eqn. 6, modified from Klein & Howes 20165 to ac-
at 7.9λ . Thecorrelationwiththeionsmaintains count for drifting equilibrium distributions, are ap-
De
±
an even velocity space structure such that the ions plied to a set of simulations where fluid and kinetic
continue to neither lose nor gain net energy from instabilitiesarepresent. Thestructureofthe result-
6
x10-3 CE(fe1) (a) CE(fe2) (b) CE(fi) (c) Run 1 (d)
2 30 30 30 30
f
e1
f
1 e2
20 20 20 fi 20
e
p
0 ω
10 10 10 10 t
-1
0 0 0 0
-2 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8
v/vte v/vte v/vti 10-3 x ∫dvCE(fs)
x10-2 CE(fe1) (e) CE(fe2) (f) CE(fi) (g) Run 2 (h)
2 30 30 30 30
1
20 20 20 20
e
p
0 ω
10 10 10 fe1 10 t
-1 fe2
f
i
0 0 0 0
-2 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 -2 -1 0 1 2
v/vte v/vte v/vti 10-1 x ∫dvC (f )
E s
x10-3 CE(fe1) (i) CE(fe2) (j) CE(fi) (k) Run 3 (l)
4 30 30 30 30
2
20 20 20 20
e
p
0 ω
10 10 10 fe1 10 t
-2 fe2
f
i
0 0 0 0
-4 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 -3 -2 -1 0 1 2 3
v/vte v/vte v/vti 10-2 x ∫dvCE(fs)
FIG. 5. The velocity dependentfield-particle correlation CE(fs) at x=0 for runs1(panels a-c),2 (panels e-g),and
3 (panels i-k) as well as the velocity integrated correlation for fe1, fe2, and fi (black, red, and blue lines in panels
d, h, and l), for τωpe = 5.64 (run 1), 5.15 (run 2), and 7.11 (run 3). The vertical dashed black lines indicate the
electron drift velocities vd1/vte and vd2/vte, while the green and magenta vertical lines give the resonant velocities
for the least damped and/or fasted growing Langmuir and acoustic waves respectively.
ingcorrelations,whichcanbeinterpretedasthesec- inadvanceoffutureworkapplyingsuchcorrelations
ularphase-spaceenergydensity transferredbetween to systems of higher dimensionality, where magne-
the fields and distributions, can be used to iden- tization, turbulence, and inhomogeneities may com-
tify the presenceofinstabilities aswellastocharac- plicate the interpretation of the correlation. By de-
terize the mechanisms driving the unstable growth. termining signatures of basic plasma physics phe-
The form of the correlation allows for such charac- nomena responsible for energy transfer between
terizations to be made from observationsat a single fields and distributions, such as Landau damping
point, or a few points, in co¨ordinate space, as op- andone-dimensionalinstabilities,welaythefounda-
posed to requiring knowledgeof spatially integrated tion for the determination of the velocity distribu-
quantities typically not accessible to experimental tionsignaturesofmorecomplexinteractions,suchas
measurements. cyclotron9 and transit time damping,12 heating by
largeamplitude, stochasticfluctuations,13 andmag-
We consider simplified 1D-1V electrostatic sys-
netic reconnection,14 which have all been proposed
temsinanattempttocharacterizefield-particlecor-
to play a role in the dissipation of turbulent fluctu-
relationsasmeasurementsofsecularenergytransfer
7
∫dt∫dvCE(fs)x=-12.8λDe x=-7.9λDe x=-4.9λDe x=-0.0λDe x=7.9λDe x=11.3λDe
0.1
1 0.05
n fe1
u 0
R fe2
-0.05
f
p
-0.1
0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30
2
2
n 1
u 0
R -1
-2
0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30
0.15
0.1
3
n 0.05
u 0
R-0.05
-0.1
-0.15
0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30
tω tω tω tω tω tω
pe pe pe pe pe pe
FIG. 6. The time and velocity integrated field-particle correlation R dtR dvCE(fs) at six points in coo¨rdinate space,
arranged by columns, with values for fe1, fe2, and fi given in black, red, and blue. The top, central, and bottom
rows illustrate valuesfor runs1,2, and 3.
ations. Future work will also consider the effects of netospheric Multiscale (MMS)15 missions, which
solarwindexpansion,electronconduction,andother are comprised of four or five spacecraft arranged
inhomogeneousmechanismsontheplasmatoclearly in particular geometric configurations.
identify the roleofsuchenergytransfermechanisms 3G.I.Taylor, RoyalSocietyofLondonProceedings
in the solar wind. By constructing the correlation Series A 164, 476 (1938).
to be obtained from single-point measurements, we 4K. G. Klein, G. G. Howes, and J. M. TenBarge,
allow for the identification of such damping mech- Astrophys. J. Lett. 790, L20 (2014).
anisms from in situ observation of the solar wind 5K.G. Klein andG. G. Howes, Astrophys.J. Lett.
oncurrentandfuturemissionsincludingDeep Space 826, L30 (2016).
ClimateObservatory(DSCOVR),MMS,15andSolar 6G.G.Howes,K.G.Klein,andT.C.Li, J.Plasma
Probe Plus.16 Phys. (under review).
7L.D.Landau, J.Phys.(USSR)10,25(1946), [Zh.
Eksp. Teor. Fiz.16,574(1946)].
ACKNOWLEDGMENTS 8N. A. Krall and A. W. Trivelpiece, Principles of
plasma physics, McGraw-Hill, 1973.
9T.H.Stix, Waves in plasmas, AmericanInstitute
The author would like to thank Gregory Howes,
of Physics, 1992.
Justin Kasper, and Jason TenBarge for insightful 10R.D.HazeltineandF.L.Waelbroeck, The frame-
discussions regarding aspects of this work. This
work of plasma physics, Westview, 2004.
research was supported by the NASA HSR grant 11B. D. Fried and S. D. Conte, The Plasma Disper-
NNX16AM23G.
sion Function, Academic Press, 1961.
12A. Barnes, Phys. Fluids 9, 1483 (1966).
13B. D. G. Chandran, B. Li, B. N. Rogers,
REFERENCES E.Quataert,andK.Germaschewski,Astrophys.J.
720, 503 (2010).
1R.BrunoandV.Carbone,LivingRev.SolarPhys. 14M. Yamada, R. Kulsrud, and H. Ji, Reviews of
2, 4 (2005). Modern Physics 82, 603 (2010).
2Notable exceptions to the single-point limitation 15J.L.Burch,T.E.Moore,R.B.Torbert,andB.L.
are the are the Cluster,17 THEMIS,18 and Mag- Giles, Space Science Reviews 199, 5 (2016).
8
16N. J. Fox et al., Space Sci. Rev. (2015). 18V. Angelopoulos, Space Science Reviews 141, 5
17C. P. Escoubet, M. Fehringer, and M. Goldstein, (2008).
Annales Geophysicae 19, 1197 (2001).
9