PHYSICS OF PLASMAS 15, 052108 共2008兲
The dispersive Alfvén wave in the time-stationary limit with a focus
on collisional and warm-plasma effects
S. M. Finnegan,1,a兲 M. E. Koepke,1,b兲 and D. J. Knudsen2
1
Department of Physics, West Virginia University, Morgantown, West Virginia 26506-6315, USA
Department of Physics and Astronomy, University of Calgary, Calgary, Alberta, Canada
2
共Received 16 October 2007; accepted 13 February 2008; published online 28 May 2008兲
A nonlinear, collisional, two-fluid model of uniform plasma convection across a field-aligned
current 共FAC兲 sheet, describing the stationary Alfvén 共StA兲 wave, is presented. In a previous work,
Knudsen showed that, for cold, collisionless plasma 关D. J. Knudsen, J. Geophys. Res. 101, 10761
共1996兲兴, the stationary inertial Alfvén 共StIA兲 wave can accelerate electrons parallel to a background
magnetic field and cause large, time-independent plasma-density variations having spatial
periodicity in the direction of the convective flow over a broad range of spatial scales and energies.
Knudsen suggested that these fundamental properties of the StIA wave may play a role in the
formation of discrete auroral arcs. Here, Knudsen’s model has been generalized for warm,
collisional plasma. From this generalization, it is shown that nonzero ion-neutral and electron-ion
collisional resistivity significantly alters the perpendicular ac and dc structure of
magnetic-field-aligned electron drift, and can either dissipate or enhance the field-aligned electron
energy depending on the initial value of field-aligned electron drift velocity. It is also shown that
nonzero values of plasma pressure increase the dominant Fourier component of perpendicular
wavenumber. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2890774兴
I. INTRODUCTION
The shear-mode Alfvén wave balances ion polarization
current in the direction perpendicular to the background
magnetic field B0 with magnetic-field-aligned 共i.e., parallel兲
current carried by electrons. For large wavelengths ␭⬜ perpendicular to B0, the parallel electric field magnitude is insignificant, and the parallel phase speed equals the Alfvén
speed VA 共VA ⬅ B0 / 冑␮0min, where mi is the ion mass and n is
the plasma density兲. When ␭⬜ becomes small, comparable to
the electron inertial length ␭e ⬅ c / ␻ pe for the case in which
␤ ⬍ me / mi 共␤ = kinetic pressure normalized to magnetic field
pressure兲 or comparable to the ion acoustic gyroradius ␳s in
higher-␤ plasmas, the parallel electric field magnitude 共balanced by electron inertia for ␤ ⬍ me / mi and by parallel electron pressure for ␤ ⬎ me / mi兲 becomes significant and the
wave becomes dispersive.1,2 Waves in the low- and high-␤
limits are termed “inertial” and “kinetic,” respectively, and
together they are known as a dispersive Alfvén wave. The
dispersive Alfvén wave has been studied extensively because
of its possible role in accelerating electrons in the Earth’s
magnetosphere, particularly the auroral ionosphere, where
the inertial wave can generate transient 共⬃1 s兲 bursts of electron energy at energies up to ⬃1 keV.3–7 The dispersive
Alfvén wave also leads to electron acceleration as a result of
interhemispheric field-line resonances having periods of
⬃10 min.8,9 See the comprehensive review by Stasiewicz
et al.10 The linear dispersion relations for the dispersive
a兲
Electronic mail: sfinnega@mix.wvu.edu.
Electronic mail: mkoepke@wvu.edu. Also at School of Electrical Engineering, Space & Plasma Physics, Royal Institute of Technology 共KTH兲,
Teknikringen 31 SE-10044 Stockholm, Sweden.
b兲
1070-664X/2008/15共5兲/052108/13/$23.00
Alfvén wave in both the inertial and kinetic limits have been
verified in laboratory experiments.11–17
The dispersive Alfvén wave can be described in the
context of two-fluid theory. Linear properties are derived by
neglecting the nonlinear term v · ⵜ of the complete timederivative operator 共d / dt = ⳵ / ⳵t + v · ⵜ兲. Seyler et al.18 predicted that inclusion of this nonlinear term leads to Alfvén
wave steepening, wave breaking, and enhanced wave dissipation. Those authors also found a solution predicting a
purely spatial perturbation in plasma density in the zerofrequency limit.
In this paper, we are concerned with the limit ⳵ / ⳵t = 0.
Knudsen19 predicted that in the presence of background, perpendicular, plasma convection through the wave fields, the
v · ⵜ term alone generates Alfvén-wave-like behavior, with
electromagnetic structures evolving in space as a result of the
apparent time variation seen by plasma particles convecting
across static 共electric and magnetic兲 field structures. Within
otherwise large-scale sheets of homogeneous parallel current, the stationary Alfvén 共StA兲 wave imposes structure both
on plasma density and on parallel electron energy.
The term “stationary Alfvén wave” was introduced by
Maltsev et al.20 to describe a stationary electromagnetic
structure resulting from plasma convection past a conducting
strip. The application of the term was broadened by
Mallinckrodt and Carlson21 to include structuring from a
moving field source in a magnetized plasma. The term
“Alfvén wing” is used to describe a stationary wave resulting
from moving conductors, for example in the case of Io orbiting within Jupiter’s magnetosphere22 or as a result of a conducting tether orbiting with a spacecraft.23 In these cases, the
structure of the stationary wave is imposed by the source,
although Chust et al.22 argue that additional filamentation
15, 052108-1
© 2008 American Institute of Physics
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052108-2
Phys. Plasmas 15, 052108 共2008兲
Finnegan, Koepke, and Knudsen
can result. In contrast, the nonlinear two-fluid model described here and by Knudsen19 predicts a self-consistent StA
wave whose structure arises from intrinsic properties of the
current sheet in which the wave forms. In other words, the
wave represents the spatial restructuring of a large-scale homogeneous current sheet into a new electromagnetic equilibrium, and does not require a spatially structured source. As
with a dispersive Alfvén wave, according to two-fluid predictions, parallel electron thermal pressure and nonzero electron mass lead to a significant parallel electric field associated with the StA wave.
The purpose of the present paper is to extend the fluid
theory predictions of a stationary inertial Alfvén wave19 to
the finite-␤ case, i.e., to the realm of the stationary kinetic
Alfvén wave, and to study its properties in the presence of
collisions.
FIG. 1. Depiction of relevant vectors. Left: direction of plasma flow due to
the convection electric field Ey = VdB0 in the polar region of the ionosphere.
Right: 2D Cartesian geometry of our stationary Alfvén wave model.
共6兲
Vix = Vix0 + vix ,
II. STATIONARY ALFVÉN WAVE EQUATIONS
In this study, we present a two-dimensional 共x-z Cartesian plane兲 wave model describing a sheetlike 共⳵ / ⳵y = 0兲, stationary 共⳵ / ⳵t = 0兲, electromagnetic structure embedded within
a warm, collisional plasma, oriented obliquely at a small
angle ␪, to a uniform background magnetic field 共ẑB0兲. This
stationary Alfvén wave pattern is supported by uniform
plasma flow Vd, due to a convection electric field Ey = VdB0,
across a magnetic-field-aligned current sheet. Wave equations describing StA waves are derived from the following
set of isotropic, time-independent, two-fluid equations:
m jn j共V j · ⵜ兲V j = q jn j关E + 共V j ⫻ B兲兴 − ⵜp j + R j ,
共1兲
ⵜ · 共n jV j兲 = 0,
共2兲
ⵜ ⫻ E = 0,
共3兲
ⵜ ⫻ B = ␮0J.
共4兲
and
Here, j denotes particle species, q j is the particle charge, V j
is the fluid velocity, n j is the particle density, p j = n jT j is the
isotropic thermal pressure, T j is the particle temperature in
energy units 共eV兲, and R j is the collisional resistivity operator. The set of equations 共1兲–共4兲 are closed through the
quasineutrality condition ne ⬇ ni. The electron and ion collisional resistivity operators are
Ri = − ␯innimiVi
共5a兲
where Vix0 is the initial ion drift at x = x0 关vix共x = x0兲 = 0兴 and
represents the sum of the E ⫻ B drift Vd and steady-state
analog of the polarization drift Vip0. The nonlinear variation
in ion velocity vix is not necessarily a small perturbation and
is, in general, comparable in magnitude to Vix0. Similarly, the
perpendicular and field-aligned electron drifts are
Vex = Vd + vex
共7a兲
Ves = Ves0 + ves .
共7b兲
and
Here, Ves0 is the initial field-aligned electron drift at x = x0
关ves共x = x0兲 = 0兴. The perpendicular electron drift is assumed
to result from the E ⫻ B drift alone, since the electron polarization drift is smaller than that of the ions by a factor of
me / mi and may be neglected.
A graphical representation of the assumed Cartesian geometry, including vector directions of background magnetic
field B0 and initial particle drifts, for our two-dimensional
StA wave model is shown in Fig. 1. StA wave patterns are
assumed to be nearly field-aligned, specifically, oriented at a
small angle ␪ with respect to the background magnetic field
such that
␪ ⬇ tan共␪兲 =
Re = − Ceineme␯ei共Ve − Vi兲 − neme␯enVe ,
共5b兲
where ␯in is the ion-neutral collision frequency, ␯en is the
electron-neutral collision frequency, and ␯ei is the electronion Coulomb collision frequency. The coefficient Cei is a
constant with Cei = 0.51, in accordance with Braginskii.24 The
fluid velocity of the neutrals is assumed to be negligible with
respect to the fluid velocities of the charged species.
Perpendicular ion drift is represented by
Vd
.
Vphs
共8兲
Conceptually, Vphs = −kxVd / kz = Vd / tan ␪ is the effective, but
nonexistent, parallel wave phase speed. The subscript s indicates the magnetic-field-aligned 共i.e., parallel兲 direction with
unit vector
ŝ ⬅
and
冉 冊
冉 冊 冉 冊
B2
By
␦Bz
B
= ŷ + ẑ + O
+ O 2y .
B0
兩B兩
B0
B0
共9兲
The total magnetic field is composed of an assumed uniform
background magnetic field in the z direction B0, the magnetic
field structure associated with the background current channel By0, and the wave field ␦B, so that B = ŷ共By0 + ␦By兲
+ ẑ共B0 + ␦Bz兲. Only terms to first order in By / B0, the y component of the total field, are retained. Terms of order ␦Bz / B0
can be shown to be of order B2y / B20 and are consequently
neglected. An important correction to the field-aligned com-
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052108-3
Phys. Plasmas 15, 052108 共2008兲
The dispersive Alfvén wave…
ponent of electric field, proportional to By / B0 关see Eq. 共16兲兴,
is retained by including terms to first order in By / B0. Using
Eq. 共9兲, the differential operators ⳵x = ⳵ / ⳵x and ⳵z = ⳵ / ⳵z are
transformed into the field-aligned direction,
⳵z ⬇ ⳵s
冉 冊
冋 冉 冊 册
冉 冊册
冋
冉 冊册
冋
␭e⳵xĒx = 1 +
⫻
共10a兲
+
and
⳵z ⬇ − ␪⳵x .
共10b兲
An expression that relates the density of species j to its
fluid velocity is obtained from the continuity equation, Eq.
共2兲. Equations 共10a兲 and 共10b兲 are used to reduce both the
electron and ion continuity equations to differential equations in x,
⳵x关ni共Vix0 + vix兲 − ␪niviz兴 = 0
共11a兲
⳵x关neVd − ␪ne共Vez0 + vez兲兴 = 0.
共11b兲
and
Integrating over the interval 关xo , x兴 yields expressions for the
total particle density n j in terms of both the initial particle
density N j = n j 共x = x0兲 and the components of the particle
fluid velocities,
1
1
ni
=
=
Ni 共1 + vix/Vix0兲 − ␪viz/Vix0 ui − a
共12a兲
共12b兲
Normalized functions of the electron and ion fluid velocities
have been defined as ue ⬅ 1 − 共ves + Ves0兲 / Vphs and ui ⬅ 1
+ vix / Vix0, respectively. Also, Ue ⬅ 1 − Ves0 / Vphs is the initial
condition at x = x0 for the function of electron velocity
关ue共x = x0兲 = Ue兴. In this paper, we investigate situations
in which the initial ion drift in x is dominated by E ⫻ B
motion, and viz Ⰶ VA. Under these assumptions, the term
a ⬅ ␪viz / Vix0 can be neglected with respect to ui in Eq. 共12a兲.
Particle densities are necessarily positive for physically
meaningful solutions, thus only solutions for which both
Ue / ue ⬎ 0 and ui ⬎ 0 are considered.
The stationary analog of the ion polarization drift is calculated from the ion momentum equation 共1兲. Retaining
terms to first order in By / B0, the ion momentum equation can
be written as
␻2ci
=
冊
+ 1 关Vix0 + vix兴
冉
− ¯␤
1
¯ 2ci p2U2e
␻
+ ¯␤
␶ pUe
␭ e⳵ xu e
␶ + 1 u2e
1 2
␶
␭ 共⳵ u 兲2
␶ + 1 u2e e x e
1
1
␶
− ¯␤
u ␭ 2⳵ 2u .
¯ 2ci p2U2e
␶ + 1 u2e e e x e
␻
共14兲
Equation 共14兲 is written in terms of the dimensionless vari¯ ci ⬅ ␻ci / 共Vd / ␭e兲, ¯␯in ⬅ ␯in / 共Vd / ␭e兲,
ables Ēx ⬅ Ex / 共␭e␻ciB0兲, ␻
␶ ⬅ Ti / Te, and ¯␤ ⬅ ␤共mi / me兲. The factor p is the initial polarization drift variable and is defined as
p ⬅ 共1 + Vip0/Vd兲−1 = Vd/Vix0 .
共15兲
In this paper, we limit our study to cases in which the initial
ion drift is assumed to be dominated by E ⫻ B motion, and
so we only consider cases for which p = 1.
The parallel electron motion is a response to the parallel
component of the electric field,
Es ⬅ ŝ · E ⬇ Ez + VdBy ,
共16兲
which is calculated from the electron momentum equation
共1兲,
共17兲
1 − Ves0/Vphs
Ue
ne
=
=
.
Ne 1 − 共ves + Ves0兲/Vphs ue
关共Vi · ⵜ兲 + ␯in兴2
2
¯ 2ci pUe
␻
mene共ve · ⵜ兲Ves = qeneEs − Te⳵sne − mene共Cei␯ei + ␯en兲Ves .
and
冉
+
¯␯2in
pUe
−
共1 + ¯␯inĒx兲 + ¯␯in
2
¯ ci
ue
␻
冊
1
Ti
关共Vi · ⵜ兲 + ␯in兴 ⫻ Ex −
⳵xni + Vd , 共13兲
␻ciB0
q in i
where ␻ci = qiB0 / mi is the ion gyrofrequency. The definitions
of the functions of electron and ion fluid velocity, along with
Eq. 共12a兲 and 共12b兲 and the quasineutrality condition, are
used to solve Eq. 共13兲 for the stationary analog of the polarization drift,
Solving Eq. 共17兲 for the parallel component of electric field,
and reducing it to a one-dimensional differential equation in
x using Eqs. 共8兲 and 共12b兲 yields
Es =
冉
冊
me
me
␪关Vphs − Ves兴⳵xVes + 关Cei␯ei + ␯en兴 2 qeneVes
qe
n eq e
−
2
VTe
me
␪
⳵xVes .
qe 关Vphs − Ves兴
共18兲
Here, VTe = 冑Te / me is the electron thermal speed. We see
above that the parallel electric field, balanced by electron
inertia 共first term on the right-hand side兲, has the opposite
sign of the parallel electron pressure term 共last term on the
right-hand side兲. This difference was first identified by
Goertz and Boswell.1 Also, we see that the finite-electron
conductivity term, expressed as ␩Js in Eq. 共18兲 共middle term
on the right-hand side兲, is out of phase with both electron
inertia and electron pressure terms. This phase difference
plays a fundamental role in the spatial decay and growth of
StA waves. Equation 共18兲 is normalized, and rewritten as a
function of ue,
Ēs = ␭e⳵xu2e − 2¯␯e共1 − ue兲 − 2
冉 冊
VA2 ¯ 1 ␭e
⳵u.
2 ␤
␶ + 1 ue x e
Vphs
共19兲
Equation 共19兲 is expressed in terms of the normalized
parameters Ēs ⬅ Es / 关−meV2d / 共2␪qe␭e兲兴 and ¯␯e ⬅ 共Cei␯ei
+ ␯en兲 / 共Vd / ␭e兲. As a note, representing the electron collision
frequency in this manner allows for the role of parallel elec-
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052108-4
Phys. Plasmas 15, 052108 共2008兲
Finnegan, Koepke, and Knudsen
tron resistivity to be generalized. If we were to consider an
effective electron collision frequency as the result of all resistive electron interactions, not just particle collisions, then
we could equivalently use
¯␯e =
␭e
兺 ␯ej ,
Vd j⫽e
共20兲
plasma E ⫻ B drift and thus the ion polarization parameter,
Eq. 共15兲, is p = 1 for all solutions. Furthermore, we only consider solutions for strongly magnetized plasma and ignore
ion gyrofrequency effects by letting ␻ci → ⬁. With these two
assumptions, the coupled set of differential equations describing the stationary Alfvén wave, Eqs. 共24兲 and 共14兲, respectively, reduce to
without altering the resulting StA wave equation.
The field-aligned current density is assumed to be carried by the electrons and is given by
Js ⬅ ŝ · J = qeneVes .
0=
共21兲
冉 冊
冋
冋
+ ¯␯e − ¯␯in
Using Eqs. 共8兲 and 共12b兲, the current density is rewritten in
terms of the function ue,
1
Vphs
J̄s =
Ue
−1 .
Ves0
ue
␭2e 2 2 VA2 ¯ 2 2
⳵u −
␤␭ ⳵ ln共ue兲
2 x e V2phs e x
−
共22兲
VA2
2
Vphs
冉 冊 册
VA2 ¯
␶ Ue
␭ e⳵ xu e
2 ␤
Vphs ␶ + 1 u2e
共1 + ¯␯inĒx兲 − 1
册 冋
V2
Ue
− Ue − 2A
ue
Vphs
册
共25兲
and
Here, the current density has been normalized to the current
density at x = x0 关Js共x = x0兲 = qeNeVes0兴.
Differentiating Eq. 共18兲 with respect to x and combining
Eqs. 共3兲, 共4兲, 共10a兲, 共10b兲, 共12b兲, and 共21兲 and the quasineutrality condition yield a second-order, nonlinear, differential
equation for the function ue,
冉 冊
冉 冊
␭e⳵xĒx = 1 −
− ¯␤
冉 冊
␶ Ue
Ue
共1 + ¯␯inĒx兲 − ¯␯in¯␤
␭⳵u
ue
␶ + 1 u2e e x e
冉 冊
␶
␭2⳵2 ln ue .
␶+1 e x
共26兲
V2
␭2e 2 2
1
⳵x ue + Ue
− 1 + 2A ⳵xĒx + ¯␯e␭e⳵xue
2
ue
Vphs
−
VA2 ¯ 1
␭2⳵2 ln ue = 0.
2 ␤
␶+1 e x
Vphs
共23兲
Substitution of Eq. 共14兲 into Eq. 共23兲 yields the wave equation for a resistive stationary Alfvén wave,
冉
0= 1+
再
冊
2
VA2 /Vphs
␭2e 2 2 VA2 ¯ 2 2
⳵ u − 2 ␤␭e ⳵x ln共ue兲
2 2 2
¯ ci p Ue 2 x e Vphs
␻
+ ¯␯e + ¯␯in
冋
冋
− p
VA2
2
Vphs
− Ue −
VA2
2
Vphs
冋
2
¯ 2ci pUe
␻
册
冉 冊册
共1 + ¯␯inĒx兲 − 1
VA2
2
Vphs
− ¯␤
1+
¯␯2in
¯ 2ci
␻
冉 冊 册冎
␶ pUe
␶ + 1 u2e
␭ e⳵ xu e
Ue
ue
.
共24兲
Equation 共24兲 is coupled to Eq. 共14兲 through the polarization
field Ex and, together, represent the closed set of equations
describing StA wave modification of field-aligned electron
drift. It is important to note that Eqs. 共24兲 and 共14兲 contain
the full ion and electron convective nonlinearities and are
valid to first order in the magnetic perturbation By / B0. We
see in Eqs. 共24兲 and 共14兲 that the characteristic scale length
for these waves is the electron inertial length, ␭e ⬅ c / ␻ pe.
The characteristic time scale is identified through the dimensionality of the collision frequency and is the time required
for a particle to E ⫻ B drift the distance of one electron inertial length, ␭e / Vd.
For simplicity and without loss of generality, the numerical results presented in this paper have been obtained with
the assumption that the initial perpendicular ion drift is bulk-
III. NUMERICAL SOLUTIONS
A. General StIA wave characteristics
The general properties of the collisionless StIA wave
were first presented in detail by Knudsen19 and so only a
brief discussion is given here. Knudsen found that uniform
plasma convection across field-aligned current sheets supports the StIA wave, which can accelerate electrons parallel
to a background magnetic field and cause large, timeindependent, plasma-density variations having spatial periodicity in the direction of the convective flow over a broad
range of spatial scales. Figure 2 shows numerical solutions
of Eq. 共30兲 for the perpendicular wavelength associated with
the dominant spectral wave number ␭o / ␭e, the maximum
deviation from the initial value in electron velocity
共ves,max / Vphs兲, and total density 共ni / Ni兲 as a function of the
initial field-aligned electron drift speed 共Ves0兲 for three different values of effective parallel wave phase speed 共Vphs兲.
The top panel in Fig. 2 shows that, for a fixed effective
parallel wave phase speed, the perpendicular wavelength of
StIA waves spans several orders of magnitude, depending on
the value of the initial parallel electron drift. It is clear in the
top panel of Fig. 2 that a wave “resonance” 共␭o → 0兲 occurs
at Ves0 = Vphs. The middle panel shows that the parallel electric StIA wave field either accelerates or decelerates electrons
in the parallel direction, depending on the magnitude of the
initial parallel electron drift. For initial values of parallel
electron drift smaller than the effective wave phase speed
共Ves0 ⬍ Vphs兲, the StIA wave accelerates electrons in the direction of their initial drift, while for Ves0 ⬎ Vphs, electrons
are decelerated. Note that no enhancement in electron energy
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052108-5
2
10
Ves0=0.1VA ; Vphs=0.5VA
(a)
parallel electric field: Es/Es0
perpendicular wavelength: λo/λe
0
10
−2
−1
0
(b)
0
−0.5
ion density: ni/Ni
−1
−1
6
4
−0.5
0
0.5
1
= 0.5V
phs
V
phs
A
= 0.25V
A
2
V
phs
0
−1
−1
−2
5
(a)
10
5
(b)
10
5
(c)
10
1.4
1.2
1
0
1.15
(c)
V
0
−3
0
1.6
1
electron velocity: Ves/Ves0
10
ion density: ni/Ni
electron velocity: ves,max/Vphs
Phys. Plasmas 15, 052108 共2008兲
The dispersive Alfvén wave…
1.1
1.05
= 0.1V
0
A
1
electron drift: Ves0/VA
FIG. 2. StIA: Dependence on electron drift velocity −VA ⬍ Ves0 ⬍ VA. 共a兲 The
dominant perpendicular wavelength ␭o asymptotically approaches zero as
Ves0 → Vphs. 共b兲 For 0 ⬍ Ves0 ⬍ Vphs, electrons are accelerated 共ves,max ⬎ 0兲
from their initial parallel velocity, while for Ves0 ⬍ 0 ⬍ Vphs and 0 ⬍ Vphs
⬍ Ves0, electrons are decelerated 共ves,max ⬍ 0兲. 共c兲 The direction of the initial
parallel electron drift determines whether the density is depleted 共Ves0
⬍ Vphs兲 or enhanced 共Ves0 ⬎ Vphs兲 from its initial value. The vertical dotted
lines in 共a兲 and 共c兲 indicate where the perpendicular wavelength and normalized ion density are undefined, respectively.
occurs in the absence of the field-aligned current Ves0 = 0 and
when Ves0 = Vphs. The maximum density perturbation, seen in
the bottom panel of Fig. 2, shows that the StIA wave depletes
the particle density for antiparallel electron drift 共APED兲,
and enhances the particle density for parallel electron drift
共PED兲. Reference 19 characterizes the general properties of
the StIA wave in more detail.
1
0
x−position: x/λ
e
FIG. 3. StIA: Electron collisions spatially damp the amplitude of fluctuations in the fluid wave variables for Ves0 ⬍ Vphs. ¯␯e = 0 共solid line兲, ¯␯e = 0.1
共dashed line兲, ¯␯e = 1 共dashed-dotted line兲, and ¯␯e = 5 共dotted line兲. Here,
Ves0 = 0.1VA and Vphs = 0.5VA.
B. Collisional StIA waves
To investigate the effects of electron collisional resistivity on the StIA wave, Eqs. 共25兲 and 共26兲 are solved numerically for ue, with ¯␯in = 0 and ¯␤ = 0. In Fig. 3, the normalized
parallel electric field Ēs = Es / Es0, parallel electron velocity
Ves / Ves0, and particle density ni / Ni are plotted as a function
of x / ␭e for initial electron drift Ves0 = 0.1VA and Vphs = 0.5VA.
A family of curves is shown representing solutions for ¯␯e
= 0 共solid line兲, ¯␯e = 0.1 共dashed line兲, ¯␯e = 1 共dashed-dotted
line兲, and ¯␯e = 5 共dotted line兲. We see in Fig. 3 that sufficiently frequent electron collisions damp the spatial pattern
Downloaded 27 Apr 2009 to 128.97.43.16. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
Phys. Plasmas 15, 052108 共2008兲
Finnegan, Koepke, and Knudsen
C. Finite plasma-␤ effects on StIA waves
The inclusion of nonzero plasma-␤ gives important
warm-plasma corrections to the behavior of the StIA wave.
In this paper, we have restricted our study to low-beta plasma
in the range 0 ⬍ ␤ Ⰶ 1. Numerical solutions to the StIA wave
=0.8V ; V
es0
2
A
=0.5V
phs
A
(a)
s
parallel electric field: E /E
s0
V
1
0
−1
−2
0
electron velocity: Ves/Ves0
of fluid variables to dc values equal to the mean of the undamped patterns, i.e., waveforms. Likewise, spatial wave
damping increases with increasing electron collisionality until the wave is critically damped spatially. For Ves0 ⬍ Vphs, the
phase of the parallel component of electric field balanced by
finite-electron conductivity leads the phase of the electron
inertia contribution at x = x0, as seen in Eq. 共18兲, causing
damping of the spatial pattern and the spatial mean of the
field-aligned electric field to reach more-negative values.
In contrast, for Ves0 ⬎ Vphs, the phase of the resistive
force on the electrons lags behind that of the inertial response
at x = x0, causing the spatial pattern to grow in the perpendicular direction, as seen in Fig. 4. In Fig. 4, a family of
curves is shown that represent solutions for ¯␯e = 0 共solid line兲,
¯␯e = 0.01 共dashed line兲, ¯␯e = 0.05 共dashed-dotted line兲, and
¯␯e = 0.1 共dotted line兲. The nonlinear evolution of the phase
relationship between the inertial and resistive contributions
to the parallel electric field causes the total parallel electric
field spatial pattern to steepen. The amplitude of the pattern
increases for increasing values of x / ␭e until the wave crests
and subsequently collapses. Figure 5 shows the spatial
growth rate of the StIA pattern for several values of electron
collision frequency, plotted as a function of Vphs for a fixed
value of Ves0. We see, in Fig. 5, that the spatial growth rate
becomes discontinuous at Ves0 = Vphs, corresponding to the
asymptotic approach to zero of the dominant perpendicular
wavelength as Ves0 → Vphs, as seen in Fig. 2. For antiparallel
electron drift, electron collisions only damp the amplitude of
the modulated fluid variables, with the growth rate exhibiting
a local minimum in its Vphs dependence.
By neglecting electron collisions and retaining only the
ion collisional resistivity, Eqs. 共25兲 and 共26兲 are solved nu¯ 2ci → ⬁. Normerically for ue with ¯␯e = 0, ¯␤ = 0, p = 1, and ␻
malized Es, Ves / Ves0, and ni / Ni are plotted in Fig. 6 for
¯␯in = 0 共solid line兲, ¯␯in = 0.001 共dashed line兲, ¯␯in = 0.01
共dashed-dotted line兲, and ¯␯in = 0.05 共dotted line兲. We see, in
Fig. 6, that ion-neutral collisions, for Ves0 ⬍ Vphs, damps the
wave amplitude while introducing an increasing dc offset in
the fluid variables Ves and ni. This dc offset results from ion
collisions continuously reducing the perpendicular wavenumber, broadening the k-space wave spectrum. This spreading in k space is most clearly visible in the top panel of Fig.
6. Unlike the case of electron collisional damping, the fieldaligned component of electric field is damped to a singular
negative dc value, irrespective of the ion-collision frequency.
Ion-neutral collisional damping of the StIA-wave amplitude
is observed for all values of Ves0 and Vphs. Figure 7 is a plot
of the spatial growth rate as a function of Vphs for a fixed Ves0
and several values of ␯in. The growth rate increases in the
negative direction for increasing ion-neutral collision frequency and exhibits a local minimum in its Vphs dependence
for antiparallel electron drift.
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
(b)
1
0.8
0.6
0
15
(c)
ion density: ni/Ni
052108-6
10
5
0
0
x−position: x/λe
FIG. 4. StIA: Electron resistivity modifies the field-aligned component of
electric field Es, inducing a spatial pattern in the fluid variables for Ves0
⬎ Vphs. ¯␯e = 0 共solid line兲, ¯␯e = 0.01 共dashed line兲, ¯␯e = 0.05 共dashed-dotted
line兲, and ¯␯e = 0.1 共dotted line兲. Here, Ves0 = 0.8VA and Vphs = 0.5VA.
equations are obtained for ¯␯e = 0 and ¯␯in = 0. In Fig. 8, the
dominant perpendicular spectral wavelength is plotted as a
function of Vphs for several values of Ves0 and ¯␤. We see in
Fig. 8 that, for antiparallel electron drifts, increasing finite
plasma-␤ significantly reduces the dominant wavelength ␭o
at small values of Vphs, whereas ␭o remains relatively unaffected at large Vphs. For increasing parallel electron drift, the
dominant perpendicular wavelength also reduces, with the
most significant reduction occurring as Ves0 approaches the
value of Vphs. The inclusion of finite plasma-␤ was found to
produce an increase in the amplitude of the nonlinear modulation of the fluid variables by less than a few percent over
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052108-7
Phys. Plasmas 15, 052108 共2008兲
The dispersive Alfvén wave…
V
es0
growth rate
(a)
=−V
A
−4
−10
−2
−10
0
−10
0
0.2
0.4
parallel electric field: Es/Es0
−6
−10
0.6
parallel wave phase speed: V /V
growth rate
V
=0.1V
es0
A
0
−5
0
0.1
0.2
0.3
0.4
es0
(b)
es
5
A
electron velocity: V /V
phs
several orders of magnitude change in ¯␤. The role of the
parameter ␶ on the spatial growth and decay rates of StIA
waves is beyond the scope of this paper and will be the topic
of future work.
ion density: ni / Ni
parallel wave phase speed: Vphs/VA
V
=0.1V ; V
es0
A
=0.5V
phs
A
0.1
0
−0.1
−0.2
0
2.5
(b)
5
10
15
20
5
10
15
20
10
15
20
2
1.5
1
0
1.6
(c)
0.5
FIG. 5. StIA: Electron collisional effects on the dependence of the spatial
growth on the wave phase speed. 共a兲 For antiparallel electron drift 共APED兲,
the spatial damping rate increases with increasing electron collisionality,
exhibiting a local minimum in the growth rate dependence on the parallel
wave phase speed Vphs. 共b兲 For parallel electron drift 共PED兲, electron collisional resistivity causes either spatial growth 共Ves0 ⬎ Vphs兲 or spatial damping
共Ves0 ⬍ Vphs兲 of fluid wave variables. ¯␯e = 1 共solid line兲, ¯␯e = 0.1 共dasheddotted line兲, and ¯␯e = 0.01 共dotted line兲.
0.2 (a)
1.4
1.2
1
0
5
x−position: x/λ
e
D. Kinetic StA waves
FIG. 6. StIA: Ion collisions spatially damp the ac modulation of fluid variables while introducing an increasing dc offset. ¯␯in = 0 共solid line兲, ¯␯in
= 0.01 共dashed line兲, ¯␯in = 0.05 共dashed-dotted line兲, and ¯␯in = 0.1 共dotted
line兲. 共a兲 Parallel electric field, 共b兲 electron velocity, and 共c兲 ion density.
Here, Ves0 = 0.1VA and Vphs = 0.5VA.
The stationary kinetic Alfvén 共StKA兲 wave is a solution
to the StA wave equations, for which ¯␤ ⬎ 1 共i.e., the sound
gyroradius ␳s = Cs / ␻ci is larger than the electron inertial
length ␭e = c / ␻ pe兲 and Vphs ⬎ VA. In this section, the general
properties of the StKA wave are discussed and several key
differences between the StKA wave and the StIA wave are
highlighted. Similar to the StIA wave, the StKA wave carries
a nonzero parallel component of electric field. This parallel
electric field is balanced by the parallel electron pressure
gradient 关see the last term in Eq. 共18兲兴, and can energize
particles along magnetic field lines. Figure 9 shows solutions
to the collisionless StA wave equations for normalized quantities ␭o / ␭e, ves,max / Vphs, and ni / Ni as a function of Ves0 with
¯␤ = 100. In the top panel of Fig. 9, we see that the dominant
perpendicular wavelength increases with increasing Ves0 and
decreases with increasing Vphs. The functional dependence of
␭0 on Ves0 and Vphs is the opposite of what was found for the
case of the StIA wave, cf. Fig. 2. The middle panel of Fig. 9
shows that the StKA wave can accelerate electrons to parallel
speeds much larger than the local Alfvén speed. Unlike the
StIA wave, however, the StKA wave modulates the direction
of the electron flow. Thus, the StKA wave creates adjacent,
alternating polarity, filamentary structures of field-aligned
current. The StKA wave also produces large density depletions or enhancements, like the StIA wave, which are dependent on the direction of the initial parallel electron drift, cf.
the bottom panel in Fig. 9. It is important to note that studying kinetic effects using a two-fluid model is limited in its
range of applicability. For the two-fluid model to be valid for
our study of the StKA wave, we require that k⬜␳s ⬍ 1, which
is equivalent to the condition that ¯␤ ⬍ ␭ / ␭ . For all cases
冑
o
e
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052108-8
Phys. Plasmas 15, 052108 共2008兲
Finnegan, Koepke, and Knudsen
−3
2
perpendicular wavelength: λ /λ
e
×10
growth rate
−0.5
Ves0=0.1VA
−1
0.4
0.6
0.8
parallel wave phase speed: V /V
phs
−3
A
e
0 ×10
−0.4
0
10
V
=−0.1V
es0
A
e
β=0
i
0
10
β=5×10−3(m /m )
e
i
−2
−1
10
0
1
β=10 (m /m )
e
0.2
(a)
i
0.4
0.6
V
=V
es0
10
0
0.8
A
β=0
A
Ves0=0.1VA
−2
β=10 (me/mi)
10
−1
β=10−4(m /m )
e
0.2
0
0.3
0.4
0.5
parallel wave phase speed: V /V
phs
1
parallel wave phase speed: Vphs/VA
i
(b)
=−V
es0
0.1
V
β=0.25(me/mi)
β=0.5(m /m )
o
−0.2
−0.6
(b)
0.2
1
10
perpendicular wavelength: λ /λ
growth rate
(a)
Ves0=−VA
o
0
10
0.1
0.2
0.3
0.4
0.5
parallel wave phase speed: V /V
phs
A
A
FIG. 7. StIA: Ion collisional effects on the spatial growth. 共a兲 For parallel
electron drift 共PED兲, ion-neutral collisional resistivity causes only spatial
damping of fluid wave variables. 共b兲 For antiparallel electron drift 共APED兲,
the spatial damping rate increases with increasing ion-neutral collisionality,
exhibiting a local minimum in the growth rate dependence on the parallel
wave phase speed Vphs. ¯␯in = 0.1⫻ 10−3 共solid line兲, ¯␯in = 0.2⫻ 10−3 共dashed
line兲, ¯␯in = 0.5⫻ 10−3 共dashed-dotted line兲, and ¯␯in = 1 ⫻ 10−3 共dotted line兲.
of the StKA wave presented here, we let ¯␤ = 100. Thus, our
study is restricted in 共Vphs , Ves0兲 parameter space to StKA
wave solutions for which ␭o / ␭e ⬎ 10.
E. Collisional StKA waves
For nonzero electron collisions, numerical solutions to
the StA wave equations with ¯␯in = 0 for Ēs, Ves / Ves0, and
ni / Ni are plotted in Fig. 10. Solutions for ¯␯e = 0 共solid line兲,
¯␯e = 0.1 共dashed line兲, ¯␯e = 0.25 共dashed-dotted line兲, and
¯␯e = 0.5 共dotted line兲 are shown. It is clear from Fig. 10 that
parallel electron conductivity causes the amplitude of the
StKA wave pattern to grow spatially. This wave growth is
being generated by the initial phase lag between the parallel
electric field, balanced by finite electron conductivity, and
the parallel thermal pressure. This mechanism is similar to
the Ves0 ⬎ Vphs case for the StIA wave. The growth rate
clearly increases with increasing collision frequency. Similar
to a spatially growing StIA wave, StKA wave growth continues until the parallel electric field steepens and crests, at
which point the wave collapses 共not shown兲. Figure 11
shows the spatial growth rate of the StKA wave as a function
of Vphs for both parallel and antiparallel electron drift. In
both cases, we see that the growth rate increases with in-
FIG. 8. StIA: The effect of warm plasma is to exaggerate the reduction of
perpendicular wavelength ␭o for both the PED and APED wave cases near
the minimum wavelength.
creasing Vphs while exhibiting only a weak dependence on
Ves0. No wave damping is observed for nonzero values of
electron collisionality.
Ion-neutral collisions can also produce growth in StKA
wave amplitude, as seen in Fig. 12. In Fig. 12, electron collisions are neglected, and ␶ is set to unity. Numerical solutions to the StA wave equations for Ēs = Es / Es0, Ves / Ves0, and
ni / Ni are shown for ¯␯in = 0 共solid line兲, ¯␯in = 0.01 共dashed
line兲, ¯␯in = 0.025 共dashed-dotted line兲, and ¯␯in = 0.05 共dotted
line兲. As in the case of electron collisions, the StKA wave
amplitude grows until the parallel component of the electric
field crests, collapsing the wave. Ion-neutral collisional enhancements to the polarization current, for large Vphs / VA in
the kinetic Alfvén wave regime, can be sufficient to cause
dissipation of the StKA pattern, as shown in Fig. 13. Like the
case of ion-neutral collisional damping of the StIA wave, the
value of the dominant perpendicular wavelength in Fig. 13
varies continuously in space, resulting in a continuously
changing k-space wave spectrum. For the StKA wave, however, this effect is much weaker. In Fig. 14, the growth rate
of the StKA wave is plotted as a function of the normalized
effective parallel wave phase speed Vphs / VA. For antiparallel
electron drift, we find that the growth rate is always positive
and decreasing with increasing Vphs / VA. The growth rate in
the case of parallel electron drift, however, becomes negative
at large values of Vphs / VA. The ratio ␶ of ion to electron
temperature does not change the general characteristics of
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Phys. Plasmas 15, 052108 共2008兲
The dispersive Alfvén wave…
2
V
1/2
=2 V
phs
V
A
parallel electric field: Es/Es0
10
=2V
phs
A
1
−0.5
0
0.5
0
−1
0
0.5
1
i
−0.5
0.5
0
20
40
60
80
100
120
20
40
60
80
100
120
20
40
60
80
100
120
(b)
0
−5
0
1.2
(c)
1.1
i
ion density: n /N
i
i
ion density: n /N
es,max
electron velocity: v
1
−2
−1
6
(a)
−1
0
5
1
electron velocity: Ves/Ves0
10
−1
2
Ves0=0.1VA ; Vphs=2VA
1
−0.5
Vphs=5VA
/V
phs
perpendicular wavelength: λo/λe
052108-9
4
2
1
0.9
0.8
0
0
−1
−0.5
0
0.5
electron drift: V /V
es0
x−position: x/λ
e
1
A
FIG. 9. StKA: Dominant perpendicular wavelength ␭o decreases with increasing parallel wave phase speed Vphs for both PED and APED wave
conditions 共top panel兲. StKA waves accelerate electrons to many times their
initial drift speed in the direction opposing their initial drift 共middle panel兲.
Antiparallel initial electron drifts produce an enhancement of the initial
particle density, while parallel initial electron drifts reduce the initial particle
density 共bottom panel兲. The vertical dotted line in the top panel indicates
where the perpendicular wavelength is undefined.
the StKA wave. The effects of ␶ on the collisional StKA
wave is beyond the scope of this paper, and will be the topic
of future work.
FIG. 10. StKA: Electron collisional resistivity generates a field-aligned electric field which modifies the field-aligned StKA wave electric field producing spatial growth in the amplitude of oscillating fluid variables. ¯␯e = 0 共solid
line兲, ¯␯e = 0.1 共dashed line兲, ¯␯e = 0.25 共dashed-dotted line兲, and ¯␯e = 0.5 共dotted line兲. Here, Ves0 = 0.1VA and Vphs = 2VA. Here, Ves0 = 0.1VA and Vphs
= 2VA.
are stable in the absence of the background field-aligned current, we write Eq. 共25兲 in a generalized form assuming that
␯in = 0,
f共ves兲
冉
冊
⳵2ves
⳵ves ⳵ves
2
+ kx0
ves = F,
2 + g ves,
⳵x
⳵x ⳵x
共27兲
IV. DISCUSSION
A. Source of free energy and wave growth
The source of free energy which is responsible for the
stationary Alfvén wave is the background magnetic-fieldaligned current Js0 = qenVes0. As seen in Figs. 2 and 9, in the
absence of the field-aligned current Ves0 = 0 the wave is stable
and no wave perturbation forms. To show that these waves
2
where the characteristic pattern kx0
is
2
=
kx0
冉
VA2
2
Vphs
− Ue
冊冉
␭2e U2e −
VA2
2
Vphs
␳s2
冊
−1
,
and the driving term F on the right-hand side of Eq. 共27兲 is
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Phys. Plasmas 15, 052108 共2008兲
Finnegan, Koepke, and Knudsen
−3
20 ×10
(a)
s0
Ves0=0.1VA ; Vphs=2VA
Ves0=−VA
growth rate
15
10
5
3
4
parallel wave phase speed: Vphs/VA
0
electron velocity: V /V
growth rate
×10
(b)
V
=0.5V
es0
15
A
10
40
60
80
100
120
0
20
1.1
(c)
40
60
80
100
120
40
60
80
100
120
2
−2
FIG. 11. StKA: The spatial growth rate increases with increasing electron
collisionality and parallel wave phase speed Vphs for both the PED and
APED cases. ¯␯e = 0.01 共solid line兲, ¯␯e = 0.025 共dashed line兲, ¯␯e = 0.05
共dashed-dotted line兲, and ¯␯e = 0.1 共dotted line兲.
i
5
i
4
ion density: n /N
3
parallel wave phase speed: Vphs/VA
1
0.9
0.8
0
冉
F = Ves0 1 −
(b)
0
5
0
2
20
es
20
−0.5
5
−3
(a)
0
es0
0
2
0.5
s
parallel electric field: E /E
052108-10
Ves0
tan ␪
Vd
冊冉
␭2e U2e −
冊
VA2 2 −1
.
2 ␳s
Vphs
Here, we see that the driver is spatially and temporally constant and depends on the sources of free energy, i.e., the
initial parallel current in the form of Ves0 and the perpendicu2
is inlar plasma drift Vd. Also, the characteristic pattern kx0
fluenced by the free energy source Ves0 in terms of Ue = 1
− Ves0 / Vphs. When the initial parallel current is turned off,
i.e., Ves0 = 0 and hence Ue = 1, the driving term is zero and
without an initial perturbation, the StA wave pattern will not
form. Thus, in the absence of the background-parallelcurrent free-energy source, the StA wave is stable. This is not
strictly true, however. For p ⫽ 1, an initial ion polarization
current also provides a source of free energy capable of supporting the StA wave.
The damping term shows explicitly that the spatial
wave-form is damped when Vphs ⬎ Ves and grows when
Vphs ⬍ Ves. In order to understand how parallel electron conductivity leads to wave growth, it is useful to use the parallel
electric field, Eq. 共18兲,
20
x−position: x/λ
e
FIG. 12. StKA: Ion collisions modify the polarization drift 共current兲 producing growth in amplitude of the modulated fluid variables. ¯␯in = 0 共solid
line兲, ¯␯in = 0.01 共dashed line兲, ¯␯in = 0.025 共dashed-dotted line兲, and ¯␯in = 0.05
共dotted line兲. Here, Ves0 = 0.1VA and Vphs = 2VA.
qeEs = me␪关Vphs − Ves兴⳵xVes − me␪
2
VTe
⳵xVes
关Vphs − Ves兴
+ ␯emeVes .
For inertial waves, i.e., the case VTe = 0, the electric field,
balanced by electron inertia, leads the resistive term when
Ves0 ⬎ Vphs. The nonlinear relative-phase difference causes
the electrons traveling along the magnetic field to spend
more time in the accelerating field than in the decelerating
field, leading to a spatially growing field-aligned current
and a spatially growing StIA wave. Conversely, when
Ves0 ⬍ Vphs, the electrons spend more time in the decelerating
field than in the accelerating field, leading to a spatially
damped parallel current. In the kinetic regime, the parallel
electric field balanced by the parallel electron pressure dominates over, and is 180° out of phase with, the electric field
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052108-11
Phys. Plasmas 15, 052108 共2008兲
The dispersive Alfvén wave…
−3
15 ×10
0.1
0.05
0
−0.05
−0.1
(a)
40
60
80
5
0
2
100
3
V
4
−1
20
40
60
80
100
(c)
5
(b)
=0.5V
es0
A
2
0
2
3
4
5
parallel wave phase speed: Vphs/VA
0.9
FIG. 14. StKA: Ion collisions produce positive growth rates with a local
extremum in the parallel wave phase speed Vphs dependence, for APED
StKA waves. PED StKA waves are damped by ion collisions above a critical
value of parallel wave phase speed Vphs. ¯␯in = 0.01 共solid line兲, ¯␯in = 0.02
共dashed line兲, ¯␯in = 0.05 共dashed-dotted line兲, and ¯␯in = 0.1 共dotted line兲.
0.8
0
4
parallel wave phase speed: Vphs/VA
×10−3
0
1
A
10
(b)
0
ion density: ni/Ni
20
(a)
=−V
es0
growth rate
electric field: Ves/Ves0
0
1
V
growth rate
parallel electric field: Es/Es0
Ves0=0.5VA ; Vphs=5VA
20
40
60
80
100
x−position: x/λe
FIG. 13. StKA: Spatial damping of fluid StKA wave variable for ¯␯in = 0
共solid line兲, ¯␯in = 0.1 共dashed line兲, and ¯␯in = 0.5 共dotted line兲. Here, Ves0
= 0.5VA and Vphs = 5VA.
冉
冊
⳵ xE x
␯inEx
pUe
=1−
1+
.
␻ciB0
ue
Vd␻ciB0
共28兲
Substitution into Eq. 共23兲 for the case p = 1 and neglecting
electron collisions ␯e = 0 yields
␭2e ⳵2 2 VA2 2 ⳵2
VA2 ␯in Ue Ex
u
−
␳
ln共u
兲
−
e
s
2
2
2 ⳵x2 e Vphs
⳵x2
Vd ue ␻ciB0
Vphs
balanced by the electron inertia. Again this phase difference
was first identified by Goertz and Boswell.1 So, when
Ves0 ⬍ Vphs, the pressure force leads the resistive force and
the nonlinear relative phase difference causes the electrons to
spend more time in the accelerating field, leading to a fieldaligned current that grows spatially in amplitude and a spatially growing StKA wave. The source of free energy available for the spatial growth of the StKA wave amplitude is the
bulk plasma E ⫻ B drift Vd, generated by the convection
electric field Ey.
Spatial damping and growth of the StA wave pattern by
ion-neutral collisions occurs through damping/growth of the
ion polarization current. The steady-state analog of the ion
polarization drift is given by Eq. 共14兲. For simplicity, we
assume that Vd / ␭e Ⰶ ␻ci, and ignore the ion thermal pressure
␶ = 0 reducing Eq. 共14兲 to
冉
+ 1−
VA2
2
Vphs
冊
V2
Ue
= Ue − 2A .
ue
Vphs
共29兲
Here, ␳s2 = ¯␤␭2e is the ion acoustic gyroradius. In the inertial
regime 共␳s = 0兲, the ion polarization drift is in the opposite
direction to the bulk plasma E ⫻ B drift Vd, and the ionneutral collisional drag force damps the polarization current.
This spatial damping of the ion polarization current damps
the StIA wave. Hence ion-neutral collisions are purely dissipative in the inertial regime. In the kinetic regime 共␭e = 0兲,
the ion polarization drift is in the same direction as Vd. During the first half-period of the wave, ion-neutral collisional
drag acts to damp the spatial pattern 共see Fig. 12兲. As position increases in the E ⫻ B direction, ion-neutral collisions
inevitably reduce the net value of ion flow below Vd changing the direction of the cross-field current, and in doing so,
the direction of the field-aligned component of electric field.
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052108-12
Phys. Plasmas 15, 052108 共2008兲
Finnegan, Koepke, and Knudsen
This results in growth during the second half-wavelength of
the wave, and causes net spatial growth of the amplitude of
the StKA wave-form. The free-energy source for this wave
growth is the plasma E ⫻ B drift Vd.
B. Linear limit of StA wave equation
To illustrate the connection between our model and previous work, we compare some limiting cases of our model
with related models of the dispersive Alfvén wave. First, in
the cold-plasma 共¯␤ = 0兲 collisionless limit, the wave equation
for the stationary inertial Alfvén wave, first derived by
Knudsen,19 is recovered,
冋
册 冋
册
V2
␭2e 2 2
V2
Ue
⳵x ue = p 2A − 1
+ Ue − 2A .
2
ue
Vphs
Vphs
共30兲
In the linear, or small-amplitude 共兩ves / Vphs兩 Ⰶ 1兲, limit, retaining electron collisions and neglecting initial field-aligned
electron drift Ves0 = 0, Eq. 共25兲 reduces to
冉
冊
V2
V2
␭2e 2
⳵x ves − 2A ␳s2⳵2x ves + ¯␯e␭e⳵xves = − 2A − 1 ves .
2
Vphs
Vphs
共31兲
Here, ␳s = CS / ␻ci is the acoustic gyroradius, with
CS = 冑共Te + Ti兲 / mi being the sound speed. By Fourier transforming Eq. 共31兲 and substituting the relation ␪ = −kz / kx from
the Fourier transform of Eqs. 共10a兲 and 共10b兲, Eq. 共31兲 can
be written as
−
关1 + ␳s2k2x 兴1/2
k xV d
= VA
.
2 2
kz
关1 + ␭e kx 共1 − i␯e/kxVd兲兴1/2
共32兲
For the inertial Alfvén wave in collisionless 共␯e = 0兲 plasma,
Eq. 共32兲 reduces to the zero-frequency limit of Eq. 共5兲 in
Drozdenko and Morales,25 describing the linear effects of
cross-field plasma flow on a field-aligned current channel,
−
VA
k xV d
=
.
kz
关1 + ␭2e k2x 兴1/2
Recognizing that ␻ = −kxVd is the wave frequency, as observed by the convecting plasma, allows Eq. 共32兲 to reduce
to Eq. 共18兲 of Morales and Maggs,26
关1 + ␳s2k2x 兴1/2
␻
.
= VA
kz
关1 + i共␯e/␻兲␭2e k2x 兴1/2
If collisions are neglected and electron inertia is included,
Eq. 共32兲, with ␻ = −kxVd, will represent the dispersion relation for the low-frequency dispersive Alfvén wave including
finite Larmor radius effects,10
冉
1 + ␳s2k2x
␻
= VA
kz
1 + ␭2e k2x
冊
1/2
.
Setting ␭e = 0 and ␳s = 0, we see that the parallel wave phase
speed is simply the Alfvén speed,27 i.e., Vphs = VA.
C. Applications to auroral ionosphere
Before the StA wave model can be applied to predicting
details of auroral arc formation, several important effects not
represented in the present form of the StA wave model must
be evaluated and possibly incorporated into the model.
One potentially important effect in both the auroral ionosphere and a laboratory experiment that remains unaddressed
by the present model is wave reflection at a conducting
boundary. StA wave solutions are nonlinear and, thus, wave
reflection might not involve simple superposition. There is
no obvious reason to expect that the interference produced
by a reflected StA wave would inhibit the electron acceleration and plasma density variation induced by the wave, particularly since, in the linear limit, the parallel electric field
associated with the inertial Alfvén wave may constructively
interfere upon reflection from a conducting boundary such as
the ionosphere.1
In addition to wave reflections, the present StA wave
model does not include background parallel and perpendicular inhomogeneities. The strong inhomogeneities present in
the auroral ionosphere may make the establishment of a StA
wave propagating at a fixed angle to the background magnetic field nontrivial. Of particular importance is the functional dependence of the electron and ion collision frequencies with altitude. We have shown that both Coulomb and
ion-neutral collisions can significantly alter the StA waveform and so inhomogeneities in the collision frequencies
would have to be addressed before making any direct applications of the collisional StA wave to the auroral zone. Predicting the details of interference and reflection as well as
parallel and perpendicular inhomogeneities may be accomplished more conveniently via computer simulation.
Although an auroral arc often appears as a periodic
structure, i.e., multiple parallel discrete arcs, it can also appear as a solitary structure. The solutions to the StA wave
equations, presented previously, pertain to a plasma that is
unbounded and predict the periodic nature of the StA wave.
Solitary solutions to the StA wave equations have not been
found. One explanation for the absence of a solitary wave
solution is the possible restriction in the range of wavenumber values imposed by the finite size of a naturally occurring current sheet. Another explanation is model incompleteness, which motivates our future efforts on the StA
wave.
V. SUMMARY
In this paper, we predict how the inclusion of ion-neutral
collisional resistivity, parallel electron resistivity, and nonzero plasma ␤ affects the wave properties of the stationary
Alfvén wave. The parallel component of electric field associated with the stationary Alfvén wave accelerates electrons
parallel to the magnetic field and creates either density enhancements or depletions depending on the direction of the
initial parallel electron drift. Ion-neutral and electron-ion collisional resistivity are predicted to both significantly alter the
perpendicular ac and dc structure of magnetic-field-aligned
electron drift and either dissipate or enhance the field-aligned
electron energy depending on the initial value of parallel
Downloaded 27 Apr 2009 to 128.97.43.16. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
052108-13
electron drift velocity. Nonzero plasma ␤ is predicted both to
reduce the dominant perpendicular wavelength of the StIA
wave structure and to have minimal effect on the wave amplitude. With the inclusion of resistive effects and nonzero
plasma ␤, the conclusions of Knudsen’s cold, collisionless
case are unchanged. However, unlike the StIA wave, the
StKA wave, occurring at higher beta, modulates the direction
of parallel electron flow, creating adjacent, alternating polarity, current filaments. The electric field produced by finite
parallel electron conductivity supports the parallel StKA
wave electric field, and enhances the parallel electron energy.
Ion-neutral collisional resistivity is predicted to either enhance or dissipate the parallel electron energy, depending on
the magnitude of the initial parallel electron drift.
ACKNOWLEDGMENTS
Useful discussions with S. Vincena, W. Gekelman, J.
Maggs, G. Morales, and M. Goldman are gratefully acknowledged. This work is part of a dissertation to be submitted by S. M. Finnegan to the Eberly College of Arts and
Sciences, West Virginia University, Morgantown, WV, in
partial fulfillment of the requirements for the Ph.D. degree in
physics.
This work is supported by a U.S. NSF/DOE grant at
WVU 共Grant No. PHYS-06兲. D. J. Knudsen is supported by
the National Research Council of Canada. This work is part
of the Space-Plasma Campaign being carried out by M. E.
Koepke, S. M. Finnegan, D. J. Knudsen, R. Rankin, R.
Marchand, C. Chaston, and S. Vincena at the Basic Plasma
Science Facility, University of California, Los Angeles and
thus receives part of the facility support from the NSF and
the DOE.
1
Phys. Plasmas 15, 052108 共2008兲
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