PHYSICS OF PLASMAS 16, 092102 共2009兲
On generation of Alfvénic-like fluctuations by drift wave–zonal flow system
in large plasma device experiments
W. Horton,1 C. Correa,1 G. D. Chagelishvili,2 V. S. Avsarkisov,2 J. G. Lominadze,2
J. C. Perez,3 J.-H. Kim,3 and T. A. Carter4
1
Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712, USA
Georgian National Astrophysical Observatory, The Chavchavadze State University, Tbilisi 0160, Georgia
and M. Nodia Institute of Geophysics, Tbilisi 0193, Georgia
3
Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
4
Department of Physics, University of California, Los Angeles, California 90095, USA
2
共Received 26 February 2009; accepted 3 August 2009; published online 3 September 2009兲
According to recent experiments, magnetically confined fusion plasmas with “drift wave–zonal flow
turbulence” 共DW-ZF兲 give rise to broadband electromagnetic waves. Sharapov et al. 关Europhysics
Conference Abstracts, 35th EPS Conference on Plasma Physics, Hersonissos, 2008, edited by P.
Lalousis and S. Moustaizis 共European Physical Society, Switzerland, 2008兲, Vol. 32D, p. 4.071兴
reported an abrupt change in the magnetic turbulence during L-H transitions in Joint European Torus
关P. H. Rebut and B. E. Keen, Fusion Technol. 11, 13 共1987兲兴 plasmas. A broad spectrum of
Alfvénic-like 共electromagnetic兲 fluctuations appears from E ⫻ B flow driven turbulence in
experiments on the large plasma device 共LAPD兲 关W. Gekelman et al., Rev. Sci. Instrum. 62, 2875
共1991兲兴 facility at UCLA. Evidence of the existence of magnetic fluctuations in the shear flow
region in the experiments is shown. We present one possible theoretical explanation of the
generation of electromagnetic fluctuations in DW-ZF systems for an example of LAPD experiments.
The method used is based on generalizing results on shear flow phenomena from the hydrodynamics
community. In the 1990s, it was realized that fluctuation modes of spectrally stable nonuniform
共sheared兲 flows are non-normal. That is, the linear operators of the flows modal analysis are
non-normal and the corresponding eigenmodes are not orthogonal. The non-normality results in
linear transient growth with bursts of the perturbations and the mode coupling, which causes the
generation of electromagnetic waves from the drift wave–shear flow system. We consider shear flow
that mimics tokamak zonal flow. We show that the transient growth substantially exceeds the growth
of the classical dissipative trapped-particle instability of the system. © 2009 American Institute of
Physics. 关doi:10.1063/1.3211197兴
I. INTRODUCTION
In magnetically confined fusion plasmas, the reduction
in cross-flow transport by shear flow is now widely utilized.
A variety of measurements have confirmed the role of flow
shear in impeding the transport and diminishing the intensity
of turbulence of fusion plasmas in all geometries and
regimes.1–5 Moreover, record values of confinement time and
the ratio of fusion power output to input heating power have
been achieved using flow-shear-induced transport barriers in
tokamak/Joint European Torus 共JET兲 plasmas.6 Consequently, the importance of sheared E ⫻ B flows to the development of the L-H transition and internal transport barriers is
now widely appreciated. As a result, for more than a decade,
the study of shear-flow-induced phenomena has been central
in fusion research.
It is now accepted that zonal flow 共ZF兲 is a crucial player
in the mechanism for self-regulation of drift wave 共DW兲 turbulence and transport in nearly all cases and regimes of fusion plasmas, so this type of turbulence is now commonly
referred to as “drift wave–zonal flow 共DW-ZF兲 turbulence.”
Both theoretical work and numerical simulation made important contributions to this paradigm shift and to the study of
the dynamics of DW-ZF turbulence systems. Specifically,
1070-664X/2009/16共9兲/092102/11/$25.00
numerical simulations confirm the experimental observation
that DW turbulence and transport levels are reduced when
the ZF is properly generated. In contrast, the fluctuations
themselves lack such universality, changing character from
device to device and within any given device in different
regimes.7–11 Specifically, the recent JET 共Ref. 7兲 and large
plasma device 共LAPD兲 共see below Sec. III兲 experiments indicate that DW-ZF turbulence gives rise to a broadband spectrum of electromagnetic waves, which is the subject of the
present study. ZF is different from externally imposed shear
E ⫻ B flow in that ZF is a self-organized turbulence-driven
phenomena and it is bipolar. However, since locally shear
E ⫻ B flow acts much like tokamak ZF, the two will be
treated interchangeably.
Sharapov et al.7 reported abrupt changes in magnetic
turbulence during L-H transitions in JET plasmas. The fluctuation data show a broad spectrum of electromagnetic
waves in the presence of large ZF. Alfvénic-like fluctuations
appear from E ⫻ B flow driven turbulence in experiments on
the LAPD facility at UCLA. The LAPD 共see Fig. 1 of Refs.
10–12兲 is 18 m in length, with a singly ionized helium
plasma column approximately 0.8 m in diameter created by a
pulsed discharge 20 ms long from a barium oxide coated
emissive cathode. A radial electric field is established by bi-
16, 092102-1
© 2009 American Institute of Physics
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092102-2
Phys. Plasmas 16, 092102 共2009兲
Horton et al.
asing the chamber wall with respect to the anode and cathode
for approximately 5 ms during the discharge. The radial electric field profiles can be varied by changing the bias voltage
and the magnetic field boundary conditions within the
LAPD. The experiment is described further in Sec. III and
Ref. 11. A strong radial electric field, in turn, induces an
E ⫻ B plasma rotation in this region of the plasma. This wallto-cathode biasing results in different sheared E ⫻ B flows
and, consequently, different turbulent regimes at the edge of
the plasma column. For some parameters of the LAPD, it
appears that the DW-ZF system is enriched by high frequency Alfvénic-like fluctuations.10,11 Generation of broadband electromagnetic fluctuations in DW-ZF systems is a
significant phenomenon because electromagnetic fluctuations
can modify the anomalous transport. In addition, the characteristics of these fluctuations indirectly give information
about the dynamics of the DW-ZF system.
In the LAPD experiments, the ZF is generated by external mechanisms and may be stronger than ZF generated by
self-regulation mechanisms, such as in tokamaks. Specifically, ZF amplitude is reported as UE ⯝ 4 ⫻ 105 cm/ s, the
radial size of the ZF region is LE ⯝ 5 cm, and the Alfvén
speed is VA ⯝ 9 ⫻ 107 cm/ s. The reported Alfvénic-like fluctuations occur at high shear rates of the ZF velocity 共see Sec.
III兲. It will be shown that fluctuations in flows with high
shear rates are strongly non-normal. The strong nonnormality results in linear transient growth with bursts of the
perturbations and mode coupling, which is how electromagnetic waves are generated from a DW-ZF system. For the
above values, the flow-shear parameter is A = UE / LE ⯝ 0.8
⫻ 105 s−1. Considering perturbations with characteristic size
along the axial magnetic field of the order of the length of
the device L储 ⬅ 1 / k储 ⯝ 103 cm, the normalized shear parameter is of the order of unity,
S⬅
A
⯝ 0.9.
k储VA
共1兲
The aim of this paper is to describe a possible mechanism for the generation of electromagnetic waves in DW-ZF
systems on an example of LAPD experiments. We will show
that this phenomenon is universal at high shear rates of ZF
and should also take place in tokamaks. In tokamaks, the S
parameter is peaked on the rational surfaces. This study extends the hydrodynamic stability theory of nonuniform
flows, which saw a great deal of development in the 1990s,
to DWs. The regime, where the dimensionless shear parameter S = A / k储VA ⬇ 1, sees strong shear modification of the DW
turbulence. The non-normality of sheared flow and the inadequacy of the modal/spectral approach are addressed by
some plasma physicists. There is an interesting study by
Cooper,13 for instance, that notes shear flow non-normality
and numerically describes the transient character of linear
time evolution of azimuthal perturbations for a case of
sheared toroidal rotation where the flow is parallel to the
background magnetic field. In contrast, the present study addresses the case where the shear flow is perpendicular to the
background magnetic field. By solving for the Floquet modes
of the fluctuations, Cooper showed an example of bursts of
wave energy that are periodic in time, which resemble
fishbone-type instabilities in tokamaks. However, by examining the review articles,4,5 it seems that the breakthrough of
the 1990s of the hydrodynamic community had limited impact on the plasma and magnetic fusion communities.
To demonstrate the relevance of this novel approach of
the hydrodynamic community to the dynamics of fusion
zonal/shear flows and to join a bibliography of the subject,
some salient results are presented in Sec. II. In Sec. III the
essence of the LAPD experiments and some important
graphs of experimental results are presented. Section IV
gives dynamical equations describing local linear dynamics
of perturbations sheared E ⫻ B flow of LAPD experiments,
introduces the physical approximations and the mathematical
formalism, and describes characteristics of perturbations in
the absence of shear flow. In Sec. V we present the qualitative and quantitative analyses of the linear dynamics of perturbations at small and high shear rates of ZF. We show how
the shear flow generates the Alfvénic fluctuations through
共the shear flow-induced兲 coupling to DWs. We summarize
and discuss the results in Sec. VI. The set of model equations
that describe the linear dynamics of perturbations on LAPD
experiments is derived in Appendix A. The initial conditions
for numerical simulations of the basic equations are derived
in Appendix B.
II. SHEAR FLOW “NON-NORMALITY”
AND ITS CONSEQUENCES
Investigating shear flow dynamics implies the following
steps: introduction of perturbations into a mean flow, linearization of the governing equations, and description of the
dynamics of perturbations and flow using the solutions of the
initial value problem 共i.e., following temporal balances in the
flow兲. This can be done, in principle, but in practice it is a
formidable task. Therefore, a common simplification in
many stability calculations is the assumption of exponential
time dependence, also referred to as the normal-mode 共or
modal兲 approach. This ansatz transforms the linear initial
value problem into a corresponding eigenvalue problem. The
imaginary part of the computed eigenvalues represents an
exponential growth rate. The basic flow is labeled unstable if
an eigenvalue is found in the upper half of the complex
plane. In time, this approach became the canonical methodology and resulted in a focus on the asymptotic stability of
the flow with a lack of attention to initial values or to the
finite time period dynamics. The finite time period dynamics
were considered insignificant and usually left to speculation.
According to classical fluid mechanics,14 the existence
of an inflection point in the equilibrium velocity profile is a
necessary condition for the occurrence of a linear 共spectral兲
instability, corresponding to exponentially growing solutions,
in hydrodynamic flows. Thus smooth shear flows 共flows
without an inflection point兲 are spectrally stable. However, it
is well known from laboratory experiments and from numerical simulations that perturbations may cause a transition
from laminar to turbulent state in spectrally stable flows
共e.g., Couette flow兲. Specifically, difficulties in the modal
analysis of linear processes in shear flows were rigorously
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092102-3
revealed in the 1990s.15 It has been shown that the operators
from the modal analysis of linear processes in shear flows are
exponentially far from normal. 共The norm of the flow nonnormality increases with the shear rate.兲 Consequently, the
eigenfunctions are not orthogonal to each other and strongly
interfere. As a result, even when all the eigenfunctions decrease monotonically in time 共i.e., all eigenfrequencies are in
the lower half of the complex plane兲, a particular solution
may indicate a large relative growth in a limited time interval. That is, a superposition of decaying normal modes may
grow initially, but will eventually decay as time goes by.
Thus, analysis of eigenmodes and eigenfunctions as linearly
independent is misleading. The physics of the shear flow
non-normality induced transient growth of vortical perturbations is presented in Ref. 16 共by means of conservation
laws—the “lift-up” mechanism兲 and Ref. 17 共by means of
dynamical equations on an example of fluid parcel/particle
dynamics兲. The potential for this transient growth has been
recognized for more than a century 共see Refs. 18–20兲. However, the importance of this phenomenon was only understood in the 1990s. It was shown that transient growth can
be significant for spectrally stable flows and that its interplay
with nonlinear processes can result in the transition to
turbulence.
The modal approach dominated the study of disturbances for many decades. Only until the 1990s were its limitations recognized and its shortcomings linked to the abovementioned behavior. As a result, novel ways of describing
fluid stability emerged 共see Refs. 21–24兲, which allow the
quantitative study of short-term disturbance behavior. These
approaches have enjoyed substantial success in furthering a
more complete understanding of instabilities and in providing the missing linear and nonlinear dynamics in a variety of
shear flows. As an alternative to the modal approach, the
system is treated as an initial value problem. Two different
formulations of the nonmodal approach will be described.
The first formulation of the nonmodal approach, the Kelvin
mode approach 共stemming out of the 1887 paper by Lord
Kelvin18兲 has become well established and has been extensively used since the 1990s.25–35 The Kelvin modes represent
the “simplest element” of shear flow physics17 and have also
been referred to as “flowing eigenfunctions”36 in expanding
fluctuations. Strictly speaking, the Kelvin mode approach describes systems with constant shear flow, but, nonetheless, it
also guides the understanding and qualitative description of
smooth shear flow phenomena. In particular, the Kelvin approach correctly describes transient exchange of energy between basic shear flow and perturbations. It exposes two
novel channels of linear coupling of perturbation modes in
shear flows.
共a兲
Phys. Plasmas 16, 092102 共2009兲
On generation of Alfvénic-like waves…
The first energy transfer channel is resonant by nature
and leads to energy exchange between different wave
modes.33,37,38 The mutual transformation of different
kinds of waves is studied numerically33 and
analytically38 in detail for magnetohydrodynamic
waves. The mutual transformation occurs at small
shear rates if the dispersion curves of the wave
branches have pieces nearby one another.
共b兲
The second energy transfer channel is nonresonant
共vortex and wave mode characteristic times are significantly different兲 and nonsymmetric 共a vortex mode is
able to generate a wave mode but not vice versa兲. This
channel leads to energy exchange between vortex and
wave modes, as well as between different wave
modes.32,35 Chagelishvili35 discussed flow nonnormality induced phenomena, including the nonresonant coupling mechanism, in detail and presented several simpler examples. We concentrate on this channel
of mode coupling because it is important at high shear
rates, like the shear rates observed in the LAPD experiment 关see Eq. 共1兲 in Sec. I兴. In the present study, the
Kelvin mode approach will be used.
The second formulation of the nonmodal approach is a
generalized stability theory 共GST兲39,40 that extends modal
stability theory to comprehensively account for all transient
growth processes in nonuniform flows by including the interaction among modes and the mean flow regardless of
whether the modes are unstable or not. GST introduces a
finite time horizon over which an instability is observed.
Finite-time stability analysis markedly deviates from the traditional Lyapunov stability concept,41 and it is not surprising
that the disturbance that grows the most over a short time
scale differs significantly from the least stable mode. Given a
finite time horizon, GST can be employed to quantitatively
describe the perturbation dynamics.21,23,24 GST reveals a remarkably rich picture of linear disturbance behavior that differs greatly from the solutions of the modal approach. GST is
easily extendable to incorporate time-dependent flows, spatially varying configurations, stochastic influences, nonlinear
effects, and flows in complex geometries.
One should also mention that during the 1990s, a new
viewpoint emerged in the hydrodynamics community on understanding the onset of turbulence in spectrally stable shear
flows, labeled as “bypass” transition 共cf. Refs. 27, 30, and
42–52兲. Although the bypass transition scenario involves
nonlinear interactions, which intervene once the perturbations have reached finite amplitude, the dominant mechanism
leading to these large amplitudes appears to be linear. This
bypass transition concept is based on the linear transient
growth of vortical perturbations. The bypass concept implies
that the energy extracted from the basic flow by linear transient mechanisms causes the increase in the total perturbation energy during the transition process. The nonlinear
terms are conservative and only redistribute the energy 共produced by the linear mechanisms兲 in the wavenumber space
repopulating transiently growing perturbations.
III. MAGNETIC FLUCTUATIONS IN THE LAPD
SHEARED FLOW EXPERIMENTS
This work focuses on the LAPD experiments with a
step-down electric field profile with velocity and density
scale length of comparable sizes, in which magnetic fluctuations appear. Measurements of this regime using a vorticity
probe have been reported in Ref. 11 and studied with reduced
two-fluid model simulations. However, so far only measurements of density, temperature, and potential have been ana-
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092102-4
Phys. Plasmas 16, 092102 共2009兲
Horton et al.
FIG. 1. 共Color online兲 Background flow, measured from Mach probe and
E ⫻ B measurements, and vorticity background.
lyzed and modeled. Here, new measurements of magnetic
field fluctuations are reported. Although magnetic fluctuations in the plane perpendicular to the external field are low,
of the order ␦B / B0 ⬃ 10−5, there is a clear increase in the
magnetic fluctuation levels in the shear layer region, which
indicates that they may be driven by the drift wave-shear
flow system.
Diagnostics of this experiment were made using a triple
probe to measure density temperature and floating potential,
a magnetic probe that measures all three components of the
fluctuating magnetic field, the vorticity probe, introduced in
Refs. 10 and 11, and a Mach probe.
Experiments show the steady state background profiles
of density, temperature, and plasma potential during the bias
pulse. The plasma is very well confined to a column with a
diameter of 60 cm and a steep density gradient of scale
length Ln = −n / 共dn / dx兲 ⬃ 5 cm is observed. Electron temperature is fairly flat except that it slightly rises at the edge of
the plasma column. The electric field profile, obtained from
plasma potential shows a radially inward equilibrium electric
field that generates the E ⫻ B sheared rotation. The velocity
shear length is comparable to the density scale length
共LE ⬇ Ln兲. Figure 1 shows the E ⫻ B flow from the triple
probe measurements compared to flow measurements using a
Mach probe.
Figure 2 shows a significant increase in magnetic fluctuations in the shear flow region. Sections IV and V present
the analysis of the generation of the magnetic fluctuations in
this DW-ZF system.
IV. BASIC EQUATIONS AND MATHEMATICAL
FORMALISM
LAPD has a cylindrical geometry. The axial background
magnetic field B0 will be taken to lie along the z-direction.
The sheared E ⫻ B flow region is located some distance
r = r0 from the main axis of the device. From Fig. 1, it follows that the size of the ZF 共LE ⯝ 5 cm兲 is much less than
FIG. 2. 共Color online兲 Fluctuations of the three components of magnetic
field as a function of radius.
r0共⯝25 cm兲. Therefore, the dynamical processes may be
studied in the local Cartesian frame at the particular point
r = 共r0 , 0 , 0兲 with the x-axis directed along the radius and the
y-axis along the azimuth. The set of model equations that
describe the linear dynamics of perturbations on LAPD experiments is derived in Appendix A 关see Eqs. 共A17兲, 共A22兲,
and 共A26兲兴. The equations are linear as the generation of
Alfvénic-like fluctuations in the experiment takes place due
to the linear mode coupling. In a regime with no dissipation,
the set may be rewritten as
˜ VA2 ⳵
⳵2v0y共x兲 ⳵ ␾
d 2˜
共ⵜ2 ˜␺兲,
共ⵜ⬜␾兲 −
=
dt
⳵ x2 ⳵ y
c ⳵z ⬜
冉冊
2 2
˜ vTe
␦e ⳵ 2 ˜
⳵␾
kBTe d ñ
=
共ⵜ ␺兲,
+ vde
⳵y
e dt n0
c ⳵z ⬜
冉
冊
⳵ ˜ kBTe ñ
⳵ ˜␺
d
2 ˜
兲␺兴 = c
关共1 − ␦2e ⵜ⬜
␾−
− vde ,
⳵z
⳵y
dt
e n0
共2兲
共3兲
共4兲
where
⳵
⳵
d
⬅ + v0y共x兲
dt ⳵ t
⳵y
共5兲
˜ , −˜␺, and ñ are perturbations of the scalar electrostatic
and ␾
potential, the axial vector potential, and the density, respectively. The corresponding ZF velocity, magnetic field, and
density are
c
c
˜ 兲,
共ez ⫻ ⵜ⬜␾兲 = v0y共x兲ey + 共ez ⫻ ⵜ⬜␾
B
B
共6兲
B = B0ez + ez ⫻ ⵜ˜␺ ,
共7兲
n = n0共x兲 + ñ.
共8兲
The other quantities used are the electron temperature and charge, Te and e; the electron drift velocity,
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092102-5
Phys. Plasmas 16, 092102 共2009兲
On generation of Alfvénic-like waves…
vde ⬅ ckBTe / eB0Ln = Cs␳s / Ln; the Alfvén velocity, VA
= B20 / 4␲min0; the electron thermal velocity, vTe = 冑kBTe / me;
the ion sound velocity, Cs = 冑kBTe / mi; the density inhomogeneity scale length 共the ZF radial size兲, Ln; the ion inertial
scale length, ␳s = Cs / ␻ci; and the electron inertial length,
␦e ⬅ c / ␻ pe.
According to Fig. 1, the ZF is approximately piecewise
linear, v0y = Ax with A ⯝ 0.8⫻ 105 s−1 and therefore one can
neglect the second term on the left-hand side of Eq. 共2兲.
Hereafter, we use nondimensional variables and physical
quantities. Spatial scales are normalized to the length scale
of perturbations along the magnetic field 共L储 = 1 / k储兲 and time
is normalized to the Alfvén wave time 共␶A = 1 / k储VA兲,
共x,y,z兲k储 = 共X,Y,Z兲,
vde
= Vde,
VA
tk储VA = ␶,
vTe
= VTe,
VA
A
= S,
k储VA
␳s2k2储 = ␳ˆ s2,
␦2e k2储 = ␦ˆ 2e ,
共9兲
␾ˆ
= n̂ .
␺ˆ
Cs ñ
VA n0
Cs e ˜
␺
c k BT e
In this notation, Eqs. 共2兲–共4兲 reduced to
冉
⳵
⳵
⳵ 2ˆ
2 ˆ
+ SX
共ⵜ⬜
␾兲 = 共ⵜ⬜
␺兲,
⳵Y
⳵␶
⳵Z
冉
⳵ ␾ˆ
⳵
⳵
2 ˆ2 ⳵
2 ˆ
= VTe
+ SX
n̂ + Vde
␦e 共ⵜ⬜
␺兲,
⳵Y
⳵Y
⳵␶
⳵Z
冉
⳵
⳵ ␺ˆ
⳵
⳵ ␾ˆ ⳵ n̂
2 ˆ
−
− Vde ,
+ SX
关共1 − ␦ˆ 2e ⵜ⬜
兲␺兴 =
⳵Y
⳵Y
⳵␶
⳵Z ⳵Z
冊
共10兲
冊
共11兲
冊
共12兲
冢
n̂共X,Y,Z, ␶兲
␺ˆ 共X,Y,Z, ␶兲
冣冢 冣
␾ k共 ␶ 兲
= nk共␶兲 eiKx共␶兲X+iKyY+iZ + c.c.,
␺ k共 ␶ 兲
共13兲
共14兲
The wavenumbers of the SFH 共Kelvin modes兲 vary in time
along the flow shear. In the linear approximation, SFH
“drift” in the K-space 共in wavenumber space兲.
Substitution of Eqs. 共13兲 and 共14兲 into Eqs. 共10兲–共12兲
results in
共15兲
共16兲
⳵ ␺k
2
共␶兲兴
− 2S␦ˆ 2e Kx共␶兲Ky␺k
关1 + ␦ˆ 2e K⬜
⳵␶
= i␾k − ink − iKyVde␺k ,
共17兲
where
2
共␶兲 ⬅ K2x 共␶兲 + K2y .
K⬜
共18兲
This system of equations corresponds to spectrally stable
DWs. In fact, in accordance to Ref. 53, low frequency DWs
are subject to the trapped-particle instability. To account for
such instability, Eq. 共16兲 becomes
共19兲
where ⑀ Ⰶ 1. We do not fix a concrete value of ⑀, as the
dynamical timescale is considerably shorter than the characteristic timescale of the trapped-particle instability in all
cases 共see below兲. Although the ⑀ term is formally inserted in
the system, in fact, the trapped-particle instability has no
significant influence on the DW dynamics.
In the simulations below, the quadratic form 共spectral
energy density兲 for a separate SFH as a measure of its intensity is
储
with
Kx共␶兲 = Kx共0兲 − SKy␶ .
⳵ nk
2 ˆ2 2
+ iKyVde␾k = − iVTe
␦ e K ⬜共 ␶ 兲 ␺ k ,
⳵␶
⬜
共␶兲 + Ekin共␶兲 + En共␶兲 + Em共␶兲,
E共␶兲 = Ekin
which permits the use of the nonmodal 共or Kelvin mode兲
analysis. Perturbations cannot have a simple plane wave
form in the shear flow due to the shearing background. Accordingly, the ansatz consists of spatial Fourier harmonics
共SFH兲 of perturbations with time-dependent phases,
␾ˆ 共X,Y,Z, ␶兲
⳵ ␾k
2
− 2SKx共␶兲Ky␾k = iK⬜
共␶兲␺k ,
⳵␶
⳵ nk
2 ˆ2 2
+ iKyVde共1 + i⑀兲␾k = − iVTe
␦ e K ⬜共 ␶ 兲 ␺ k ,
⳵␶
冢 冣冢冣
Cs e ˜
␾
V A k BT e
2
K⬜
共␶兲
共20兲
with widely used definitions 共cf. Ref. 54兲 of the energies that
satisfy conservation of energy in the absence of shear flow
and dissipation. The perpendicular and parallel kinetic, electron thermal, and magnetic energy spectral densities of the
perturbations, respectively, are
⬜
2
共␶兲 = ␳ˆ s2K⬜
共␶兲兩␾k兩2 ,
Ekin
共21兲
4
Ekin共␶兲 = ␳ˆ s2␦ˆ 2e K⬜
共␶兲兩␺k兩2 ,
共22兲
En共␶兲 = 兩nk兩2 ,
共23兲
2
Em共␶兲 = ␳ˆ s2K⬜
共␶兲兩␺k兩2 .
共24兲
储
A detailed study of how increased levels of the biased
electric field change the particle flux and fluctuation spectrum is reported by Carter and Maggs.12 Of particular relevance to the present work is their measurement, showing
the stretching of the two-dimensional cross-correlation function in the direction of the sheared flow with increasing bias
levels. In the present analysis, this stretching physics is contained in the wavenumber vector K共␶兲 time dependence induced by the shear flow parameter S. The effect is relatively
easy to understand: convection of the initial structures
stretches them in the direction of the sheared flow. This oc-
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092102-6
Phys. Plasmas 16, 092102 共2009兲
Horton et al.
TABLE I. Parameters of the numerical simulations of the dynamic equations 共15兲, 共17兲, and 共19兲.
Simulation
Ky
Kx共0兲
Vde
␦ˆ e
2
VTe
⑀
1
2
3
4
103
103
104
103
3 ⫻ 104
3 ⫻ 104
3 ⫻ 105
2 ⫻ 10−3
2 ⫻ 10−3
2 ⫻ 10−3
5 ⫻ 10−4
5 ⫻ 10−4
5 ⫻ 10−4
2
2
2
0.1
0
0.1
1
1
1
5
3
3 ⫻ 104
0.9⫻ 104
2 ⫻ 10−3
2 ⫻ 10−3
5 ⫻ 10−4
5 ⫻ 10−4
10
2
0.1
0.1
1
0.3
10
S
curs for structures of all three fields: vorticity, density, and
magnetic flux. This time-dependent stretching induces a coupling between the three fields.
A. Perturbation spectrum in the shearless limit
The dispersion equation of our system may be obtained
in the shearless limit 共S = 0兲 using the full Fourier expansion
of the variables, including time. Although the roots of the
dispersion equation obtained in the shearless limit do not
adequately describe the mode behavior in the shear case, we
use this limit to understand the basic spectrum of the considered system. Hence, using Fourier expansion of the field vector 关e.g., nk共␶兲 ⬃ exp共−i␻␶兲兴, we derive for the shearless limit
the cubic dispersion relation,
␻ 关1 +
3
␦ˆ 2e 共K2x
+
K2y 兲兴
− ␻ KyVde − ␻关1 +
2
+ 共1 + i⑀兲KyVde = 0.
2 ˆ2
VTe
␦e 共K2x
+
K2y 兲兴
共25兲
This third order dispersion equation describes three different modes of perturbations: two high frequency kinetic
Alfvén waves and a low frequency DW. Due to the nonzero
electron skin depth scale 共␦ˆ e ⫽ 0兲, the Alfvénic-like fluctuations are dispersive with ␻ dependent on Kx. This fact is
very important for mode coupling since Kx is time dependent
in nonuniform flow, which, in turn, makes ␻ also time
dependent.
The dispersion equation 共25兲 is solved numerically for
the parameters shown in Table I 共taking S = 0兲 and the real
and imaginary parts of the dispersive curves, respectively,
are plotted 关␻R = ␻R共Kx兲, ␻I = ␻I共Kx兲兴 in Fig. 3. The plots
show that the magnitude of the frequencies of Alfvénic-like
fluctuations differ substantially from the DW frequency for
all values of Kx. Consequently, the Alfvénic-like and DWs
are linearly coupled solely by the second/nonresonant channel at sizeable shear flow rates described in Sec. II.
共d兲
Figure 3 shows that maximum values of frequency 共␻R
兲
共d兲
and growth rate 共␻I 兲 for the least stable DW mode are 0.9
and 0.09, respectively, and are achieved at 兩Kx / Ky兩 ⱗ 1. Thus,
␻I共d兲 Ⰶ S ⯝ 1 and the trapped-particle instability has no significant influence on the dynamical phenomena.
共d兲
According to Eq. 共15兲 in the shearless limit, at ␻R
⬇ 1,
when the axial vector potential is comparable to the electrostatic potential 共i.e., at 兩Kx / Ky兩 ⱗ 1兲, DWs have a small electromagnetic component that arises from the parallel plasma
current from the finite eneE储 ⬇ −kBTeⵜ储ne. However, as
共d兲
兩Kx / Ky兩, ␻R
→ 0, DWs become electrostatic. As to the other
two modes, they are Alfvénic-like with high frequency
FIG. 3. 共a兲 Real parts of dispersive curves of the three wave modes in Eq.
共25兲. The dotted line is for the DWs. The solid line and the dashed line relate
to the relatively high frequency Alfvénic-like wave modes. 共b兲 Imaginary
part of the DW dispersive curve. The dispersion curves are plotted for the
2
six dimensionless parameters Ky = 103, Vde = 2 ⫻ 10−3, ␦ˆ e = 5 ⫻ 10−4, VTe
= 2,
⑀ = 0.1, and S = 0.
共+兲
共−兲
共␻R
, ␻R
⬎ 1兲 and have dominant magnetic fluctuations for
all Kx. Next we will analyze the coupling of the DW mode
with these Alfvénic-like modes.
V. QUANTITATIVE ANALYSIS OF THE LINEAR
DYNAMICS: TRANSIENT GROWTH
AND MODE COUPLING
Spectral Fourier harmonics dynamics are studied by numerically solving the three complex time evolution equations
共15兲, 共17兲, and 共19兲. Separating the fields into the real and
imaginary parts with
␾k = ␾k⬘ + i␾k⬙ ,
nk = nk⬘ + ink⬙ ,
␺k = ␺k⬘ + i␺k⬙ ,
共26兲
gives the six real coefficient differential equations,
⳵ ␾k⬘
2
− 2SKx共␶兲Ky␾k⬘ = − K⬜
共␶兲␺k⬙ ,
⳵␶
共27兲
⳵ nk⬘
2 ˆ2 2
− KyVde␾k⬙ − ⑀KyVde␾k⬘ = VTe
v␦e K⬜共␶兲␺k⬙ ,
⳵␶
共28兲
2
共␶兲
K⬜
2
共␶兲兴
关1 + ␦ˆ 2e K⬜
⳵ ␺k⬘
− 2S␦ˆ 2e Kx共t兲Ky␺k⬘
⳵␶
= − ␾k⬙ + nk⬙ + KyVde␺k⬙ ,
2
共␶兲
K⬜
⳵ ␾k⬙
2
− 2SKx共␶兲Ky␾k⬙ = K⬜
共␶兲␺k⬘ ,
⳵␶
⳵ nk⬙
2 ˆ2 2
+ KyVde␾k⬘ − ⑀KyVde␾k⬙ = − VTe
␦e K⬜共␶兲␺k⬘ ,
⳵␶
共29兲
共30兲
共31兲
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092102-7
Phys. Plasmas 16, 092102 共2009兲
On generation of Alfvénic-like waves…
FIG. 4. 共Color online兲 The dynamics of relations 兩nk / ␾k兩 and 兩␺k / ␾k兩 for
initial conditions that correspond to the pure DW with Ky Ⰶ Kx共0兲. Here
2
Kx共0兲 = 3 ⫻ 104, Ky = 103, Vde = 2 ⫻ 10−3, ␦ˆ e = 5 ⫻ 10−4, VTe
= 2, ⑀ = 0.1, and
S = 1 共Table I, simulation 1兲. Initial conditions are defined by Eqs.
共B9兲–共B11兲 of Appendix B.
⳵ ␺⬙
2
关1 + ␦ˆ 2e K⬜
共␶兲兴 k − 2S␦ˆ 2e Kx共t兲Ky␺k⬙
⳵␶
= ␾k⬘ − nk⬘ − KyVde␺k⬘ .
共32兲
Equations 共27兲–共32兲, together with the appropriate initial
values, pose the initial value problem describing the dynamics of a perturbation SFH. The character of the dynamics
depends on which mode SFH is initially imposed in the
equations: pure DW SFH 共␻共d兲兲, one of the Alfvénic wave
SFH 共␻共+兲 or ␻共−兲兲, or a mixture of these wave SFHs. Let us
concentrate on the linear dynamics when we initially insert
in Eqs. 共27兲–共32兲 a SFH nearly corresponding to a DW perturbation with wavenumbers satisfying the condition
Kx共0兲 / Ky Ⰷ 1. The numerical simulations55 are performed using the MATHEMATICA numerical ordinary differential equation solver, an implementation of the nonstiff Adams
method, and a stiff Gear backward differentiation method.
Note that the action of the flow shear on the dynamics of
DW SFH at wavenumbers Kx共0兲 / Ky Ⰷ 1 is negligible. This
fact permits writing the initial conditions for the pure DW
SFH 共see Appendix B兲.
The simulations reveal a novel linear effect—the excitation of Alfvénic-like fluctuations—that accompanies the linear evolution of DW mode perturbations in the ZF. 共For a
physical description of this excitation refer to Refs. 32, 34,
and 35.兲 Mathematically, the problem is equivalent to timedependent scattering theory in quantum mechanics. By using
a 3 ⫻ 3 matrix representation of the system, formal solutions
can be written in terms of the time ordering operator and
exponentials of matrices.56
The evolution of the initial DW SFH according to the
dynamic equations 共27兲–共32兲 for the LAPD parameters in
Table I is presented in Figs. 4–8. Recall that Kx共␶兲 changes
in time according to Eq. 共14兲: the shear flow sweeps Kx共␶兲 to
low values and then back to high values but with negative
FIG. 5. 共Color online兲 The evolution of a single SFH 关logarithms of real and
imaginary parts of normalized fields 共⌿1, ⌿2, and ⌿3兲兴 is shown for the
same case as in Fig. 4.
Kx共␶兲 / Ky. As shown in graph 共a兲 of Fig. 4, while Kx共␶兲 / Ky
⬎ 1, the DW SFH undergoes substantial transient growth
without any oscillations and the magnetic fluctuations are
small 共兩␺k / ␾k兩 Ⰶ 1兲. Significant magnetic field fluctuations
共兩␺k / ␾k兩 ⯝ 1兲 appear when Kx共␶兲 / Ky ⬇ 1. While Kx共␶兲 / Ky
⬍ 0, the DW SFH generates the related SFH of Alfvénic-like
wave modes through the second channel of the mode coupling outlined in Sec. II. This generation of Alfvénic-like
wave modes is especially prominent in Fig. 5, where significantly higher frequency oscillations of all the fields are
clearly seen at times when Kx共␶兲 / Ky ⬍ 0. Figure 6 shows the
related dynamics of the different energies. It indicates a substantial transient burst of the electron thermal energy 共elec-
FIG. 6. 共Color兲 The evolution of different kinds of energies of perturbation
SFH 关Ek共␶兲 / E共0兲, En共␶兲 / E共0兲, Em共␶兲 / E共0兲, and E共␶兲 / E共0兲兴 for simulation 1
with initial conditions that correspond to a pure DW with
Ky Ⰶ Kx共0兲. Here, as in Fig. 4, Kx共0兲 = 3 ⫻ 104, Ky = 103, Vde = 2 ⫻ 10−3,
2
␦ˆ e = 5 ⫻ 10−4, VTe
= 2, ⑀ = 0.1, and S = 1 共simulation 1兲. To enlarge “the dynamically active region,” the evolution is presented for times ␶ ⬎ 20, as
initially, the growth of different energies is quite trivial: small and
monotonic.
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092102-8
Horton et al.
Phys. Plasmas 16, 092102 共2009兲
2
FIG. 7. 共Color兲 Same as Fig. 6, but at 共a兲 ⑀ = 0 共simulation 2兲, 共b兲 Kx共0兲 = 3 ⫻ 105 and Ky = 104 共simulation 3兲, 共c兲 VTe
= 10 共simulation 4兲, and 共d兲
S = 0.3 and Kx共0兲 = 0.9⫻ 104 共simulation 5兲. The evolution is presented for times ␶ ⬎ 10.
tron density兲 of fluctuations and an appearance of Alfvéniclike fluctuations.
Comparing Figs. 6 and 7共a兲, one can see that the dynamics only slightly vary with variation in ⑀: the amplification is
just twice as strong at ⑀ = 0.1 than at ⑀ = 0. Consequently, the
transient burst substantially exceeds the growth of the classical dissipative trapped-particle instability of the system.53
Thus, the growth is governed by the flow non-normality due
to the shear flow as described at the end of Sec. IV. The
Alfvénic-like fluctuation generation is intensified with increasing Ky 关cf. Figs. 6 and 7共b兲兴, i.e., the generation increases with scaling down of SFHs in the zonal direction.
2
changes the energy evolution 关cf. Figs. 6
Increasing VTe
and 7共c兲兴: the total energy growth increases due to an increase in the electron thermal energy transient growth. However, the magnetic energy growth somewhat decreases.
The phenomenon strongly depends on the value of
the normalized shear parameter S 关cf. Figs. 6 and 7共d兲兴:
the Alfvénic fluctuation generation becomes appreciable at
S = 0.3, pronounced at S ⬎ 0.5 and dominant at S = 1,
such as in the LAPD experiment 共Vde = 2 ⫻ 10−3 ; ␦ˆ e = 5
2
⯝ 2 ; S ⯝ 1兲. Note that simulation 5 starts with
⫻ 10−4 ; VTe
Kx共0兲 = 0.9⫻ 104 to retain the same dynamical time as in the
previous simulations ␶dyn = Kx共0兲 / KyS.
The curves in Fig. 8 compare the perpendicular kinetic
energy transient growth according to Eq. 共15兲 to that of the
pure vortex mode 关i.e., when the right-hand side of Eq. 共15兲
equals zero兴: the pure vortex 共solid line on the graph兲 undergoes transient growth, while in the considered case, the dynamics of the perpendicular kinetic energy 共dashed line on
the graph兲 are complicated and reduced due to the action of
other fields.
VI. CONCLUSIONS AND DISCUSSION
Drift-Alfvén waves are investigated in plasma with a
significant level of background sheared flow. Magnetically
confined plasmas in both laboratory experiments, space
physics, and coronal loops are examples where sheared flows
occur. As part of the LAPD basic physics experiments, detailed measurements of the fluctuating plasma density, electrostatic potential, and magnetic fields were carried out. We
have been guided by the data from the LAPD 共see Sec. III兲
experiments in developing the theory given in this work.
FIG. 8. The evolution of perpendicular kinetic energy of perturbation SFH
关Ek共␶兲 / E共0兲兴 in two different cases. The dashed line relates to the considered
system for the parameters presented in Fig. 6. The solid line relates to the
pure vortex mode case 关i.e., when the right-hand side of Eq. 共15兲 equals
zero兴. The evolution is presented for times ␶ ⬎ 20, as initially, the growth of
energies in both cases is quite identical.
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092102-9
Phys. Plasmas 16, 092102 共2009兲
On generation of Alfvénic-like waves…
We show that the linear dynamics of DWs, such as those
in the LAPD experiments, are qualitatively changed by the
presence of sheared flows when the shear normalized parameter S 关see Eq. 共1兲兴 approaches unity and, consequently, there
is strong excitation of magnetic fluctuations by the drift
wave–shear flow system. The shear flow induces transient
growth/bursts along with complex temporal wave forms and
generates Alfvénic-like fluctuations from DWs. We show
that the trapped-particle or any classical DW instability is far
slower than these transient bursts and has no notable influence on the dynamic processes for the parameter values of
the LAPD experiments. The frequency of the bursts is determined by the frequency of the generated Alfvénic-like waves
关cf. Figs. 5 and 7共b兲兴. In the case presented by Cooper,13
where the flow is parallel to the background magnetic field,
the frequency of the bursts depends only on the value of
velocity shear parameter.
The complex linear dynamics are a result of the shear
flow continually sweeping the wavenumber of the DW SFH
Kx共␶兲 to low values and then back to high values. In this
time-dependent sweeping of Kx共␶兲, the DW SFH undergoes
substantial transient growth and, when Kx共␶兲 / Ky ⬍ 0, it generates the related SFH of Alfvénic-like wave modes, as explained in Sec. V and illustrated in Figs. 4 and 5. The linear
mode coupling channel at work is nonresonant 共see Refs. 32
and 35 for details兲 and universally leads to energy exchange
between different perturbation modes at high shear rates.
This unambiguously occurs in the LAPD experiments with
S ⯝ 1 and may account entirely for the observed magnetic
fluctuations. Flow non-normality induced mode coupling
should also occur in tokamaks and may be related to the
abrupt changes in magnetic turbulence during L-H transitions reported in Ref. 7. This work focuses on the LAPD
configuration where the flow is externally imposed. Future
work will extend this study to the tokamak/toroidal geometry
and ZF configuration, which is bipolar and generally weaker.
The energy evolution is easily produced by integrating
the linear system of coupled field equations given in Sec. IV.
For sufficiently low values of the shear flow, the coupling
becomes weak and the usual Doppler-shifted, well-separated
modes of the linear system are recovered. We find that for
the level of shear flow in the LAPD laboratory experiments,
the magnetic turbulence level increases significantly. This
increase in the magnetic turbulence level in the regions
where an internal transport barrier is produced is also observed in the large JET tokamak, as reported recently by
Sharapov et al.7
In space physics the effect could be associated with
sheared Earthward flows in the nightside plasma sheet that
are driven by enhanced solar winds with southward embedded solar magnetic field components. Spacecraft in the
plasma sheet measure high speed sheared flows driven by the
convection electric field. Enhanced magnetic fluctuations associated with these flows are also observed. It remains to
make a quantitative analysis of the magnetospheric problem.
Finally, note that nonlinear simulations showing the growth
of vortex structures out of the linear transients are reported in
Ref. 57. Further nonlinear studies are being planned for the
laboratory and space physics settings of this qualitatively
new phenomenon.
ACKNOWLEDGMENTS
This work was supported by ISTC Grant No. G-1217
and by NSF Grant No. ATM-0638480.
The authors would like to thank Dr. R. Bengtson for his
involvement in the LAPD experiment and the reviewer for
constructive feedback. G.D.C. would like to acknowledge
the hospitality of the Institute for Fusion Studies at the
University of Texas at Austin.
APPENDIX A: DERIVATION OF REDUCED PLASMA
EQUATIONS: IN CGS
We derive the set of model equations that describes the
linear dynamics of perturbations on LAPD experiments
where the generation of Alfvénic-like fluctuations in DW-ZF
systems is observed. This set of three model equations are
written for perturbations of the scalar electrostatic potential
˜ , the component of the electromagnetic vector potential
␾
parallel to the mean magnetic field −˜␺, and the density ñ.
˜ , ˜␺ , ñ and
Therefore, we express all physical quantities via ␾
linearize the equations in parallel.
The electrostatic and magnetic potentials are represented
˜ and ˜␺, and the fields as
by ␾ = ␾0共x兲 + ␾
E ⬜ = − ⵜ ⬜␾ ,
共A1兲
E储 = − b̂ · ⵜ␾ +
1 ⳵ ˜␺
,
c ⳵t
B = B0ez + ␦B = B0ez + ez ⫻ ⵜ⬜˜␺ ,
共A2兲
共A3兲
where ␾0共x兲 is the mean electrostatic potential and B0 is the
value of the background magnetic field.
The cross electron and ion drift velocities have the form
ve⬜ = vE + vde,
vi⬜ = vE + v pi ,
共A4兲
where vE and v0y are full and mean sheared E ⫻ B flow
velocities
vE = c
v0y =
E⬜ ⫻ B
B20
=
c
共ez ⫻ ⵜ⬜␾兲,
B0
c
c ⳵ ␾0
ey ,
共ez ⫻ ⵜ⬜␾0兲 =
B0
B0 ⳵ x
共A5兲
共A6兲
vde and v pi are electron diamagnetic drift and ion polarization
drift velocities,
vde = − c
v pi =
b̂ ⫻ ⵜpe
ckBTe 1 ⳵ n0
ey ,
=−
en0B0
eB0 n0 ⳵ x
m ic 2 d
c dE⬜
=−
ⵜ ⬜␾ ,
␻ciB0 dt
eB20 dt
共A7兲
共A8兲
n = n0共x兲 + ñ, pe, and Te are the electron density, pressure, and
temperature, and Ln is the ZF radial size. The basic ZF ve-
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092102-10
Phys. Plasmas 16, 092102 共2009兲
Horton et al.
locity v0y is directed along the y-axis and defines the convective time derivative,
d ⳵
= + v0y · ⵜ.
dt ⳵ t
共A9兲
In Alfvén DW turbulence, there are two important nonlinear directional derivatives given by
1
1
␦B · ⵱f = 共ez ⫻ ⵜ⬜˜␺兲 · ⵜf = 关␺, f兴,
B0
B0
vE · ⵱f =
c ⳵ ␾0
c
˜兲
ey + 共ez ⫻ ⵜ⬜␾
B0 ⳵ x
B0
= v0y⳵y fey +
c
关␺, f兴.
B0
共A10兲
共A11兲
共A12兲
These directional derivatives are Lie derivatives along
the dynamical vector fields. The derivatives are strongly nonlinear when the fluctuation amplitudes reach the levels
␦B / B0 = 兩k储兩 / k⬜ and vE = 兩␻兩 / k⬜. In the nonlinear vortices,
both nonlinearities reach their limiting values, which are
large for the lowest values of k⬜.
The first equation is derived from the charge conservation equation ⳵共qini − qene兲 / ⳵t + ⵜ · 共qinivi − qeneve兲 = 0 for a
quasineutral inviscid incompressible fluid
ⵜ · 关en共vi − ve兲兴 = 0,
ve · ⵜ关n0共x兲 + ñ兴 = v0y
共A14兲
Taking into account the axial component of Ampere’s
2˜
␺ = 4␲J储 / c兲 and the parallel electron current expreslaw 共ⵜ⬜
sion 共J储 = −env储兲, one can express the parallel electron velocity via ˜␺,
− 关n0共x兲 + ñ兴共ⵜ · ve兲 = − n0共x兲
c
ⵜ2 ˜␺ .
4␲en ⬜
共A15兲
冉
冊
⳵
nmic2 ⳵
+ v0y
ⵜ ⬜␾ .
J⬜ = env pi = −
2
⳵
y
B
⳵t
冉
Substituting expressions of J⬜ and J储 from Eqs. 共A15兲
and 共A16兲 into Eq. 共A17兲, one obtains
冊
˜ VA2 ⳵
⳵
⳵2v0y ⳵ ␾
⳵
2˜
共ⵜ2 ˜␺兲,
+ v0y
=
ⵜ⬜
␾−
⳵y
⳵t
⳵ x2 ⳵ y
c ⳵z ⬜
共A17兲
where vA = B0 / 冑4␲min0 is the Alfvén velocity.
The second equation is derived from the electron continuity equation,
⳵ ne
+ ⵜ共neve兲 = 0
⳵t
共A18兲
冊
⳵ ñe
+ ve · ⵜ共n0共x兲 + ñ兲 = − 共n0共x兲 + ñ兲共ⵜ · ve兲.
⳵t
共A19兲
共A22兲
The third equation is derived from the linearized electron
parallel momentum equation in the absence of resistivity,
m en 0
冉
冊
⳵
⳵
+ v0y
v储e = − enE储 − ⵜ储 pe .
⳵y
⳵t
共A23兲
From Eqs. 共A1兲 and 共A10兲 in the linear limit follows
E储 = −
冉
冊
˜ 1 ⳵
˜
⳵ ˜
1 ⳵ ˜␺ 1 ⳵ ˜␺
⳵␾
⳵␾
=−
␺,
−
+
+
+ v0y
⳵y
⳵ z B0 ⳵ y c ⳵ t
⳵z c ⳵t
共A24兲
ⵜ 储 p e = k BT eⵜ 储 n = k BT e
冉
= n 0 k BT e
冉
⳵ ñ 1 ⳵ n0 ⳵ ˜␺
−
⳵ z B0 ⳵ x ⳵ y
冊
冊
⳵ ñ evde ⳵ ˜␺
+
.
⳵ z n0
c ⳵y
共A25兲
Taking into account Eqs. 共A15兲, 共A24兲, and 共A25兲, Eq.
共A23兲 becomes
2
共1 − ␦2e ⵜ⬜
兲
冉
冊
冉
冊
⳵ ˜
⳵ ˜ kBTe ñ
⳵ ˜␺
⳵
+ v0y
␺=c
␾−
− vde ,
⳵y
⳵z
⳵y
⳵t
e n0
共A26兲
␦2e ⬅ c2 / ␻2pe.
APPENDIX B: INITIAL CONDITIONS
Consider a SFH for which Kx共0兲 / Ky Ⰷ 1 and, thus, the
action of the shear S is negligible. In this case, the instability
term 共⑀ ⯝ 0兲 can be neglected. Assuming
冢 冣冢冣
␾ k共 ␶ 兲
␾k
nk共␶兲 = nk exp共− i␻R␶兲,
␺ k共 ␶ 兲
␺k
共B1兲
Eqs. 共15兲–共17兲 become
− ␻R关Kx共0兲,Ky兴␾k = ␺k ,
or
⳵ v储
c ⳵ 2˜
ⵜ ␺ . 共A21兲
=
⳵ z 4␲e ⳵ z ⬜
˜
⳵ ñe
⳵
c ⳵ 2˜
c ⳵␾
=
+ v0y
ⵜ ␺.
+
⳵ y n0 BLn ⳵ y 4␲en0 ⳵ z ⬜
⳵t
where
共A16兲
共A20兲
Introducing the density inhomogeneity scale length
Ln = −n / 关⳵n0共x兲 / ⳵x兴, finally, the second equation reduces to
The perpendicular current is supplied by the ion polarization drift,
冉
⳵ ñ c ⳵ ˜␺ ⳵ n0共x兲
−
,
⳵y B ⳵z ⳵x
共A13兲
ⵜ⬜ · J⬜ = − ⵜ储J储 .
v储 = −
Taking into account Eqs. 共A4兲–共A7兲 and 共A15兲, in the
linear regime, the second terms on the left-hand side and the
right-hand side of Eq. 共A19兲 become
共B2兲
2 ˆ2
␦e 关K2x 共0兲 + K2y 兴␺k ,
− ␻R关Kx共0兲,Ky兴nk + KyVde␾k = − VTe
共B3兲
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092102-11
Phys. Plasmas 16, 092102 共2009兲
On generation of Alfvénic-like waves…
15
− 兵1 + ␦ˆ 2e 关K2x 共0兲 + K2y 兴其␻R关Kx共0兲,Ky兴␺k
= ␾k − nk − KyVde␺k
共B4兲
in terms of ␾k, nk, and ␺k are
nk =
再
冎
KyVde
2 ˆ2
− VTe
␦e 关K2x 共0兲 + K2y 兴 ␾k ,
␻R关Kx共0兲,Ky兴
␺k = − ␻R关Kx共0兲,Ky兴␾k .
共B5兲
共B6兲
Solving the dispersion equation 共25兲 at Kx = Kx共0兲 and
defining the DW frequency 共i.e., the smallest 兩␻R兩兲 as
共d兲
␻R
关Kx共0兲 , Ky兴 ⬅ ␻0, Eqs. 共B5兲 and 共B6兲 take the form
nk =
再
冎
KyVde
2 ˆ2
− VTe
␦e 关K2x 共0兲 + K2y 兴 ␾k ,
␻0
␺ k = − ␻ 0␾ k .
共B7兲
共B8兲
At Kx共0兲 / Ky Ⰷ 1, ␻0 Ⰶ 1 and, consequently, ␺k Ⰶ ␾k.
Finally, taking into account Eq. 共26兲, one can write initial conditions for numerical solutions of Eqs. 共27兲–共32兲 as
1
␾k⬙ = 0,
␾k⬘ = 1,
nk⬙ = 0,
nk⬘ =
␺k⬙ = 0,
␺k⬘ = − ␻0 .
KyVde
2 ˆ2
− VTe
␦e 关K2x 共0兲 + K2y 兴,
␻0
共B9兲
共B10兲
共B11兲
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