PHYSICS OF PLASMAS 15, 082101 共2008兲
Cherenkov radiation of shear Alfvén waves
B. Van Compernolle,a兲 G. J. Morales, and W. Gekelman
Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA
共Received 16 April 2008; accepted 18 June 2008; published online 1 August 2008兲
A calculation is presented of the radiation pattern of shear Alfvén waves excited by a burst of
charged particles propagating along the confinement magnetic field. The characteristic wake patterns
are obtained for the inertial and kinetic regimes of wave propagation. In the inertial regime, the
waves are only excited by particles moving slower than the Alfvén speed. The radiated wake
exhibits an inverted V-shape due to the backward-wave nature of the modes. In the kinetic regime,
particles moving faster as well as slower than the Alfvén speed can radiate propagating modes. The
super Alfvénic particles, however, excite modes with relatively short transverse scales. The
motorboat type of wake of the kinetic modes is more typical of the Cherenkov process obtained in
scalar dielectrics. The predictions are in agreement with experimental observations 关B. Van
Compernolle et al., Phys. Plasmas 13, 09112 共2006兲兴 and computer simulations 关F. S. Tsung et al.,
Phys. Plasmas 14, 042101 共2007兲兴 in which a burst of fast electrons generated by resonant
absorption in a magnetized plasma excites a pulse of large-amplitude Alfvén waves. © 2008
American Institute of Physics. 关DOI: 10.1063/1.2956334兴
I. INTRODUCTION
The excitation of shear Alfvén waves by a burst of
charged particles is a basic process that can arise in magnetized plasmas ranging from the laboratory to the astrophysical environment. For instance, in the auroral ionosphere, fast
electron bursts can be generated by time-varying, localized
potential structures.1 Such electrons are expected to radiate
Alfvén waves with a characteristic signature that can be
sampled by spacecraft at locations far from the acceleration
event. An analytical study2 suggests that an initially localized
electron burst generated during a solar flare propagates over
large distances as a whole structure with nearly monochromatic speed. Clearly, such an object would excite shear
Alfvén waves by the Cherenkov mechanism. Alpha particles
generated by fusion reactions in a magnetic confinement device would also trigger a similar process that enhances the
level of magnetic fluctuations.
While many interesting scenarios leading to the aforementioned process can be identified, the present study is directly motivated by our experimental results3,4 and the associated computer simulation.5 The experiment consists of the
irradiation of a magnetized plasma by a high-power pulse of
high-frequency electromagnetic waves, which results in the
generation of shear Alfvén waves. As has been illustrated in
the computer simulation, nonlinear modifications in the density profile give rise to large, field-aligned electric fields that
generate a burst of fast electrons. The spatial propagation of
the electron burst along the confinement magnetic field has
been clearly mapped in the laboratory. It is also found that
after the pump wave is shut off, a train of shear Alfvén
waves is detected far from the resonant absorption region.
These waves exhibit a particular spatio-temporal pattern that
constitutes the focus of this manuscript.
It should be mentioned that several investigations have
a兲
Electronic mail: bvcomper@gmail.com.
1070-664X/2008/15共8兲/082101/11/$23.00
explored the consequences of shear Alfvén waves driven unstable by the presence of a beam. Noteworthy among them
are investigations related to pulsar emissions6 and studies of
electromagnetic waves generated by auroral electron
precipitation.7 While sometimes this interaction is referred to
as a Cherenkov interaction, such studies typically add a
beam term to the dispersion relation for shear modes and
proceed to deduce the resulting growth rate. This is in contrast to the classic Cherenkov process8 in which a moving
charge plays the role of an antenna that excites waves whose
properties are determined by a background dielectric. It is the
classic Cherenkov process that is considered here.
An earlier study9 that explored the magnetohydrodynamic 共MHD兲 turbulence generated by time-varying, fieldaligned currents considered some elements that form part of
the present article. Although the authors of Ref. 9 described
their work as addressing the excitation of kinetic Alfvén
waves, in reality what was considered was the limit in which
the Alfvén speed, vA, is much larger than the electron thermal speed of the plasma, i.e., vA Ⰷ v̄e. This is what is properly described as the inertial regime. In the present paper,
both regimes of the shear Alfvén wave are examined: The
proper inertial regime and the other extreme, correctly labeled the kinetic regime, in which v̄e Ⰷ vA. Because Ref. 9
focused on the signature sampled by a spacecraft crossing
over a current sheet in which one direction is assumed to be
ignorable, no consideration was given to the threedimensional 共3D兲 spatial structure of the wake pattern radiated by moving charges. In contrast, the present study illustrates the topology of the 3D wakes for both regimes of shear
wave propagation. Also, since the formulation used was
based on the MHD description, the important features associated with cutoffs at the ion-cyclotron frequency were
absent.
The paper is organized as follows. Section II presents a
formulation of the radiation of electromagnetic waves by a
15, 082101-1
© 2008 American Institute of Physics
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082101-2
Phys. Plasmas 15, 082101 共2008兲
Van Compernolle, Morales, and Gekelman
burst of charges moving through a background plasma along
the magnetic field. The properties of the shear Alfvén waves
radiated are examined in Sec. III for conditions corresponding to the inertial regime, and in Sec. IV for the kinetic
regime. Section V discusses the connection of the results to
published laboratory experiments and computer simulations.
Conclusions are given in Sec. VI.
k共k · Ẽ兲 − k2Ẽ = −
where
J̃ext共k, ␻兲 = 2␲
4␲i␻
␻2
J
2 J̃ext − 2 ⑀ · Ẽ,
c
c
冉
冊冉
共4兲
冊
k2 d2
␻ − k储v − i␯
vqNb
exp − ⬜ ⬜ ␦
ẑ,
4
⍀i
⍀i
共5兲
II. FORMULATION
An infinite, uniform magnetized plasma having a single
ion species is considered. A Cartesian coordinate system
共x , y , z兲 is used in which the z direction is along the direction
of the confinement magnetic field, and 共x , y兲 are mutually
perpendicular to it. It is envisioned that some external process gives rise to the generation of a charged particle burst at
z = 0. This results in a field-aligned current that proceeds to
expand ballistically through the otherwise unperturbed background plasma. The spatio-temporal evolution of the current
density of the burst is modeled by
冉
冊 冉
冊
x2 + y 2
⍀i
⍀i
␦
共z
−
t兲
exp
−
ẑ,
Jext = vqNb
v
2
2
vA
␲ v Ad ⬜
d⬜
4␲ ⳵
关Jext共r,t兲 + J p共r,t兲兴,
c2 ⳵t
共1兲
冕
drdtf共r,t兲e−i共k·r−␻t兲 .
Accordingly, Eq. 共2兲 becomes
共6兲
h共x,y兲 = 共kx,y共k2 − k20⑀⬜兲 ⫾ ky,xk20⑀xy兲,
共7兲
hz = 关共k2储 − k20⑀⬜兲共k2 − k20⑀⬜兲 + k40⑀2xy兴/kz ,
共8兲
2
− k20⑀储兲关共k2储 − k20⑀⬜兲共k2 − k20⑀⬜兲 + k40⑀2xy兴
D共k, ␻兲 = − 共k⬜
2 2 2
+ k⬜
k储 共k − k20⑀⬜兲,
B̃x =
4␲i
k y⑀⬜
,
J̃ext,z
2
c
⑀⬜k⬜ + ⑀储k2储 − ⑀储⑀⬜k20
B̃y = −
共3兲
共9兲
in which k0 = ␻ / c.
The dispersion relation for the collective modes supported by the background plasma is obtained from the solutions of D共k , ␻兲 = 0. Since the interest here is in the excitation of shear Alfvén waves, the coupling to the
compressional mode of polarization is not included. This implies that in Eqs. 共7兲–共9兲, the off-diagonal component of the
dielectric tensor, ⑀xy, is neglected. In this limit, the factor
共k2 − k20⑀⬜兲, associated with the compressional polarization,
and appearing in the numerator and the denominator of Eq.
共6兲, exactly cancels out.
Using Faraday’s law, the components of the Fouriertransformed magnetic field excited by the charged particle
burst are found to be
共2兲
where c is the speed of light.
Since a linear problem is being considered, the solution
to Eq. 共2兲 can be extracted by implementing a Fourier transform in which every relevant quantity f共r , t兲 has a transform
given by
f̃共k, ␻兲 =
4␲i␻
kz
h共x,y,z兲 ,
2 J̃ext,z
D共k, ␻兲
c
Ẽ共x,y,z兲 = −
with
where q is the charge, t ⬎ 0 is the time, and v is the velocity
that the charges acquire during the burst event. Note that Nb
is the total number of particles experiencing the impulsive
acceleration, and not their density. For mathematical simplicity, the transverse shape of the burst is represented by a symmetric Gaussian having scale length d⬜. The choice of a
delta function for the burst velocity implies that the effective
Green’s function response is being considered. The more realistic problem can be treated by integrating the basic wake
pattern obtained here over an appropriate velocity distribution function. The scaling factors ⍀i / vA are introduced into
the delta function in anticipation of the relevant axial scale
length associated with Alfvén waves. Here, ⍀i refers to the
ion cyclotron frequency of the background plasma.
As the charge burst propagates through the background
plasma, a self-consistent electric field E共r , t兲 develops. This
field drives currents along and across the confinement magnetic field, which are represented by the current density
J p共r , t兲. The connection between the electric field and the
currents is described by the vector wave equation obtained
from Maxwell’s equations,
ⵜ ⫻ ⵜ ⫻ E共r,t兲 = −
in which a small imaginary part ␯ is included to keep track of
causality in inverting the transform. The quantities k储 and k⬜
correspond to the parallel and perpendicular components of
the wave number, respectively. The linearized response of
the background plasma appears in Eq. 共4兲 through the plasma
⑀.
dielectric tensor J
The algebraic solution of Eq. 共4兲 results in the following
expressions for each of the components 共x , y , z兲 of the Fourier amplitudes of the electric field:
B̃z = 0.
4␲i
k x⑀ ⬜
J̃ext,z
,
2
c
⑀⬜k⬜ + ⑀储k2储 − ⑀储⑀⬜k20
共10兲
共11兲
共12兲
The z component of the magnetic field vanishes because the
coupling to the compressional mode has been dropped. The
different regimes of the shear Alfvén mode, i.e., inertial or
kinetic, are obtained by making suitable approximations for
the parallel component of the dielectric tensor ⑀储, as is considered in the following sections.
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082101-3
Phys. Plasmas 15, 082101 共2008兲
Cherenkov radiation of shear Alfvén waves
III. INERTIAL REGIME
In the limit in which vA Ⰷ v̄e, the parallel component of
the dielectric tensor can be approximated as
⑀储 ⇒ −
␻2pe
,
␻2
共13兲
and for the case of relatively cold ions, as is of interest here,
the transverse component is well-represented by
⑀⬜ ⇒
c2
vA2
1
,
␻2
1− 2
⍀i
共14兲
in which it is assumed that for the shear Alfvén modes, the
following frequency ordering holds: ␻ 艋 ⍀i Ⰶ ␻ pi Ⰶ ⍀e , ␻ pe,
where ␻ pi and ␻ pe are the ion and electron plasma frequencies, respectively, and ⍀e is the electron cyclotron frequency.
Evaluating Eq. 共10兲 for this inertial regime results in
B̃x =
4␲i ⍀2i
c vA2
k2储 −
⍀2i
vA2
冉
ky␭2e
2 2
k储 vA
2 −
␻
2 2
1 − k⬜
␭e
冊
J̃ext,z ,
共15兲
␬=
in which the electron skin-depth is represented by ␭e
= c / ␻ pe. A similar expression is obtained for the y component
in Eq. 共11兲.
It should be noted that the vanishing of the denominator
in Eq. 共15兲 corresponds to the finite-frequency dispersion
relation for a shear Alfvén wave in the inertial regime, i.e.,
1−
␻2
⍀2i
␻2
= vA2
2 2.
k2储
1 + k⬜
␭e
共16兲
It is evident from this expression that propagating inertial
waves exist only in the band ␻ ⬍ ⍀i. In this regime, the
parallel phase velocity becomes smaller than the Alfvén
speed, vA, as the frequency and/or the transverse wave number increases. All of these features are of significance in the
Cherenkov radiation of inertial modes by a charge burst because this effect arises from the matching of the waveparticle resonance ␻ = k储v. It should be mentioned that the
investigation reported in Ref. 9 does not include the finite
frequency dependence of the shear mode, hence the Cherenkov matching condition is not properly represented there.
To deduce the parameter regime where Cherenkov radiation of inertial modes is possible, it is useful to consider the
relationship among the squares of the scaled wave number,
frequency, and burst velocity, i.e.,
␬=
冉 冊
k储vA
⍀i
2
,
w=
冉 冊
␻ 2
,
⍀i
u=
冉 冊
v
vA
FIG. 1. Constraints on Cherenkov radiation in inertial regime. The vertical
and horizontal axes correspond to the quantities ␬ and w of Eq. 共17兲. Radiation arises when the conditions of Eq. 共18兲 are met in the upper righthand quadrant. The solid straight line represents the wave-particle resonance
condition for a value v / vA = 0.5. The broken curves correspond to the dispersion relation of the inertial mode for 共k⬜␭e兲2 ⬍ ␬⬜,M 共long dashes兲 and
共k⬜␭e兲2 ⬎ ␬⬜,M 共dot-dashed兲.
2
,
共17兲
2 2
␭e . Wave propagation occurs
for a fixed value of a = 1 + k⬜
when ␬ ⬎ 0 and w ⬎ 0, simultaneously, for values that satisfy
the pair of constraints
aw
,
1−w
␬=
w
,
u
共18兲
which correspond to Eq. 共16兲 and the wave-particle resonance condition. From Eq. 共18兲 it follows that the possible
solutions require that 1 / u ⬎ a, which implies that only bursts
with velocities smaller than the Alfvén speed lead to Cherenkov radiation of inertial modes. Furthermore, for a fixed
burst velocity, the spectrum of perpendicular wave numbers
that can be excited extends up to a maximum value given by
2 2
␬⬜,M = max共k⬜
␭e 兲 =
vA2
v2
− 1.
共19兲
The constraints on Cherenkov radiation are illustrated
graphically in Fig. 1. The vertical and horizontal axes correspond to the quantities ␬ and w, respectively. The ranges
cover positive and negative values, and radiation arises when
the conditions of Eq. 共18兲 are met in the upper right-hand
quadrant of the figure. The solid straight line represents the
wave-particle resonance condition for a value v / vA = 0.5 and
the broken curves correspond to the dispersion relation of the
inertial mode. The curve with large dashes has a transverse
scaled wave number 共k⬜␭e兲2 ⬍ ␬⬜,M while the dot-dashed
curve has 共k⬜␭e兲2 ⬎ ␬⬜,M . As can be seen, in the upper-right
quadrant the straight line intercepts the curve with 共k⬜␭e兲2
⬍ ␬⬜,M and not the other. This leads to Cherenkov radiation.
The intercept in the lower-left quadrant corresponds to the
generation of a near-field signal that remains attached to the
charge burst.
Figure 2 complements Fig. 1 by showing a possible
range of intercepts leading to Cherenkov radiation for values
of the burst velocity in the interval 0.3⬍ v / vA ⬍ 0.7 for a
fixed 共k⬜␭e兲2 ⬍ ␬⬜,M . Note that in this display, the axes used
are the linear, scaled wave number and frequency, not their
squares, as is done in Fig. 1. It is seen from Fig. 2 that as the
burst velocity decreases, the frequency of the radiated waves
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082101-4
Phys. Plasmas 15, 082101 共2008兲
Van Compernolle, Morales, and Gekelman
cited by the burst is obtained by performing the inverse Fourier transform, i.e.,
Bx,y共x,t兲 =
1
共2␲兲4
冕 冕
d 3k
d␻B̃x,y共k, ␻兲eik·x−i␻t .
共20兲
Because the burst is assumed to be symmetric about the confinement magnetic field, it is useful in performing the integration in Eq. 共20兲 to use cylindrical coordinates in wavenumber space, i.e., using a representation 共k⬜ , k储 , ␣兲 with ␣
the azimuthal angle. The angular integration is readily done
using the identity
冕 冉 冊
2␲
d␣
0
sin ␣
cos ␣
eik⬜␳ cos共␣−␾兲 = 2␲i
冉 冊
sin ␾
cos ␾
J1共k⬜␳兲,
共21兲
where the complementary cylindrical coordinates in configuration space are 共␳ , z , ␾兲.
Representing the excited magnetic field in this natural
coordinate system of the problem yields a single component,
namely the azimuthal component B␾. The radial component
cancels out exactly by symmetry, and the z component is
dropped once the compressional mode is neglected in the
dispersion relation.
The frequency integration in Eq. 共20兲 is readily done
using the factor ␦共␻ − k储v − i␯兲 in Eq. 共5兲. This implies that
the signals generated are driven by the wave-particle resonance, i.e., the burst particles play the role of a moving antenna. The resulting magnetic field is given by
FIG. 2. Range of intercepts leading to Cherenkov radiation for values of the
burst velocity 0.3⬍ v / vA ⬍ 0.7 for a fixed 共k⬜␭e兲2 ⬍ ␬⬜,M . The axes are the
linear, scaled wave number and frequency, not their squares, as in Fig. 1. For
burst velocities less than 0.5vA, the wave frequencies are within 15% of ⍀i.
approaches the ion cyclotron frequency. For burst velocities
less than 0.5vA the wave frequencies are within 15% of ⍀i.
Thus, a signature of Cherenkov radiation by sub-Alfvénic
bursts is the clustering of the shear Alfvén wave spectrum
around ⍀i. Of course, the range of velocities considered must
be larger than the electron thermal velocity in order to be
consistent with the approximation leading to Eq. 共13兲.
The spatio-temporal structure of the magnetic fields ex-
B␾共x,t兲 =
2
vqNb ⍀i
␲c vA2
冕
冉
k⬜␭2e J1共k⬜␳兲exp −
k⬜dk⬜dk储eik储共z−vt兲
2
k储 −
⍀2i
vA2
冉
2 2
d⬜
k⬜
4
冊
k2储 vA2
2 2
− 1 − k⬜
␭e
共k储v + i␯兲2
The next step consists of using contour integration to perform the k储 integral. The essential point is the determination
of the topology of the roots of the denominator term in the
limit in which the quantity ␯ → 0. Due to the presence of ␯,
the roots are shifted away from the real axis. For small k⬜,
the roots are primarily real with a small imaginary part proportional to ␯. These roots correspond to propagating shear
Alfvén waves. As k⬜ increases, the roots move toward the
imaginary axis. For k⬜ ⬎ 共1 / ␭e兲冑vA2 / v2 − 1, the roots become
purely imaginary; they correspond to evanescent waves.
Since the focus of this study is the radiation of Alfvén waves,
only the roots corresponding to the propagating signals are
retained. Therefore, the range of contributions to the integration is limited to k⬜ ⬍ 共1 / ␭e兲冑vA2 / v2 − 1. As mentioned earlier, these roots 共labeled ks兲 have a small imaginary part. It is
found that the imaginary part is negative for both roots. By
writing ks as Re共ks兲 + iIm共ks兲 with 兩Im共ks兲兩 Ⰶ 兩Re共ks兲兩 and sub-
冊
共22兲
.
stituting this into the expression in the denominator in Eq.
共22兲, one obtains, after separating the real part and the imaginary part,
Re共ks兲2 =
冉
冊
⍀2i vA2
2 2
2
2 − 1 − k ⬜␭ e ,
v
vA
⍀2i
␯
Im共ks兲 = −
.
v Re共ks兲2v2
共23兲
Since ␯ and v are positive quantities, both poles in the
integrand of Eq. 共22兲 lie below the real axis. For small k⬜,
the poles are just below the real axis, with opposite real part.
As k⬜ increases, the poles move toward the imaginary axis,
until k⬜ reaches 共1 / ␭e兲冑vA2 / v2 − 1. For larger values of k⬜,
these roots become imaginary.
Evaluating the contribution from these poles yields
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082101-5
Phys. Plasmas 15, 082101 共2008兲
Cherenkov radiation of shear Alfvén waves
B共x,t兲 = 0
z
t⬍ ,
v
ˆ
B共x,t兲 = B␾␾
共24兲
z
t⬎ ,
v
共25兲
with
vqNb ⍀i
c vA2
2
B␾共x,t兲 = 2
冉
⫻exp −
冕
k⬜,M
0
2 2
d⬜
k⬜
4
冊
2 2
dk⬜k⬜
␭e
J1共k⬜␳兲
sin共ks共z − vt兲兲
,
ks
共26兲
where
ks =
⍀i 2 2
冑v /v − 1 − k⬜2 ␭2e ,
vA A
k⬜,M =
1 2 2
冑v /v − 1.
␭e A
共27兲
The field structure described by the integral in Eq. 共26兲
is analogous to that found in the study10 of inertial Alfvén
waves radiated by a stationary disk antenna.11 In such a situation, the relevant integral takes the form
冕
⬁
0
sin共ka兲
J1共kr兲eik储共k兲z ,
dk
k
共28兲
where k储共k兲 corresponds to the parallel wave number of a
shear wave with transverse wave number k. The main differences here are in the Doppler shift of the pattern and the
truncation of the transverse spectrum.
To illustrate the characteristics of the wake pattern, it is
convenient to display the self-consistent, parallel component
of the low-frequency, total current density associated with
the shear wave, namely
Jz = 共c/4␲兲 ⵜ ⫻ B,
共29兲
which yields an integral similar to that of Eq. 共26兲 but with
the factor J1共k⬜␳兲 replaced by J0共k⬜␳兲k⬜c / 4␲. Figure 3 displays the result obtained by performing the relevant numerical integration for d⬜ / ␭e = 3 and v / vA = 0.5. This is a colorcontour display in the scaled 共x , z兲 plane at a fixed time. Note
that the spatial scales are compressed by the square root of
the mass ratio. The distance across the magnetic field is
scaled to c / ␻ pi while along the confinement field it is scaled
to c / ␻ pe. The color scale is normalized to the maximum
absolute value. In this figure, the charge burst is moving
upwards 共positive z direction兲 and at the chosen time it is
found at the position indicated by the arrow. The reader
should imagine this display to be a snapshot of a pattern that
is being dragged upwards, behind the charge burst. A characteristic signature of the pattern is that it takes the shape of
an inverted V, which is the opposite of the classic Cherenkov
pattern obtained in a scalar dielectric in which the source is
located at the tip of the V.
The physics behind the inverted-V pattern is sketched in
Fig. 4. In the inertial regime, the shear Alfvén wave is a
backward mode in the transverse direction, i.e., the direction
FIG. 3. 共Color兲 Example of an inertial wake. Spatial pattern of current
density Jz / 兩Jz兩max for d⬜ / ␭e = 3 and v / vA = 0.5 in the inertial regime. Colorcontour display in the scaled 共x , z兲 plane at a fixed time. Perspective is
compressed by the square root of the mass ratio. The distance across the
magnetic field is scaled to c / ␻ pi while along the confinement field it is
scaled to c / ␻ pe. Color scale is normalized to its maximum absolute value.
The charge burst is moving upwards 共positive z direction兲 at the position
indicated by the arrow.
of the perpendicular group velocity vg,⬜ is opposite to that of
the perpendicular phase velocity v p,⬜. Because the wave energy spreads out, away from the charge burst, the perpendicular group velocity must point away from the x = 0 line.
Due to the backward feature, this implies that the constant
phase fronts must converge toward this line.
It can be seen from the integrand in Eq. 共26兲 that due to
2
, no magnetic-field waves with k⬜ = 0 are radithe factor k⬜
ated. A similar conclusion is also obtained for the current
density because the corresponding integrand contains a fac3
. Therefore, in the far-field region there are no signals
tor k⬜
on axis, at x = 0. However, in the near-field region the peak of
the current density Jz occurs at x = 0. This on-axis peak in Jz
FIG. 4. Physics of the inertial wake. The inertial shear Alfvén wave is a
backward mode in the transverse direction, i.e., the direction of the perpendicular group velocity vg,⬜ is opposite to that of the perpendicular phase
velocity v p,⬜. Because the wave energy spreads out, the perpendicular group
velocity must point away from the x = 0 line. Due to the backward feature,
this implies that the constant phase fronts must converge toward this line.
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082101-6
Van Compernolle, Morales, and Gekelman
FIG. 5. Temporal signature of the radiated magnetic field corresponding to
the current-density pattern of Fig. 3. The signal is sampled at a fixed radial
location ␳ / d⬜ = 0.66. Top panel displays the dependence of the magnetic
field 共in arbitrary units兲 on the scaled time ⍀it. Zero of time corresponds to
the arrival time of the charge burst at the axial position sampled. Bottom
panel is the power spectrum of the temporal signal in the top panel. It shows
a distinct peak slightly below the ion cyclotron frequency.
arises because the source current Jext,z has a finite radial extent, and waves are radiated with nonzero k⬜ from different
radial locations. The interference of these waves causes the
signal to peak on-axis, in the near-field region. For z locations far from the source, the field peaks at radial positions
off-axis. This feature is somewhat apparent in the display of
Fig. 3. Increasing d⬜ would cause the near-field effects to
become more important, and accordingly the peak radiation
would appear on-axis over a greater axial distance from the
location of the burst.
The temporal signature of the radiated magnetic field,
obtained from Eq. 共26兲, is illustrated in Fig. 5 for the same
parameters used in Fig. 3. The fixed radial location where the
signal is sampled corresponds to ␳ / d⬜ = 0.66. The top panel
displays the dependence of the azimuthal component of the
magnetic field 共in arbitrary units兲 on the scaled time ⍀it. The
zero of time corresponds to the arrival time of the charge
burst at the axial position being sampled. The bottom panel
corresponds to the power spectrum associated with the temporal signal in the top panel. It shows a distinct peak slightly
below the ion cyclotron frequency. The frequency at which
this peak appears is determined by the velocity of the burst,
as is illustrated next.
The dashed curve in Fig. 6 displays the dependence on
the scaled burst velocity, v / vA, exhibited by the maximum
current induced on the plasma after the passage of the burst.
This curve corresponds to the scaled plasma current density.
It is obtained by evaluating Eq. 共29兲 as a function of time at
an arbitrary z position, at the center of the burst, i.e., ␳ = 0.
The maximum value is extracted by examining the time series at this fixed location. For the chosen value of d⬜ / ␭e = 3 it
is found that the strongest perturbation to the background
plasma is produced by a burst having a speed v / vA = 0.65.
The reason for the drop in efficiency at the higher velocities
is a consequence of the maximum allowed wave number
given in Eq. 共27兲. As the velocity increases, this maximum
Phys. Plasmas 15, 082101 共2008兲
FIG. 6. Dependence of plasma response on the scaled burst velocity in the
inertial regime for d⬜ / ␭e = 3. The solid curve is the maximum current density induced after passage of the burst. The dashed curve is the value of the
scaled frequency for which the spectral amplitude is maximum, for a given
value of the burst speed.
wave number decreases in value, eventually going to zero as
v → vA. But since the coupling strength, as seen from the
2
, this results in
integrand in Eq. 共26兲, is proportional to k⬜
vanishing efficiency as the velocity of the burst approaches
the Alfvén speed, which is somewhat counterintuitive. The
drop in efficiency for slow bursts is a consequence of the
correspondingly slower waves excited. The waves in this region have relatively large wave number and thus the spectral
content available for their excitation, due to the transverse
shape of the burst, is small. Also, there is a phase-mixing
effect in the integrand of Eq. 共26兲 that leads to a more effective destructive interference as the value of k⬜ increases.
The solid curve in Fig. 6 is obtained by taking the Fourier transform of the time series used to generate the dashed
curve in the same figure. It represents the value of the scaled
frequency for which the spectral amplitude is maximum, for
a given value of the burst speed. The shape of this curve is a
direct consequence of the inclusion of finite ␻ / ⍀i in the
dispersion relation for inertial Alfvén waves. Slow speeds
correspond to frequencies close to the ion cyclotron frequency.
To complete the discussion of the inertial regime, it is
appropriate to consider the plasma response to a burst moving faster than the Alfvén speed, i.e., v ⬎ vA. In this case, the
waves are evanescent and only near-fields are excited. Nevertheless, the situation is interesting because the resulting
structure corresponds to a localized, closed-current system
propagating through the plasma. This is the magnetic analog
of Debye shielding of the electrostatic field associated with a
moving charge. In obtaining the solution in this regime, the
previously neglected contributions from the poles along the
imaginary axis must be retained because for this case all the
excited waves are evanescent. The expression obtained for
B␾ is analogous to that given by Eq. 共26兲, but now the term
sin共ks共z − vt兲兲 / ks is replaced by exp共−兩ks兩共z − vt兲兲 / 2兩ks兩 for t
⬍ z / v, and by exp共兩ks兩共z − vt兲兲 / 2兩ks兩 for t ⬎ z / v. The typical
spatial pattern obtained in this regime is illustrated in Fig. 7
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082101-7
Phys. Plasmas 15, 082101 共2008兲
Cherenkov radiation of shear Alfvén waves
the center of the charge burst. The solid curve is the scaled
current density along the confinement magnetic field and the
dashed curve is the transverse current density 共magnified by
a factor of 10兲. The shielding effect arises from the parallel
return-currents indicated by the shoulders of the solid curve.
IV. KINETIC REGIME
In this regime, the Alfvén speed is assumed to be much
smaller than the electron thermal speed, i.e., v̄e Ⰷ vA, but the
same frequency ordering appropriate for the inertial regime
holds. Accordingly, the parallel component of the dielectric
tensor is approximated as
⑀储 ⇒
FIG. 7. Example of magnetic shielding of a super-Alfvénic burst in the
inertial regime for v / vA = 2 and d⬜ / ␭e = 3. Arrows indicate the direction and
magnitude of the total current density, at a fixed time, at selected spatial
locations in the 共z , x兲 plane. A few “field lines” connecting the current density vectors are drawn. The confinement magnetic field points up 共positive z
axis兲. For the time sampled, the center of the burst coincides with z / ␭e
= 140.
␻2pe
k2储 v̄2e
while the perpendicular component remains as given by Eq.
共14兲.
The methodology used previously in the calculation of
the fields now yields the following expression for Eq. 共15兲:
B̃x共k, ␻兲 =
for v / vA = 2 and d⬜ / ␭e = 3. This figure shows arrows indicating the direction and magnitude of the total current density at
selected spatial locations in the 共z , x兲 plane. The confinement
magnetic field points up 共positive z axis兲 in the figure. For
the arbitrarily chosen time, the center of the burst coincides
with z / ␭e = 140. It is seen that closed current loops surround
the burst. The pattern consists of a radially extended,
coaxial-current structure formed by the return current of
plasma electrons flowing along the field lines. Current closure across the confinement magnetic field arises from the
polarization current due to the plasma ions.
Figure 8 shows a line-cut of the pattern of Fig. 7 along
共30兲
,
4␲i
c
ky␳s2
␻
␻2
2 2
+
k
␳
−
⬜ s
⍀2i
k2储 vA2
2
1−
J̃ext,z ,
共31兲
and a similar expression for the y component. Here, ␳s
= cs / ⍀ci is the ion sound gyroradius with the sound speed
given by cs = 冑m / M v̄e.
As before, the structure of the radiated fields in the region far from the burst is determined by the poles of the
expression in Eq. 共31兲 and its y-component partner. The location of these poles can be expressed in the form
冉
冊
␻2
␻2
2
2 2
+ k⬜
␳s ,
2 = vA 1 −
k储
⍀2i
共32兲
which is the dispersion relation for shear Alfvén waves in the
kinetic regime.
In analogy with Eq. 共18兲, the radiation of kinetic Alfvén
waves requires that the dispersion relation and the waveparticle resonance be satisfied simultaneously. In terms of the
scaled quantities defined in Eq. 共17兲, these conditions now
take the form
␬=
FIG. 8. Line-cut of the pattern of Fig. 7 along the dotted line in Fig. 7. The
solid curve is the scaled current density along the confinement magnetic
field. The dashed curve is the transverse current density 共magnified by a
factor of 10兲. Shielding effect arises from the parallel return-currents indicated by the shoulders of the solid curve.
w
,
1−w+b
␬=
w
,
u
共33兲
2 2
␳s . The intercept of these two curves for positive
where b = k⬜
values of ␬ and w 共i.e., the wave-propagation region兲 requires that 1 / u ⬎ 1 / 共1 + b兲. In terms of physical parameters,
2 2
␳s . The consequence is that
this implies that vA ⬎ v / 冑1 + k⬜
for bursts that are slower than the Alfvén speed, all the transverse wave numbers are radiated. However, for superAlfvénic bursts, only transverse wave numbers that satisfy
k⬜␳s ⬎ 冑v2 / vA2 − 1 are accessible. This behavior is in contrast
with that of the inertial regime, in which only particles
slower than the Alfvén speed emit Cherenkov radiation, and
the transverse wave-number spectrum is bounded by the
value given by Eq. 共19兲.
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082101-8
Phys. Plasmas 15, 082101 共2008兲
Van Compernolle, Morales, and Gekelman
FIG. 9. 共Color兲 Example of a sub-Alfvénic wake in the kinetic regime.
Color-contour display of parallel component of the current density for
d⬜ / ␳s = 3 and v / vA = 0.5. Now v̄e / vA = 12, as is appropriate to satisfy the
approximation leading to Eq. 共30兲. To be compared with Fig. 3.
The magnetic field generated by the burst in the kinetic
regime is obtained from manipulations analogous to those
leading to Eq. 共25兲. The result is
z
t⬍ ,
v
B共x,t兲 = 0
共34兲
z
t⬎ ,
v
ˆ
B共x,t兲 = B␾␾
共35兲
where
vqNb ⍀i
c v2
2
B␾共x,t兲 = − 2
冉
⫻exp −
but now
ks =
⍀i
v
k⬜,m =
冑
冑
v2
vA2
+⬁
k⬜,m
2 2
dk⬜k⬜
␳s J1共k⬜␳兲
冊
2 2
d⬜ sin共ks共z − vt兲兲
k⬜
,
4
ks
2 2
1 + k⬜
␳s −
1
␳s
冕
v2
vA2
共36兲
,
− 1⌰共v − vA兲,
共37兲
where ⌰ represents the Heaviside function.
The spatial pattern associated with the radiation of kinetic Alfvén waves by a sub-Alfvénic burst is shown in Fig.
9. It is to be contrasted with Fig. 3, which shows the analogous pattern obtained in the inertial regime. Again, in Fig. 9
the color-contour display corresponds to the self-consistent
parallel component of the current density induced in the
plasma by the moving burst for d⬜ / ␳s = 3 and v / vA = 0.5.
However, now v̄e / vA = 12, as is appropriate to satisfy the approximation leading to Eq. 共30兲. It is seen that in the kinetic
regime, the wake radiated by a sub-Alfvénic burst exhibits
the characteristic motorboat pattern associated with Cherenkov radiation in scalar dielectrics.
FIG. 10. 共Color兲 Pattern transition as the burst velocity becomes superAlfvénic for the kinetic regime for d⬜ / ␳s = 3, v̄e / vA = 12. Scaled velocities
increase from top to bottom panel over the range v / vA = 0.8, 1.2, 1.5, 2.0.
The classic Cherenkov pattern shown in Fig. 9 is continuously distorted as the velocity of the burst becomes
super-Alfvénic, as is shown in Fig. 10 for v / vA
= 0.8, 1.2, 1.5, 2.0 and for the same parameters used in Fig. 9.
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082101-9
Phys. Plasmas 15, 082101 共2008兲
Cherenkov radiation of shear Alfvén waves
FIG. 11. Dependence of plasma response on the scaled burst velocity in the
kinetic regime. Analogous to Fig. 6. The dashed curve is the maximum
current density induced after passage of the burst. The solid curve is the
value of the scaled frequency for which the spectral amplitude is maximum,
for a given value of the burst speed. v̄e / vA = 12 and d⬜ / ␳s = 3 are held fixed
while v / vA is varied.
It is seen that as the velocity of the burst increases, the pattern evolves into a periodic array of field-aligned current
filaments.
The dependence of the plasma response on scaled burst
velocity, analogous to that shown in Fig. 6 for the inertial
regime, is extended to the kinetic regime in Fig. 11. However, in the kinetic survey v̄e / vA = 12, and d⬜ / ␳s are both
kept fixed while v / vA is varied. It is found that there is no
optimum velocity and the frequency decreases as the velocity increases. The increase in the scaled amplitude response
共dashed curve兲 as the velocity becomes smaller can be traced
mathematically to the radiated wake evolving into a spatial
delta function in the limit v → 0.
V. DISCUSSION
The theoretical model described in the previous sections
is found to be in general agreement with our experimental
observations3 and computer simulations.5 For technical reasons, however, at present the experiments and simulations
have explored only the inertial regime of wave propagation.
Thus, the interesting new features related to super-Alfvénic
bursts in the kinetic regime await future studies.
For completeness and further perspective on the analysis
reported in this paper, an abbreviated account of the experimental and computational results is included.
The experiment was performed in the upgraded large
plasma device 共LAPD-U兲12 operated by the Basic Plasma
Science Facility at the University of California, Los Angeles.
A pulse of high-power microwaves in the O-mode polarization was delivered across the confinement magnetic field to
an afterglow helium plasma with a cross-field density gradient. The parameters were such that the Alfvén speed was
larger than the electron thermal velocity. At a location in the
plasma where the frequency of the microwaves matched the
FIG. 12. Typical temporal signature of By detected in the experiment, at two
different axial locations, z⍀i / vA = 2.2 and z⍀i / vA = 6.5 共see Fig. 18 in Ref.
3兲.
local electron plasma frequency, nonlinear processes generated a burst of field-aligned, fast electrons. An analogous
situation was considered in the simulation study.
The burst of fast electrons was followed by a wake of
magnetic oscillations whose properties were conclusively
demonstrated to correspond to shear Alfvén waves. A typical
temporal signature of the oscillations detected in the experiment is shown in Fig. 12. The top panel displays two temporal traces of the perturbed transverse component of the
magnetic field measured at two different axial locations from
the region where the burst was generated. The bottom trace
was closer to the source region than the top one. The bottom
panel shows the frequency spectrum associated with the temporal signals. It is seen that for the location closer to the
source, the spectrum peaks near the ion cyclotron frequency,
while farther away the spectrum shifts to lower frequencies.
This is a consequence of the dispersive property of shear
Alfvén waves in the inertial regime, as is illustrated in Fig. 1.
The simulation study uses an electromagnetic particlein-cell 共PIC兲 code to model the experimental scenario. Figure
13 presents a display of simulation results akin to those seen
in Fig. 12, but only sampled at one axial location from the
burst source. The left panel corresponds to the temporal behavior of the transverse magnetic field while the right panel
(a)
(b)
FIG. 13. Left panel: Temporal signature of By at ␳ / ␭e = 40, z⍀i / VA = 20.
Right panel: Wigner frequency-distribution of By, 兩W共t , ␻兲兩, showing the
increase in frequency as time progresses. Courtesy of F. S. Tsung 共Ref. 5兲.
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082101-10
Phys. Plasmas 15, 082101 共2008兲
Van Compernolle, Morales, and Gekelman
VI. CONCLUSIONS
FIG. 14. Upper panel: Beam distribution function, as a function of v / vA.
Lower panel: Calculated Jz traces at ␳ = 0, and for different axial positions.
Further away from z = 0, the initial peak becomes broader, and the dominant
Alfvén wave frequency decreases.
is the Wigner transform of the signal. As in the experiment, it
is found that the wake signal consist of low frequencies early
and frequencies close to the ion cyclotron frequency later.
Clearly, the experiment and the simulation exhibit the
characteristic temporal signature of the theoretical model
shown in Fig. 5, namely a signal resembling a damped oscillator with a frequency peak close to the cyclotron frequency. However, on closer scrutiny it is seen from the data
of Fig. 12 that in the experiment, the spectral peak broadens
and its center progressively shifts to lower frequencies as one
moves away from the source of the burst. It is also seen that
the amplitude of the first temporal peak decreases with distance and its width increases with distance. This is a behavior expected from the ballistic spreading of the initial charge
burst resulting from particles with different velocities 共i.e.,
the source strength decreases and becomes spatially
smeared兲. This implies that the burst generated by the nonlinear processes is not monoenergetic, as is assumed in Eq.
共1兲.
Because the analytic results discussed in the previous
sections are equivalent to the Green’s function solutions for
Cherenkov radiation of shear Alfvén waves, the effects produced by a burst of particles having an energy spectrum can
be found by integrating the results obtained over the appropriate velocity distribution function. An illustration of such a
calculation is given in Fig. 14. The top panel shows a model
velocity distribution function introduced into Eq. 共26兲 and
subsequently leading to the axial current according to Eq.
共29兲. The bottom panel displays the currents obtained at three
different axial locations, z = 2vA / ⍀i , 6.5vA / ⍀i , 11vA / ⍀i,
from the source of the burst. It is now seen that the experimentally observed features are more closely reproduced. The
amplitude of the peak signal decreases with increasing distance, and the width of the peak broadens. Also, the frequency of the oscillations decreases with distance.
This study has examined the excitation of shear Alfvén
waves by a burst of charged particles propagating along the
confinement magnetic field. The characteristic 3D wave patterns have been obtained for both the inertial and the kinetic
regimes of wave propagation.
In the inertial regime, the radiation pattern takes the
shape of an inverted V due to the backward-wave nature of
the mode, as illustrated in Fig. 3. Propagating inertial waves
are only excited by charged particles moving at speeds below
the Alfvén speed. Slow particles with v / vA ⬍ 0.5 excite shear
waves that exhibit peaked power spectra, with the dominant
frequency close to the ion cyclotron frequency, i.e., 0.85
⬍ f / f ci ⬍ 1.0. Particles with speeds above the Alfvén speed
do not excite propagating inertial modes. This results in localized, closed-current systems propagating through the
plasma with the speed of the burst. The behavior corresponds
to the magnetic analog of Debye shielding of an electrostatic
field associated with a moving charge, as shown in Fig. 7.
The radiation patterns in the kinetic regime are characterized by a motorboat-type wake, similar to the classic
Cherenkov pattern obtained in a scalar dielectric in which the
source is located at the tip of the wake. In the kinetic regime,
propagating waves can be excited by particles moving both
slower and faster than the Alfvén speed. The Cherenkov
wake pattern has been shown to continuously develop collimated filaments as the velocity of the charged particles becomes super-Alfvénic. At high speeds, the pattern evolves
into a periodic array of field-aligned current filaments with
short transverse scale lengths, as shown in Fig. 10.
The theoretical results for the inertial regime are in
agreement with experimental observations3,4 and corresponding computer simulations.5 In these earlier investigations, a
burst of fast electrons, generated by resonant absorption at
the electron plasma frequency, was observed in conjunction
with a wake of shear Alfvén waves. Both in the experiment
and in the simulation, the shear waves exhibit temporal signatures and peaked power spectra consistent with the calculations reported here, and summarized by Fig. 5. It has been
found that improved agreement with experimental observations is obtained when a velocity spread is included in the
model of the burst, as shown by Fig. 14.
The applicability of the results obtained is not limited to
the experiment that motivated the investigation. The excitation of shear Alfvén waves by means of a burst of charged
particles is a basic process that can occur in magnetized plasmas ranging from the laboratory to the astrophysical environment. Electron bursts generated by time-varying, localized potential structures in the auroral ionosphere or fast
electrons generated during a solar flare are expected to radiate Alfvén waves. The theoretical results for the 3D wave
pattern obtained in the present paper can aid in the interpretation of spacecraft data. Also, alpha particles generated by
fusion reactions in a magnetic confinement device would
trigger enhancements of the magnetic fluctuations, in a similar fashion.
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082101-11
ACKNOWLEDGMENTS
The authors thank Dr. J. E. Maggs and Dr. F. S. Tsung
for valuable technical discussions. Dr. Tsung provided the
materials in Fig. 13, the details of which are reported in Ref.
5. This work was funded by the Department of Energy under
Grant No. DE-FG03-98ER54494, and more recently by the
National Science Foundation. The work was performed at
the Basic Plasma Science User Facility at UCLA, which is
funded by NSF/DOE.
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Phys. Plasmas 15, 082101 共2008兲
Cherenkov radiation of shear Alfvén waves
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