PHYSICS OF PLASMAS 19, 092109 (2012)
n waves in plasmas with two ion species
Cherenkov radiation of shear Alfve
W. A. Farmer and G. J. Morales
Physics and Astronomy Department, University of California Los Angeles, Los Angeles,
California 90095, USA
(Received 26 June 2012; accepted 20 August 2012; published online 14 September 2012)
A calculation is presented of the radiation pattern of shear Alfven waves generated by a burst of
charged particles in a charge-neutral plasma with two-ions of differing charge-to-mass ratios. The
wake pattern is obtained for the inertial and kinetic regimes of wave propagation. Due to the
presence of two ion-species, the Alfven waves propagate within two different frequency bands
separated by a gap. One band is restricted to frequencies below the cyclotron frequency of the
heavier species and the other to frequencies between the ion-ion hybrid frequency and the cyclotron
frequency of the lighter species. The radiation pattern in the lower frequency band is found to exhibit
essentially the same properties reported in a previous study [Van Compernolle et al., Phys. Plasmas
15, 082101 (2008)] of a single species plasma. However, the upper frequency band differs from the
lower one in that it always allows for the Cherenkov radiation condition to be met. The methodology
is extended to examine the Alfvenic wake of point-charges in the inertial and adiabatic regimes.
The adiabatic regime is illustrated for conditions applicable to fusion-born alpha particles in
C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4751462]
ITER. V
I. INTRODUCTION
The excitation of shear Alfven waves by energetic particles is a phenomenon that can occur naturally in space plasmas as well as in magnetically confined fusion devices. It
has been shown that fast electron bursts occurring in the
auroral ionosphere can be generated by interactions with
time-varying, localized potential structures.1 Further, fast
particle bursts have been measured in the magnetosphere
even during magnetically quiet periods.2 In a tokamak environment, the fusion process produces energetic alpha particles. In each of these cases, the fast particles can directly
radiate energy in the form of Alfven waves through the Cherenkov process. This mechanism enhances the level of the
ambient magnetic fluctuations and contributes to the slowing
down of the fast particles. In basic laboratory experiments,
energetic particle bursts can be generated by the ablation of
solid targets with powerful laser pulses3 and the ensuing
emission of waves can be investigated. The controlled injection of ion beams into tokamaks and basic magnetized devices4 can also result in Cherenkov radiation of shear Alfven
waves.
A previous theoretical investigation5 evaluated the pattern of shear Alfven waves excited by the Cherenkov process
in a single species plasma. The results were related to a basic
laboratory experiment6 in which an electron burst arises
from the nonlinear absorption7 of a powerful pulse of highfrequency electromagnetic waves. The present study extends
the previous calculations of the radiation pattern to plasmas
with two-ions of differing charge-to-mass ratios for a
charged particle burst. Further, it also computes the wake
pattern and expected temporal signal for a point-particle in
an attempt to understand the role that fast alpha particles
would play in the excitation of shear Alfven waves in ITER.
For a single species plasma, it is found5 that, in the inertial regime of wave propagation, the wake pattern of the
1070-664X/2012/19(9)/092109/11/$30.00
resulting Cherenkov radiation exhibits an inverted V-shape.
This shape arises from the backwards-propagating nature of
shear Alfven waves in this regime; the direction of the perpendicular group velocity is opposite to that of the perpendicular phase velocity. Another predicted5 feature, observed
in the laboratory6 and in particle-in-cell (PIC) simulations,7
is that the frequency of the radiated waves clusters around
the ion cyclotron frequency. Additionally, in this regime, no
radiation occurs for particles moving faster than the Alfven
speed. In the kinetic regime, the wake exhibits the familiar
Cherenkov cone pattern for particles moving slower than the
Alfven speed. But for particles moving faster than the Alfven
speed, the pattern develops field-aligned stripes associated
with multiple current filaments induced on the background
plasma.
The presence of a second ion species leads to an additional characteristic frequency, widely known8,9 as the ion-ion
hybrid frequency, xii . The important feature is that at this frequency the perpendicular component of the cold-plasma
dielectric, ? , vanishes. In the cold plasma limit, and at frequencies on the order of the cyclotron frequency of the ions
2
2
1 xx2
c
ii
:
? )
2
2
vA
1 x2 1 x2
X1
(1)
X2
Here, vA is the Alfven speed, c is the speed of light, and X1
and X2 are the cyclotron frequencies of the two ions. Without loss of generality, the frequency ordering used throughout this paper is X2 > X1 or m2 < m1 . The ion-ion hybrid
frequency and Alfven speed are related to the ion-plasma
and ion-cyclotron frequencies through the relations
19, 092109-1
x2ii ¼
x2p1 X22 þ x2p2 X21
x2p1 þ x2p2
;
(2)
C 2012 American Institute of Physics
V
092109-2
W. A. Farmer and G. J. Morales
2
x2p1 x2p2
c
¼ 2 þ 2;
vA
X1
X2
Phys. Plasmas 19, 092109 (2012)
(3)
with xp1 and xp2 being the respective ion plasma frequencies. For perpendicular scales that are small compared to the
ion skin-depth, the dispersion relation for the shear Alfven
wave takes the approximate form
x2
k 2 c2
:
(4)
kjj2 ¼ 2 ? 1 ?2
c
x jj
In addition to the usual wave solutions for x < X2 , the presence of a second ion species, of mass m1 , causes another
band of propagating waves to exist for frequencies that satisfy xii < x < X1 . If the magnetic field strength is allowed
to vary along a field line, reflections can occur in the upper
frequency band at locations where the frequency matches the
local value of xii .10 The neglect of longer-wavelength features eliminates from consideration other wave modes that
can lead to important consequences, as illustrated by Ganguli
and Rudakov11,12 in connection to the generation of strong
turbulence at MHD scales. In contrast to these studies, the
process considered here involves features with short perpendicular scale and small amplitudes.
The consequences of the additional upper-frequency
band have been explored in numerous studies within the space
physics community. Narrow-banded extremely low frequency
(ELF) waves generated by the auroral electron beam were
shown to excite frequencies above the local value of xii determined by the local magnetic field and the densities of Hþ and
Heþ ions.13 Further, due to the topology of the earth’s magnetic field, and that of general planetary magnetospheres, the
possibility of natural shear Alfven wave resonators exists.14
Various mechanisms for exciting such a resonator in the
earth’s magnetosphere have been explored,15 and a recent laboratory experiment16 in the large plasma device (LAPD) at
the University of California, Los Angeles (UCLA)17 has verified the resonator concept. Cherenkov emission by fast particles, of the type considered here, is an additional source that
can excite the Alfven ion-ion hybrid resonator in planetary
magnetospheres, as well as in laboratory and fusion plasmas.
Since the generation of fusion power requires a mix of
deuterium and tritium ions, in the operation of burning plasmas in magnetic confinement devices such as ITER,
multiple-ion effects will inherently be present. Evidence for
the reflection of Alfven waves at the ion-ion hybrid location
has been obtained in a tokamak experiment.18 Additionally,
it has been shown that the upper frequency band can be
unstable in the presence of superthermal alpha particles.19
Thus, it is of interest to explore the properties of Cherenkov
radiation in such environments.
The manuscript is organized as follows. Section II summarizes the previous model5 of Cherenkov emission by a
field-aligned, charged particle burst. It is reproduced here for
convenience, as the details are especially important to the development of the remainder of the paper. Section III illustrates the properties of the radiated waves in the inertial
regime, and Sec. IV examines the corresponding results in
the adiabatic regime. Explicit displays of the patterns are
made for experimental conditions that can be realized in
LAPD. Section V calculates the wake due to a point particle
in the inertial regime for parameters relevant to the LAPD
and in the adiabatic regime for parameters relevant to an
alpha particle in ITER. Conclusions are given in Sec. VI.
II. FORMULATION
In this investigation, the plasma is considered to be
spatially uniform and infinite in extent. The confinement
magnetic field is aligned with the z-direction of a Cartesian
coordinate system (x,y,z). A charged particle burst is
assumed to propagate along the magnetic field. The specific cause of the burst is irrelevant here, as the primary
focus of this manuscript is to examine the collective
response of the plasma after the burst is created. The particle burst is assumed to travel at a constant speed, i.e., it is
treated as a test particle. In reality, there is a radiative reaction that, in principle, can be treated in a perturbative
scheme. Additionally, the burst is assumed to travel with
sufficient speed such that collisions with the background
plasma are negligible. Under these assumptions, the burst
of charged particles is modeled as a source with current
density given by
2
1
x þ y2
dðz vtÞ^z :
(5)
jS ¼ vqNb 2 exp 2
d?
pd?
Here, v is the speed of the burst, q is the charge of an individual particle, and Nb is the number of particles associated
with the burst. A generic Gaussian spatial profile is used in
the transverse direction to simplify the calculations. A delta
function is chosen in the z direction, so that the particles are
all assumed to be traveling at the same speed. In reality,
there will exist some velocity spread. For situations in which
the consideration of velocity spread is important, the results
of the present investigation can be integrated over the relevant velocity distribution function to obtain the desired
quantity.
To solve Maxwell’s equations for the source given by
Eq. (5), a Fourier representation is implemented for all quantities. For a generic quantity, f(r, t), the relation to its transform is
ð
(6)
f~ðk; xÞ ¼ f ðr; tÞeiðkrxtÞ drdt;
ð
dkdx
:
f ðr; tÞ ¼ f~ðk; xÞeiðkrxtÞ
ð2pÞ4
(7)
In Fourier space, the plasma current density is obtained from
the dielectric tensor, and upon solving for the electric field in
Maxwell’s equations, the following equation results
4pix
ðki kj k2 dij þ k02 ij ÞE~j ¼ 2 ð~j S Þi :
c
(8)
Here, k0 ¼ x=c and ij is the ij component of the plasma
dielectric tensor. The Fourier transform of the current density
associated with the burst is
092109-3
W. A. Farmer and G. J. Morales
Phys. Plasmas 19, 092109 (2012)
k2 d2
~j ¼ 2pvqNb e ?4 ? dðx kjj v iÞ^z :
S
(9)
A small imaginary part has been added to the frequency as a
mathematical aid, to ensure that the fields vanish as
t ! 1. In performing contour integrals later, the limit
! 0 is taken. This procedure can be considered as a condition used to enforce causality. The quantity kjj is the component of the wave vector parallel to the background magnetic
field, i.e., along the z-direction. Similarly, k? is the magnitude of the waveqvector
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiin
ffi the direction perpendicular to z,
2
2
and is equal to kx þ ky . The solution to the problem, as
formulated by Eqs. (8) and (9), requires the inversion of the
~ and then performing the relevant inteoperator acting on E
grals to invert the Fourier transforms. Two integrations can
be performed analytically, while the final integral is performed numerically.
At this point, it is necessary to specify the explicit form
of the dielectric tensor. As in the previous study,5 the interest
here is primarily in radiation of shear Alfven waves. The
inclusion of the off-diagonal component of the dielectric tensor, xy , results in coupling between shear and compressional
polarizations, and for this reason, it is neglected. With this
choice, upon solving for the Fourier amplitudes of the electric field, the following equations result
~ ¼ 4pix j~S;z kjj h;
E
Dðk; xÞ
c2
0
1
kx
A;
ky
h¼@
ðkjj2 k02 ? Þ=kjj
2
Dðk; xÞ ¼ k02 ðk02 ? jj k?
? kjj2 jj Þ:
(10)
(11)
e
0
sin a
0
da ¼ 2pi
J ðk rÞ:
cos a
1 1 ?
ð
1
? J1 ðk? rÞeiðkjj zxtÞ ~
2
k?
dk? dkjj dx 2
j :
2 k2 S;z
2
2p c
k0 ? jj k?
?
jj jj
(15)
Since in the sections that follow, contour plots of the
z-component of the plasma current density are presented to
illustrate the wake pattern, an expression for this quantity is
obtained from Ampère’s law. Away from the source current,
and neglecting the displacement current (as is appropriate
for the frequencies of interest here), the current density of
the plasma satisfies
jp ¼
c
r B:
4p
(14)
(16)
Substituting the result of Eq. (15) yields
jz ¼ ð
1
? J0 ðk? rÞeiðkjj zxtÞ ~
3
dk
dk
dx
j : (17)
k
?
jj
2 k 2 S;z
8p3 ?
k02 ? jj k?
?
jj jj
III. INERTIAL REGIME
In the inertial regime, the burst speed is assumed to be
much larger than the electron thermal speed, v v e . In this
limit, the parallel component of the dielectric tensor can be
approximated as
(12)
Because of the azimuthal symmetry of the source given by
Eq. (5), it is useful to express the magnetic field in terms of
cylindrical coordinates ðr; /; zÞ. In the cylindrical representation, B~x ¼ B~r and B~y ¼ B~/ when y ¼ 0, and since the radial
and azimuthal components of the wave magnetic field are independent of the azimuthal angle, the variable x can be
replaced by r in the resulting expressions. Representing the
wave vector in cylindrical coordinates yields, kx ¼ k? cos a
and ky ¼ k? sina, where a is the azimuthal angle of the wave
vector. In inverting the Fourier transform, the integration
over a can be performed independently of either the form of
the dielectric tensor, or the specific type of source. This property leads to the following integral over a which is readily
evaluated
ik? r cos a
B/ ¼ jj ) From Faraday’s law, the Fourier amplitude of the magnetic
field is
0
1
ky
4pi
?
~ ¼
(13)
j~ @ kx A:
B
2 k2 S;z
c k02 ? jj k?
?
jj
jj
0
ð 2p
Upon performing this integral, the only non-vanishing component of the magnetic field is B/ . This results in the expression
x2pe
;
x2
(18)
and for the case of relatively cold ions, the transverse component is given in Eq. (1). The plasma density and magnetic
field are assumed to be sufficiently large to satisfy the frequency ordering: x < X2 xpi xpe ; Xe .
It is useful to introduce the dimensionless quantities:
w¼x=X1 , a¼k? de , j¼kjj vA =X1 , l¼X2 =X1 , r¼xii =X1 ,
t0 ¼ X1 t, z0 ¼ X1 z=vA , r 0 ¼ r=de , and u ¼ v=vA , where
de ¼ c=xpe . Substituting these definitions into Eq. (15)
obtains
ð
X21
a2 dadjdw
2p2 cde vA
2
0
0
1 wr2 J1 ðar0 Þeiðjz wt Þ
2
j~S;z ; (19)
j
2 Þ 1 w2 ð1 þ a2 Þ 1 w2
ð1
w
2
2
2
w
l
r
B/ ¼ a2 d2
?
vA qNb 4d2
ue e dðw ju i 0 Þ:
j~S;z ¼ 2p
X1
(20)
Before evaluating the integrals, it is useful to first understand
the poles associated with the dispersion relation of the
waves. Due to the delta function associated with the source
current, the denominator in the integrand of Eq. (19) vanishes for values of w and j that satisfy the simultaneous
relations
092109-4
W. A. Farmer and G. J. Morales
w
¼ u;
j
j2
w2
w2
2
2
ð1 w Þ 1 2 ð1 þ a Þ 1 2 ¼ 0:
w2
l
r
Phys. Plasmas 19, 092109 (2012)
(21)
(22)
Solving these relations for j yields two solutions. One solution corresponds to a propagating wave with frequency
smaller than X1 , while the other is in the frequency range
xii < x < X2 . The solutions are
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
1
4b
2
bs 6 jbs j 1 þ 20 ;
(23)
j6 ¼
2
bs
with
bs ¼
1 2
l2
ðl
þ
1Þ
ð1 þ a2 Þ;
r2
u2
(24)
l2
l2
2
ð1
þ
a
Þ
:
u2
u4
(25)
b0 ¼
Since 1 < r < l, the discriminant is always positive. Further, it can be shown that j2þ is always positive and that j2
is positive provided that
rffiffiffiffiffiffiffiffiffiffiffiffiffi
1
a < aM ¼ Re
1:
(26)
u2
To develop intuition for the meaning of the two solutions,
Eqs. (21) and (22) are graphically displayed in Fig. 1 for a
helium-hydrogen mixture and parameters typical of experiments that can be performed in the LAPD.10,16 The vertical
and horizontal axes correspond to j and w, respectively. The
solid straight line represents Eq. (21) for u ¼ 0.5; it corresponds to the Cherenkov matching condition. The singledashed curves correspond to Eq. (22) for a value of “a” that
satisfies the condition of Eq. (26). It is seen that there are
two intersections between these curves and the solid straight
line. These intersections are the two values represented in
Eq. (23); the negative root corresponds to intersection at a
smaller scaled frequency, and the positive root, to a larger
scaled frequency. The dot-dashed curves are, again, Eq. (22),
but now for a value of the parameter “a” that violates Eq.
(26). Here, the intersection at higher frequency is seen, but
the intersection at lower frequency is not possible. A solution
exists for which j is purely imaginary, but this corresponds
to an evanescent signal and would not exhibit Cherenkov
radiation.
Figure 2 complements Fig. 1 by illustrating possible
solutions for various burst velocities in the range
0:3 u 0:7. As u decreases, it is apparent that the solutions move closer to the respective cyclotron frequencies,
especially for the smaller values of “a” (which is expected
due to the radial envelope of the source). For u ¼ 0.3, a clustering of the radiated frequencies occurs near the cyclotron
frequencies of both species; the signal generated exhibits
beating between these two frequencies, as shown below. If
the source is sharply peaked radially, then large values of
“a” admit Cherenkov solutions with frequencies close to the
ion-ion hybrid frequency. This is explored in Sec. V, where
the wake due to a single alpha particle for conditions anticipated in the ITER device, is considered.
With this appreciation for the properties of the possible
wave features, the magnetic field given by Eq. (19) is evaluated. First, the integral over frequency is performed.
Although an imaginary part has been included in the delta
function, at this stage its value is considered to be vanishingly small. The finite imaginary part causes the poles to
move away from the real axis in such a manner as to preserve causality. Upon completion of the frequency integral,
the expression for the magnetic field becomes
a2 d 2
2?
2 2
0
0
j
u
0
ð
1 r2 J1 ðar Þe 4de eijðz ut Þ
qNb X1 l2 2
a dadj
:
B/ ¼ pcde u
ðj2 j2þ Þðj2 j2 Þ
(27)
Next, the integral over j is evaluated. A distinction must be
made between the two cases, z0 ut0 < 0, and z0 ut0 > 0.
FIG. 1. Constraints on Cherenkov radiation in the inertial regime. Vertical
axis is the scaled wave number, j, and horizontal axis, the scaled frequency,
w, as defined above Eq. (19). The solid, straight line corresponds to Eq. (21)
and reflects the wave-particle resonance condition for a value of v=vA ¼ 0:5.
The broken curves correspond to the two branches of the dispersion relation
of the inertial mode in Eq. (22) for a 50% Hþ, 50% Heþ plasma, for two
values of the transverse width “a.” The upper frequency branch always possesses a real
solution,ffi but the lower branch only yields a real solution if
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a < aM ¼ 1=u2 1. x=X1 ¼ 1 is the helium cyclotron frequency,
x=X1 ¼ 2, the ion-ion hybrid frequency, and x=X1 ¼ 4, the hydrogen cyclotron frequency.
FIG. 2. Range of intercepts leading to Cherenkov radiation. The blue line is
for u ¼ 0.3, green u ¼ 0.4, red u ¼ 0.5, and cyan u ¼ 0.7. The dashed curves
are the same shown in Fig. 1. As u decreases and at smaller perpendicular
wave numbers, the radiative solutions cluster around the individual cyclotron frequencies.
092109-5
W. A. Farmer and G. J. Morales
Phys. Plasmas 19, 092109 (2012)
For the former case, the contour is closed below the real
axis, and for the latter case, above. Causality dictates that the
pole due to jþ be treated as if it lies below the real axis. It
only contributes to the integral when z0 ut0 < 0, or for
times after which the burst has passed the axial position in
question. For times such that z0 ut0 > 0, the pole is not
2
B/ ¼ 2
qNb X1 l2 6
6
cde u 4
ð1
0
enclosed by the contour, and thus, it does not contribute to
the integral. The second root, j , has the same properties for
values of “a” that satisfy Eq. (26). For a > aM , j is purely
imaginary, and makes a contribution regardless of which
direction the contour is closed. Upon completion of the integral over j, the magnetic field becomes
2
j2þ u2
a2 d?
0
0
0
J
1
ðar
Þsin
j
ðz
ut
Þ
1
þ
2
r2
daa2 e 4de
jþ ðj2þ j2 Þ
2
j2 u2
a2 d?
0
0
0
J
1
ðar
Þsin
j
ðz
ut
Þ
1
2
2
r
daa2 e 4de
þ
j ðj2 j2þ Þ
0
3
2
2
j u2
a2 d?
0 jj jðz0 ut0 Þ
ð1
J
1
ðar
Þe
1
2
7
1
r2
7; z0 ut0 < 0;
daa2 e 4de
þ
2
5
2
2 aM
j ðj jþ Þ
ð aM
tity can be evaluated from the dispersion relation in Eq. (12).
In the inertial regime, the corresponding angle is given by
and
ð
a2 d 2
?
qNb X1 l2 1
2 4d2e
daa e
B/ ¼ cde u aM
0
0
j2 u2
1 r2 J1 ðar 0 Þejj jðz ut Þ
;
j ðj2 j2þ Þ
(28)
tan h ¼ 0
0
z ut > 0; (29)
where the evanescent part of the response is included for completeness. This component of the signal has been previously
examined,5 and it can be thought of as the magnetic analog of
Debye shielding. Since the focus of this paper is the additional
radiative solution associated with a second ion species, the
evanescent part of the signal is neglected in what follows.
In describing the wake pattern, it is illuminating to display the z-component of the plasma current density. This
quantity is given in Eq. (17), and its calculation amounts to
replacing the Bessel function, J1 ðar 0 Þ, in the expressions for
the magnetic field with the quantity J0 ðar0 Þac=4pde . Figure 3
illustrates the resulting pattern, after performing the relevant
numerical integration, for the values l ¼ 4, r ¼ 2, u ¼ 0.5,
d? =de ¼ 3, again, corresponding to a hydrogen-helium
plasma in LAPD. The color scale in the contour displays is
normalized to the maximum amplitude of the signal. The
charge burst is located at the origin and is moving upward at
speed v. The pattern is shown at a time designated t ¼ 0. At
future or earlier times, the whole pattern simply translates
with the charge at the upper end. In addition to exhibiting
the inverted-V pattern, as in the single species case, it is clear
that two distinct waves are propagating and interfere with
each other. Further, in addition to having different wavelengths and frequencies, the Cherenkov cone angle is different for each of the waves that make up the total signal. This
is understood by examining the group velocity for shear
Alfven waves. The angle at which the group velocity propagates is given by tan h ¼ vg? =vgjj ¼ @kjj =@k? . This quan-
k? kjj d2e
2 d2
1 þ k?
e
;
(30)
or in terms of scaled quantities
tan h0 ¼
d e X1
aj
tan h ¼ :
vA
1 þ a2
(31)
From Fig. 1 and Eq. (23), it is clear that jþ > j for a given
value of “a.” This implies that the group velocity angle is
FIG. 3. Wake radiated by a particle burst in the inertial regime. Spatial pattern of parallel current density jz =jjz jMax for d? =de ¼ 3, u ¼ 0.5, r ¼ 2, and
l ¼ 4. Color contours represent the current wake at t ¼ 0 with the charge
burst centered at the origin and moving upwards. Two separate patterns are
clearly visible. The broader, longer wavelength pattern corresponds to the
lower frequency branch and the narrower, shorter wavelength pattern, to the
upper branch.
092109-6
W. A. Farmer and G. J. Morales
Phys. Plasmas 19, 092109 (2012)
the time signal in Fig. 5(a), again normalized to its peak value.
The radiated frequencies are clustered close to the cyclotron
frequencies of the individual ion species. This is due to the
associated increase in transverse group velocity as the cyclotron frequency is approached, in accordance with Eq. (30).
The relative amplitude of the power that appears into the
lower and higher frequency components, however, depends
on the radial position sampled. At small radial positions, the
higher frequency spectral peak is dominant.
FIG. 4. Relationship between angle formed by group velocity, h, and the
cone angle of the Cherenkov emission, v. A signal is excited in the plasma
after the charge passes-by. As h increases, v decreases. This explains the
two cone angles visible in Fig. 3.
much larger for the higher frequency band than for the lower
one. Schematically, the resulting situation is illustrated in
Fig. 4. Geometrically, the cone angle of the radiation, v, is
related to the angle formed by the group velocity by
v ¼ p=2 h. Thus, for a larger angle, h, the cone angle, v, is
smaller as is shown in Fig. 3.
The temporal signature of the radiated magnetic field,
obtained by evaluating Eq. (28) at a fixed spatial position, is
shown in Fig. 5(a) for the same parameters used in Fig. 3.
Time t ¼ 0 corresponds to the burst being located at the same
axial position at which the field is sampled. The radial position is r 0 ¼ 10, chosen to allow for sufficient temporal separation between the two components of the signal. The displayed
signal is normalized to its peak value. It is seen from Fig. 5(a)
that the lower frequency component arrives first and the larger
frequency component arrives later. This feature is primarily
due to the larger cone angle exhibited by the lower-frequency
propagation band. Figure 5(b) displays the power spectrum of
IV. ADIABATIC REGIME
In the extreme kinetic regime, the burst speed is
considered to be much smaller than the electron thermal
speed, v v e , while all other assumptions made in the
analysis of the inertial regime are presumed to remain
valid. In this limit, the parallel dielectric coefficient is
approximated by
jj )
x2pe
;
kjj2v 2e
(32)
while the perpendicular dielectric coefficient, ? , remains as
in Eq. (1) (i.e., kinetic effects are negligible for the ions). The
cold ion contribution is neglected in Eq. (32), which eliminates the ion-acoustic wave from the dispersion relation. This
is a valid approximation when the speed of the source is much
greater than the ion-acoustic sound speed, cs . Further, to
neglect the coupling between the Alfven and acoustic modes,
it is assumed that vA cs which is appropriate for low-beta
plasmas. If either of these conditions is violated, the ion contribution must be retained in Eq. (32) in order to properly
include the ion-acoustic wave in the formulation.
The same methodology used in the inertial case applies
here. All quantities are scaled as before, except for the radial
variable. Now the relevant radial scale is the ion-sound gyroradius qs ¼ cv e =vA xpe . Accordingly, the scaled transverse
direction becomes r 0 ¼ r=qs and b ¼ k? qs . With these scaled
quantities and the adiabatic expression for the parallel dielectric, Eqs. (9) and (15) become
ð
X21
b2 dbdjdw
2p2 cqs vA
2
0
0
1 wr2 J1 ðbr0 Þeiðjz wt Þ
j~S;z ;
2
w
2 1 w2 ð1 w2 Þ 1 w2
b
j2
r2
l2
B/ ¼ (33)
b2 d2
?
vA qNb 4q2
ue s dðw ju i 0 Þ:
j~S;z ¼ 2p
X1
(34)
As before, the solution is determined by the poles associated
with the dispersion relation. In analogy with Eqs. (21) and
(22), the poles are now determined by the relations
FIG. 5. Temporal signature of the wake pattern of Fig. 3 at a fixed radial
position, r=de ¼ 10. (a) Time dependence of the magnetic field normalized
to its peak value. t ¼ 0 corresponds to the arrival of the source at the same
axial position of the location sampled. The lower frequency arrives sooner
than the upper frequency due to the difference in cone angles, as illustrated
in Fig. 4. (b) Power spectrum normalized to the peak value. Two frequencies
bands are clearly visible just below the individual cyclotron frequencies.
w
¼ u;
j
w2
b2
j2
w2
1 2
r
(35)
w2
ð1 w Þ 1 2
l
2
¼ 0;
(36)
092109-7
W. A. Farmer and G. J. Morales
Phys. Plasmas 19, 092109 (2012)
which yield two solutions, j6 , that take the same form as
Eq. (23), but with the new definitions for bs and b0
1
l2 2
2
2
bs ¼ 2 1 þ l 2 ðu b Þ ;
(37)
r
u
b0 ¼
l2 2
ðu 1 b2 Þ:
u4
(38)
As in the inertial case, the discriminant is always positive.
To prove this, first observe that the discriminant can only be
negative if b0 < 0. With this restriction in mind, and recalling that 1 < r < l
b2s
b2s >
2
1
l4
l2
¼ 4 1 þ l2 2 b0 2 ;
r
r
u
2
1
u4
1
1
b
> 4 ð1 u4 b0 Þ2 :
r2 0
u4
u
(39)
(40)
With this relation, it can be shown that b2s þ 4b0 by the following inequalities
b2s þ 4b0 >
1
1
½ð1 u4 b0 Þ2 þ 4u4 b0 ¼ 4 ð1 þ u4 b0 Þ2 > 0:
u4
u
(41)
2
B/ ¼ 2
qNb X1 l2 6
6
cqs u3 4
ð1
0
Next, it can be shown that jþ is always real. This can be
seen by showing that if bs < 0, then b0 > 0. The former con2
dition implies that b2 < u2 rl2 ð1 þ l2 Þ. This relation leads
2
2
to the condition that b0 > lu4 rl2 ð1 þ l2 Þ 1 > u14 , where
the condition that r > 1 has been used.
Examining next the conditions on j , it is seen that if
bs < 0, then j is imaginary. If instead it is assumed that
bs > 0, then the condition that j be real is equivalent to
b0 < 0. This leads to the condition that
pffiffiffiffiffiffiffiffiffiffiffiffiffi
b > bM ¼ Re u2 1;
(42)
which is the same condition obtained for a single species
plasma.
With this understanding of the nature of the poles of the
integrand, the frequency integration in Eq. (33) yields
b2 d2
2?
0
0
j2 u2
0
ð
1 r2 J1 ðbr Þe 4qs eijðz ut Þ
qNb X1 l2 2
b dbdj
:
B/ ¼
pcqs u3
ðj2 j2þ Þðj2 j2 Þ
(43)
The integrations over j are next performed in the same manner as in Sec. III for the inertial regime to obtain
2
j2þ u2
b2 d?
J1 ðbr 0 Þsinðjþ ðz0 ut0 ÞÞ
1
2
2
r
dbb2 e 4qs
jþ ðj2þ j2 Þ
3
2 2
j2 u2
b
d
?
ð1
1 2 J1 ðbr 0 Þsinðj ðz0 ut0 ÞÞ
7
2
r
7;
dbb2 e 4qs
þ
2
5
2
j ðj jþ Þ
bM
in which the evanescent fields have been neglected. If
included, they result in a form similar to those found in the
inertial regime, shown in Eqs. (28) and (29). The remaining
integrals are evaluated by numerical integration.
The wake patterns associated with the z-component of the
plasma current density are illustrated in Fig. 6 for values of
u ¼ 0.5, 1.0, 1.5, and 2.0. The parameters are l ¼ 4, r ¼ 2,
and d? =qs ¼ 3, corresponding to LAPD conditions. In each
panel, the current density is normalized to the maximum amplitude in the contour. As before, the charge is located at the
origin and is traveling upward with velocity v. Figures 6(a) and
6(b) not only display, overall, the characteristic Cherenkov
radiation pattern but also exhibit spatial interference between
two modes with different wavelengths, corresponding to simultaneous excitation of the two propagation bands. Figure 6(c)
shows that as the velocity of the burst increases beyond u ¼ 1,
a periodic array of field-aligned current filaments develops
from the magnetic structure associated with the second integral
in Eq. (44). This behavior is in agreement with the singlespecies result found previously.5 The effect can be understood
z0 ut0 < 0;
(44)
from Eq. (42), which reflects the fact that as u increases, the
lower-branch is dominated by short perpendicular wavelengths. Also, when the limit of large u is considered,
j 2r2 =u2 , indicating that the parallel wavelength becomes
large. However, the limiting spectral decrease in the source
current, due to the assumed Gaussian radial shape of the burst,
causes the signals with large values of “b” to become less significant, until the pattern becomes dominated by the first integral in Eq. (44). This is the reason why the field-aligned
currents are no longer visible in Fig. 6(d) for u ¼ 2.0, since the
display format is normalized to the peak value. At these higher
speeds, the pattern is dominated by the radiation of the higher
frequency band, which does not develop filamentary structures.
V. WAKE DUE TO A POINT PARTICLE
To illustrate the Alfvenic wake produced by a point particle, two situations are considered: a fast particle propagating in LAPD with inertial electron response, and a fusionborn alpha particle propagating in ITER-like conditions for
092109-8
W. A. Farmer and G. J. Morales
Phys. Plasmas 19, 092109 (2012)
which the electrons exhibit an adiabatic response. In evaluating these two situations, all of the previously discussed concepts are appropriate. However, in this context the moving
source (i.e., the fast particle) must be considered as a point
source, and this can be understood as taking the limit
d? ! 0, in which the transverse Gaussian profile approaches
a delta function. In taking this limit, care must be taken to
ensure numerical convergence of the resulting integrals.
To illustrate the limiting pattern, the inertial regime is first
considered. The z-component of the wave current becomes
2
j2þ u2
1 2 J0 ðar 0 Þsin jþ ðz0 ut0 Þ
2 6ð 1
qNb X1 l 6
r
daa3
jpz ¼ 2 u 4
jþ ðj2þ j2 Þ
2pde
0
þ
ð aM
0
daa3
3
j2 u2
1 2 J0 ðar0 Þsin j ðz0 ut0 Þ
7
r
7;
2
5
2
j ðj jþ Þ
(45)
z0 ut0 < 0:
The second integral is easily performed numerically. To perform the first integral, the asymptotic form of the integrand
must be examined. In the limit of large “a”
j2 j2þ
l2 2
a ;
r2
(46)
r2 r2
r2 2
1
2
2 4 1 þ 2 ðr 4l 4Þ 2 :
u
u
4l
a
(47)
If the numerically evaluated portion of the first integral in
Eq. (45) is truncated at a value, aMax , that satisfies both
aMax 1 and aMax r 0 1, then for the asymptotic contribution, the Bessel function can also be replaced by its asymptotic form. In this case, the asymptotic portion of the first
integral is proportional to
ð1
cosðar0 p=4Þ
2cosðaMax r 0 p=4Þ
da ¼
pffiffiffiffiffiffiffiffiffi
3=2
aMax
a
aMax
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!#
0
pffiffiffiffiffiffi
2armMax r
2armMax r 0
0
C
;
þ2 pr S
p
p
(48)
where the functions C(x) and S(x) are the Fresnel integrals
CðxÞ ¼
ðx
cos
0
pu2
du;
2
SðxÞ ¼
ðx
sin
0
pu2
du: (49)
2
Upon expanding the Fresnel integrals for large argument,
Eq. (48) becomes
!
cosðar0 p=4Þ
4cosðaMax r 0 p=4Þ
1
da þ O 3=2 :
pffiffiffiffiffiffiffiffiffi
aMax
a3=2
aMax
aMax
ð1
(50)
FIG. 6. Wake radiated by a burst in the adiabatic regime. Spatial pattern of
parallel current density jz =jjz jMax for d? =qs ¼ 3, r ¼ 2, and l ¼ 4. From top
to bottom, u ¼ 0.5, 1.0, 1.5, and 2.0. The pattern of the lower branch transitions into the vertical striped pattern as u increases above one. This is most
apparent in (c) for the u ¼ 1.5 case. In (d), u ¼ 2.0, the upper branch is dominant. The upper branch does not develop the current filament structure
because it has no lower limit on the integral over k? .
Thus, if aMax is chosen such that the integrand (including the
Bessel functions) is asymptotic and the leading order term in
Eq. (50) vanishes, then the integral converges relatively
quickly and the oscillatory part of the integral at infinity can
be neglected. Similar results hold for the magnetic field with
092109-9
W. A. Farmer and G. J. Morales
the leading order term of the asymptotic portion being pro3=2
portional to cosðaMax r 0 3p=4Þ=aMax .
Figure 7(a) illustrates the Cherenkov wake pattern for
the single particle case in the inertial limit for a H-He mixture of equal concentrations, as can be realized in LAPD
experiments. The general features seen earlier in Fig. 3 are
present: there are two distinct patterns and the lower branch
has a larger Cherenkov angle than the upper branch. However, Fig. 7(a) differs from Fig. 3 in that the wake pattern is
dominated by the contributions from the upper frequency
branch with the lower branch only barely visible just outside the cone of the upper branch, as seen in Fig. 7(b),
which is a blow-up of the region bounded by the dashed
rectangle in Fig. 7(a), and locally normalized to the peak
value. To further illustrate these differences, Fig. 8 shows
the temporal evolution of the magnetic field at radial position r 0 ¼ 10, readily comparable to Fig. 5. In the top panel,
the signal is again normalized to the peak value, and in the
bottom panel, the frequency spectrum is normalized to the
peak value in the upper frequency branch. From Fig. 8(a), it
is apparent that the upper branch does not arrive until
roughly t0 15. Although, not evident from the figure, the
signal associated with the upper band chirps wherein the
signal starts at frequencies close to the hydrogen cyclotron
frequency and then gradually moves towards the ion-ion
hybrid; this is due to the narrowing of the Cherenkov coneangle at frequencies close to the ion-ion hybrid frequency.
In Fig. 8(b), it is seen that there is a much larger frequency
spread than in Fig. 5. This effect is mainly caused by the
contribution of larger transverse wave numbers, which
allows solutions to the Cherenkov matching condition at
frequencies away from the cyclotron frequencies. The clus-
Phys. Plasmas 19, 092109 (2012)
FIG. 8. Temporal signature of the wake pattern of Fig. 7 at a fixed radial
position, r=de ¼ 10. (a) Magnetic field. (b) Power spectrum. The lower
branch signal arrives first, followed by the upper branch, which starts around
t0 ¼ 15. x=X1 ¼ 1 is the helium cyclotron frequency, x=X1 ¼ 2, the ion-ion
hybrid frequency, and x=X1 ¼ 4, the hydrogen cyclotron frequency. A clear
frequency separation between the upper and lower band is apparent.
tering of the spectrum around the cyclotron frequency is
less apparent for the lower branch than it is for the Gaussian
beam case. For the upper branch, the radiated signals now
extend from the ion-ion hybrid frequency to the ion cyclotron frequency.
Next, it is useful to illustrate the Cherenkov radiation
by fusion-born alpha particles in ITER. In the fusion
environment expected for ITER, the thermal speed of
10 keV electrons is v e ¼ 4 109 cm/s, and the initial
speed of a 3.5 MeV alpha particle, va ¼ 1:3 109 cm=s.
The Cherenkov condition is that the parallel phase velocity
of the wave be equal to the velocity of the particle considered. With these conditions, the kinetic, parallel dielectric
is
jj ) FIG. 7. (a) Wake pattern in the inertial regime due to a single particle traveling at u ¼ 0.5 in a background plasma of equal concentrations of hydrogen
and helium. The dominant signal possesses a narrower cone angle and corresponds to the upper branch. The longer wavelength pattern with a larger
Cherenkov cone angle corresponds to the lower branch and is barely visible.
(b) Expanded view of region within dashed-line rectangle in (a).
x2pe 0 va
p
ffiffi
ffi
Z
;
2v 2e
2v e
(51)
where Z 0 ðxÞ is the derivative of the plasma dispersion function. For relevant parameters, the argument of the plasma
dispersion function is on the order of 0.1, placing the electron response in the adiabatic regime, i.e., Z 0 2. Furthermore, as the alpha particle slows down, the adiabatic
approximation is better satisfied.
Taking the same point-charge limit as in the inertial regime, the wave current is described by the relation
092109-10
W. A. Farmer and G. J. Morales
Phys. Plasmas 19, 092109 (2012)
2
jpz ¼
qNb X1 l2 6
6
2pq2s u3 4
ð1
1
dbb3
0
j2þ u2
J0 ðbr0 Þsin jþ ðz0 ut0 Þ
2
r
jþ ðj2þ j2 Þ
3
j2 u2
ð1
1 2 J0 ðbr 0 Þsin j ðz0 ut0 Þ
7
r
7;
dbb3
þ
2
5
2
j ðj jþ Þ
bM
Again, the asymptotic regime of the integral is considered.
In the limit of large “b”
j2þ j2
l2 2
b ;
u2 r2
r2
r4
l2 2
1
2
2
2 2 2 1þl 2u r
;
u
l u
r
b2
(53)
z0 ut0 < 0:
limitation, a low-pass filter function is introduced that isolates the contribution from shear Alfven waves in the integration over “b” in Eq. (52). A filter function possessing the
desired properties is
f ðbÞ ¼
(54)
The second integral is amenable to the same type of analysis
as done in the inertial regime, and the same results apply.
The first integral is more problematic. The asymptotic form
of this integral is proportional to
ð1
l
ðz0 ut0 Þb db:
b3=2 cosðbr 0 p=4Þsin
(55)
ur
bMax
From the form of this integral, it is clear that it is not convergent. From Eq. (55), it is apparent that the frequencies of the
waves associated with the divergence are, w ¼ lb=r, which
increase without limit. This is unphysical and reflects the
neglect of the displacement current in the expression for the
perpendicular dielectric. The equivalent expression in the
analysis for the one-species plasma5 shares this same divergent property, which shows that it is not a multi-ion effect.
The breakdown in the formulation results from trying to
extend the integration to frequencies above the cyclotron frequency of the lighter ion species. At these higher frequencies, the dispersion relation used to explicitly isolate the
shear Alfven mode (i.e., Eq. (12)) is not valid. To rectify this
FIG. 9. Wake pattern in the adiabatic regime due to a single charge traveling
at 1.5 times the Alfven speed. Background plasma consists of equal concentrations of deuterium and tritium, a situation analogous to a fusion-born
alpha particle traveling parallel to the magnetic field in ITER. The parallel
wave current along the field line is shown, normalized to its peak value. The
pattern exhibits the type of filamentary structure shown in Fig. 6(c) for a
charge burst.
(52)
1
:
1 þ exp bf
n
(56)
Here, f is the value of “b” that satisfies the equation,
jþ l ¼ u, and n is a parameter which determines the sharpness of the filter. In order to have a reasonably sharp filter,
the value n ¼ 0:01 is used in performing the numerical
integration.
Figure 9 displays the wave current along the confinement
magnetic field for a background plasma of equal
pffiffiffiffiffiffiffi D-T concentration, corresponding to l ¼ 1:5 and r ¼ 1:5 1:22. The
source particle is traveling at 1.5 times the Alfven speed. The
wake pattern displays the type of current filaments seen earlier
FIG. 10. Temporal signal corresponding to the wake pattern of Fig. 9 at a
fixed radial position, r=qs ¼ 10. (a) Temporal dependence of the magnetic
field normalized to its peak value. Time t ¼ 0 corresponds to the arrival of
the source at the same axial position of the location sampled. (b) Power
spectrum, normalized to the peak value. x=X1 ¼ 1 corresponds to the tritium cyclotron frequency, x=X1 1:22, the ion-ion hybrid frequency, and
x=X1 ¼ 1:5, the deuterium frequency. The propagation seen above the individual cyclotron frequencies is a feature of the adiabatic approximation for
shear waves.
092109-11
W. A. Farmer and G. J. Morales
in Fig. 6(c); this is a consequence of the lower cutoff of the
lower band. The contribution from the upper frequency band
results in a slight bending of the current filaments.
Figure 10 shows the corresponding temporal evolution
of the magnetic field at a position r=qs ¼ 10 from the trajectory of the particle. Time t ¼ 0 corresponds to the particle
passing the axial position of this location. The signal in Fig.
10(a) is normalized to its peak value, while the power spectrum shown in Fig. 10(b) is normalized to the peak below the
ion-ion hybrid frequency. Figure 10(a) shows a large signal
detected around t0 ¼ 0 which corresponds to the leading
edge of the current filament at that position. The power spectrum exhibits a frequency gap below the ion-ion hybrid frequency (i.e., x 1:22) and peaks at a frequency that is upshifted from the cyclotron frequency. This shift is a consequence of the adiabatic approximation for shear Alfven
waves that is also present in single species plasmas.
VI. CONCLUSIONS
This theoretical investigation has extended the results of
a previous study5 of Cherenkov radiation of shear Alfven
waves by a burst of charged particles, in magnetized plasmas
with a single charge species, to the more general case where
two ion species are present. By taking the limit of vanishing
transverse dimension of the burst, the Alfvenic wake of a
point-charge has been obtained for the inertial and adiabatic
regimes of wave propagation. The methodology has been
used to illustrate the properties of the magnetic perturbations
associated with fusion-born alpha particles for conditions
expected in ITER.
The essential physics arising from the presence of two
ion species is that Alfven waves propagate within two different frequency bands separated by a gap determined by the
value of the ion-ion hybrid frequency. In the inertial regime,
one band is restricted to frequencies below the cyclotron frequency of the heavier species, and the other to frequencies
between the ion-ion hybrid frequency, and the cyclotron frequency of the lighter species. In the adiabatic regime, the frequency bands extend slightly beyond the individual cyclotron
frequencies for large perpendicular wave numbers. The Cherenkov radiation pattern in the lower frequency band is found
to exhibit essentially the same properties reported in the previous single species study. However, the upper frequency band
differs from the lower one in that the Cherenkov radiation
condition can be satisfied for all particle velocities.
The observable consequences of the simultaneous excitation of two different propagation bands are that a mixture
of spatial patterns arises in the wake, and that the temporal
signal exhibits beats. Typically, at a fixed spatial location the
frequencies corresponding to the lower frequency band
arrive earlier and those in the higher band, later. This arises
because each band propagates along a different cone angle.
Phys. Plasmas 19, 092109 (2012)
In the inertial regime, two separate spatial patterns arise. A
broader, longer wavelength pattern corresponds to the lower
branch and a narrower, shorter wavelength pattern, to the
upper band. The pattern for the adiabatic regime is significantly altered by the presence of two ion species. The development of a periodic array of filamentary currents seen in
Fig. 10 of Ref. 5, for super-Alfvenic bursts in a single species plasma, is tempered by the stronger excitation of the
upper frequency band, as shown in Fig. 6.
In summary, intrinsic features associated with the presence of two ion species have been identified in the process of
Cherenkov radiation of shear Alfven waves in magnetized
plasmas. This information should be useful in interpreting
observations in space, laboratory, and fusion plasmas where
such ion populations are present. The gap structure in the frequency spectrum may provide a useful diagnostic signature
in fusion environments.
ACKNOWLEDGMENTS
This work is sponsored by DOE Grant No. DESC0007791.
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