Structure
of Alfvb
waves at the skin-depth
scale
G. J. Morales, R. S. Loritsch, and J. E. Maggs
Department of Physics, University of California, Los Angeles, Los Angeles, California 90024-1547
(Received 5 May 1994; accepted 9 August 1994)
This analytical study demonstrates that shear Alfvin waves having transverse scale on the order of
the electron skin depth exhibit a collisionless divergence determined by propagation cones that
emanate from the edges of the exciting structures. Axial current channels are found to spread
radially due to the skin effect up to the cone trajectories and at distances of a few wavelengths from
the exciter develop radial diffraction patterns. For values of the collision frequency slightly larger
than the wave frequency resistive diffusion allows the axial currents to expand beyond the cone
trajectories. 0 1994 American Institute of Physics.
1. INTRODUCTION
It is widely recognized that shear (or slow) Alfven waves
are responsible for the propagation of low frequency electromagnetic disturbances along the ambient magnetic field in
magnetized plasmas. The majority of the studies related to
shear modes address global problems in which the transverse
scale is so large that the approximation kl-+O is routinely
made. In this limit low frequency disturbances propagate
without divergence, and the propagation problem essentially
reduces to the mapping of boundary information along the
ambient field. Of course, if the excitation conditions allow
for a coupling to the compressional (or fast) branch, then an
isotropic spread results.“2
The present analytical study focuses on the properties of
shear waves having small transverse scale, i.e., on the order
of the electron skin depth cIw~~, where c is the speed of
light and w,,~ the electron plasma frequency. In this limit
parallel electron currents cause the spreading of the signals
in the absence of collisions and result in unusual interference
patterns not commonly associated with Alfven waves.
The consideration of small-scale Alfvdn waves is motivated by several topics of broad interest. For instance, it is
presently being considered3v4 that small-scale filamentation
may spontaneously develop in tokamak plasmas, and it is
conjectured5’6 that filamentary Alfvdn waves play a fundamental role in cross-field energy transport. In space plasmas
nonuniformities in the ambient density having small transverse scale are present in regions of field aligned current and
may result in the generation of Alfvin waves having skindepth scale. The basic question of how a microscopic current
channel spreads is the essence of the present study.
The present study shows that Alfvin wave patterns exhibit an intrinsic collisionless divergence determined by
propagation cones that emanate from the edges of the exciting structures. The spreading cones follow an angle 8
relative to the ambient magnetic field given by tan 8
=(w/Q)(mlM)“*[
1 -(w/~i)2]-“2,
where w is the wave
frequency, ni is the ion gyrofrequency, m and M are the
electron and ion mass, respectively. Within a few wavelengths of the exciter, a radial diffraction pattern develops
due to coaxial currents induced by the skin effect. The transition from collisionless divergence along cone trajectories to
radial diffusion occurs for values of v,Iw slightly larger than
Phys. Plasmas 1 (12), December 1994
unity, where v, is the electron collision frequency and o the
wave frequency.
The propagation properties of Alfvdn waves with small
transverse scale can be studied in basic laboratory experiments by using disk exciters that stimulate parallel current
flows. An experimental study that tests some of the theoretical predictions made here has been performed in the Large
Plasma Device7 (LAPD) at the University of California, Los
Angeles, and the results are found in a companion paper.8
An extensive study of the localized propagation of shear
Alfven waves has been previously carried out by Cross and
collaborators.‘79*‘0 Since the experimental environment motivating their work consisted of plasmas having very small
skin depth and large collisionality, a majority of their results
focus on the resistive spreading of filamentary structures
much larger than the skin depth. This is to be contrasted to
the present study which emphasizes radial spread and interference features arising solely from transverse scales on the
order of the skin depth, and in the near absence of collisions.
The manuscript is organized as follows. In Sec. II, a
mathematical formulation is presented for an idealized disk
exciter. The general features of the spatial structures generated are presented in Sec. III. Displays of the spatial patterns
obtained for different parameter values are shown in Sec. IV.
The effects of collisions and Landau damping are discussed
in Sec. V. Conclusions and relationships to the LAPD experiments are established in Sec. VI. The Appendix addresses a
conceptual issue related to the importance of skin-depth currents at finite values of o/~i.
II. FORMULATION
This study considers the simplest model of localized excitation of axial currents in a cold plasma. It consists of an
ideal disk exciter of radius a, placed across the confining
magnetic field. The surface charge of the disk is prescribed to
oscillate at frequency w, i.e.,
(1)
where (r’ is the experimentally measurable quantity, and r
represents the radial distance from the center of the disk. The
useful coordinates in this problem consist of a cylindrical
1070-664X/94/1 (12)/3765/l O/$6.00
Q 1994 American Institute of Physics
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and correspondingly
a{{
j~=[j,(r,z,w)~+jr(r,z,u)F]e-‘W’+c.c.
z=o
FIG. 1. Schematicof geometry of the problem. A transparentdisk of radius
a carries an oscillatory surface charge u(r). A cylindrical coordinatesystem
whose origin is at the center of the disk is used.The .? unit vector is along
the ambient magnetic field B,, which is perpendicularto the disk surface.
system (i, 2,:) centered on the disk, as illustrated in Fig. 1.
The axial coordinate is z and is aligned with the confining
magnetic field Bn.
The function CTin Eq. (1) is in general difficult to determine self-consistently in the presence of a plasma. The underlying mathematical structure of this issue (an inverse scattering problem) is rather involved, and has been solved only
approximately in the pioneering studies of Levine and
Schwinger ” (for scalar waves), and of Bethel2 (for vector
waves). While this is an interesting and fundamental topic,
its resolution is not essential to the determination of the intrinsic structure of small-scale Alfven waves. Consequently,
we proceed to idealize the exciter response by representing
the surface charge by its vacuum value in the w-+0 limit,
which is shown in the early edition of the textbook by
Jackson” to be
(7)
Since we are interested in describing features that become prominent as o--+Q , the question arises as to the need
to allow for an azimuthal component of the plasma current,
j,, that results from the relative slippage between the EXB,
drift of ions and electrons at finite values of o/Ri, The answer to this question is found in the Appendix, where it is
shown that for perpendicular wave numbers k, that satisfy
the condition
kl J-2m ( W/~i)3’2
M[1-(iSl~i)2]“2’
;I;-’
s
with k, = wpelc, the effect of the parallel electron current is
larger than that of the current imbalance resulting from the
E xBo slippage. Thus, with the exception of a small frequency interval close to Q, the structure of Alfvin waves
with small transverse scale (i.e., large k,) is determined (in a
cold plasma) by the current flow pattern determined by the
radial polarization current due to the ions, and the axial current carried by the electrons, This azimuthally symmetric
current pattern implies the electric field components of Eq.
(6) and the corresponding magnetic field
B’=Bg(r,z,w)e-iUf+c.c.
(9)
The oscillatory currents, obtained from the fiuid equations of motion, are the radial polarization current,
and the parallel electron current,
fl= [&;)2]*12? rca*
with co a constant.
This surface charge increases sharply at the edge of the
disk (as expected for a zero-width object), but it is integrable.
The carresponding radial Bessel representation, to be used
later, is
atr
Here vA denotes the actual phase speed of the wave at
frequency w and k, =0,
v*=cp)[l-[$‘]“‘:
(12)
the
dk iF(k)
conventional Alfvdn
speed being the factor
with Wpi the ion plasma frequency.
The system of coupled partial differential equations governing the propagation of the shear mode is obtained from
Eq. (5), in cylindrical coordinates, with the currents given by
Eqs. (10) and (1 l),
C(fijlOpi)‘VAO*
with
sin ka
CT(k)= croa k .
(4)
The self-consistent electric field is determined by combining Faraday’s law with Ampere’s law in which the displacement current is neglected, as is appropriate for low frequencies
VxVxE’=-Fzj’
E,. =-k;E,.,
!
(13)
(14)
47r d
P’
where E’ and j; are the experimentally measurable electric
field and plasma current. Assuming axial symmetry, the harmonic exciter generates a linear response given by
E’=[EZ(r,z,w)~+E,(r,z,w)i]e-iWc+c.c.,
3766
- f
Phys. Plasmas, Vol. 1, No. 12, December 1994
(6)
where kA=u/vA I
Equations (13) and (14) are decoupled by the following
radial Bessel representation (for ~0):
E,(r,z,w)
=
m
dk kE,(k)Jo(kr)e”kllck)Z,
I0
05)
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E,(r,z,o)
dk
=
k~‘,(k)J*(kr)e’kll’k)z,
(16)
where the parallel wave number, $1, depends explicitly on
the effective perpendicular wave number k due to the finite
skin depth,
k,,(k)=k*[
1 + (klk,)2]“2.
where la= - ioaa~~a* is the complex amplitude of the AC
current that flows to the disk from an external generator.
To simplify subsequent discussions, it is useful to introduce the scaled quantities
(26)
(17)
Inserting (15) and (16) into Eqs. (13) and (14) yields the
self-consistent relation
k*+k*
=i s
&W,
(18)
kkll
which indicates that, in principle, once g,(k) is determined
the structure of the wave pattern can be obtained.
W e proceed to determine l?,(k) by matching the jump in
E, at the surface of the disk at z=O. For this purpose we
envision the disk exciter to be nearly transparent (as can be
realized in the laboratory by using a fine-mesh screen), so
that
p=
f ;
[=kAZ,
(27)
cw
i,(k)
AE,(r,z=0)=47r[a(r)+a,(r)],
(19)
with ‘TJ r) representing the oscillatory surface charge pileup
generated by the field-aligned electron flow that intersects
the disk boundary. This charge pileup is determined selfconsistently from E, , i.e.,
2
qAr)=
&
i
z
i
E,(r,z=O”).
-(o/w,,)*2mc+(r),
J~(~~)ew[~EGGGPl.
b(p,.$)= lowdK y
(21)
where the appropriate wQo~, limit is assumed.
Equation (21) represents the effective alternating current
(AC) screening produced by the cold plasma in response to
the charge, n, delivered to the surface of the disk by an
external generator.
Combining Eqs. (21) and (4) yields i,(k) from which
the following integral expression is obtained.
Ez(r,z,w)
(29)
Equation (29) describes the physical quantity which is
most amenable to direct experimental investigation with
small induction-coil detectors, i.e., the scaled azimuthal magnetic field. The spatial structure depends on a single parameter, p, which represents the ratio of the radius of the disk
exciter to the skin depth.
The effects of collisions can be incorporated into Eq.
(29) through the substitutions
1
g+u
+qg
$9
(30)
b-+(1 +ir,b5
Using the antisymmetry of E, on the disk surface, i.e.,
AE,=2EZ(r,z=O+)
results in
EZ(r,z=O+)=
Equation (25) then takes the form
where rll= v,l w, and r, = (Vi/O)
+
(mlM)(V,lO)
X [ 1 - ( o/Ri)2] “*, with vi and v, the ion and electron collision frequencies, respectively. These results are obtained by
adding a collision frequency to the equation of motion and
essentially represent the imaginary parts of the parallel (the
correction to p-*) and of the perpendicular (the correction to
.$) components of the dielectric tensor of a magnetized, cold,
collisional plasma.
Next we proceed to outline the spatial structure predicted
by Eq. (29).
III. GENERAL FEATURES
= -2111 c)*coaJfdk
sin ka Jo(kr)eikll(k)z.
(22)
The corresponding azimuthal magnetic field is represented as
cc
dk kbo(k)J1(kr)eikll(k)z.
Bdr,z,w)=
(23)
I0
Using the z component of Ampere’s law results in
In the limit p--‘m, i.e., large disk exciter, a universal
pattern is obtained from Eq. (29),
bht)-
it
I
sin K
mdK yJI(KP).
0
It consists of a wave propagating along the magnetic field
with wave number kA . The radial structure is determined
from the K integral, which is the Weber-Schafheitlin integral that results inI
P
l+(l-p*)“z’
b(p,5)-ei5
from which the integral expression is determined,
BB(r,z,~)=
2
I
m sin ka
dk k
Jl(kr)e ikII(k)z,
0
Phys. Plasmas, Vol. 1, No. 12, December 1994
p<1,
(32)
1
P>l,
(25)
i P’
and whose behavior is illustrated in Fig. 2.
Morales, Loritsch, and Maggs
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p=1
1
Ibl
FIG.3. Different propagationregionshoundedby the cone structuresresulting from finite skin depth. p=rla,
q=k&p,
p=aw&c.
0
2
1
r/a
FIG. 2. Radial dependenceof magnitudeof scaled magnetic field b correspondingto the universal pattern obtainedin the limit p--+~ i.e., large disk
exciter. Axially, b propagatesas a wave with wave number w/VA.
Physically, Eq. (32) indicates that a parallel current
channel is established along those ambient field lines that
intersect the disk exciter. Outside the disk shadow (p>l) no
axial current flows, as indicated by the l/p falloff of the
oscillating magnetic field b. Thus, the universal pattern obtained for ~--+a can be visualized as arising from a rigid
plasma wire directly tied to the surface of the disk.
The other extreme situation is p--+0, which corresponds
to a very small exciter. In this limit, Eq. (29) becomes
b(p,c)-+
sin K
dK 7
J,(Kp)eiKYP,
A key feature of the resulting spatial pattern is that it is
governed by propagation cones emanating from the edges of
the exciter along the positive characteristic lines p= (i&)-i- 1
(the outer cone) and p=(</p)- 1 (the inner cone), the later of
which physically contributes for s>p. There is also a characteristic with negative slope p= 1 -(Qp) which extends
from &=O up to the physical location t=p. The boundaries
formed by the characteristic lines separate four regions (labeled I-IV in Fig. 3) within which the magnetic field exhibits a different functional dependence on the scaled radius p,
for fixed & the scaled axial position. In this limit the spatial
pattern is also universal (i.e., parameter independent) since
the factor p can be used to rescale the quantity 6, i.e., by
introducing v= t/p.
The behavior of the scaled magnetic field b(p,rl) in the
various regions outlined in Fig. 3 is as follows:
Region I (O< rl< 1, p< 1 - 7):
1
P
2 i (~+1)+[(~+1~2-P211f2
which can be put in the more transparent form (exhibiting
important characteristic lines)
bhE1-t
;
with 1 = 2 1. A closed-form expression for the integral in Eq.
(34) can be constructed by combining the relationsi
3:
s
dK Jl(Kp)
~
K
0
1
+ (l-
Region II (+l,
p<v--1):
P
Z( i
1
1
sin /3K
- (T-
1 ~+[(T--
Region III (+l,
(35)
P
(381
U2-p21if2 i ’
v-l<p<@l):
1
2 i (rl+1)+rhw2-P21’12
and
-~ (TP2
1)
i
(39)
cos PK=
o<p<p
aver the proper physical ranges of 5 and p.
3768
(37)
u2-P21'fz 1 '
rl)f[(17-
Phys. Plasmas, Vol. 1, No. 12, December 1994
’
(36)
Region IV (@ 1, p> q+ 1):
(40)
Morales, Loritsch, and Maggs
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Next we identify a diffraction pattern that is contained in
Eq. (29) for &l, i.e., far from the exciter. In this limit the
dominant contribution to the integral in (29) arises from the
small K region,
0.6
b(~~l,p)-+e’*
03
c
JO
dK J1(Kp)e’5K2’2p2,
(42)
which can be expressed in closed form
Ibl
b(Pl+)-t ( p2p)
25 ( ~)ei(*-f$+wi2),
0.2
(43)
0.0
0
5
10
r/a
FIG. 4. Radial dependenceof magnitude of scaled magnetic field in the
limit p+O for scaled axial position v=kAzlp=r, p=ao~~Jc. The labels
I-IV refer to the regions defined in Fig. 3.
with c$=p*p*/(45).
This pattern consists of an obliquely propagating wave
whose effective radial wave number increases with position
(at fixed &l), i.e., k,--tp2p/2& while the axial wave number is k,. Surprisingly, the amplitude of the magnetic field
exhibits a radial diffraction pattern,
Ib(&I,p)l+:
The pattern described in Eqs. (37)-(40) is illustrated in Fig.
4 for v=rr. It is to be contrasted with the p-+a case illustrated in Fig. 2.
From Eq. (40), it is found that for ~91, at fixed v,
b-t I/p and for ~$1, p> v+ 1 (outside the outer cone),
b-+1/p. For 741 and p<~- 1 (inside the inner cone), Eq.
(38) yields b+-2p/$.
These features indicate that the pattern generated by a very small exciter can be viewed as a
radially expanding axial current channel whose spread is
bounded by the outer cone, p= v+ 1, emanating from the
edge of the exciter. The spread of the current channel causes
the magnetic field to become progressively weaker in the
axial direction, i.e., a hollow pattern develops far from the
disk.
The angle 8 relative to the ambient magnetic field that
bounds the spreading current channel (i.e., the cone angle) is
given by
tan 0= ~=~=~(~)[l-(~)z]-l’“,
(41)
where m and M refer to the electron and ion mass, respectively. The corresponding axial spacing between the outer
and the inner cone at a fixed radial position is Ah5=2p.
Equation (41) reduces in the limit OK&~ 1 to the expression identified by Borg et a1.9 as a resonance cone phenomenon associated with a Green’s function solution.
The cone-like features clearly elucidated in the extreme
p--+0 limit are present also in the finite p case of experimental relevance. The reason is that the contributions arising
from the large K components in Eq. (29) (i.e., due to the
sharp edges) have the same mathematical structure as the
integrand of Eq. (33) since the effect appears through the
form ( K/P)~. Of course, at finite p there is also an important
contribution from the small K region giving rise to the axial
propagation at the Alfvdn speed.
Phys. Plasmas, Vol. 1, No. 12, December 1994
sin ‘$
.
I(
)I
Consequently, far from the exciter and near the axis
IW-Lp+$
P,
(44)
(45)
which indicates that the axial current channel becomes progressively hollow. The radial location of the maximum value
of I bl for ,$+ 1 occurs at p= (2m$/p2)“*, which implies that
IbI<2pl(2#“.
It is also possible to obtain closed-form expressions that
incorporate higher-order (in K) contributions to the diffraction pattern. Although it is not worthwhile reproducing those
expressions here, it is found that the approximate Eq. (45) is
valid for peJ65fp2, for fixed @=-1.
IV. NUMERICAL RESULTS
In this section we present displays of the spatial patterns
obtained by direct numerical integration of Eq. (29). The
various features identified in Sec. III as limiting cases of Eq.
(29) are clearly seen in Fig. 5. This figure exhibits the radial
dependence of the magnitude of the scaled azimuthal magnetic field, I bl, for different axial positions away from the
exciter. This case corresponds to a disk whose radius is equal
to the skin depth, i.e., p = 1.
The two sharp edges present in each of the curves in Fig.
5 correspond to the locations of the cones at t-la = k,z + 1.
For small axial distance, t=rr/2, the pattern consists of two
sharp rises; from the center of the disk to the inner cone
(region II in Fig. 2), and then from the inner to the outer cone
(region III). Outside the outer cone the magnetic field falls
off as l/r for all values of kAz. It is seen from Fig. 5 that as
k,z increases, the location of the sharp edges moves to larger
r/a, as follows from the cone trajectories. Simultaneously,
the slope of the linearly rising field near the origin decreases,
as expected from Eq. (45). Finally, as one moves away from
the exciter (e.g., k,z>6v),
a radial diffraction pattern
Morales, Loritsch, and Maggs
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0.6
0.2
b
Ibl
0.2
-0.2
-0.6
FiG. 5. Radial dependenceof magnitudeof scaledmagneticfield for different axial positions away from exciter for a disk radius equal to the skin
depth.p = 1. The radial spacing betweencone featuresis the disk diameter
2a. Diffraction pattern develops continuously from the inner current channel.
emerges. The maximum of lb/ moves to a larger r/a, as
predicted by Eq. (44). Eventually the diffraction pattern becomes squeezed against the cone boundaries where the
spreading current channel ends.
For larger disk exciters, the features elucidated in Fig. 5
are also present, but as expected from the p-scaling they
become contracted radially, as is illustrated in Fig. 6 for a
p =4 case.
0
1
2
3
r/a
15
r/a
FIG. 7. Radial dependenceof real and imaginary parts of scaled magnetic
field correspondingto the [b[ curve for p = 1 and k,z = d2 in Fig. 5.
Figures 7 and 8 display the radial structure of the real
and imaginary parts of the scaled magnetic field b corresponding to the lb] curves shown in Fig. 5 for <= rri2, and
e=27r, The radial coordinate has been expanded to illustrate
the sharp changes across the cone boundaries.
Figure 9 exhibits, for the small disk case (p= l), the
axial dependence corresponding to the radial features displayed in Fig. 5. It shows the real and imaginary parts of b,
as well as the amplitude I bl, for two different radial positions
outside the shadow of the disk exciter (r/a=3, 10). The pattern in Fig. 9 consists of an axially propagating wave whose
amplitude changes upon crossing the intersection with the
0.4
Ibl
b
0.0
r/a
5
rfa
FIG. 6. Radial dependenceof magnitudeof scaledmagnetic field for different axial positionsaway from exciter for a disk with p =4 (larger than that of
Fig. 5).
3770
Phys. Plasmas, Vol. 1, No. 12, December 1994
FIG. 8. Radial dependenceof real and imaginary parts of scaled magnetic
field correspondingto the 161 curve for p= 1 and k,Z=2n in Fig. 5.
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0.2
cones .‘$
,“‘*“..“““‘....“.““‘
.Ibl
0.2
0.0
- 0.2
n
E
2Ji 0.4
0.0
n
!i
&i -0.2
2
0.2
0.0
0.0
-0.4
0
5
10
15
20
25
-0.2
30
0
15
kG
30
kAZ
FIG. 9. Axial dependenceof the real and imaginary parts of scaledmagnetic
field for two radial positions outside the disk shadow.p = 1.
cone boundaries. In this axial cut the outer cone is crossed at
a value of z smaller than that for the inner cone. The decay in
amplitude seen at r/a=3
corresponds to crossing from region III into region II of Fig. 3, where the radially spreading
diffraction pattern causes lb\ to decrease.
The axial wave number of the propagating signal shown
in Fig. 9 has a value which is about 10% larger than k, . This
behavior results from summation over the k, spectrum of
the disk, with each component contributing a kll= k,
[ 1 +(k,lk,)*]“*.
The axial behavior corresponding to the larger disk
@=4) examined in Fig. 6 is shown in Fig. 10, for the same
scaled radial locations considered in Fig. 9. It is seen now
that for r/a = 10, the signal propagates without axial decay.
The reason is that this position is in region IV where the field
FIG. 11. Effect of moderate collisionality on axial dependenceof real and
imaginary parts of scaled magnetic field. Radial position is r/a= 10 for
p=l; r=v,/w.
falls off as l/r for all z. The pattern obtained for r/a =3 also
contains a part of this behavior in the region of small z.
However, as the cones are crossed, the signal builds up and,
eventually at large z, decays since the pattern corresponds to
region II.
V. DISSIPATIVE EFFECTS
A. Collisions
In this section we discuss the modifications produced by
electron collisions on the patterns generated by the collisionless skin depth. We focus our attention on the major features
arising from rll in Eq. (30), i.e., due to resistive dissipation of
the axial currents. The damping associated with cross-field
drifts, i.e., due to r, in Eq. (30), is in general smaller and
can be easily incorporated by making 5 a complex quantity.
Figure 11 illustrates the effect of moderate collisionality,
I?ll=O. 1, on the axial propagation for a p = 1 exciter. It is seen
by comparison with the rll=O pattern that the sharp phase
changes across the cone structures are smoothed out, but
their presence is still evident. The magnitude of the signal,
1b I, does not exhibit a significant axial decay at this level of
collisionality.
The effect of collisional dissipation on the diffraction
pattern can be described in the limit of 5 +l by modifying
Eq. (43) according to the prescription of Eq. (30), i.e.,
p*+-i
(i+rII)
p2,
(46)
2
1+rII
~....,....,....,....,....,....I
0
15
kAz
30
which results in the expression
pt
b(‘+l,p)+(l-e-*s(‘+rll)),
P
FIG. 10. Axial dependenceof the real and imaginary parts of scaled magnetic field for two radial positions outside the disk shadow.p =4
Phys. Plasmas, Vol. 1, No. 12, December 1994
P2P2
s=4g-(r;f+iy
(47)
For arbitrary values of I’ll the magnitude of the scaled magnetic field far from the exciter is given by
Morales, Loritsch, and Maggs
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Ibl
Ibl
0.1
0.0
0.0
15
0
15
rta
r/a
FIG. 12. Transition from skin-depthdivergencedeterminedby cone angle to
diffusive spreaddue to collisions at k,z. = 3?r/2 for a p = 1 exciter; I’= v,/w.
IhI+;t?-Srli[ sin* s + sinh*( srll)]
FIG. 13. Effect of increasing collisionality on the asymptotic diffraction
pattern for a p = 1 exciter at k,z = 8% lY= v&o.
still at the same radius as predicted by the collisionless result. For I’ll=5, the behavior is entirely diffusive, as predicted
I”,
by Es. (49).
which reduces to Eq. (44) for I’ll-to.
The role of moderate collisionality (i.e., Fll-0.1) on the
diffraction pattern is seen from Eq. (48) to produce two effects: a decay in the overall amplitude of the signal and a
filling-in of the sharp minima. The reduction in amplitude
represents the loss of energy while the filling-in is due to
radial diffusion of the current channel.
Equation (48) also recovers the limit investigated by
Borg et a1.,9 i.e., strong collisionality (in their displays
r/l-30). In this limit
The sensitivity of the magnetic field patterns in Figs. 12
and 13 to the value of l?ll makes them useful in determining
the electron collision frequency, i.e., they can be used as a
diagnostic tool. For this purpose it is useful to consider the
p4 1 behavior of lb I, i.e., the linearly increasing portion near
the axis of the exciter. In this region Eq. (48) yields
I+
p*p
I-exp(
-$$)I,
which can be put in the form
Phys. Plasmas, Vol. 1, No. 12, December 1994
Ibl)-*-l]*".
(511
(49)
which represents a purely diffusive spread of the current
channel.
The transition from skin-depth divergence, determined
by the cone angle of Eq. (41), to the diffusive spreading
caused by parallel resistivity is illustrated in Fig. 12 for p = 1
and axial location k,z =3~/2, i.e., before the diffraction pattern develops. It is seen that the current channel does not
spread significantly beyond the radial location of the outer
cone for r,lGl. For rll=S, however, the cone features are
destroyed by radial diffusion and the maximum value of the
magnetic field occurs at a radius beyond the outer cone. It is
found from direct comparison that the prediction of Eq. (49)
is indistinguishable from the curves in Fig. 12 for rl125. For
l?ll= 1, the qualitative shape predicted by Eq. (49) holds, but
there are observable deviations from the exact curve shown
in Fig. 12.
The effect of increasing collisionality on the asymptotic
diffraction pattern is illustrated in Fig. 13 for p = 1 and axial
position k,z=8m Again, it is seen that for rll=l the interference effects disappear, but the location of the maximum is
3772
(50)
2~(rl;ff)*'*'
q=[($$
b--+G[
30
Equation (51) can be used to yield a range of independent values of ve by measuring the radial slope of lb]. In
practice, the measured quantity, 1~~1,is not absolutely normalized, so one needs to compare relative changes in slope
for different values of 4. This can be accomplished by altering &k,z
in two different ways, either by changing the
frequency, or by changing the axial location.
B. Landau damping
Another source of dissipation is the resonant acceleration of electrons by the parallel electric field, i.e., Landau
damping. This kinetic effect can be incorporated into the
analysis by replacing the fluid expression for the electron
current, given by Eq. (1 l), with the expression obtained from
the linearized Vlasov equation,
*-
JZ-
*
”
(52)
where 6 is the electron thermal velocity, 5= ONcllU,and Z’(J’)
is the derivative of the well-known plasma dispersion funcMorales, Loritsch, and Maggs
Downloaded 11 Aug 2006 to 128.97.43.7. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp
tion. Using the same model of disk coupling to plasma current used to obtain Fq. (21), it is found that the spatial dependence of the magnetic field is again given by Eq. (25).
Now, however, the dependence of the parallel wave number
on k [i.e., +1(k)] is found by solving the dispersion relation
(53)
In the general case, the roots of this equation are complex, so
that the parallel wave number has both a real and imaginary
part. The imaginary part, of course, represents the asymptotic
(i.e., within one wavelength of the exciter) spatial Landau
damping of the radiated wave. A discussion of the spatial
features resulting from Landau damping is found in the companion experimental paper.*
VI. CONCLUSIONS
The results presented in this analytical study demonstrate that shear Alfven waves having transverse scale on
the order of the skin depth exhibit a collisionless divergence
determined by propagation cones that emanate from the
edges of the exciting structures. The cone angle is given by
the expression tan 8=(mlM)“2( ~/a,)[ 1 - ( o10J2]-‘”
in
which the (w/Q)~ dependence is a direct consequence of the
overwhelming effect of axial electron currents for small
transverse scales, as discussed in the Appendix.
The axial currents induced by a small disk exciter have
been found to exhibit a collisionless radial expansion associated with the skin effect. The radial expansion of the current
channel is bounded by the cone trajectories. At axial distances on the order of a few parallel wavelengths, the collisionless radial expansion gives rise to an azimuthal magnetic
field whose shape is analogous to that of a diffraction pattern, as exemplified by the bottom two curves in Fig. 5.
The inclusion of parallel electron resistivity allows the
parallel current channel to diffuse radially beyond the collisionless boundary set by the outer cone trajectory, i.e.,
r/a = (k,zlp) + 1. For moderate values of electron collisionality (i.e., u,Iws l), it is found that the cone features and the
diffraction pattern remain well resolved, as demonstrated by
Figs. 12 and 13. For values of u,Iw slightly larger than unity,
the diffusive behavior becomes dominant and results in the
patterns predicted by Eq. (49), which have been the focus of
attention of earlier theoretical and experimental studies.“”
The sensitivity of the spatial pattern of small-scale Alf&n waves to the electron collisionality suggests the possibility of a useful diagnostic which allows the determination
of Z&J through Eq. (51).
The experimental studies performed in the LAPD facility, reported in a companion paper,8 are able to sample several of the general properties of the theoretical study presented here. Primarily the experiments are able to sample the
channel divergence along cone trajectories, and clearly establish the outer l/r region. The scaling with w and p have been
well documented. As of this writing, however, the asymptotic
diffraction pattern has not been sampled due to the small
collisionality (l?ll<O.l) and large axial distances required,
i.e., about four Alfvdn wavelengths.
Phys. Plasmas, Vol. 1, No. 12, December 1994
Direct experimental observation of features in the radiation pattern associated with cone-like propagation of the
shear Alfvin wave as illustrated in various figures in the
manuscript is difficult because dissipative effects smooth o&
the sharp features associated with the edge of the disk as
illustrated in Figs. 1 l- 13. The most direct method to find the
location of the outer Alfven cone in experimental data is to
identify the radial position at which the r-’ decay in the
radiated magnetic field pattern begins. However, it must be
emphasized that the entire spatial pattern of the magnetic
field radiated by a disk is determined by the cone-like propagation of Alfven waves, and not just the sharp features associated with the edge of the disk. Therefore, the most meaningful test of the cone-like propagation properties of the
shear wave consists of detailed, quantitative comparisons of
the predicted spatial profile of the radiation pattern with experimental observation as a function of frequency. Consequently, the purpose of the arrows labeled “cones” in various
figures is to help in the visualization of the path followed by
the boundary that encloses the current channel, and should
not be interpreted as the only manifestation of cone-like behavior.
ACKNOWLEDGMENT
The authors thank Professor W. Gekelwn for valuab&
discussions of the measurements made in the LAPD facility
at the University of California at Los Angeles.
This work is sponsored by the Office of Naval Research
and the U.S. Department of Energy.
APPENDIX: FINITE FREQUENCY EFFECTS
In describing shear Alfvin waves whose frequency o is
comparable to the ion gyrofrequency, ni, the question tises
as to what is the importance of the relative slippage between
the EXB, drift of the electrons and that of the ions. As is
well known for akin 1, the two charge species drift together and no net current results from this flow. In this limit,
the perpendicular plasma current is entirely due to the ion
polarization current. However, for finite WIsli an additional
contribution to this current arises from the EX&, slippage.
The question of relevance to the present study is the following: What effect does this extra current have for Alfvdn
waves whose transverse scale is on the order of cIwpe? In
this limit a significant parallel electron current is present and
competes with the slippage current due to finite dfii.
To illustrate the competition between these two effects, it
is useful to examine the exact dispersion relation for the
shear Alfvin wave for a uniform, cold plasma,
(Al)
K2+K2X2(l-W2)
2
+Wz(lfKz),
(A21
where W=Wl.ni, kAo=WIUAo, UAo=CslilOpi, K=klCItipe,
X=k,lkAo=(Mlm)“2/W;
kll and k, are the wave numbers
parallel and perpendicular to the ambient magnetic field.
Morales, Loritsch, and Maggs
3773
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In the limit K-+0, i.e., zero skin depth (and correspondingly no contribution from the parallel electron current),
(Al) reduces to
2
kAO
&(l+w)=-
1-W’
(A3)
It should be noted that in this limit the denominator contains
a term W and not W2. The W2 dependence is canceled by the
1 + W term that arises from the inclusion of the relative slippage between the ions and electrons. Mathematically, the
dominance of the slippage effect implies F-+ W2 in the dispersion relation.
Next we consider the other extreme case, W+O, but K
finite. This corresponds to no EXB, slippage, but retaining
the contribution of the parallel electron current. In this limit,
(A 1) yields
k+ki,(
I+
K2),
(A4)
which mathematically implies that
F--t
K2+K2h2(1-W2)
2
i
2
1.
(A5)
Having identified the two opposite limits of importance
for these two currents, we then determine the condition for
dominance of the finite skin-depth currents at finite values of
w/f&. This requirement implies that in the quantity F given
by 642)
K2+K2X2(1-W2)
2
%-W2( 1 -FK2),
(A61
which can be solved for a condition on K (small) that implies
3774
Phys. Plasmas, Vol. 1, No. 12, December 1994
klc d-2m
-s
WPe
M[
( WlL2i)3’2
1 - (oln,)y
*
This condition is well satisfied for the physical situation of
except for a
interest in the present study, i.e., k, --w~Jc,
narrow frequency interval close to fii. Consequently, the
correct dispersion relation for the structures investigated here
is
kAo2] ,2[l+($)2].
k~f=[l-(dq
’
648)
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Morales, Loritsch, and Maggs
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