ADAlO7 269 NAVAL RESEARCH LAB WASINGTON DC F/6 4/1THE LOWER-HYBRID-DRIFT INSTABILITY IN NOW-ANTIPARALLEL REVERSED--ETC (U)OCT 81 J D HUBA, N T #LADD, ~J F DRAKE

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CONTENTS

1. INTRODUCTION ................. 1

Ii. THEORY ..........................................

A. Assumptions and Plasma Configuration .............. . 5. . 6

B. Dispersion Equation ............................... 10C. Application to Reversed Field Plasmas .................. 1

III. DISCUSSION .......................................... 16

ACKNOWLEDGMENTS ..................................... 18

APPENDIX ............................................... 19

REFERENCES.......................................

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THE LOWER-HYBRID-DRIFT INSTABILITY IN NON-ANTIPARALLEL REVERSED

FIELD PLASMAS

I. INTRODUCTION

An important set of problems in plasma physics consists of

understanding the physical processes which can occur in reversed

field plasmas. For example, how can magnetic field energy be

rapidly converted into particle energy? How does the topology

of the magnetic field change? Under what conditions can

topological changes occur? An enormous amount of work has been

devoted to these (and related) questions over the past 20 years.

Generally speaking, this research has been focused on the inves-

tigation of magnetic field reconnection processes. Some parti-

cular topics of interest have been field line annihilation (e.g.,

1D Sweet-Parker models); forced reconnection (e.g., 2D Petschek

model); and tearing instabilities. In an astrophysical context,

this research has been relevant to a variety of space phenomena

such as solar flares, interplanetary D sheets and geomagnetic

substorms.

One process which can be important in reconnection physics

is plasma microturbulence. In general, plasma microinstabilities

can often produce anomalous transport of particles, momentum and

energy. This can be critical to reconnection process, especially

in the null region (B - 0), since it allows the plasma to

"decouple" from the magnetic field. Moreover, anomalous trans-

port effects can greatly enhance the rate of energy conversion

from the magnetic field to the plasma. A variety of micro-

instabilities which can lead to fine scaled turbulence have been

analyzed to determine their relevance to reconnection (see

Ibnuseript submitted July 17, 1981.1

Papadopoulos (1979) for a review). However, a drawback of many

of these analyses is the assumption of a one-dimensional magnetic

field (i.e., B = B z(x)ez); an assumption not usually justified in

space plasmas. Other components of the magnetic field leads to

the following effects. First, a component of B normal to the

neutral line (i.e., B = B eX + B e ) introduces field line cur-

vature. This effect, which is generally incorporated in micro-

instability theories via an artificial gravity, leads to addition-

al particle drifts and can be either a stabilizing or destabiliz-

ing influence depending upon the plasma conditions. Secondly,

a component of B parallel to the current (i.e., B = Byey + Bze z)

introduces magnetic shear. Physically, this corresponds to the

situation where the magnetic fields on either side of the neutral

are not anti-parallel. Magnetic shear leads to Landau resonances

of particles and waves, and the coupling of cross-field modes to

parallel propagating modes. Magnetic shear is generally a stabi-

lizing influence on instabilities. Since the magnetic shear in-

duced Landau resonances and parallel mode couplings are strongly

dependent on spatial position, the analysis of this effect re-

quires a nonlocal theory. In contrast, the magnetic curvature

induced particle drifts are not strongly dependent on spatial

position and can be accurately analyzed with a local theory.

In this paper we analyze the effect of magnetic shear on the

lower-hybrid-drift instability and discuss some of the implica-

tions of magnetic shear as it regards the dynamics of reversed

field plasmas in the magnetosphere. We choose the lower-hybrid-

2

drift instability since it is the most likely instability to be

excited in reversed field space plasmas of interest (i.e., the

earth's magnetopause and magnetotail (Huba et al., 1978; Gary

and Eastman, 1979)). Moreover, the anomalous transport properties

associated with this instability can be important to reconnection

processes (Huba et al., 1977; Drake et al., 1981; Huba et al.,

1980). We focus here on the effect of magnetic shear on the

lower-hybrid-drift instability since it has been shown to have

a strong stabilizing effect on this mode [Krall, 1978]. We will

not consider here the effect of field line curvature since it has

a much weaker influence on the lower-hybrid-drift instability

(Krall and McBride, 1977; Rajal and Gary, 1981).

A self-consistent theory of the lower-hybrid-drift instabi-

lity in a sheared, reversed field plasma is difficult to develop.

Major complications arise because of finite plasma 8 and electron

temperature effects (i.e., electromagnetic coupling, electron VB

drift-wave resonance, finite electron Larmor radius effects).

The inclusion of these effects, within the context of a weakly

nonlocal stability analysis, leads to three coupled, partial

differential equations involving complex velocity integrations.

At this early stage of analysis, we do not attempt such a com-

prehensive theory. Rather, we use a simpler theory here to ex-

amine the dominant effect of magnetic shear on the lower-hybrid-

drift instability with emphasis on understanding the implications

of this mode for reversed field plasma dynamics. Specifically,

we simplify the analysis by using electrostatic theory with

cold electrons. However, we do include an Appendix which

3

discusses the effect of electromagnetic coupling and electron VB

drift-wave resonances in the shear stabilization criterion. A

more comprehensive theory will be presented in a future report.

The scheme of the paper is as follows. In Section II we

present a general discussion of the lower-hybrid-drift instabi-

lity and the effect of magnetic shear on it; showing specific

results of our analysis, i.e., a marginal stability curve show-

ing the stabilizing influence of shear. We also apply these re-

sults to a particular sheared, reversed magnetic field configu-

ration. In Section III we discuss the implications of these re-

sults on reconnection processes in the earth's magnetosphere.

Finally, an analysis incorporating finite 8 effects is presented

in the Appendix.

4

U. THEORY

A. Assumptions and Plasma Configuration

The plasma configuration and slab geometry we consider is

described as follows. The ambient magnetic field is in the y-z

plane (B = B ey + B e ) and is only a function of x. The densityydyenit

is also a function of x but we take the ion temperature to be

finite and constant. For simplicity, we take the electron tem-

perature to be zero (i.e., T e/Ti 0). Finite electron temperature

effects are discussed in the Appendix. Equilibrium force bal-

ance on an ion fluid element in the x direction requires Viy=Vdi

where Vdi = (vi2/22i)a in n/ax is the ion diamagnetic drift velo-

city. Here, vi = (2Ti/mi) is the ion thermal velocity and

Qi = eB o/mic is the ion Larmor frequency. We can relate the

ion diamagnetic velocity to the mean ion Larmor radius and scale

length of the density gradient by Vdi/Vi = rLi/ 2Ln where

r Li m v./2 and L n= (a a n/ax) . The electrons are assumed to

be magnetized, while the ions are treated as unmagnetized. This

is reasonable since, in treating the lower-hybrid-drift instabi-

lity, we are considering waves such that a i << w << 9e and

k2r Li2 >> 1. Only electrostatic oscillations are considered

(electromagnetic coupling is discussed in the Appendix) and we

assume that the plasma is weakly inhomogeneous in the sense that

r 2 (a in n/ax)2 << 1 and r Le2 (a k B/aX)2 << 1. We assume thatrLe L

the local equilibrium magnetic field is

B(x) = B (x (e + (x-x )/Lsy) ()0 0 Z 0 s3y

in the vicinity of x = x0 (i.e., (x-x )/L 8< 1) where

5

Ls = (O/Dx)-land 0 = tan-(ByBz). Thus, L is the scale length

characterizing the magnetic shear.

If, for the moment, we consider B = Boe z , then the plasma

configuration just described is unstable to the kinetic lower-

hybrid-drift instability when 1 >Vdi/vi >(me/mi) (Davidson et

al., 1977). The instability is driven by the cross-field cur-

rent and is excited via an ion-wave resonance (i.e., inverse

Landau damping). The waves are characterized at maximum growth

by y d h' - k V k P 1 and-k.B = 0 where

= (Ti/me)Q e For modes such that k.B # 0 (i.e., k1 0Pes = /m ) eI;# 0),

electron Landau damping reduces their growth rate or stabilizes

them, depending on the magnitude of k,,. We now let B = By(x)e +^y

Bzez as in Eq. (1), which introduces magnetic shear as shown in

Fig. 1. The magnetic field rotates in the y-z plane as a func-

tion of x so that k is also a function of x. At x = x0 we note

that k 0 (k.B = 0) but at x = xl, ki, # 0 (k.B # 0). Thus, the

dispersive properties of the plasma are also a function of x

(e.g., electron Landau damping and parallel mode coupling can

occur at x1 but not at x ). Making use of Eq. (1), we use the

prescription kz (x) = kzo + ky (x-x )/Ls to incorporate magnetic

shear into the analysis where we choose kzo = kz (x=x 0 0.

B. Dispersion Equation

Within the context of the assumptions outlined in the pre-

vious sub-section, the equation which describes the lower-hybrid-

drift instability in a sheared magnetic field is given by

[Davidson et al., 1978].

6

F--

A q2O + [B-C(x-xo)j2 , 0 (2)ox2

where

k y A k2Vi 2 [k es - Z(Ci) (3)

2

B [k_2P 2 _ Wi + 1+ Z(r (4)

2 L 2(5)s

and w 2 = 41Tne2/m vi2 = 2Ti/mi' Pes = (T i/mee 2),

Vdi = (vi 2/20i )o 2n n/ax and Ci = (w-kyVdi)/kv In writingk2XD 2 < an>~p

Eq. (2) we have assumed 2 and wpe here

xD 2 = vi2/2wpi 2

Equation (2) is in the form of Weber's equation and the

eigenfrequency is defined by

B = (2m + 1) (AC)h (6)

where m is the mode number (i.e., m = 0, 1, 2, ... ). The branch is

chosen according the outgoing energy prescription of Pearlstein

and Berk (1969). The associated eigenfunction is

= 'o Hm(ux)exp[ - a 2x 2/2] (7)

where a = (C/A) and Hm is the Hermite polynominal of order m.

In general, a numerical analysis is required to solve Eq. (6)

(using Eqs. (3)-(5)). However, we first consider a limiting case

to obtain an analytical solution to Eq. (b).

8

We consider the weak drift limit, Vdi << vi , so that the Z

functions in Eqs. (3) and (4) can be approximated by Z(Ci) i/.

The dispersion equation then becomes

2 2 kyVdi [ w-kyVdD(w,k) = 1 + k2 2e yd + i Ivr(kv ii) +kkdi 1

= es - w k i /+ k~iw ky~

k pes (2m + 1)] = 0 (8)I(8)

The first imaginary term is a destabilizing term due to inverse

ion Landau da-iping. The second imaginary term is the stabilizing

effect of magnetic shear. It's origin in the analysis is a termk

-(:1)2 in the magnetized electron response. Physically, magnetic

shear leads to stabilization since it allows wave energy to propa-

qate away from the excitation region (i.e., where kl = 0).

The real frequency is given by

2 2r = kVdi/( + k Pes (9)

where shear corrections to wr have been neglected. The mode

is stabilized when Im D(w,k) = 0 or

LS v di k2 2(Ln\ _ __ Vdi kpes

L - (M ) v (+k2Pes 2 ) 2 (10)

where Eq. (9) has been used and the subscript cr refers to the

critical value of L n/L . The maximum value of the RHS of Eq.22

(10) occurs for k Pes = 1.0 so that all wavenumbers are stable

whenLn > /-T (2m+l)-i Vdi l > - (v i

rLi\L- - r >m2m l (r L n

9

Note that the higher order modes (m ' 0) are more easily

stabilized by shear than the lowest order mode (m = 0) (i.e.,

L (m # 0) > L (m = 0).s s

We now relax the weak drift assumption and solve Eq. (6)

numerically. Figure (2) is a plot of (L n/L s ) vs. Vdi/v i for~ cr d

W pe2 /e2 = 100, 0 = 0.0, m = 0 and Te = 0. The growth rate is

maximized with respect to kpes so that the maximum value of

(L n/L s ) necessary to stabilize the mode is obtained. Twonscr

regions are labeled: stable (L n/Ls > (L n/L s ) ) and unstablen s n cr

(L n/Ls < (Ln/Ls) ). For Vdi/v i = 1.0 we find that (Ln/L s )n cr cr

0.325 which, surprisingly, is in good agreement with our

analytical result [Eq. (11)] even though the drift is not weak.

C. Application to Reversed Field Plasmas

In order to apply these results to a reversed field plasma,

we consider the following magnetic field profile

B= B [sin ey + tanh e (12)

Equation (12) describes a reversed field plasma (i.e., the z-

component reverses direction at x = 0) with the field undergoing

a total directional change n + 8 in the y-z plane. Thus, if

6 = 0 then the standard one-dimensional Harris profile is

recovered (i.e., anti-parallel field) with a discontinuous

rotation at x = 0. On the other hand, for 0 p 0 the field con-

tinuously rotates in the y-z plane and remains finite every-

where. This profile (Eq. (12)) is a rough approximation to the

10

0.4-

0.3

STABLE

0.2

UNSTABLE

0.1

////

/I I I

0.25 0.50 0.75 1.00

Vdi/V i

Fig. 2 - Plot of (L./L,)cr vs. VdI/Vi for wPe21ae = 100 and Te 0. Thelower-hybrid-drift instability is stable (unstable) for L,/L. >L/L,) (L n / L,< (Ln/li)cr). The curve is dashed for VdI/VI < (me/m)4 f 0.16 because the ionsare magnetized in this regime and our theory, strictly speaking, is not applicable.

11

magnetic field of the nose of the magnetosphere of the earth.

It also approximates the distant geomagnetic tail when no normal

field component is present.

We plot L n/Ls vs. x/X for several values of 0 (a = 50, 100,

15-, 200, 25', 30', 450) in Figure 3. We note that Ln/Ls varies

2-3 orders of magnitude over the range of x/X shown. Also, as

increases so does L n/Ls , as anticipated. In the region

x/A ! 0.25, the shear is strong (i.e., Ls 2 Ln) and the theory

discussed in this paper is not adequate. Thus, we restrict

our attention to regions such that Ln > Ls .

We combine the results of Figs. (2) and (3) in Fig. 4

which plots 0 vs. xp/X for several values of Vdi/v i (Vdi/v = 0.25

0.50, 1.00 which correspond to X/rLi = 4.0, 2.0, 1.0, respectively).

Here, x /X represents the linear penetration distance of thep

lower-hybrid-drift instability. That is, we expect the modes to

be stable for x < xp because of shear stabilization. We find

that (1) as the ion diamagnetic drift increases (i.e., the cur-

rent sheet becomes thinner) the mode can penetrate closer to the

"null" region (i.e., x - 0) and (2) as 0 increases, which

increases the magnetic shear, the penetration distance x p/

becomes larger. Nonlocal analysis of the lower-hybrid-drift

instability in a field reversed plasma (Huba et al., 1980) has

found that the dominant mode is localized at a position

x/A 1.25 for Te = 0. Thus, even for e = 450 the fastest

growing mode is not stabilized due to shear. However, higher

order modes, which penetrate closer to x = 0, are expected to be

12

10-

L5 10-1-

45- 30

25- 20

15

10-2 10

5c

0.25 0.50 0.75 1.00 1.25 1.50

XxFig. 3 - Plot of L,1/L5 vs. OAX for the magnetic field profile given by Eq. (12) andseveral values of 0. Note that the shear parameter Ln L varies over 3 orders ofmagnitude.

13

450-

40°-!Vdi

Vdi 1. W =0.25v- 1.00 V.350- i =0.50

300- STABLE

250-

0ol

200- UNSTABLE

150 -

100 -

50

0.25 0.50 0.75 1.00 1.25 1.50

xP/Xxp/

Fig. 4 - Plot of 8 vs. x A for Vdi/Vi = 1.00, 0.50, 0.25. Here, 0 represents the

strength of the magnetic shear in a reversed field plasma (i.e., 0 - 0: no shear,

fields anti-parallel; 0 * 0: shear, fields non-antiparallel) and xp/X is the linear

penetration distance of the lower-hybrid-drift instability to the neutral line. The

mode is linearly stable (unstable) in the region x <xp (x >Xp). As the amount of

shear increases, i.e., 0 becomes larger, the mode is stabilized further away from

the neutral line (x = 0).

14

affected by shear. These results are, however, highly profile

dependent. If the dominant shearing of the field occurred in

the region where the LHD was localized, rather than in the null

region (the situation implied by Eq. (12)), then the stabilizing

effect of shear would be much more pronounced.

15

II. DISCUSSION

The purpose of this brief report has been to study the in-

fluence of magnetic shear on the lower-hybrid-drift instability

in reversed field plasmas. As mentioned in the introduction,

microturbulence associated with such modes can play an important

role in the dynamic evolution of a reversed field plasma via

its anomalous transport properties. The lower-hybrid-drift in-

stabiltiy, although linearly stable in the field reversal region

near the neutral line (Huba et al., 1980), does play a dramatic

role in the evolution of an anti-parallel field reversed plasma

(Drake et al., 1981; Winske, 1981; Tanaka and Sato, 1981). It

has been shown both theoretically and by computer simulations

that the mode causes magnetic flux to diffuse towards the neutral

line which leads to an enhanced current density at the neutral

line. Eventually microturbulence penetrates to the null region

and permits field line reconnection/annihilation to occur, there-

by dissipating magnetic energy. In the case where the reversed

field is non-antiparallel (i.e., the field is sheared) we have

found here that the lower-hybrid-drift instability will be

linearly unstable further away from the neutral line than the

case where the field is completely anti-parallel (Fig. 4). Thus,

magnetic shear has an inhibiting effect on the penetration of

the lower-hybrid-drift mode toward the neutral line. We anti-

cipate that this means that the evolution of a non-antiparallel

reversed field plasma will differ from that of an anti-parallel

one. For example, if the shear is sufficiently strong the mode

16

may take substantially longer to penetrate to the neutral line

or may not penetrate at all. This conjecture, of course, is

based on our linear analysis and must be substantiated by further

(nonlinear) analysis. Also, we reemphasize that the results

presented in this paper are not comprehensive since we have

neglected important finite 8 and Te effects (i.e., coupling of

electromagnetic and electrostatic fluctuations and VB resonances).

Nonetheless, our results are qualitatively correct although they

can be improved quantitatively. We are presently developing

a more comprehensive theory for finite B and Te plasmas and

will report our results in a future publication.

Two regions of the magnetosphere in which the magnetic

field can reverse direction (in one component) and is also sheared

are the nose and magnetotail. In both these regions reconnection

processes are believed to occur and can be important to the dyna-

mic interaction of the solar wind and the magnetosphere. In the

case of the nose, the angle between the incoming IMF and the

earth's geomagnetic field varies from 0 to r. A thin magneto-

pause boundary layer exists (whose width is roughly r i) overLi

which the B field undergoes a directional change. In the magne-

totail it is known that a substantial crosstail magnetic field

(By 15y) can exist at tines [Akasofu et al., 1978]. Such a

magnetic field can introduce a strong shear, i.e., the magnetic

field undergoes a rotation as one passes from the north to south

lobe.

Crooker (1978) has suggested that the nose reconnection only

17

occurs in regions where the IMF and geomagnetic field are anti-

parallel (no shear). Classical reconnection theories predict

reconnection even if the fields are non-antiparallel [Ugai,

1981] so that some Nanomalous" process may be responsible for

inhibiting reconnection in this instance. Moreover, for recon-

nection events observed at the nose, the time scale for the

energization of the plasma appears to be slower for more strongly

sheared cases (non-antiparallel fields) than for non-sheared

fields (anti-parallel fields) (Russell, private communication).

We suggest one possibility for such an effect is the influence

of magnetic shear on the lower-hybrid-drift instability (or other

inqtabilities) and its associated microturbulence.

ACKNOWLEDGMENTS

This research has been supported by ONR and NASA

18

APPENDIX

We derive an expression for (Ln/Ls) which includes finiten cr

0 effects (i.e., electrostatic-electromagnetic coupling, elec-

tron VB drift-wave resonance). In order to make the analysis

tractable, we consider the limit 0i << 1 and Te << T i The non-

local dispersion equation is given by [Davidson et al., 1977]

A 2 + [B-C(x-x ) ] = 0 (Al)X 2 0

where

2A =-1 pe (A2)

k y2 ne 2

2w *2 (1r 2 kVdi(B I I+ kp 2 +l~~-k2 2 es c 2k 2 ) 2

1

2 2 -i

C wpe 1pe (4C 2 Ls2 1+c2k2(M

2 42 2 2where w pa= 4nea/alv 2 = 2T alma, Pes2 (T i/me)/"e,

Oi = 8 nT1/B 2 ' Vdi = (vi 2/2Qi)a 2 n/ax, se = r 2/Ve 2 , Pi = (w-

2kVdi)/kv i , vr = 20ew /k yB and cB = a n B/ax. The finite 8

corrections included in (Al) are as follows. First, the coupling

of electromagnetic and electrostatic perturbations arises from

terms proportional to wpe2 /C2k2. The 0 dependence can be seen by

19

noting that k2pes 2(Wpe 2/C k2 ) i/2. Second, non-resonant

electron VB effects are contained in the term proportional to

i (kVdi/w). This is a fluid-like response of the electrons due

to the inhomogeneous magnetic field. Finally, resonant electron

VB effects are contained in the final term of B in Eq. (A2).

Equations (A2)-(A4) are similar to those derived by Davidson et

al. (1977) but, in addition, contain electron resonance terms

(Huba and Wu, 1977).

We now assume kvr/Qe >> 1 which corresponds to

2r eOi_ -< kpes 1 and rewrite Eqs. (A2) - (A4) as follows

W2

A L p (A5)

y e

2 -i

+ i YITk i +T ex( A6

w Le 2 2k2pes2 (A7)

The dispersion equation is again given by Eq. (1). The real

frequency is the same as Eq. (9) but with e replacing pes

where Pes = Pes/1(1 + Pi/2). To lowest order in Oi, the criti-

cal shear length is

20

(Ln) V di k2 2_2 T ~ s eexpcr 7i 1+k 2P 2(2 2 2es es l+kpes

(AS)

where s e can be written as

2 1S- (A9)

Maximizing (L n/L ) with respect to wavenumber yieldscr

(n) = di v i ( +) l( Ti)2 exp(-i/Oe)Ls r - 4 4 T T e (20i) (AI0)

where kp es kpe s = 1.

Two interesting points concerning Eqs. (A7) and (A9) are

the following. First, the finite Oi dependence in the first

term of Eqs. (A7) and (A9) arise from the electromagnetic cor-

rection due to 6AI1 (i.e., the transverse magnetic field fluctua-

tions). The influence of this correction is to increase the

amount of shear necessary to stabilize the mode (Davidson et

al., 1978). That is, as 0i increases then the shear length Ls

necessary for stabilization decreases so that the mode is

harder to stabilize. Physically this occurs because the fluc-

tuating electric field associated with 6A inhibits free stream-

ing electron flow along the magnetic field which reduces the

rate at which energy can be convected away from the localization

region (Pearlstein and Berk, 1969), Secondly, the second term

in Eq. (A8) represents the resonant VB correction which is a

21

=_Z

damping effect. This term tends to decrease the amount of

shear necessary to stabilize the mode. Thus, the finite a cor-

rections have different influences on the shear stabilization

criterion. Loosely speaking, electromagnetic effects are de-

stabilizing (i.e., a stronger shear is needed to stabilize the

mode from the =0 situation) while the resonant VB effects

are stabilizing (i.e., a weaker shear is needed to stabilize

the mode from the =0 situation). As to which effect dominates,

a more careful analysis is needed which we are presently develop-

ing and will be presented in a future report.

22

REFERENCES

Akasofu, S.-I., A.T.Y. Lui, C.I. Meng and M. Haurwitz, Need

for a three-dimensional analysis of magnetic fields in the

magnetotail during substorms, Geophys. Res. Lett. 5, 283,

1978.

Crooker, N.U., The half-wave rectifier response of the magneto-

sphere and anti-parallel merging, J. Geophys. Res. 85,

575, 1980.

Davidson, R.C., N.T. Gladd and Y. Goren, Influence of magnetic

shear on the lower-hybrid-drift instability in toroidal

reversed-field plasmas, Phys. Fluids 21, 992, 1978.

Drake, J.F., N.T. Gladd and J.D. Huba, Magnetic field diffusion

and dissipation in reversed field plasmas, Phys. Fluids

24, 78, 1981.

Gary, S.P. and T.E. Eastman, The lower-hybrid-drift instability

at the magnetopause, J. Geophys. Res. 84, 7378, 1979.

Huba, J.D. and C.S. Wu, Effects of a magnetic field gradient

on the lower-hybrid-drift instability, Phys. Fluids 19,

988, 1976.

Huba, J.D., N.T. Gladd and K. Papadopoulos, The lower-hybrid-

drift instability as a source of anomalous resistivity for

magnetic field line reconnection, Geophys. Res. Lett. 4,

125, 1977.

Huba, J.D., N.T. Gladd and K. Papadopoulos, Lower-hybrid-drift

wave turbulence in the distant magnetotail, J. Geophys. Res.

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23

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24

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