The growth of electrostatic waves near the plasma frequency (fp) due to an unstable electron beam is investigated by solving the unmagnetized electrostatic dispersion equation numerically. The numerical solutions are compared with analytic theories for reactive (or fluid-type) and kinetic versions of the beam instability, and for the O'Neil/Malmberg connection of the beam and Langmuir modes. Conditions for growth significantly above or below fp are given. Three general results are found: (1) The unstable waves do not grow on a mode with Langmuir dispersion except in the limit of a very dilute beam with growth on O'Neil/Malmberg's connected mode; (2) The properties of the unstable mode depend strongly on beam parameters such as beam density, speed and temperature; and (3) The frequency of maximum growth frequently lies significantly above or below fp, and differs significantly from that predicted by the Langmuir dispersion relation. Results imply important consequences for theories of strong turbulence and nonlinear wave-wave processes, and observational identification of fp from observed wave frequencies

Cairns, I. H.

English

NASA Technical Reports Server

N89- 2491' 
53
THE ELECTRON BEAM INSTABILITY REVISITED: GROWTH ABOVE AND BELOW fp
I H Cairns
Physics & Astronomy Dept., The University of Iowa, Iowa, USA
ABSTRACT
The growth of electrostatic waves near the plasma frequen-
cy fp due to an unstable electron beam is investigated by
solving the unmagnetized electrostatic dispersion equation
numerically. These numerical solutions are compared with
analytic theories for reactive (or fluid-type) and kinetic ver-
sions of the beam instability, and for the O'Neil/Malmberg
connection of the beam and Langmuir modes. Conditions
for growth significantly above or below f_ are given. Three
general results given are found: (1) The unstable waves do
not grow on a mode with Langmuir dispersion except in the
limit of a very dilute beam with growth on O'Neil/Malm-
berg's connected mode. (2) The properties of the unstable
mode depend strongly on beam parameters such as beam
density, speed and temperature. (3) The frequency of max-
imum growth frequently lies significantly above or below fp,
and differs significantly from that predicted by the Lang-
muir dispersion relation. These results imply important
consequences for theories of strong turbulence and non-
linear wave-wave processes, and observational identification
of fp from observed wave frequencies.
1. INTRODUCTION
Electrostatic waves excited near the electron plasma fre.
quency fp by electron beams are of widespread interest
in plasma physics; specific applications include planetary
foreshocks (Ref. 1), strong Langmuir turbulence (Ref. 2)
and pulsar magnetospheres (papers this volume). Theoret-
ical concepts for these electron "beam instabilities" have
changed relatively little since their initial development
(Refs. 3-5), in part because most research effort has been
concentrated on the non-linear and strong turbulence evo-
lution of the waves. This paper is, however, specifically
concerned with the linear properties of the waves and the
instabilities themselves.
The two primary characterizations of the beam instability
are due to Bohm and Gross (Ref. 3) and Briggs (Ref. 4),
and to O'Neil and Malmberg (Ref. 5). This first character-
ization corresponds to the physical growth mechanism of
the waves: viz. the reactive (or non-resonant or fluid-like)
and kinetic (or resonant or resistive) instabilities in which
growth occurs due to particle bunching or by inverse Lan-
dau damping, respectively. These different growth mecha-
nisms imply different wave spectra for the two instabilities
(e.g, Ref. 6). The second characterization (Ref. 5) corre-
sponds to the dispersive character of the growing mode: the
waves might be "beam modes" with wr "_ k v_, or might lie
on a mode with approximately Langmuir dispersion formed
after connection of the ordinary Langmuir mode and the
beam mode. Although analytic theory (Ref. 7) implies
that the reactive instability occurs on a beam mode, the
detailed connection between these two characterizations of
the beam instability is not clear. One aim of this paper
is to clarify the connection between these two character-
izations of the beam instability. This is done by solving
the exact electrostatic unmagnetized dispersion equation
numerically, and comparing the numerical solutions with
analytic solutions. It is shown that the reactive instability
does tend to occur on the beam mode, but that the insta-
bility becomes kinetic on a modified extension of the beam
mode (the 'modified- beam mode') long before the O'Neil
and Malmberg connection between the Langmuir mode and
the beam mode takes place. The unstable mode is shown to
be intrinsically beam-associated (i.e., dependent on beam
parameters, at least) unless the beam is extremely dilute,
i.e., relative beam densities nb/no,£10-4; that is, growth
does not usually occur on a normal mode of the plasma
in the absence of the beam, such as the Langmuir mode.
Consequently, there is no a priori reason for growth to oc-
cur above fp at Langmuir-like frequencies. Conditions for
reactive or kinetic growth well above and well below fp, as
observed in the Earth's foreshock (Refs. 8, 9) are therefore
investigated here.
Previous numerical investigations of this dispersion equa-
tion (e.g., Ref. 10) have paid relatively little attention to
the dispersive characteristics of the growing waves. Yet the
detailed dispersion relation of the waves is vital in theo-
retical descriptions of non-linear wave processes and strong
turbulence phenomena (e.g., Ref. 2): differences between
the actual dispersion relation and the assumed dispersion
relation imply differences in the governing equations for the
wave fields and possible kinematic difficulties for the wave
processes, as discussed in Section 4 below. This paper pro-
ceeds as follows: Section 2 contains a brief description of
previous analytic theory, while Section 3 contains the re-
sults of the paper. A more complete version of this work is
described in Ref. 6.
2. RELEVANT ANALYTIC THEORY
Given the longitudinal part of the unmagnetized, spatially
homogeneous dielectric tensor K L = 1+ Ki + Ke +K,, where
the subscripts, i, e, and b refer to the contributions of ions,
Proceedings of the Joint Varenna-Abastumani International School & Workshop on Plasma Astrophysics, held in Varenna, Italy,
24 Aug.-3 Sept. 1988 (ESA SP-285, Vol. L Dec. 1988).
https://ntrs.nasa.gov/search.jsp?R=19890015546 2020-03-20T01:38:40+00:00Z
54 I H CAIRNS
background electrons and beam dectrons, respectively, the
dispersion equation is K L = O. Assuming that all species
have Maxwellian distributions, Ka is of the form
i) (i)g.= , k-V- )-
where@ is related to the Friede-Conte function, and wp_, Va
and _ are the angular plasma frequency, thermal speed
and drift velocity of species a, respectively. Writing w =
wr + i7, a wave is said to be resonant with species a if
_,, = (w,.-k.go)/v_ k V,, _ 1. The beam instability is said
to be resonant if _ < 2, corresponding to the distribution
function of species a at the phase velocity of the wave being
2% of its value at v, = v_.
The reactive instability is derived by ignoring Im K,, (i.e.,
Im Ka = 0), whence the dispersion equation is a real
equation for a complex variable w, thereby implying the
presence of complex conjugate solutions (Ref. 7). Ignor-
ing Im K,, corresponds to ignoring Landau (and inverse
Landau) damping; growth is instead due to particle bunch-
ing. Assuming I_bl >> 1 (non-resonant) one finds complex
conjugate roots with maximum growth with wp = k_- _.
w,. = k " V_b(1 -- 1. nb )i'_ x/_ ( nb )}
_ ,  tT .o ) , (2)
The kinetic instability is usually derived by ignoring
Re(Ks, Ki), assuming I_bl <<1 (resonant), and taking
Im Kb (3)
with w_ determined by the equation Re K_ = O. Growth
is then due intrinsically to inverse Landau damping, with
maximum growth rate
_e nb , Vb ,2 (4)7k = n--:t_) _p
where e = exp(1), on the Langmuir mode. A commonly
used criterion (e.g., Ref. 7) for determining whether the
instability is reactive or kinetic is
_oo Vbp=( )i _ b >1 (5)
or P < 1, respectively. A more accurate empirical criterion
is given below.
O'Neil and Malmberg's (Ref. 5) criterion for whether the
instability is on the beam-mode or on the connected
Langmuir-beam mode is
s = 2_IP < 1.47 (6)
or s > 1.47, respectively. Gary's results (Ref. 10) and the
results below verify this criterion empirically.
3. SUMMARY OF RESULTS
3.1 Reactive/kinetic, beam/Langmuir-beam, and disper-
sion relations
Figure 1 shows the evolution of the dispersion curves for
a beam with fixed relative density nb/no = 10 -3 and drift
speed vb/V_ = 10 as the beam temperature T,/T_ is var-
ied (TdTi = 3 is fixed). The two complex conjugate roots
corresponding to the reactive instability are clearly visi-
ble in Figure l(a), together with their beam dispersion
wr '_' kvb and non-resonant character. It is clear that
the reactive/kinetic transition takes place long before the
O'Neil and Malmberg (Ref. 5) connection of the Langrnuir
and beam mode takes place in Figure l(f); furthermore,
the instability does not take place on the pure Langmuir
mode, or on a mode with Langmuir-like dispersion until
greater beam temperatures than those in Figure l(f) are
reached. The following changes occur as Tb increases in
Figures l(b) through (d): (1) the complex conjugacy of
the beam roots disappears, (2) the dispersion curve of the
growing mode consists of a portion with beam-type disper-
sion wr .,, k Vb for wavenumbers smaller than some k. and
slowly-varying dispersion w_ ... A for k > k., where A is of
order wp. (3) The wavenumber km and real frequency wm
corresponding to maximum growth increase with Tb, corre-
sponding to the peak in the growth rate curve (as a function
of wavenumber) moving continuously towards and along the
Wr "_ A portion of the dispersion curve. The instability is
marginally resonant in Figure l(b) and strongly resonant
by Figure l(c), corresponding to a kinetic instability. By
Figure l(d) the portion of the growing mode with beam
dispersion (i.e., small wavenumbers) is heavily damped, as
is the Langmuir mode for higher wavenumbers. In contrast,
the Langmuir mode at small wavenumbers, and the modi-
fied beam mode at larger wavenumbers are lightly damped
or growing. As Tb increases further, the two lightly damped
and two heavily damped portions of these modes connect,
leading to O'Neil and MMmberg's 'Langmuir-beam' mode
(Gary's terminology) upon which the instability is found,
and a heavily damped mode of no further interest. This
connection is shown in Figures l(e) and (f). For complete-
ness note that a progression of figures analogous to Fig-
ure 1 could also be constructed by either decreasing the
beam density or decreasing the beam speed, keeping other
parameters constant.
The strong modification of the beam root's dispersive char-
acteristics as the instability becomes resonant (and so ki-
netic rather than reactive) necessitates a clarification of
terminology: (1) Portions of dispersion relations with wr "_
kvo are called 'beam' type. (2) Due to its origin with the
beam the strongly modified mode (Figures X(b)-(e)), on
which growth occurs following modification of the complex
conjugate beam mode, is called the "modified-beam" mode.
(3) The connected O'Neil and Malmberg mode is called the
"Langmuir-beam" mode following Gary (Ref. 10). With
this terminology the instability is resonant (kinetic) on the
modified-beam mode and Langmuir-beam mode, and non-
resonant (reactive) on the beam mode. These identifica-
tions of the reactive and kinetic instabilities are consistent
with the variations in the maximum growth rate 7,- as a
function of relative beam density nb/no in equations (2)
and (4). Finally, values of P corresponding to the reactive
instability are P>_6, while the kinetic instability is strongly
resonant when P%2. Reactive growth occurs on a beam
mode which strongly obeys beam-type dispersion. Quali-
tatively, the modified-beam mode's dispersion curve con-
sists of a beam-type portion at low wavenumbers and a
portion for which w_ varies slowly; it is inappropriate to
approximate the modified-beam mode with the Langmuir
dispersion relation, particularly as the maximum growth
rateusually occurs below fp.
The dispersion relation, near where the growth rate is max-
imum, of the Langmuir-beam mode is also often consider-
ably different from the Langmuir dispersion relation (Fig-
ures l(f) and 2). Figure 2 shows the dispersion relation near
ELECTRONBEAMINSTABILITY 55
maximum growth for beams with various beam densities
nb/no but constant beam speed vb/Ve = 10 and tempera-
ture Tb/T¢ = 1; the Langmuir-beam connection takes place
near rib no = 5 x 10 -4. Stringent conditions on rib no are
apparent for the dispersion relation to be closely Langmuir-
like: nb/no should be at least a factor of 10 less than that
beam density satisfying the O'Neil and Malmberg criterion
(6), i.e., nb/no < 5 x 10 -5. This rule should be of great
practical importance: for the above warm, relatively small
velocity beam one requires nb/no < 5 x 10 -5, and for faster,
colder beams one requires even more dilute beams; however,
in most laboratory and space science applications (e.g., the
Earth's foreshock, but not Type III solar radio bursts) one
expects nb/no>_10 -3.
This sub-section may be summarized as follows: (1) The
unstable waves do not have the Langmuir dispersion rela-
tion, except in the limit of a very dilute beam with growth
on the Langmuir-beam mode: n_/no < 5 x 10 -5 for vb/V_
._10 and T,/T_%l. (2) In most applications growth should
occur on the beam mode (reactive) or modified-beam mode
(kinetic). (3) The properties of the unstable waves depend
strongly on the beam parameters, such as beam density,
speed, and temperature. (4) The frequency of maximum
growth frequently lies either significantly below or above fv,
and is significantly different from the frequency predicted
by the Langmuir dispersion relation.
3.2 Growth significantly above and below fp
Figures 1 and 2 show that the frequency of maximum
growth for the beam instability often lies below fp, despite
the conventional belief (due to the idea that growth oc-
curs on the unmodified Langmuir mode) that growth oc-
curs above fp. Effective growth well above or below fp
is due to the strong dependence on beam parameters of
the dispersion relations of the beam, modified-beam and
Langmuir-beam (unless the beam is very dilute) modes,
and the non-Langmuir nature of these modes' dispersion
relations. Figure 3 illustrates the roles of beam speed, tem-
perature and density in determining the real frequency at
maximum growth wm (Ref. 9). Growth significantly be-
low wp generally (except as in Figure 2) requires growth
to occur on the beam (reactive) or modified-beam (kinetic)
modes. For high beam speeds (vb/V_ > 20) wm becomes
approximately constant, equal to w, say, and dependent
only on nb/no. For fixed beam density and temperature,
decreasing the beam speed causes Wm to increase above w,
once va/V_%lO, and subsequently to decrease towards zero
frequency once vs/Ve %2, provided the beam temperature is
sufficiently small. This increase in Wm above w, increases
with decreasing rib no until nn/no " 1O-s. Increasing the
beam density corresponds to moving the entire curve in
Figure 3, especially w,, to lower frequencies: note that
for nn/no:_lO -a ,w,%0.99wp. For warm beams (e.g., the
dashed curves) damping overcomes growth before Wm can
increase substantially above w,, and certainly before wm
decreases towards zero frequency. This Figure provides a
natural explanation for waves observed at frequencies sig-
nificantly above and below w v in the Earth's foreshock (see
Refs. 8, 9).
There are therefore two primary ways to obtain growth sig-
nificantly below wv: dense beams or cold, slow, relatively
dilute beams. Growth significantly above wr requires rela-
tively slow (2%vb/Vt%lO) and dilute (since otherwise w, is
well below wt) beams which may be either cold or warm.
Detailed conditions for reactive or kinetic growth well above
or below w v are given in Ref. 9.
4. DISCUSSION
Due to length restrictions the discussion is restricted to
the following two points: (A) The results of this pa-
per indicate that, due to the assumption of the Langrnuir
dispersion relation in derivations of the Zakharov equa-
tions (e.g., Ref. 2), and the neglect of particle bunching
effects therein, the Zakharov equations are valid only un-
der stringent circumstances: ns/no%lO -4 for vb/V_ > 10
and Ts/T_%IO. (B) The generally non-Langmuir disper-
sion relations, and tendancy for maximum growth to oc-
cur below wp, of waves growing by the beam instability
imply kinematic difficulties for non-linear wave processes.
For example, frequency and wavevector conservation for the
Langmuir wave decay L ---+L' + S (L denotes the electron
.plasma wave, while S is an ion acoustic wave) is impossi-
ble when nb/no>_lO -a since wm is below wv, yet w_ (the
backward propagating wave) is greater than Wp (Ref. 6). A
more detailed discussion is given in Ref. 6.
6. ACKNOWLEDGEMENTS
The numerical code used is based on one generously pro-
vided by Peter Gary. Discussions with Peter Gary, Marty
Goldman, and Steve Spangler are gratefully acknowledged,
as is financial support from NASA Grant NAGW-831 and
NSF Grant ATM-8716770.
5. REFERENCES
1. Klimas A J 1985, The electron foreshock in Collisionless
Shocks in the Heliosphere: reviews of current research
(eds. B T Tsurutani and R G Stone), Washington,
AGU-35, 237-252.
2. Goldman M V 1984, Strong turbulence of plasma
waves, Rev. Mod. Phys. 56, 709-725.
3. Bohm D and Gross E P 1949, Theory of plasma oscilla-
tions, A, Origin of medium-like behaviour, Phys. Rev.
75, 1851-1864.
4. Briggs R J 1964, Electron-stream interaction with plas-
mas, Cambridge, M.I.T. Press.
5. O'Neil T M and Malmberg J H 1968, Transition of the
beam roots from beam-type to Landau-type solutions,
Phys. Fluids 11, 1754-1759.
6. Cairns I H 1988, Electrostatic wave generation above
and below the plasma frequency by electron beams,
Phys. Fluids [in press].
7. Mikhailovskii A B 1974, Theory of plasma instabilities,
Vol. 1, New York, Consultants Bureau.
8. Fuselier S Aet al 1985, The downshift of electron
plasma oscillations in the electron foreshock region, J.
Geophys. Res. 90, 3935-3946.
9. Cairns I H and Fung S F 1988, Growth of electron
plasma waves above and below fp in the electron fore-
shock, J. Geophys. Res. 93, 7307-7317.
10. Gary S P 1985, Electrostatic instabilities in plasmas
with two electron components, J. Geophys. Res. 90,
8213-8221.
56 I H CAIRNS
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0 8 .5
o17
0.08 03 0.12
'.'t:_
0.7 i,I/ ...... ",'" , , , , ,
0.08 0.I 0.(2 0.08 0 I 0.12
_. 0 ---
>-.
-o.i -o.,_ , , i , , , I -o.,_I I
0.08 0.1 0.12 0.08 03 0.12 0.08 OI 032
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3
3"
1.05
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0 _... 0 --_ - 0 "
-0.I -0.I LX--. t , l i -0.I I L I I
008 03 0.12 0.08 0.1 012 0.08 0.1 012
kXDE kkDE kkDE
Figure 1. Plots of the dispersion relation and growth rate for the growing mode (full line),
damped beam mode (dashed line), Langmuir mode (dotted line), and Langmuir-beam
mode (dash-dot line) for various beam temperatures. For parts (a) to (f) Tb/T, equa/s
0.017, 0.067, 0.33, 1.0, 1.5, and 1.67, respectively. The fixed beam parameters are
nb/no = 10 -s, ,b/V, = 10, and Te/Ti = 3.
0-C,_848
I I I i I I 8- G87- 472 i
1.2
Tb= 0"I Te __ Tb=T e
I 0.8
i i i , i I i I
0.06 0.08 0.( 0.12 0.14
kX_
/ Tb : I0" Te
1.04 [ , I ' I ' I ' I I I
/
0.02 _-
V
-0.04 L , I ,Y I , I , I , t , t
0.06 0.08 0.1 0.12 0.14 0.16 0.18
kxo, ,
Figure 2. The dispersion relation of the Langmuir-beam
mode as a function of rib no: 5 × 10 -4 (full
line), 10 -4 (dashed), 5 × 10 -5 (dash-dot), and
10 -5 (dotted). Here vb/V_ = 10, Tb/T, = 1,
and T,/Ti = 3. Connection of the Langmuir
and beam modes occurs near nb/no = 5 x 10 -4.
The mode has the dispersive characteristics of
the Langmuir mode for rib no < 10 -s.
0.16 038 _ 0.6
E %/no: IO-z' Te : Ti
3
0.4
0.2 t
0
iO
%1%
Figure 3.
1.0
tO n
100
The frequency of maximum growth, win, as a
function of vb/V, for nb/n o = 10 -2 and various
beam temperatures. For high beam speeds tOm
is approximately constant, equal to w,, and pri-
marily dependent only on the beam density. If
the beam is sufficiently cold, oJ,,_ first increases
above and then decreases below w, as vb/Ve de-
creases.


The electron beam instability revisited: Growth above and below f sub p

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