Nonlinear Landau damping of resonantly
in nonuniform plasmas
excited fields
S. Srivastava, G. J. Morales, and J. E. Maggs
Department of Physics, University of California, Los Angeles, Los Angeles, California 90024
(Received 6 October 1993; accepted 29 November 1993)
In this study the resonant excitation of a nonuniform plasma by two electromagnetic waves at
closely spacedfrequencieso1 , o2 with w1> o2 is considered.Each of the pump waves excites a
Langmuir wave at the point in the density protile, where plasma resonanceis achieved. For
profiles having scale lengths L large compared to the characteristic Airy scale length (i.e.,
,&L) 1) the linear Landau damping of the Langmuir waves is quite small close to their cutoff,
hence the beat interaction with slow electrons plays an important role. The problem is
formulated in terms of differential equations in configuration space for the waves at wl, w2,
which are coupled to the idler field at w1-w2. The idler is describedkinetically by a differential
equation in Fourier space. The interaction is governed by the parameter i3s= ( 1 - 02/w1) k,L
and the scaled pump strength. For Z&(1 the ponderomotive nonlinearity is recovered and a
dissipative contribution is obtained. For Z3 > 1 plasmon transfer from wi to os causesstrong
depletion of the high-frequency wave before significant ponderomotive profile changes set in.
The fractional power absorbedthrough the idler is smaller than the power transferred to w2 by
a factor of (1 --w2/01).
I. INTRODUCTION
In uniform plasmas, nonlinear Landau damping’” (a
processmore accurately named beat damping) is known to
play a key role in transferring energy and momentum of
large phase velocity waves to bulk plasma particles. The
underlying concept consists of the beat excitation of an
idler (not a collective mode) electric field having frequency
w1-02 and wave number kl - k,, arising from the bilinear
interaction of collective modes having frequencies and
wave numbers (o,,kl) and (w2,k2), respectively. In addition to the generation of a strong wave-particle interaction
for those particles whose velocity u satisfies the beat resothe idler field alnance condition u- (q-~~)/(k~-k~),
lows a transfer of wave action (or quanta) from the higherfrequency mode, say wl, to the lower frequency one, w2. It
is this cascading aspect of the process that is thought to
contribute to the spectral broadening of narrow spectrum
instabilities and launched coherent signals within the
framework of weak turbulence theory.
Although nonlinear Landau damping has been exteninvestigated,
both
theoretically4
and
sively
experimentally,s-7for various plasma collective modes, the
majority of the studies have concentrated on its properties
in uniform plasmas. Most applications of this process to
nonuniform plasmas have consisted of Wentzel, Kramers,
and Brillouin (WKB) generalizations of the uniform medium results. By contrast, it is the purpose of the present
study to illustrate a physical situation in which the nonuniformity of the plasma plays a key role, namely the resonant excitation of a nonuniform plasma. Aside from its
intrinsic basic interest, this problem has relevanceto laserplasma experiments, ionospheric heating by highfrequency (HF) waves, and electron acceleration in the
aurora1 ionosphere.
Specifically, the present study considers a warm unPhys. Plasmas 1 (3), March 1994
magnetized plasma whose zeroth-order density Q, varies
linearly with position z, as illustrated in Fig. 1. We envision
the plasma to be externally excited by two coherent electromagnetic waves at closely spacedfrequencieso1 and w2
Near plasma resowith 04 >02, and (ot--~~)/w~(l.
nance, the density profile is taken to be a linear function of
where wP is the local
position, with 1--wi(z)/o:=z/L,
plasma frequency and L is the density scale length. As can
be shown rigorously,8Y9the mode conversion process that
occurs at plasma resonancecan be described by replacing
the electromagnetic fields by uniform (capacitor-like)
pumps of the form Eoj exp( --iwit), j = 1, 2, and t is the
time. The complex amplitudes Eoi can be related to the
incident power of the external electromagnetic signals.”
The linear responseof the warm plasma consists of the
resonant excitation of two Langmuir waves at frequencies
o1 and 02, whose amplitudes are sketched in Fig. 1 by the
solid curves whose peaks appear close to the points
Wj=Wp. Since we choosethe origin (z=O) to coincide with
the cutoff for the Langmuir wave at frequency wi, the wave
pattern of the second wave at the lower frequency, w2, is
shifted by a distance z,===
L[ 1 - ( w2/wl) ‘1. To obtain a better appreciation of the relevant parameter dependencesin
the problem, we note that each of the driven-Airy patterns
exhibits a maximum phase velocity that scales as
(uA)j=~j/kA, where the characteristic Airy-wave number
is kA= ( k,L/G)2’3L-*,
and the Debye wave number is
defined as k,=q/iT,
with i7 the electron thermal velocity
defined as r = G.
This implies that the effective
phase velocity of the individual electric fields driven near
resonances
as
the
local
plasma
scale
(VA)j/F=(wj/o1)
(3kDL) 1’3* Consequently, for plasmas
having large values of (k,L) *‘3( > 5), as may be encountered in laboratory and space applications, the resonantly
driven Langmuir waves experiencerelatively small Landau
1070-664X/94/1(3)/567/12/$6.00
@ 1994 American Institute of Physics
567
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wp:w,
z:o
wp:w 2
t=z2
FIG. 1. Schematic of the geometry considered. The plasma configuration
in this study consists of a linear background density profile Q,(Z) and two
incident electromagnetic pumps Ec, and Eo2 at frequencies ret and w,
(w, Z ws) with resonances at z=O and z=zs. Near resonance, the pumps
are approximated by capacitor-like fields.
damping, unless a fast electron tail population is present.
O f course, since the WKB phase velocity of Langmuir
waves decreasesas they propagate in the direction of decreasing density, eventually the waves experience strong
Landau damping, provided nonlinear processesdo not play
a role at distancesclose to their generationpoints, i.e., z=O
and z=z,. One such nonlinear processis nonlinear Landau
damping.
Figure 2 illustrates the spatial dependenceof the WKB
phasevelocity of each of the resonantly excited Langmuir
and
waves,i.e., wj/kj(z), with kj(z) =kD[(z-zj)/3L]“2,
zj is the respectivecutoff position. The phasevelocities are
scaled to uA and the relevant scaled spatial coordinate is
c= k,z. Figure 2 also shows the spatial dependenceof the
Group vetocity
0 ____ -------5
0
I
10
J
__________
15
20
--_
2.5
F
FIG. 2. Spatial dependence of the WKB phase velocities (scaled on uA)
of the resonantly excited Langmuir waves. The WKB group velocity of
the wave at frequency ot is a good estimate of the phase velocity of the
idler field in the small-frequency separation limit.
568
Phys. Plasmas, Vol. 1, No. 3, March 1994
WKB group velocity, i+e., u,(z) =3(kJk,)E.
The latter
quantity being of relevance in describing the beat resonance velocity (ol -~~)/[k~(z) -k2(z)] for closely separated frequencies.The WKB picture that emergesfrom
Fig. 2 is that each of the resonantly excited waves undersmall
goes negligibly
Landau
damping for
{<(&I (k&/ti)2’J
(where Oj/kj>4F), while the beat resonance sweepsacrossthe bulk of the electron distribution
function. The consequenceof this behavior is that at sufficiently large pump amplitudes, significant beat coupling
results, and is accompaniedby strong depletion of the wave
having higher frequencytit. Simultaneously, the amplitude
of the lower frequency wave, 02, must increasein order to
conservethe plasmon number. O f course,beyond this stage
the second wave can trigger other strong nonlinear processessuch as particle trapping and phase independent
acceleration” of both background electrons and particles,
whose initial velocity is boostedby beat resonancewith the
idler field.
Although the results of the present study indeed corroborate the general qualitative features expectedfrom the
WKB picture suggestedin Fig. 2, several detailed features
are determined by the non-WKB properties of the drivenAiry pattern, which are most pronounced where the field
amplitude is largest, i.e., within the first lobe of the Airy
pattern. In fact, becausestrong nonlinear Landau damping
occurs within the first lobe, it is relatively difficult to realize
a situation in which two propagting waves interact in the
manner describedfor uniform plasmas.
While the WKB picture shown in Fig. 2 is not useful in
making a direct quantitative prediction of the role of nonlinear Landau damping in the resonantabsorption process,
it is quite valuable in suggestingwhat approximations can
be made in the correct mathematical description of the
process.First of all, for ( kDL) 1’3+1, linear Landau damping of the primary waves at w1 and w2 is negligible in the
region where the beat resonancesweepsacrossthe electron
velocity distribution function, so that the collective response at WI and w2 can be accurately representedby a
warm-fluid description. This implies that the spatial evolution of the electric fields at o1 and w2 is determined by a
differential equation in configuration space, in which
pumping by both the external source and beat charge oscillations must be included, but kinetic linear damping effects can be ignored. However, the responseof the plasma
to the idler field doesnot admit a fluid description since the
beat-resonancevelocity is smaller than the thermal velocity
of electrons. Consequently, the response at the lowfrequency w3=w1 - w2 must be described kinetically, and
since one is dealing with a spatially nonuniform plasma,
this requires a ditferential equation in Fourier transform
space. The full problem then consists of three coupled
equations: one in Fourier space and two in configuration
space,
The system of scaled equations [Eqs. (19)-(21)] that
govern this problem depend explicitly on three scaled parameters: the frequency separation 6.3,= ( 1 - w;?/wt ) kAL,
the pump strength pj= (k,LEoj)2/24pnoT,,
and the gradient scale length k,L. However, for large scale lengths
Srivastava, Morales, and Maggs
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(as assumedin this study) the explicit dependenceon kAL
does not play an important role. In the limit Gs=O, the
system of equations reduces to a single equation (depending on a single pump parameter p) introduced by Morales
and Lee12 to describe the role of ponderomotive density
changes on resonant absorption. Consequently, the present
study provides a generalization of this process to nonzero
frequencies. For ++O the ponderomotive effect is no
longer adiabatic and energy absorption by slow electrons
results. The efficiency of this absorption is found to be
proportional to G3. However, this is not the main process
that causesthe nonlinear damping of the higher-frequency
wave. It is plasmon transfer from w1 to w2 that dominates
the interaction. For Z3 > 1 the higher-frequency Langmuir
wave predominantly interacts with the first lobe (where k
is nearly zero) of the lower-frequency wave. In passing
through this lobe, the w1 wave is strongly damped for values of p,zO.4, which is slightly below the required pump
amplitude to cause significant ponderomotive modifications.
The paper is organized as follows. In Sec. II the mathematical formulation is presented. In Sec. III analytic models pertaining to the problem are developed. The conservation laws for the system are discussed in Sec. IV.
Numerical results are described in Sec. V and conclusions
are presented in Sec. VI.
-iiW3f3e+u
af,,
e
----az
,E3W$
W3i
e
--g+~E3(z)
4’3f3i+v
af0i
-=o
au
.
The distribution function of the j speciesat frequency o3 is
written in the form f3j(z,u)exp( -iiw,t) +c.c., and
fsj(z,V) is the zeroth-order (unperturbed) distribution.
The quantities fiaf2, are the linearized complex amplitudes of the electron distribution function oscillating at
frequencies o1 and 02. The electron and ion massesare m
and Iw, respectively, and e is the quantum of charge (a
Z= 1 ion is assumed for simplicity). The nonlinear terms
in Eq. (2) affecting the electron distribution arise from the
mutual interaction of the high-frequency waves, while the
low-frequency ion distribution in Eq. (3) is unaffected by
the high-frequency electron dynamics.
The inverse of the spatial Fourier transform of the
electric fields is defined as
E3(z) =F-‘[E3(k)]
=&
JTI
dk exp( +ikz)i3(k),
(4)
II. IWATHEMATICAL
FORMULATION
The geometry of the problem being considered is
sketched in Fig. 1. 4 plasma with a linear density gradient
of scale length L, i.e., nZ=no( 1--z/L) is irradiated by two
electromagnetic waves at frequencies o,, c+, with o, 2 wa.
The location z=O corresponds to the plasma resonance at
oi . In the neighborhood of the resonance the electromagnetic waves are approximated as uniform (capacitor-like)
pumps. The self-consistent spatial structure of the primary
electric fields at frequencies o1 and w2 are determined by
Poisson’s equation, whose exact form is an integral equation of the type
and similarly for the complex amplitudes of the distribution functions. The spatial Fourier transform is applied to
Pqs. (2) and (3) to solve for the un)gnown complex amplitudes of the distribution functions fsj( k,v). From these
we determine the charge density that results in the following Poisson equation in k space,
; [l-Ad41
d&(k)
F+c(os,k)E,(k)
=,!$(k),
(5)
where e(w,,k) is the exact kinetic dielectric coefficient including ion and electron contributions and S,(k) is the
Fourier transform of the bilinear source term at frequency
W3=WI-W2,
ek”D
S”3W)=--mpF
+m
(1)
s-* dZ’K(z’,z,o/)Ej(z’
I )=Eqi+~.~(z),
where Eoj and T(z)
are the pump amplitude and the
nonlinear current at frequency .~j. The experimentally observable fields are represented as Ej(z) exp( -iiojt) +c.c.
and K(z’J,tij) is a kernel obtained by integrating the linearized Boltzmann equation, but whose detailed structure
need not be analyzed here.
As explained in the Introduction, the beat-wave phase
velocity is smaller than the electron thermal velocity,
hence the evolution of the idler field at o3 = o1 - w2 must
be described kinetically, i.e., in Fourier transform space.
To obtain the idler field equation at frequency w3, we
start with the linearized Vlasov equation for’the electron
and ion response:
Phys. Plasmas, Vol. 1, No. 3, March 1994
-+?a
[ (I-;)E,(z)]
1.
(6)
In Eq. (5)) the derivative in k space arises from the spatial
nonuniformity of the zeroth-order density. In obtaining
Eq. (6), it has been assumed, as is appropria_te @ this
problem, that the contributions arising from f le,f2, are
well approximated by the cold plasma limit (i.e., large
phase velocity).
In our model, the ratio of the plasma electron thermal
velocity IYto the typical phase velocity u, of the primary
waves (wi and oZ) is considered to be small. Hence, for
these waves (j = 1,2), Eq. ( 1) can be well approximated in
the lowest order by
Srivastava, Morales, and Maggs
569
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E~(z,O/)EI(Z)=E~j+~~(z)~
(7)
i
where E, is the warm fluid dielectric operator (in configuration space),
EUJ(Z,mj)=g
3 a2 z-z.
g+++irj
(8)
+n3W~2(z)l,
(9)
~L(z)=--e[nl(z)~f(~)+nj(~)~I(~)l,
(10)
where the subscripts 1, 2, and 3 represent the quantities at
frequencies ol, 02, and w3= o1 -02, respectively. The
cold plasma approximation can be used to obtain the perturbed velocities and densities at high frequencies,
uj(z)=~~,
G3=kAL;,
@2
*
The normalized collision frequency for electrons is
rj=Y/Wj and Zj is the position of the resonancelayer at
frequency Oj. For 54 vA , the linear Landau damping contribution to the structure of the primary wave fields is
negligible near plasma resonance,the region of interest to
this study.
The nonlinear current q(z)
in Eq. (7) is driven by
the beat of the idler electric field Es and one of the primary
electric fields, i.e.,
Jr;lL(z) = -4n2(zb3(z)
l=hz,
e Ej(z)
c&=-,
WI
yj=kALI’j;
‘j”
j= 1,2,
(k,LEoj12;
24moT,
Ej=k,LEoi%‘j;
(181
j=l,2,
j=1,2,3,
where T, is the electron temperature.
The corresponding system of scaled equations that describe nonlinear Landau damping of resonantly excited
fields in a nonuniform plasma consists of three coupled
equations for the electric fields. The high-frequency fields
are described by differential equations in configuration
space,whereas the idler field is obtained from a differential
equation in k space,
(11)
J
(~+b+ir~)a,(E)=i-p2~l(p),
TZj(ZJ=-sg
(j=l,2).
[ (I.-y)Ej(Z,]
a2
(12)
To obtain the perturbed density n3(z) and velocity Q(Z) at
w3, a kinetic description is required. The procedure then
consists of determining n3( k) and r3( k) using Eq. (2) and
then transforming to configuration space,
iil(+&Q’(~)(
+T;
Z(s) =
1
d-IT
+6(k)
G;(k) =kn,
[1-+3,K)]
(20)
%+&3,K)g3(K)=g3(K).
(21)
(22)
(13)
exp( -3)
(t-s)
&
~+PIS~CE>,
h=kAL( l-w”,,,
[ ( l-y)g(z,]
[ (+W])),
+dt
--co
82(C)=
;jsz+G422)+‘Y’
The quantities appearing in the system of Eqs. ( 19)-(21)
are
(l-;-)$(k)
+;--(yg
(19)
’
0
n3(z)=cF-1[%3(k)],
(23)
(14)
(15)
(16)
u3(z)=F-1[ii3(k)].
(17)
It is useful to incorporate the natural Airy scaling associated with a nonuniform plasma, as follows:
K=k/kn,
570
Phys. Plasmas, Vol. 1, No. 3, March 1994
+iF,W*&,
(2%
Srivastava, Morales, and
Maggs
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1
&,=;z(
&;)(3kAL(
+F.f%&g
(31)
I&(z) I29
4~( T, + Ti)
which are identical to the expressionsobtained by the usual
ponderomotive calculations.
Furthermore, the nonlinear source term S1(c) in the
limit of o3 + 0 takes the form
ni(Z)
+-&;)@3(K)
[ (l-$+YE)]
+g;(C> $ [(l--$-+%5)]
I),
(26)
= -
s,(g) ay(~)
d~l(O
2Yl
J
&(
3k,L8df)
+;
I go
Ii),
(32)
(27)
where .Fs and F;’ are Fourier transform and inverse
Fourier transform operators defined in the dimensionless
conjugate space (&K), Ti is the ion temperature, and Fi is
the corresponding ion thermal velocity. Equations (19>(21) are solved numerically, and the results are presented
in Sec. V. However, we first discuss the results expectedin
two limiting cases that can be treated analytically, and
derive conservation laws that can be used as checks of
numerical accuracy.
and using the corresponding limit of the idler field given in
Eq. (21), reduces Eq. (32) to
S1(‘)=(T,+Tj)2Te
MODEL
A. Relation to ponderomotive
force
As the frequency differencebetween the primary waves
is decreased,i.e., w3/01+0, the two Langmuir waves excited at plasma resonancebecome indistinguishable within
the coherencelength of the wave packet. Then, the waves
can be treated as a single high-frequency field and the beat
interaction can alternatively be viewed as a ponderomotive
force problem. Here we show that the kinetic beat-wave
interaction model reduces to the well-known ponderomotive limit as w3+0, and obtain the generalization of the
ponderomotive source term (including dissipation) at nonzero frequency.
We consider the limit L-+ 00 to compare the result
with the uniform plasma case. In the limit of w2-‘01, Eq.
(5) using Eq. (6) reduces to
(28)
The quasineutrality condition of low-frequency oscillations
implies that the unity term in the dielectric coefficient E in
Eq. (23) is small compared to the ion and electron susceptibilities. Since both ions and electrons are adiabatic in the
limit w3 -t 0, the dielectric reduces to
(29)
Using Eqs. (28) and (29) and
n,(z)=nj(z)
yields
IE,(z) I21
Phys. Plasmas, Vol. 1, No. 3, March 1994
(30)
(33)
Finally, inserting this result in Eq. ( 19) yields
a2
p+S+iy1+2pl
(
where
go
I2 01G)=l,
)
(krA2E;
‘=24rno(
III. ANALYTICAL
8*(C) I8z,(O 12.
(34)
(35)
T,+ Ti) *
Equation (34) is the model equation first introduced by
Morales and Lee12to investigate the nonlinear modification of resonant absorption caused by profile changesdue
to the ponderomotive force. It should be noted that the
system of equations ( 19)-( 2 1) constitutes the nonzero frequency generalization of this interaction; it requires a kinetic description associatedwith wave-particle resonances
mediated by the idler field, i.e., nonlinear Landau damping.
B. Limit of well-separated
peaks
As the frequency difference w3 is increased, the location of the plasma resonancefor the wave at frequency w2
moves in the direction of decreasingdensity according to
Eq. (22), i.e., in scaled coordinates c2Y 2Gs. This implies
that for sufficiently large Z3, the beat coupling is dominated by the interaction of a traveling Langmuir wave at
or with the resonant peak (Airy peak) of the electric field
at w2, as sketched in Fig. 3.
Next, we present a simple calculation of the nonlinear
damping expected in this limit. The value of Z3 that naturally separatesthe small values of Z3 from large values is
given by the condition that the two driven-Airy principal
lobes be resolved in space.This occurs when the first peak
of the pattern at w2 falls on the second peak of I $I I 2, i.e.,
g2‘,--3.6.This yields a characteristic frequency separating
the small and large Z3 regimes,
AwCtit” 1.8.
(36)
For i3,( AwCtiit,the beat interaction becomesadiabatic, and
results in the ponderomotive density modification previously described, except that for nonzero Z3 there is always
a resistive contribution. To obtain an analytic estimate of
Srivastava, Morales, and Maggs
571
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-
I-(3kAL/4i$[2+Z’(s)]
i - (3kAL/@)Z’(S)
X
’
f41)
The nonlinear damping arises from the imaginary part
of the source, Im S, (6). Thus, neglecting Re S1({) (which
represents the nonlinear frequency shift) in I$. (41), a
simplified equation suitable for large &j3is obtained for the
field at wl,
z
.t
x
5
-0
.-s
0
E
s
z
a2
T@+g+iy1+irE”’ g1=1.
1
(
(421
The effective nonlinear damping coefficient #” associated
with idler dissipation is approximately given by
~L(~)=-P:!I~P2max121mZ’(s) ( l+---+
3L)
w2I%2 m m I 2E
FIG. 3. The amplitudes of the unperturbed electric fields are shown for
large frequency separation (is3 > AuCtiit). The beat interaction is mostly
confined to the region of overlap of the two primary waves, shown shaded
in the figure. The model presented in Sec. III approximates the field at o2
as a constant over a region and treats the field at wI as a W K B traveling
wave.
the nonlinear damping in the large ijs lim it we adopt a
simple model in which gz(t) is localized to the region of
the first Airy peak at c&,
and the electric field Z?r(<) is
approximatedby its WKB (asymptotic) form consisting of
a traveling wave with scaled wave number ~~(6) =: &
=:@g.
The scaled effective phase velocity of the idler field is
i$/~~ (LJ becausein the first lobe of ?Yz(0, K~zO. Motivated by this WKB behavior we approximate the argument
of the electron 2’ function (the ion contribution is negligible in this lim it) by its local value, i.e.,
(37)
where
(38)
Thus, from Eq. (23 ),
3k,L
d&,K)zl--Z’(S).
4%
Substituting ECq.(39) in the idler field, Eq. (21) yields, in
the lim it L-t 00,
&(K)
=,=
S3(K)
dQ’3,K)
-
(~:2*/2~33>3-,(a~‘/ag
1 - (3k,L/d&)z’(s)
’
(4o)
The nonlinear source term S, (5) in Eq (24) can then be
simplified to the form
s,m=572
I ~212W5)
G2
z,(s)
Phys. Plasmas, Vol. 1, No. 3, March 1994
(l+$),
(431
in the neighborhood of the peak of the field CY(zj,) at
c-2iZ,, with $Yy being the peak amplitude. The corresponding fractional power per unit area transferred from
the high-frequency wave to the lower-frequency wave, and
the plasma is then given by
YYLgo
%=l-exp
4
(-7r 1
(4)
For small values of eL, using Eq. (43)) we can then write
z=P2g&
(I+$9
(45)
l~~“~“~oo,
where s;i is the effective width of the interaction region in
this model and Pj is the unperturbed power flux of the
high-frequency waves, given by (for both waves)
To estimate the amplification of the lower-frequency
wave we note that the power transfer from the external
pump to the field ?Y2(6) occurs where the field is resonant,
i.e., within the first lobe of the Airy pattern. Consequently,
we expect that the processof plasmon transfer near cutoff
(i.e., near plasma resonance)simply acts as an additional
pump whose amplitude can be found by manipulating
S,( 5) to be &(g=i2). With the approximation K~ZO discussed earlier, the coupling term S,(f) in Eq. (25) becomes
~2G=62)
=i
3m3
J
k,L
syy
% ‘1(g=g2)
12,
Since we have identified that the interaction is strongestin
the region of the first Airy lobe of g2, we set g)2 equal to
its peak value in Eq. (47) in the same spirit of the approximation used to obtain Eq. (43)) as illustrated by Fig. 3.
in Eq. (47) then
Setting 82 equal to n[G~(~2i)-iAi(f;z)]
results in the source term of Eq. (20) acting as an enhanced pump. This altered pump strength implies that the
amplitude of field ?Y2is now multiplied by the factor
Srivastava, Morales, and Maggs
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(48)
with the unity term representing the contribution of the
external pump at frequency w2. Since in the large i3s limit,
%‘r(~=&) is well represented by its WKB form, i.e.,
the amplification factor becomes
Igl(5>12 ;= r/a,
A=l+a
with
3n-
a = --PI=
i
-Im(8?).
2kAL
Thus, for small a, nonlinear damping of the higherfrequency wave at wr results in a fractional power transfer
to the lower-frequency wave at w?, given by
@2
-=-plr
p2
~IdO~G22)l.
s
dz E&
z,(s)
e
(50)
i--
Re
I g212gl(o
z2
S,(5)=-
67-r
Finally, to compute the idler dissipation, we write
P3(i3j) =i
model the first Airy lobe of g2. A similar comment applies
to the comparison between the idler dissipation term (proportional to &/k,L)
contained in Eq. (45)) and the corresponding Joule-heating calculation leading to Eq. ( 53).
If a closure is desired for further applications of these formulas, then it would be appropriate to equate the different
expressionsto determine the consistent values for the effective width and amplitude of the modeled g2 field.
In arriving at the expressiongiven in Eq. (41) for the
nonlinear source term S,(g), it has been assumedthat the
ion contribution to the Z’ function is negligible, as is appropriate for G3 large. For completeness,-we point out that
the general form of Eq. (41) is
,
(51)
and use Eq. (40) with
J3=
(52)
to obtain
z=Pz’&&
,B;‘“i’(&)&
(53)
X
l- (3k,L/4&)
[2+2;(s)]
1 - ( 3kAL/4&)Z;(s)
’
(54)
where Z:(s) refers to the electron contribution and
Z;(s) to the sum of the electron and ion contributions, as
defined by Eq. (23).
From Eq. (54) it is then possibleto calculate that as Gj,
is reduced, the contribution from the Im Zi becomesvery
small, while the resistive contribution from the beat resonance of the idler field with the ions begins to play a role.
The picture that emergesas G, -to from the large Gj, side is
that the electrons becomeadiabatic, while the ions go from
being cold to contributing a peak in the dissipation to eventually becoming adiabatic as well, and the conventional
ponderomotive result given by Eq. (33) is recovered directly from Eq. (54).
where A& is the typical width of the idler field structure.
IV. CONSERVATION LAWS
The dissipated power is smaller by a factor of
iG3/kAL=oJo1
compared to the power carried by the priIt is important to identify the conservation laws for
mary fields.
this
problem. These equations provide a clearer underNote that the power lost from the high-frequency field
standing
of the physical processesinvolved, and also give
gr , as given by Eq. (45)) includes both the power transan
independent
check of the numerical procedure used to
ferred to the field at frequency zi, and the power dissipated
solve
the
field
equations.
through the idler field. The term independent of i3s can be
Multiplying Eq. (7) with q(z) and taking the imagidentified as the contribution from plasmon transfer, while
’
inary
part results in the energy conservation equation for
the term proportional to &/kAL is the dissipation. The
the
individual
Langmuir waves,
dissipation can be considerably smaller [by the factor
( 1 -oz/wl)] than the plasmon transfer.
aw,
---j-y-(z) +$(z) +$<z >+q ‘(Z)=o,
(55)
The simple analytical models presentedin this section
are useful becauseexplicit formulas can be obtained for the
various power transfer processesin terms of the scaled where
parametersof the problem. This provides a deeperphysical
3Wj
insight as well as an independent check of the numerical
(56)
wj=K
studies, to be discussed in Sec. V. Of course, in an exact
calculation it would be rigorously found that the loss of
(57)
power through the plasmon transfer term obtained in Eq.
(45) would be identical to the power increasein the lowerfrequency wave, i.e., Eq. (50). In the approximate models
(58)
discussedhere, however; thesetwo results are not identical,
but differ only by a numerical factor of order unity. The
discrepancy arises from the method used to estimate the
(59)
effective amplitude and width of the localized field used to
Phys. Plasmas, Vol. 1, No. 3, March 1994
Srivastava, Morales, and Maggs
573
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Physically, Wj represents the wave-power flux, ~jn is the
input power density provided by the external source, e is
the power density dissipated by collisions, and e is the
nonlinear power density exchange at frequency Oj. The
latter is associated with the beat-coupling process.
Substituting Eqs. (9) and (10) for flL(z)
in Eq. (59)
II
yields
-+mz) -g[ ( l-gE2(z) s-1;( )I,
&g
;g [(l-;)Ei(rq~~E,(z),
v’LZ(k,:k,)ii,).
(681
Note that the nonlinear dissipation depends directly upon
Im Zi and is proportional to w3.
V. NUMERlCAL
$ [ (1-;),,,z)],-I(;)*],
(61)
6tk)=;Z’(&)
[ (1-;-)E,(k)
+;.F
4 (W;*(z)
2
w2
E;(z)q(z)
+
(62)
One can also derive the conservation equations for the entire system. Adding Eq. (55) for o1 and w2 yields
(Z) +e%-)
+$(z)
+qL(z)
(63)
Also, dividing Eq. (55) by Oj , and summing, results in
Two separate processes, namely, plasmon transfer
from the higher-frequency wave to the one at lower frequency, and idler dissipation contribute to the nonlinear
power exchange term q/tij.
Using Eqs. (60) and (6 1) ,
it is easy to demonstrate that the plasmon transfer terms
exactly cancel out, and we are left with the nonlinear
power dissipation term
p, !p%kb
multi2
J
I-[%:
167~
xF-l($fyg
[ ( 1-T)E2(z)]
[ ( ,-t)E*(z)]
(65)
To gain insight into the meaning of Eq. (65)) it is useful to
evaluate the right-hand side in the WKB limit, in which,
sufficiently far from the resonancepoint of both waves, we
can approximate
574
Phys. Plasmas, Vol. 1, No. 3, March 1994
(67)
where k,(z) and k,(z) are the WKB wave numbers of the
fields at frequencies w1 and m2, respectively. In this limit,
Eq. (65) reduces to
1 (0)
‘(z)
=-- - @Yz)El(zF-‘
I*
(6
-+%Z)
(65)
[ ~~-32(z+32(z),
RESULTS
The coupled differential equations ( 19)-( 21) are
solved numerically using an iterative procedure. Typically
256 points with grid spacing Ag=O. 156 are used. For the
results reported here the collision frequency is set to zero in
order to accentuate the intrinsic dissipation associatedwith
nonlinear Landau damping. The pump amplitudes E,, and
Eo2 are taken to be equal in these surveys, and a plasma
with TbTi= I, kAL= 100 is considered. This reduces the
total number of parameters to the two essential quantities:
the nonlinear coupling parameter p = (k,L ) 2E$24m,T,
and the scaled frequency separation ij3 = ( 1- 02/wl ) kAL.
To ensure convergence of the Fourier transforms, a
pseudo-Landau damping term is incorporated that acts (in
configuration space) at 5225, which is far from the interaction region of interest to our study. A Green’s function
technique is used to transform the differential equations in
configuration space (19) and (20) into integral equations
for the fields E,(z) and E2(z). The integral equations are
then solved by iteration. In the first trial, the source for the
idler field in Eq. (21) is calculated using the unperturbed
driven-Airy fieldsI r[Gi(z) -tii(Z)] at frequencies 01 and
02. The idler field is then computed in k spaceand inverted
to obtain E3(z), which is used in the integral equation to
calculate the modified E, and E2. A weighted average of
the previous two iterations is used as a new estimate and
the process is repeated. In a typical run, four to ten iterations are found to be sufficient to obtain convergence.
However, as the pump amplitude increases,convergenceof
the solutions requires more iterations. Eventually, for
~20.60, convergence becomes a problem; so we limit our
study to smaller amplitudes. It should be noted, however,
that the nonlinear Landau damping interaction is essentially over before the large p regime is reached. Hence, the
modification of the density protile by the ponderomotive
force is not important for the regime under study. As
shown by Morales and Lee,” this effect becomessignificant
forp>
I.
The scaled amplitudes of the unperturbed electric
fields resonantly excited at frequencies w1 and o2 are illustrated in Pig. 4 for C&=2. These fields are driven by an
external capacitor-like pump (k,=O). The modulation in
the amplitude is due to the interference of the uniformSrivastava, Morales, and Maggs
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I
I
I
1
p=o.2
s,= 2
-;- w”
-I
- 52
i-
y<L o-
5
10
15
20
1
I
I
I
0
5
10
15
t
E
FIG. 4. Spatial dependence of the scaled amplitude of the unperturbed
(linear) resonant fields for g&=2.0 and EoI=Eoz=Eo.
pump field and the resonantly excited Langmuir wave. In
the absenceof Landau damping, these field structures are
describedby the driven-Airy waveforms, n[Gi(z) -IA,(Z)].
The modifications produced by nonlinear Landau
damping on the fields shown in Fig. 4 are exhibited in Figs.
5 and 6 for a value of p=O.2. The higher-frequency field
0 t ({) undergoes enhanced damping, while the lowerfrequency field g,(c) is amplified simultaneously with acceleration of the background plasma electrons through
idler dissipation at &. Figures 7 and 8 illustrate the effect
of increasing the frequency separation to a value S&=4. It
is seen in Fig. 7 that the onset for the damping of the
high-frequency wave occurs at a larger value of g than in
Fig. 5, becausethe Airy peaks are farther separatedfor a
FIG. 6. Spatial dependence of the scaled amplitude of the high-frequency
field at wa for the same parameters as Fig. 5. The solid curve is the
unperturbed (linear) field amplitude and the. dashed curve is the nonlinearly modified amplitude showing enhancement from plasmon transfer.
larger value of ij,. In both cases, however, most of the
energy transfer occurs in the region of the Airy peak of the
low-frequency wave.
The Fourier spectrum of the idler field associatedwith
Figs. 5 and 6 is shown in Fig. 9. The presenceof negative
and positive k contributions suggests that the idler is a
partially localized structure in configuration space, as is
illustrated in Fig. 10. The idler field has a maximum at a
location determined by the overlap of the first lobe of the
field gZ(g) with the Z?i(c) field. The idler field is largest at this location because the nonlinear source term
r
,
pco.2
Liz=4
-----i-------TI
p=o,2
s,= 2
/
/,
5
----3--
51
10
1
15
,
20
t
FIG. 5. Spatial dependence of the scaled amplitude of the high-frequency
field at oi, for p-O.2 and &=2.0. The solid curve is the unperturbed
(linear) field amplitude and the dashed curve is the nonlinearly modified
amplitude showing depletion from nonlinear Landau damping.
Phys. Plasmas, Vol. 1, ‘No. 3, March 1994
I
15
,
10
20
25
cl
FIG. 7. The spatial dependence of the scaled amplitude of the field at
frequency 0, is shown for a frequency separation larger than that of Figs.
4-6. The coupling coefficient p=O.2 and f&=4.0. The solid curve is the
unperturbed (linear) field amplitude and the dashed curve the corresponding nonlinear result. Note that the onset of nonlinear Landau damping is farther from the principal peak, in comparison to the small 5, case.
Sriiastava, Morales, and Maggs
575
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O.&-
“0
K 0.6-
FIG. 8. Spatial dependence of the scaled amplitude of the field at w1 for
the same parameters as Fig. 7. The solid curve is the unperturbed (linear)
amplitude and the dashed curve is the nonlinear result.
Si (j = 1,2) is proportional to the product of the indivrdual wave amplitudes.
Figure 11 shows the spatial dependenceof the power
dissipated in the plasma through beat resonance of the
idler field with slow electrons for G,=2.0. Again, the dissipation is maximum at a spatial location determined by
the peak amplitude of g2. The phase velocity of the idler
is approximately ( 3M/mkAL) “* times larger than the ion
thermal velocity, and for the parameter regime studied,
this ratio is 7. Hence, ions do not participate in the idler
dissipation, which is entirely due to plasma electrons.
Since the scaled pump amplitude Pi determines the
strength of the nonlinear interaction, it is expectedthat the
relative enhancementof g2 and damping of E’:1increases
as pj increases.Figure 12 shows the field amplitude of the
FIG. 10. Spatial dependence of the scaled amplitude of the idler field
associated with the nonlinear interaction illustrated in Figs. 4-6. The idler
field is localized to the region where the principal lobe of the field at o2
interacts with the field at 65,.
higher-frequency wave $9, for &&= 2.0 and p1 =O. 10, 0.20,
and 0.40. For p1=0.40, we see that most of the wave energy from gf?,has been transferred to the lower-frequency
wave, and the interaction is over before <= 10. The remaining field amplitude beyond g= 10 is too small to induce a
significant nonlinear interaction.
Figure 13 shows the dependenceof the idler dissipation
on the frequency difference between the pump fields. The
open triangles are the self-consistentnumerical results. The
solid and the dashed straight lines represent two different
predictions based on Eq. (53). The two estimates differ in
the approximation used for 1gz 12 Ag3. The dashedline is
obtained by estimating 1P2 [ * A& as the area under the
5
I
p EO.2
w”,:2
I
,
p= 0,2
w”,=2
,
I
,r\p-;;&;~---!
amin- C
A
-5
0
K
5
10
FIG. 9. Scaled modulus of the k spectrum of the idler field associated
with the nonlinear interaction illustrated in Figs. 4-6. The spectrum is
asymmetric, with relatively smaller amplitude for k vectors along the
density gradient for negative k.
576
!
%
“0x
-51 .u
tc =
- I2
Phys. Plasmas, Vol. 1, No. 3, March 1994
f
,
0
5
I
10
I
15
4
FIG. 11. Spatial dependence of the local power absorption arising from
the beat resonance of the idler field with slow electrons is shown by the
solid curve. A dashed curve shows the spatially integrated power absorption. Conditions correspond to the interaction shown in Figs. 4-6 and the
idler field shown in Fig. 10.
Srivastava, Morales, and Maggs
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I
I
p=o.4
$= 2
!
I
,i h
\
,/
I
I’,, /p,jfi~,~A~/b
\ :II\VL
y\
/
:
c\
‘“’ ‘v-‘.,-\_,\m
\ w tQNLwp
11
1
-5
0.
5
10
15
20
E
E
FIG. 12. Dependence of plasmon transfer on pump strength for various
coupling parameter values at &=2.0.
The spatial dependence of the
scaled amplitude of the field at frequency w, is shown for p=‘O.lO, 0.20,
and 0.40, respectively.
curve ] $z[’ from negative infinity to the location of the
first m inimum after the Airy-lobe peak. The solid line is for
1$z ] 2 A& estimatedas the area under the principal lobe of
I g$.
In the lim it of w3-‘0, the two primary waves merge
into one, and the interaction can be viewed as a ponderomotive process,as shown in Sec. III. In this case,the idler
field generatesa localized density depletion and the power
dissipation to the plasma goes to zero, consistent with the
prediction of Eq. (53).
The self-consistent numerical value obtained for the
asymptotic fractional power extracted from the high10-
I
I
I
p=o.2
8-
“0
a
-7
I
/
I
/
FIG. 14. Spatial dependence of the power flows associated with nonlinear
Landau damping of the higher-frequency wave at w, for Zs = 2, p= 0.4, in
the absence of collisions. Here IV, is the wave power flux [Eq. (56)], @i
is the integrated input power of the pump at w, [Eq. (69)], and @ ” is the
nonlinear power exchange [E?q. (70)]. The flat line shows the integral of
ECq.(55), given by IV, + & ‘+ @, as a ratio to the pump input power Pi
[Eq. (46)]. Its value is smaller than 10e3.
frequency field for Z&=4, and p j=0.2 (as shown in Fig.. 7)
is 0.64. For the two estimates of 1gZ I 2 A&, the corresponding result from Eq. (45) is 0.39 and 0.43. The discrepancy between the fractional power loss predicted by
Eq. (45) and the numerical results can be accountedfor by
noting that the model only estimates the damping across.
the principal lobe of g2. The numerical power loss at the
end of the first Airy lobe of g2 is 0.47, a result in much
better agreementwith the model estimates.
Figure 14 illustrates the spatial dependence of the
power flows associatedwith nonlinear Landau damping of
the wave at frequency w1 for Z3 =2, p=O.4, in the absence
of collisions. The curve labeled IV, is the wave power flux
[Eq. (56)]. The curve labeled & representsthe integrated
power input due to the pump at wl,
&cc> = Jr,
6-
pIl”cwr,
(69)
where pi” is given by Eq. (57). The curve labeled @ ” is
the integrated nonlinear power exchange,
cc
4-
@G-I=
ST, 8-%%%
(70)
2OL
0
5
FIG. 13. Dependence of the total power absorbed by the plasma on scaled
fractional frequency separation ii,= (0, -o,)k,L/o,
for p=O.2. Solid
and dashed straight lines are the analytical approximation [Es. (53)]
obtained for ij, > Aq,+ using two different estimates for the region of
interaction. Triangles are self-consistent numerical results.
Phys. Plasmas, Vol. 1, No. 3, March 1994
where flL is given by Eq. (59). The sum of the terms
IV, + @+Qp [i.e., the integral of Eq. (55)], scaledto the
unperturbed power flux Pi [Eq. (46)], appearsin Fig. 14 as
the flat solid line having nearly zero value. The actual numerical value is smaller than 10e3, which indicates the
accuracy of the numerical calculations.
Figure 14 shows that external power input occurs in a
small region around &=O, where the plasma resonanceis
excited. The oscillations in the input power of the region
,.$a5 arise due to the interference between the uniform
Srivastava, Morales, and Maggs
577
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pump and the traveling Langmuir wave. This is a linear
effect unrelated to nonlinear Landau damping. The nonlinear power exchange (dominated by the plasmon transfer
process) is seen to rise steeply, along with a strong reduction in the w1 wave power flux in the region of the Airy
peak of the low-frequency wave (c-5.0).
VI. CONCLUSION
The present study provides a specific example of a selfconsistent investigation of a kinetic nonlinearity in a nonuniform plasma. The solution to the present problem is
facilitated by the separation of fluid-like properties from
the fully kinetic features. In a nonuniform medium the
fluid-like properties are described by a differential equation
in configuration space, while the kinetic features require a
differential equation in Fourier space. This technique can
be viewed as the nonuniform medium generalization of the
powerful method used to treat beam nonlinear&& in uniform media,14and it may prove quite valuable in attacking
a wide class of problems.
It is found that although the general picture of beatwave-particle interaction developed for uniform plasmas is
not qualitatively changed, the plasma nonuniformity introduces important modifications in the waveforms that invalidate the straightforward WKB extrapolation of the results. The situation is summarized in Fig. 13, where it is
seen that the power absorbed by slow electrons increases
linearly with frequency separation. The plasma nonuniformity introduces a critical frequency separation
(A&w),= 1.8/kAL determined by the spatial width of the
first Airy lobe. For frequency differences larger than the
critical value, the interaction is essentially determined by
the beat of a relatively short-wavelength Langmuir wave at
w1 and a nearly uniform (i+e., k2zO) electric field at 02,
This implies that the resonant velocity associated with the
and not the usual group velocidler field is (or -02)/kl,
ity. Of course, the monotonic increase in electron absorption with increasing frequency separation is eventually limited by linear Landau damping of the higher-frequency
field at ol. This effect is expected to set in when the frequency separation causes the resonant fields to be spaced
by one Landau damping length of the higher-frequency
wave. This limits the maximum frequency separation to
(01-0*)/01<
4.
It has been found that the power absorbed by slow
electrons is a factor of ( 1-02/wl) smaller than the power
transferred from the high to low-frequency wave. However, as Eq. (41) shows, the dissipative response of the
slow electrons is essential to the effective plasmon transfer
578
Phys. Plasmas, Vol. 1, No. 3, March 1994
process. Strong nonlinear Landau damping of the highfrequency wave is found to occur at pump amplitudes
(p-0.2)
below the level at which ponderomotive modifications in the profile play a major role (p - 1). The value of
pump-strength parameter p for which strong nonlinear
Landau damping occurs is also smaller than the effective
value (p 2 4) used by White et al. I5 in assessingthe regime
in which the parametric decay instability is excited in
laser-plasma interactions.
For frequency differences smaller than the critical
value, the nonlinear Landau damping process has been
shown to provide a generalization of the ponderomotive
interaction to nonzero frequencies. The usual adiabatic result is recovered for Zis=O, but, as is indicated by Eq. (54),
resistive ion contributions are present for G,#O.
The features described in this study may play a role in
resonant absorption experiments in which a frequency
spread is present. The two characteristic signatures of the
nonlinear process are acceleration of slow electrons, and a
spectral shift to low frequencies. Since, under certain conditions, the accelerated electrons can e%cite secondary
waves through sideband instabilities and Cerenkov emission, these signals could be misinterpreted as arising from
parametric instabilities.
ACKNOWLEDGMENTS
This work is sponsored by the Office of Naval Research (SS and GJM) and NASA (National Aeronautics
and Space Administration) (JEM).
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Srivastava, Morales, and Maggs
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