Table Of ContentLinear wave action decay entailed by Landau damping in
inhomogeneous, nonstationary and anisotropic plasma.
∗
Didier B´enisti
5 CEA, DAM, DIF F-91297 Arpajon, France.
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(Dated: January 20, 2015)
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n Abstract
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Thispaperaddressesthelinearpropagationofanelectronwaveinacollisionless, inhomogeneous,
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nonstationary and anisotropic plasma. The plasma is characterized by its distribution function,
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h f , at zero order in the wave amplitude. This distribution function, from which are derived all
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p
- the hydrodynamical quantities, may be chosen arbitrarily, provided that it solves Vlasov equation.
m
s Then,fromthelinearizedversionoftheelectronsequationofmotion,andfromGausslaw,isderived
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p an envelope equation for the wave amplitude, assumed to evolve over time and space scales much
.
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c larger thantheoscillation periodsofthewave. Theenvelopeequation maybecastintoan equation
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y forthethewaveaction, derivedfromWhitham’svariationalprinciple,thatdemonstratestheaction
h
p decay due to Landau damping. Moreover, the Landau damping rate is derived at first order in the
[
variations of thewave numberandfrequency. As brieflydiscussed,this papergeneralizes numerous
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5 previous works on the subject, provides a theoretical basis for heuristic arguments regarding the
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4 action decay, and also addresses the propagation of an externally driven wave.
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1. PACS numbers: 52.35.Fp
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:
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∗
Electronic address: [email protected]
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I. INTRODUCTION
The linear propagation of an electrostatic wave in an inhomogeneous and nonstationaty
plasma is, clearly, an old problem in plasma physics, that has led to numerous theoreti-
cal studies for several decades (see Ref. [1] and references therein for a rather exhaustive
bibliography on the subject). Recently, it has attracted some renewed interest, due to its
relevance to issues related to plasma compression. In particular, in Ref. [1], it is shown
that Whitham’s variational principle [2] does apply to describe the linear propagation of
an electron wave in an inhomogenous and nonstationary plasma. This result is actually
not straightforward because, when the plasma density, n, varies in space, or in time, the
Lagrangian density introduced in Whitham’s theory depends on three fields, the wave am-
plitude, E , its eikonal, ϕ, and the plasma density, n, while the Lagrange equations are
p
only valid for E and ϕ. Therefore, it is not obvious that the variations in n would entail
p
variations in E as predicted by Whitham’s theory, i.e., such that the wave action,
p
∂χ
A ≡ E2d3x, (1)
∂ω p
Z
be conserved. In Eq. (1), ω ≡ −∂ ϕ is the wave frequency, while χ is the electron suscepti-
t
bility, defined precisely by Eq. (64) of Section III. In the fluid limit, ∂χ/∂ω scales as n−1/2,
so that action conservation entails that E should scale as n3/4, as explained in Ref. [1]. This
p
scaling was confirmed numerically in Ref. [3], where it was shown that plasma compression
did induce an increase in the wave amplitude, so as to keep the wave action constant. How-
ever, it was also shown in Ref. [3] that compression makes the wave phase velocity decrease
compared to the thermal one, which eventually entails the decay of the wave amplitude
(and also of the wave action) because of Landau damping. Nevertheless, to the best of
the author’s knowledge, there is no unambiguous theoretical proof of the latter result. In-
deed, in spite of the abundant literature on the subject, no kinetic derivation of an envelope
equation, accounting for Landau damping, could be found for an electron wave propagating
in an inhomogeneous and nonstationary plasma. For example, the theoretical results of
Ref. [1] relie on fluid equations and, therefore, they cannot account for Landau damping.
Moreover, the expression found for χ in Eq. (1) is the fluid one, which becomes inaccurate
at large temperatures. The most complete kinetic derivation of an envelope equation we are
aware of, for an electron wave propagating in an inhomogeneous and non stationary plasma,
is that given in Ref. [4]. This derivation stems directly from Vlasov equation, linearized
2
about the distribution function, f (x,v,t), at zero order in the wave amplitude. However,
H
the results of Ref. [4] lack of generality because f obeys a force-free Vlasov equation, which
H
is usually not the case. Moreover, and more importantly, Landau damping is not recovered
in Ref. [4]. An envelope equation that does account for collisionless dissipation (and, in
particular, Landau damping) has been derived in Refs. [5, 6] starting from first principles.
The equation found in these papers is valid in the linear and nonlinear regimes, and, in the
former one, it does show that the wave action decays at the Landau damping rate. However,
the derivations of Refs. [5, 6] only hold for a homogeneous and stationary plasma.
The present paper aims at generalizing the aforementioned articles. The main result is
Eq. (79) showing that, in the linear regime, the wave action decays at the Landau damping
rate. It is derived directly from Gauss law and from the electrons equations of motion,
linearized about the dynamics that makes the plasma evolve on hydrodynamical scales.
This dynamics may include any kind of force field, including a magnetic one (which was not
the case in previous publications). Moreover, the expression of the Landau damping rate we
provide, given by Eq. (70), is valid at first order in the time and space variations of the wave
number and frequency. Hence, it does not stem from a WKB approximation but vindicates,
a posteriori, this approach.
Among all the possible applications of these results, the present work was mainly moti-
vated by the modeling of stimulated Raman scattering (SRS). Indeed, SRS is still an issue
for inertial fusion since large Raman reflectivities were measured at the Nation Ignition
Facility [7]. As discussed in several papers (e.g. [8, 9]), correctly modeling collisionless dissi-
pation is crucial to accurately predict Raman reflectivity. This might be done in an effective
way by making use of envelope equations, since they proved in Refs. [10, 11] to provide
estimates for SRS reflectivity as accurate as a particle in cell or a Vlasov code, within much
smaller computation times. However, the results of Refs. [10, 11] were only for a plasma
with constant density. There is, currently, no kinetic modeling of SRS in an inhomogeneous
and nonstationary plasma, and the present paper is the first of a sequence of forthcoming
articles that aim at filling this gap. Therefore, in order to make the application to SRS
straightforward, we actually address here the propagation of a wave that may be driven.
The derivation and presentation of our main results are organized the following way.
Section II, provides a general expression of the charge density in terms of the plasma distri-
bution function, f , at zero order in the wave amplitude. This charge density appears to be
H
3
the sum of two contributions of distinct nature. The first contribution varies on hydrody-
namical scales, and is related to slowly-varying force field that may exist in the plasma and
may make the density and temperature vary. The second contribution is at the origin of the
electron plasma wave (EPW), and is used to define what may be viewed as a generalized
electron susceptibility, Ξ. In Section III, ℑ(Ξ) is expressed in terms of the variations of the
wave number, of the frequency, and of the electric field amplitude which, from Gauss law,
provides the wave envelope equation in a one-dimensional geometry. This equation is first
derived when there is no resonant electron in Paragraph IIIA. In this case, it is shown to
guarantee the wave action conservation, with χ in Eq. (1) explicitly expressed in terms of
f . A wave equation that accounts for the contribution of all electrons, resonant and non
H
resonant, is derived in Paragraph IIIB. Then, the expression for χ is slightly changed, and,
more importantly, it is shown that the action decays due to Landau damping. Moreover,
our expression for the Landau damping rate accounts, at first order, for the space and time
variationsofthewave number andfrequency. SectionIVgeneralizes theresults ofSectionIII
to a three-dimensional geometry, while Section V summarizes and concludes this work.
II. THE ELECTRON CHARGE DENSITY
In order to describe the propagation of an electron plasma wave that may be externally
driven, one needs to calculate the charge density, ρ , induced by the EPW and the drive.
w
As a first step of this calculation is derived, in this Section, an expression for ρ in terms of
w
the electron distribution function, f , at zero order in the wave and drive amplitudes. For
H
the sake of clarity, the plasma is first assumed to be one-dimensional (1-D), and the results
obtained in 1-D are then easily generalized to a three-dimensional (3-D) geometry. From ρ
w
is also introduced, Ξ, defined by Eq. (18) or Eq. (31), that may be viewed as a generalized
electron susceptibility, and that allows to express Gauss law in a very simple fashion.
A. One-dimensional geometry
Let us denote by −e the charge of an electron, by m, its mass, and by f(x,v,t) the
electron distribution function, such that
+∞
f(x,v,t)dv = n (x,t), (2)
e
−∞
Z
4
where n is the electron density. Let us, moreover, assume that there is only one ion specie
e
(the generalization to multi-ion species is straightforward, but would require unnecessarily
complicate notations), and let us denote by n (x,t) the ion density and by Ze its charge.
i
Then, the total plasma charge density is,
+∞
ρ = −e f(x,v,t)dv+Zen (x,t). (3)
i
−∞
Z
From Liouville theorem, the electron distribution function is conserved, so that,
f(x,v,t) = f [x (x,v,t),v (x,v,t)], (4)
0 0 0
where f is the initial distribution function, and x and v are, respectively, the initial
0 0 0
position and velocity of an electron located at x, with velocity v, at time t. Let us now
express x and v in terms of x and v the following way,
0 0
x = x +v t+δx +δx
0 0 H
≡ X +δx (5)
0
v = v +δv +δv
0 H
≡ V +δv, (6)
0
where δx and δv are induced by the EPW and the drive, while δx and δv evolve on
H H
hydrodynamical scales. The two latter quantities are specifically defined by, dδ /dt = δv
H H
and dδv /dt = F /m, where F is the force that makes all the hydrodynamical quantities,
H H H
such as he local density and temperature, vary. More precisely, the distribution function,
f (X ,V ,t), at zero order in the wave and drive amplitudes, satisfies the following Vlasov
H 0 0
equation,
∂f ∂f F ∂f
H H H H
+V + = 0. (7)
0
∂t ∂X m ∂V
0 0
Then, ifattimet = 0, thewaveanddriveamplitudesarevanishinglysmall, Eq.(7)translates
into,
f (X ,V ,t) = f (X −v t−δx ,V −δv ). (8)
H 0 0 0 0 0 H 0 H
From Eq. (4), and the definitions Eqs. (5) and (6) of X and V , Eq. (8) is equivalent to,
0 0
f(x,v,t) = f (X ,V ,t). (9)
H 0 0
5
PluggingEq.(9)intoEq. (3)forthechargedensity, andmakinguseofthechangeofvariables
v → V in the integral of Eq. (3), yields,
0
+∞ +∞ ∂δv
ρ = −e f (X ,V ,t)dV +Zen (x,t)−e f (X ,V ,t) dV . (10)
H 0 0 0 i H 0 0 0
∂V
−∞ −∞ 0
Z Z
Since we aim at deriving an envelope equation for the EPW that is only at first order in the
variations of the density, we now use f (X ,V ,t) ≈ f (x,V ,t) − δx∂ f (x,V ,t). Note
H 0 0 H 0 x H 0
that latter approximation is not a linearization with respect to the wave amplitude, but an
expansion at first order in the space variations of f . Then,
H
+∞ +∞
ρ ≈ −e f (x,V ,t)dV +Zen (x,t)+e δx∂ f (x,V ,t)dV
H 0 0 i x H 0 0
−∞ −∞
Z Z
+∞ ∂δv +∞ ∂δv
−e f (x,V ,t) dV −e ∂ f (x,V ,t)δx dV (11)
H 0 0 x H 0 0
∂V ∂V
−∞ 0 −∞ 0
Z Z
≡ ρ +ρ , (12)
H w
where
+∞
ρ = −e f (x,V ,t)dV +Zen (x,t), (13)
H H 0 0 i
−∞
Z
evolves on hydrodynamical scales, and is considered as a given quantity. As for the charge
density, ρ , induced by the wave and the external drive, from now on, it is identified with its
w
linear value. Hence, by making use of an integration by part for the third term of Eq. (11),
one finds,
+∞ +∞
′
ρ (x,t) ≈ e δx∂ f (x,V ,t)dV +e δvf (x,V ,t)dV , (14)
w x H 0 0 H 0 0
−∞ −∞
Z Z
′
where we have denoted, f ≡ ∂ f .
H V0 H
We now assume that the electric field of the EPW and the drive writes,
E = E eiφ +iE ei(φ+δφ) +c.c., (15)
p d
where c.c. stands for the complex conjugate, and where E and E are, respectively, the
p d
real amplitudes of the EPW and drive fields, that vary much more slowly than the eikonal,
φ. From φ are defined the wave number, k ≡ ∂ φ and wave frequency, ω ≡ −∂ φ. Both
x t
E and E are at the origin of the charge density, ρ , so that E will henceforth be termed
p d w
the total electric field, although the force F may include a slowly varying electric field.
H
Clearly, E, may also be written the following way,
E = E eiϕ +c.c., (16)
0
6
where E ≡ E2 +E2 −2E E sin(δφ) is the total field amplitude (which is clearly real),
0 p d d p
and eiϕ ≡ E−q1eiφ(E +iE eiδφ). We moreover assume that the wave is nearly on resonance
0 p d
with the drive, so that k ≈ ∂ ϕ and ω ≈ −∂ ϕ.
x t
Similarly, we write the charge density, ρ , as,
w
ρ = ρ eiϕ +c.c., (17)
w 0
and introduce,
iρ
0
Ξ ≡ , (18)
ε kE
0 0
which may be viewed as a generalized electron susceptibility.
In the total field E, only the EPW electric field is self-consistently related to ρ , since
w
the drive is imposed externally (for example, in the case of SRS, the drive is due to electro-
magnetic laser fields, and is just the component of the ponderomotive force along the wave
number). Then, Gauss law is just ∂ (E eiφ +c.c.) = ρ /ε , which reads,
x p w 0
(ikE +∂ E )eiφ = −ikΞE eiϕ
p x p 0
= −ikΞ(E +iE eiδφ)eiφ. (19)
p d
Since E , E and E are real, Eq. (19) is equivalent to the two following equations,
0 p d
(1+Ξ )E = E [Ξ cos(δφ)+Ξ sin(δφ)], (20)
r p d i r
k−1∂ E = Ξ E +E [Ξ cos(δφ)−Ξ sin(δφ)], (21)
x p i p d r i
where Ξ ≡ ℜ(Ξ) and Ξ = ℑ(Ξ). We henceforth assume that E ≪ E which, as shown in
r i d p
Ref. [12], is the case for SRS. Then, Eq. (20) is approximated by
1+Ξ = 0, (22)
r
which actually yields the EPW dispersion relation. The validity of the approximation that
leads to Eq. (22) is discussed in detail in Ref. [12], where it is shown that accounting for the
drive amplitude in the dispersion relation is important to calculate the nonlinear frequency
shift of the plasma wave, but not to correctly estimate the linear frequency. Since we restrict
here to linear wave propagation, Eq. (22) may be considered as exact.
As will be shown in Sections III and IV, Ξ is of the order of (kL )−1 or (ωT )−1, where L
i 0 0 0
and T are, respectively, the typical length and time scales of variation of E , k or ω. Hence,
0 0
7
for the slow variations considered in this paper, Ξ ≪ 1, so that, from Eq. (22), Ξ ≪ Ξ .
i i r
This lets us conclude that Eq. (21) may be approximated by,
Ξ E −k−1∂ E = E cos(δφ). (23)
i p x p d
B. Three-dimensional geometry
TheresultsobtainedinParagraphIIAstraightforwardlygeneralizetoathree-dimensional
geometry. Just like for a 1-D plasma, let us introduce the distribution function,
f (X ,V ,t), at zero order inthe wave amplitude, thatobeys the following Vlasov equation,
H 0 0
∂f ∂f F ∂f
H +V . H + H. H = 0. (24)
∂t 0 ∂X m ∂V
0 0
Then, Eq. (10) for the charge density becomes,
∂(V +δv)
ρ = −e f (X ,V ,t) 0 dV +Zen (x,t), (25)
H 0 0 ∂V 0 i
ZZZ (cid:12) 0 (cid:12)
(cid:12) (cid:12)
where |∂(V + δv)/∂V | is the Jacobian of(cid:12) the change(cid:12)of variables V + δv → V . The
0 0 (cid:12) (cid:12) 0 0
linear value of this Jacobian is,
∂(V +δv) ∂δv
0 n
= 1+ , (26)
∂V ∂V
(cid:12) 0 (cid:12)lin n 0n
(cid:12) (cid:12) X
(cid:12) (cid:12)
where δv and V are, respec(cid:12)tively, the nt(cid:12)h component of δv and V . Therefore, following
n 0n 0
the lines of Paragraph IIA, one straightforwardly finds that the linerarized charge density
induced by the wave and the drive is,
ρw(x,t) = e δx.∂xfH(x,V0,t)dV0 +e δv.∂V0fH(x,V0,t)dV0. (27)
ZZZ ZZZ
Let us now write ρ and the total electric field as in Paragraph IIA,
w
ρ = ρ eiϕ +c.c., (28)
w 0
E = E eiφ +E ei(φ+δφ) +c.c. (29)
p d
≡ E eiϕ +c.c., (30)
0
where E , E and E are real vectors, and let us introduce,
p d 0
iρ
0
Ξ ≡ . (31)
ε k.E
0 0
8
Then, with the same approximations as in Paragraph IIA, Gauss law yields the following
equations,
1+Ξ = 0, (32)
r
∇.E k.E
p d
Ξ − = cos(δφ). (33)
i k.E k.E
p p
III. WAVE EQUATION IN A ONE-DIMENSIONAL GEOMETRY
A. Plasma with no resonant electron
For the sake of clarity, we first derive Ξ in a one-dimensional geometry. Moreover, we
i
start by assuming that there is no resonant electron, the exact meaning of this assumption
being clarified after Eq. (46) and at the end of Subsection IIIB2.
From Eq. (14), we now need to calculate δx and δv in order to derive Ξ . This is done
i
by solving the electrons equations of motion,
dδx
= δv, (34)
dx
dδv e
= − Eiϕ +c.c. (35)
dt m 0
Eq. (35) is solved by making use of an integration by parts, which yields, at first order in
the variations of E ,
0
−m dE
0
δv ≈ E (t)I (t)− I (t)+c.c., (36)
e 0 1 dt′ 2
(cid:12)t′=t
(cid:12)
where E (t) ≡ E [x(t),t], and (cid:12)
0 0 (cid:12)
I (t) ≡ E (t′)eiϕ[x(t′),t′]dt′ (37)
1 0
Z
is one primitive of E eiϕ, and
0
′ ′ ′
I (t) ≡ I (t)dt (38)
2 1
Z
is one primitive of I . Since we only look for a first order envelope equation, we now express
1
I at first order in the variations of k and ω, and I at zero order in these variations. Then,
1 2
one easily finds,
−i d2ψ/dt′2
I (t′) ≈ eiψ(t′) − , (39)
1 dψ/dt′ (dψ/dt′)3
(cid:20) (cid:21)
eiψ(t′)
′
I (t) ≈ − , (40)
2 (dψ/dt′)2
9
′ ′ ′
where we have denoted ψ(t) ≡ ϕ[x(t),t].
Let us now provide an expression for δv that is linearized with respect to E , the zero-
0
order motion being defined by the hydrodynamical force, F . To do so, we make use
H
′ ′ ′ ′ ′ ′
of the approximations, E (t) ≈ E [x − δX (t),t] and ψ(t) ≈ ϕ[x − δX (t),t], where
0 0 0 0
′ t ′′ ′′
δX (t) ≡ V (t )dt with dV /dt = F /m. This yields,
0 t′ 0 0 H
R
′
dE0/dt|t′=t = ∂tE0 +V0∂xE0, (41)
′
dψ/dt|t′=t = −ω +kV0, (42)
d2ψ/dt′2|t′=t = −∂tω +2V0∂tk +V02∂x2k +kFH/m, (43)
where, in Eq. (43), we used ∂ k = −∂ ω and dV /dt = F /m. From Eqs. (39-43), one easily
t x 0 H
finds,
−i ∂ ω −kF /m−2V ∂ k/∂t−V2∂2k/∂x2
I (t) ≈ eiϕ(x,t) + t H 0 t 0
1 kV −ω (kV −ω)3
(cid:20) 0 0 (cid:21)
(44)
eiϕ(x,t)
I (t) ≈ − . (45)
2
(kV −ω)2
0
Plugging Eqs. (44) and(45) into the expression (36) for δv yields, δv = δv eiϕ +c.c., with
0
m −iE ∂ E +V ∂ E ∂ ω −kF /m−2V ∂ k −V2∂ k
− δv ≈ 0 + t 0 0 x 0 +E t H 0 t 0 x . (46)
e 0 kV −ω (kV −ω)2 0 (kV −ω)3
0 0 0
Note that our expansions, leading to Eq. (46) for δv, only make sense if, whatever V ,
0
(ω−kV )T ≫ 1 and (ω/V −k)L ≫ 1, where T and L are, respectively, the typical time
0 0 0 0 0 0
and length scales of variation of E , k or ω. The latter conditions are precisely what we
0
mean by the conditions for no resonant electron in the plasma.
Since, in the expression Eq. (14) for ρ , δx is multiplied by ∂ f , it only needs to be
w x H
estimated at zero-order in the variations of E , k and ω. Then, it is easily found that
0
δx = δx eiϕ +c.c. with,
0
m −E
0
− δx ≈ . (47)
e 0 (kV −ω)2
0
From the knowledge of δv and δx given by Eqs. (46) and (47), Ξ is easily derived by making
use of the expression Eq. (14) for ρ and of the relation Eqs. (17) and (18) between ρ and
w w
Ξ. The corresponding calculations are detailed in the Appendix A where Ξ is expressed in
terms of the electron susceptibility, χ, defined here as,
e2 +∞ f′
χ = − H dV . (48)
0
ε mk kV −ω
0 −∞ 0
Z
10
