Electrostatic wavepackets in the presence of
superthermal (accelerated) electrons: modulational
instability and envelope soliton modes
S. Sultana and I. Kourakis
Centre for Plasma Physics, Department of Physics and Astronomy, Queen’s University Belfast, BT7 1NN, Northern Ireland, UK
Email: ssultana02@qub.ac.uk , i.kourakis@qub.ac.uk
3. Perturbative analysis
1. Introduction
Nonthermal particles are omnipresent in various plasma environments, such as the solar wind [1, 2, 3] and the solar corona [4].
Plasmas containing these type of particles are generally characterised by a high energy trail non-Maxwellian distribution named
kappa distribution [5, 6, 7]. The parameter κ determines the high
energy power law index and approaches a Maxwellian distribution
when κ → ∞ as shown in the following figure [8].
Reductive perturbation technique: consider small deviations from
the equilibrium state S (0) = (0, 0, 1)T , i.e.
nS (n)
S = S (0) + Σ∞
n=1
The scope of this study is to trace, from first principles, the
influence of superthermality on modulated-amplitude electron
acoustic (EA) wavepackets propagating in a plasma containing
superthermal electrons in the background.
2. A Fluid model for electron acoustic waves
We consider a three-component collisionless unmagnetized plasma
consisting of:
* (species 1) inertial electrons (charge −e and mass me),
* (species 2) kappa distributed inertialess hot electrons (charge −e
and mass me), and
* (species 3) stationary ions (charge qi = Zie and mass mi).
The number density nc, the mean velocity uc of cold electrons is
governed by the continuity, momentum equation and the system
potential Φ is obtained from Poisson’s equation
∂nc
+ ∇ · (nc uc) = 0,
(1)
∂t
∂uc
e
+ (uc · ∇)uc =
∇Φ ,
(2)
∂t
me
−κ+ 1
2
eΦ
2
− Zini0 .(3)
∇ Φ = 4πe nc + nho 1 −
3
(κ − 2 )kB Th
(n)
(n)
(n)∗
where the condition S−l = Sl
holds, for reality. The wave
(5)
(6)
Normalization:
n = nc/nc0, u = uc/v0, φ = Φ/Φ0, t = t ωpeh, x = x/λDh,
1/2
−1 = (4πn e2/m )−1/2,
2
where λDh = kB Th/4πnh0e
, ωpeh
e
h0
v0 ≡ (kB Th/me)1/2.
-0.30
β = nc0/nh0 .
(1)
(1)
∂φ1
∂φ1
+ vg
=0,
∂T1
∂X1
where the group velocity vg is defined as
dω ω 3c1
vg =
= 3 .
κ − 1/2
(κ − 1/2)(κ + 1/2)
,
c2 =
,
c1 =
2
κ − 3/2
2(κ − 3/2)
(κ − 1/2)(κ + 1/2)(κ + 3/2)
c3 =
.
3
6(κ − 3/2)
3
4
5
k
-10
Figure 3. Left & right panels: green for κ = 3.5; magenta for κ = 5; red for
κ = 7; and blue for κ = 100. Here, we have assumed β = 0.5.
7.3 Bright versus dark envelope solitons
(Envelope solitons): Two types of (analytical) localized solutions:
Bright-type solitons (for P/Q > 0) and
dark solitons (for P/Q < 0):
1.0
1.0
0.5
-20
20
40
-60
-40
-20
20
-0.5
-0.5
-1.0
-1.0
40
60
Figure 4. Left panel: Bright-type modulated wavepackets (for P/Q > 0). Right
panel: Dark-type modulated wavepackets (for P/Q < 0).
We depict the ratio P/Q versus k for β = 0.5 (left panel), and
versus β for k = 3 (right panel).
dk
k β
We obtain the expression for the amplitudes corresponding to the
first, second and zeroth harmonics to order ∼ , the solution:
(1)
(2)
(2)
φ ' φ1 ei(kx−ωt) + 2 φ0 + φ2 e2i(kx−ωt) + O(3) ,
and similar expressions for n, u.
P
Q
P
Q
0.3
0.2
0.05
0.1
1
2
3
4
k
0.5
1.0
1.5
2.0
2.5
3.0
Β
-0.1
-0.2
-0.05
-0.3
→ Harmonic generation (in n, u, φ)!
Figure 5. P/Q > 0 (P/Q < 0) provide the region where bright (dark, respectively) type solitons may occur. Also, modulational instability occurs for
P/Q > 0 only (see above). Curves: green for κ = 3.5, magenta for κ = 5, red
for κ = 7, and blue for κ = 100 (Maxwellian!).
(1)
5. Nonlinear Schrödinger (NLS) equation for φ1 :
∂ 2ψ
∂ψ
2ψ = 0.
+
Q
|ψ|
+P
i
∂τ
∂ζ 2
(7)
7.4 Superthermality (via κ) and electron composition
(via β) effect
where
(1)
• ψ ≡ φ1 (ζ, τ ).
The effect of superthermality on the MI growth rate has been analysed by considering k = 3.2 and β = 0.5 (left panel). The effect
of β on the MI growth rate for representative parameter values
k = 4.5, κ = 8 is depicted in the region 0.25 ≤ β ≤ 4 (right
panel), where EAW may survive Landau damping [9]
• ζ = X1 − vg T1, τ = T2.
ω 5 c1
1
3
00
• Dispersion coefficient: P = 2 ω (k) = − 2 k4β 2 .
• Nonlinearity coefficient: Q → long expression omitted here.
6. Modulational instability (MI) of ES wavepacket
ω̃ 2 = P k̃ 2 (P k̃ 2 − 2Q|ψ̃1,0|2) ;
Gk
Gk
(8)
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
x
0.2
0.4
0.6
0.8
1.0
1.2
1.4
x
Figure 6. Left panel: κ = 3.5 (green); 5 (magenta); 7 (red); and 100 (blue) for
β = 0.5. Right panel: green for β = 0.5; magenta for 1; red for 2; blue for 4.
• If P Q > 0 then instability occurs for k̃cr = (2Q/P )1/2 | ψ̃1,0 | ;
1/2
β
• Modulational instability growth rate: Γκ = αx 2 α
;
− x2
where Γκ = Im|ω̃|/(Q∞|ψ̃1,0|2), α = P/P∞, β = Q/Q∞, and
x = k̃/(Q∞/P∞)1/2|ψ̃1,0|.
• The sign of the product P Q determines the stability profile.
7. Parametric analysis
8. Summary
• The MI wavenumber threshold kcr decreases (leading to a wider
stability window and bright solitons) in the presence of more
superthermal electrons (via β) for fixed κ.
• Short carrier wavelengths (wavenumber above the MI threshold
kcr ) lead to a bright type envelope solitary structures, while on
the other hand large wavelengths lead to a dark type soliton;
• A higher concentration of superthermal electrons (above a β
threshold, see Fig. 5b) gives rise to bright solitons (and enables
modulational instability).
The wavenumber threshold kcr (for instability) is depicted versus
– superthermality (via κ) for different density ratio β values (left
panel), and
– the density ratio β, for different κ (right panel).
• Increasing superthermality provides a wider stable region (for
fixed β; see Fig. 5a);
• The instability growth rate depends on β and κ dramatically.
References
4.0
The influence of the parameters β (hot electron concentration) and κ (superthermality) on EAW characteristics is the focus of the investigation that follows.
The coefficients entering Poisson’s equation [Eq. (6)] are
2
The equations for n = 2, l = 1 provide the compatibility condition
7.1 The critical wavenumber
The cold-to-hot electron density ratio is expressed by
1
-0.25
-40
• If P Q < 0 then: stability;
(4)
10
0.5
where the index “0” denotes the equilibrium number density values.
∂n
∂(nu)
+
= 0,
∂t
∂x
∂u
∂u ∂φ
+ u
=
,
∂t
∂x ∂x
∂ 2φ
2 + c φ3,
≈
β(n
−
1)
+
c
φ
+
c
φ
1
2
3
2
∂x
k
20
k (1)
(1)
u1 = − φ1 .
ω
• Dispersion relation (for an amplitude perturbation):
The one dimensional normalized form of Eqs. (1) - (3) reads:
5
-0.20
• Plane wave solution of (7): ψ = ψ0 exp(iQ|ψ0|2τ );
• Linear analysis: set ψ̃ = ψ̃0 + ψ̃1,0 cos (k̃ζ − ω̃τ );
We adopt a kappa distribution for the hot electrons
"
#−κ+1/2
eΦ
nh = nh0 1 −
;
3
(κ − 2 )kB Te
4
-0.15
4. Multi-harmonic solution up to order ∼ 2
Localized modulated-envelope structures are generated due to
nonlinear self interactions of the carrier wave in such plasmas.
3
-0.10
amplitude is thus allowed to depend on the stretched (slow ) coordinates of space and time as Xn = Σnnx and Tn = Σnnt,
respectively, where n = 1, 2, 3, ....., distinguished from the (fast)
carrier variables x (≡ X0) and t (≡ T0).
Substituting the above expressions into Eqs. (4) - (6) provides the
familiar EAW dispersion relation and the amplitude corresponding to the first harmonics in order for equations n = 1, l = 1,
Figure 1. Comparison of generalized Lorentzian distributions for the spectral
index of κ = 2, 6 and 25, with the corresponding Maxwellian distribution κ = ∞.
2
30
-0.05
il(kx−ωt) ,
S (n) = Σ∞
S
(X,
T
)e
l=−∞ l
k 2 + c1 (1)
(1)
n1 =
φ1 ,
β
Q
P
1
where 1 is a (real) smallness parameter. We assume that
2β
k
ω2 = 2
,
k + c1
7.2 Wavelength dependence of MI
The conditions for instability can be inferred from the diagrams:
14
3.5
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12
3.0
10
Κ
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2.5
8
Β 2.0
6
1.5
4
1.0
Proc. 10th Int. Solar Wind Conf., ed. M. Velli, R. Bruno, and F. Malara, AIP (2003).
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0.5
2
0
1
2
3
k
4
5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
k
Figure 2. Left panel: β=0.25 (green); 1 (magenta); 2.5 (red); and 4 (blue).
Right panel: κ=3 (green); 4 (magenta); 8 (red); and 100 (blue).
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