Talk at CPP Project Seminar, Queen’s University Belfast, UK, September 30, 2009, https://doi.org/10.6084/m9.figshare.13698991

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Propagation of electron-acoustic (EA) excitations
in the presence of
suprathermal background electrons:
linear and nonlinear effects
Ashkbiz Danehkar
School of Mathematics & Physics, Queen’s University Belfast, Belfast BT7 1NN, UK
Supervisors: Dr. Nareshpal Singh Saini & Dr. Ioannis Kourakis
Centre for Plasma Physics, Department of Physics and Astronomy, Queen's University Belfast,
Belfast BT7 1NN, Northern Ireland, UK
Prof. Manfred Armin Hellberg
School of Physics, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa

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We investigate Electron-Acoustic Waves (EAWs) in a collisionless and
unmagnetized plasma which consists of three species, namely:
• “cool” inertial electrons (in temperature Tc),
• inertialess “hot” suprathermal electrons (in temperature Th >Tc),
• stationary/inertial ions (separate cases, in comparison)
3. Linear effects: Dispersion Relation (DR)
2. Strategic Workplan:
• Work on a Cold One-Fluid Model (Tc=0)
• Add the Cool Electron Thermal Pressure (“Warm” One-Fluid Model; Tc>0)
• Add the Ion-fluid to make a Two-Fluid Model
Layout
5. Conclusion
1. Motivation
4. Nonlinear Pseudopotential Method: Existence of Solitary Waves

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The electron-acoustic waves occur in a plasma, where inertial cold electrons (Tc)
oscillate against inertialess hot electrons (Th).
The cold electrons provide the inertial effects maintaining the EAWs, while the
restoring force comes from the pressure of the hot electrons.
The EAWs is undamped for temperature ratio Tc/Th<0.1 and 0.2<nc/(nc+nh)<0.8.
The EAWs often occur in
• laboratory experiments
• space plasmas e.g. the Earth's bow shock
• the auroral magnetosphere
• Broadband Electrostatic Noise (BEN)
observed by satellites
1. Motivation
1/2
,0 3
2
( ) 1
h h
n n
κ
φ
φ
κ
− +
⎛ ⎞
= −
⎜ ⎟
−
⎝ ⎠
• low values of κ>3/2 are associated with
suprathermal
• For κ→∞, a Maxwellian is recovered
• Some space and laboratory plasmas
behaviors are extremely different from a
Maxwellian distribution. We describe this
suprathermal population by a κ-distribution

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2. Strategic Workplan
( )
0,
n nu
t x
∂ ∂
+ =
∂ ∂
u u
u
t x t
φ
∂ ∂ ∂
+ =
∂ ∂ ∂
0,
P P u
u P
t x x
γ
∂ ∂ ∂
+ + =
∂ ∂ ∂
( )
0,
n nu
t x
∂ ∂
+ =
∂ ∂
,
e
i
m
u u
u
t x m t
φ
∂ ∂ ∂
+ = −
∂ ∂ ∂
2
2
x
φ
∂
=
∂
n
+
Cold Electron-fluid
Thermal Pressure
1/ 2
0
3
0 2
1
h
c
n
n
κ
φ
κ
− +
⎛ ⎞
+ −
⎜ ⎟
−
⎝ ⎠
n
−
Cool Electron-fluid
Ion-fluid
1
,
c
h
T P
T n x
∂
−
∂
Hot Suprathermal Electrons
2
, 1 3,
f f
f
γ γ
= + = → =
Poisson's Equation
Blue: Model 1 (cold 1-fluid, static ions);
Blue + red: Model 2 (warm 1-fluid, static ions);
Green: Model 3 (ion inertia included)

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3. Linear effects: Dispersion Relation (DR)
Thermal Pressure
Ion-fluid
Hot Suprathermal Electrons
Poisson's Equation
Cool Electron-fluid
Linear method:
(0) (1) (0) 0
0
( , , , , , ), , 1,0,1 ,0,1,0 ,
h
c
n
S n u n u P S S S S
n
φ
⎛ ⎞
= = + = +
⎜ ⎟
⎝ ⎠
(1) (1)
,
k
n u
ω
= (1)
(1) (1)
,
c
h
k T
P
T
u φ
ω
⎛ ⎞
= − ⎜ ⎟
⎝ ⎠
−
(1) (1)
3 ,
P n
=
(1) (1)
,
k
n u
ω
=
(1) (1)
,
e
i
m k
u
m
φ
ω
=
2 (1)
k φ
− = (1)
n
+
1
(1)
0 2
3
0 2
1
h
c
n
n
κ
φ
κ
⎛ ⎞
−
+ +
⎜ ⎟
−
⎝ ⎠
(1)
0
0
h
c
n
n
n
− −

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3. Linear effects: Dispersion Relation (DR)
0
0
,
h
c
n
n
β =
2
2
2 2
D
k
k k
ω =
+
1/2
1
0 2
3
0 2
,
h
D
c
n
k
n
κ
κ
⎛ ⎞
−
= ⎜ ⎟
−
⎝ ⎠
6
2
3 k
σ
+
,
c
h
T
T
σ =
Thermal Effects
( ) ( )
2 2 2
2
2
3
1
,
D D
k
k k k k
σ
μ
+
⎡ ⎤
+ +
⎣ ⎦
+
,
e
i
m
m
μ = Inertial Ion Effects
Hot Suprathermal Effect
Frequency vs.
wavenumber k
ω
Result:
growing suprathermal distribution,
hot electron number density (nh/nc),
and decreasing cool electron
temperature (Tc/Th)
increases the linear EAW.

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4. Nonlinear Method: Existence of Solitary Waves
n
+
Thermal Pressure
1/ 2
0
3
0 2
1
h
c
n
n
κ
φ
κ
− +
⎛ ⎞
+ −
⎜ ⎟
−
⎝ ⎠
n
−
Ion-fluid
Hot Suprathermal Electrons
Poisson's Equation
Cool Electron-fluid
Pseudopotential Method:
,
x Mt
ξ = −
A traveling coordinate M is the Mach number
1
1 ,
u M
n
⎛ ⎞
= −
⎜ ⎟
⎝ ⎠
2 2
2 3 3
u M M n
φ σ σ
−
− + +
=
3
,
P n
=
1
1 ,
u M
n
β
+
⎛ ⎞
= −
⎜ ⎟
⎝ ⎠
2
2
u M M μφ
= − −
2
2
φ
ξ
∂
=
∂

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4. Nonlinear Method: Existence of Solitary Waves
1
0,
2
d
d
φ
ξ
⎛ ⎞
+ Ψ =
⎜ ⎟
⎝ ⎠
1/ 2
2
3 2
3/2
2
1
( 1
2
) 1 1
1 M
M
κ
φ
φ
β φ β
κ
− +
⎛ ⎞
⎛ ⎞
⎜
⎛ ⎞
⎛ ⎞
+ − +
⎜ ⎟
⎜ ⎟
⎟
+
Ψ = + − −
⎜ ⎟
⎜ ⎟
−
⎝ ⎠
⎝ ⎠
⎜ ⎟
⎝ ⎠
⎝ ⎠
8
Hot Suprathermal Effect
Result:
As the density of the hot suprathermal
electrons is increased, the potential
amplitude increases.

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4. Nonlinear Method: Existence of Solitary Waves
1
0,
2
d
d
φ
ξ
⎛ ⎞
+ Ψ =
⎜ ⎟
⎝ ⎠
Thermal Effects
( )
( ( )
( ) ( )
3/ 2
3
2
3 3
3/2 3/ 2
2 2
1 1
3 3
3 3
(1
1
6
2 2
)
3
M M
M M
κ
φ
β
κ
σ
σ
φ
φ
φ
β
σ
σ σ
− +
+ + ± −
⎞
⎡ ⎤ ⎡ ⎤
− + + + − ⎟
⎢
⎛ ⎞
⎛ ⎞
⎜ ⎟
+
⎥ ⎢ ⎥
⎣ ⎦
− −
⎜ ⎟
⎜
Ψ
⎟
−
⎣
⎠
⎦
⎝
⎝
= +
⎠
⎠
∓
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Result:
As the temperature of the cool electrons is increased, the potential amplitude increases.

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4. Nonlinear Method: Existence of Solitary Waves
3
M σ
<
Inertial Ion Effects
3
M σ
>
supersonic:
subsonic:
1
0
d
F
d φ
φ =
Ψ
= −
10
( )
( ( )
( ) ( )
3 3
3/ 2 3/ 2
2 2
3/ 2
3
/ 2
2
1
2
2
1 1
3 3
3
2
(1 ) 1 1
1
6
3
3
2 2
M M
M M
M
M
κ
φ
β
κ
σ σ
σ
σ
φ σ
φ
β μ
φ
μ
− +
⎛ ⎞
⎛ ⎞
⎜
+ + ± −
⎞
⎡ ⎤ ⎡ ⎤
− + + + − ⎟
⎢ ⎥ ⎢ ⎥
⎣
⎛ ⎞
⎛ ⎞
+ − −
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎟
+ − −
⎜ ⎟
Ψ
⎝ ⎠
⎦ ⎣ ⎦
⎝ ⎠ ⎜ ⎟
−
⎝ ⎠
⎝
=
⎠
⎠
∓
Result:
ion-fluid has a trivial role in modifying negative supersonic solitary waves, but it is
important to support the positive wave structures on the subsonic scale.

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4. Nonlinear effects: Supersonic domain
1 1
0
0
d
F M
d φ
φ =
Ψ
= − > → ( )
2 2
0
m
F M
φ φ −
=
= Ψ > →
low limit upper limit
( )
3
M σ
>
11
Result:
The existence domain for the
negative soliton becomes narrower
with the increase in the suprathermal
distribution, nh/nc, and Tc/Th.

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4. Nonlinear effects: Subsonic domain ( )
3
M σ
<
12
1 1
0
0
d
F M
d φ
φ =
Ψ
= − > → ( )
2 2
0
m
F M
φ
φ +
=
= Ψ > →
low limit upper limit
Result:
The existence domain for the positive soliton becomes wider with the increase in the
kappa, nh/nc, and Tc/Th.

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A. Danehkar, N. S. Saini, M. A. Hellberg and I. Kourakis, “Propagation of electron-acoustic
excitations in the presence of suprathermal background electrons,” manuscript in preparation.
5. Conclusion
• Linear Analysis: growing suprathermal distribution, and hot electron
number density and temperature (increasing nh/nc and decreasing Tc/Th)
widen the linear EAW.
• Nonlinear Analysis: The existence domain for the negative soliton
becomes narrower with the increase in the suprathermal distribution,
nh/nc, and Tc/Th.
• Ion-fluid does not affect the negative soliton existence, but is necessary
to maintain the positive solitary wave structure.
• Ion-fluid has a trivial role in the supersonic (fast) scale, but it appear to be
very important in the subsonic (slow) scale.
• Positive acoustic-waves deeply depends on suprathermal hot electron
parameters.
• Two-fluid cold model (Tc=0) cannot predict positive solitons.
• Cold electron temperature (Tc/Th) affects both negative and positive
solitons.

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Special thanks to:
• Dr. Nareshpal Singh Saini & Dr. Ioannis Kourakis (QUB, UK)
• Prof. Dr. Manfred Armin Hellberg (UKZN, Durban, S Africa)
• UK DEL funding via QUB (A.D.)
• UK EPSRC via QUB (N.S.S. & I.K.)
• S Africa NRF via UKZN (M.A.H.)
Acknowledgments:

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Thank You!