Contentsprr.hec.gov.pk/jspui/bitstream/123456789/9518/1/Sabeen_Arshad_Ph… · plasma and introduce...

1

Contents Chapter 1 ......................................................................................................................................... 4

Introduction ..................................................................................................................................... 4

1.1 Nonlinearity in Plasma Physics ............................................................................................ 4

1.1.1 Solitons ......................................................................................................................... 4

1.1.2 Korteweg-de Vries (KdV) Equation .............................................................................. 5

1.1.3 Nonlinear Schrondinger Equation ............................................................................... 6

1.1.4 Kadomtsev Petviashvili (KP) equation ......................................................................... 7

1.1.5 Zakharov-Kuznetsov (ZK) Equation .............................................................................. 7

1.1.6 Sine-Gordon Equation .................................................................................................. 8

1.2 Vortices ............................................................................................................................... 9

1.3 Sagdeev Potential .............................................................................................................. 10

1.4 Adiabatic Capture of Electron ............................................................................................ 12

1.6 Quantum plasmas .............................................................................................................. 16

1.6.1 Basic Properties of Quantum Plasmas ....................................................................... 17

1.6.2 Wigner Function ......................................................................................................... 18

1.7.3 Schrodinger-Poisson Function ................................................................................... 19

1.7.4 Quantum Hydrodynamic (QHD) Fluid Description .................................................... 20

1.7.5 Applications of Quantum Plasmas ............................................................................. 21

1.8 Alfven Waves ..................................................................................................................... 21

1.8.1 Kinetic and Inertial Alfven wave ................................................................................ 22

1.8.2 Coupled kinetic Alfven-acoustic wave ....................................................................... 23

1.9 Fluid Models ....................................................................................................................... 25

1.9.1 Multi-Fluid Model ...................................................................................................... 25

1.9.2 Single Fluid Model (Ideal Fluid Model) ...................................................................... 25

1.9.2 Hall-MHD Model ........................................................................................................ 26

1.9.3 Kinetic Model ............................................................................................................. 26

1.10 Dusty Plasmas .................................................................................................................... 26

1.11 Layout of the Thesis ........................................................................................................... 27

Chapter 2 ....................................................................................................................................... 29

2

Effect of adiabatic trapping on vortices and solitons in a degenerate plasma in the presence of a quantizing magnetic field ........................................................................................................... 29

2.1 Introduction ....................................................................................................................... 29

2.2 Basic set of Equations ........................................................................................................ 31

2.3 Linear Dispersion Relation and Bounce Frequency ........................................................... 35

2.4 KdV Type of Solution ......................................................................................................... 36


2.6 Results and Discussion ....................................................................................................... 39

2.7 Conclusion .......................................................................................................................... 40

Chapter 3 ....................................................................................................................................... 48

Finite amplitude solitary structures of coupled kinetic Alfven-acoustic waves in dense plasmas ....................................................................................................................................................... 48

3.1 Introduction ....................................................................................................................... 48

3.2 Mathematical Formulation ................................................................................................ 51


3.4 Results and Discussions ..................................................................................................... 55

3.5 Conclusion .......................................................................................................................... 57

Chapter 4 ....................................................................................................................................... 66

Coupled acoustic-kinetic Alfven waves in self gravitating dusty plasmas in the presence of adiabatic trapping ......................................................................................................................... 66

4.1 Introduction ....................................................................................................................... 66

4.2 Mathematical Formulation ................................................................................................ 68

4.3 Linear dispersion relation with negative dust charge ........................................................ 69

4.3.1 Limiting cases ............................................................................................................. 70

4.4 Stability analysis with the jeans term ................................................................................ 71


4.6 Linear dispersion relation with positive dust charge ......................................................... 74

4.6.1 Limiting cases ............................................................................................................. 75

4.7 Stability analysis with the jeans term ................................................................................ 76


4.9 Results and discussions ...................................................................................................... 77

4.10 Conclusion .............................................................................................................................. 79

3

Chapter 5 ....................................................................................................................................... 90

Summary ........................................................................................................................................ 90

References ..................................................................................................................................... 92

4

Chapter 1

Introduction

In this chapter we will discuss some fundamental properties of degenerate quantum

plasma and introduce some essential features of nonlinear wave propagation such as

solitons and vortices as these concepts will form the basis of my thesis. Since we are

going to treat nonlinear behavior, therefore different techniques to treat the nonlinear

plasma are discussed in detail.

1.1 Nonlinearity in Plasma Physics

Plasma is essentially a nonlinear medium. In the linear approximation, different types of

instabilities are investigated by taking into consideration that the growing perturbations

have small amplitude. When the amplitude becomes larger, linear theory breaks and

nonlinearities then become important. In general, in plasmas two types of nonlinearities

are studied

1. Scalar type nonlinearities (e.g., Soliton or Solitary wave)

2. Vector or Rotational type nonlinearities (e.g., Vortices etc)

We shall mainly be concerned with the first two types of nonlinear phenomena in this

thesis.

1.1.1 Solitons

John Scott Russell described solitary waves or solitons for the first time in 1834. He

observed this wave in the Union Canal in Scotland while travelling along it [1]. He saw

that the wave or perturbation moved for a long distance with no change in shape and

velocity. Russell observed this occurrence in 1834 but published this in only 1844 [1].

Later (1895) it was shown theoretically by Korteweg and de Vries [2] that for water

waves in a shallow canal that such solitary structures are produced by the balance of

5

nonlinear and dispersive effects in the medium confirming that these phenomena are

localized and preserve their shape while traveling at constant speed. This later became

known as the famous KdV equation. Much later in 1965 Zabusky and Kruskal [3]

showed through numerical investigations that these structures are very robust and have

particle like properties i.e. collisions and coined the term “solitons”

It was also later on shown that the KdV equation has an infinite number of conservation

laws showing that it is very robust in character. Thus, in summary a soliton or solitary

wave has the following properties [4]:

1. It occurs due to the balance b/w dispersion and nonlinearity.

2. is localized and propagates with no change of its properties (shape, velocity,

energy etc)

3. solitons have perfectly elastic collisions.

Different type of Solitons such as those described by the Nonlinear Schrodinger (NLS)

equation [5], Sine Gordon equation [6], Kadomtsev Petvishvilli [7] equation etc have

been observed experimentally and studied theoretically in many different branches of

physics when nonlinearity is taken into account. A brief overview of occurrences of some

of these equations is given below. Solitons are observed in: Water waves [1], oscillating

shock waves in optical fibers [8], waves in plasma physics [9] and elastic surface pulses

[10] etc. Below we describe some of the nonlinear wave equations which yield solitary

wave solutions in plasma.

1.1.2 Korteweg-de Vries (KdV) Equation

The Korteweg-de Vries equation deals with the dispersion and nonlinearity, which are

imperative plasma properties. The KdV equation was first derived by Diederik Korteweg

and Gustav de Vries [11]. Most often different type of perturbation techniques are used

to arrive at the KdV equation. The KdV equation is of the form

𝜕𝜙𝜕𝜏

+ 𝜙 𝜕𝜙𝜕𝜉

+ 𝜕3𝜙𝜕𝜉3

= 0 (1.1)

6

Where 𝜉 and 𝜏 are independent variables and 𝑎 and 𝑏 are nonzero constants. Equation is

nonlinear due to 𝜙 𝜕𝜙/𝜕𝜉, and dispersive due to 𝜕3𝜙/𝜕𝜉3. KdV equation allows soliton

solutions for one dimensional propagation of shallow water waves. This equation was

first derived by Korteweg-de Vries (1895) [12] in order to study the problem associated

with the surface water waves having constant depth. Much later in 1965 for one-

dimensional acoustic waves Kruskal and Zabusky [13,14] derived this equation for

anharmonic crystals.

Its solution is given by

𝜙 = 𝜙0 sech2[𝑘(𝑥 − 𝑣𝑡)] (1.2)

And from the above solution it can be seen that the amplitude 𝜙0, the width 1/𝑘 and the

velocity 𝑣 of propagation of the soliton are closely related. The faster solitons are

narrower and have a larger amplitude. These nonlinear equations which describe Solitons

have exact solutions. That’s why they are special class of nonlinear evolution equations.

Generally, nonlinear differential equations do not have exact solutions, but this class of

equations has exact solutions.

1.1.3 Nonlinear Schrondinger Equation

The cubic nonlinear Schrodinger (NLS) equation is

𝑖 𝜕𝜙𝜕𝑡

+ 𝛼 𝜕2𝜙𝜕𝑥2

+ 𝛽𝛷|𝛷|2 = 0 (1.3)

It has the structure of the quantum Schrodinger equation with 𝛽|𝛷|2 as a potential. It may

be noted that this is a complex equation. The solution of the above equation with

𝜉 = 𝑥 − 𝑏𝑡 (under the boundary conditions that Φ → 0 as |𝑥| → ∞) is

𝜙(𝑥, 𝑡) = 2𝑎𝛽 𝑒𝑥𝑝 𝑖 1

2𝛼𝑏𝑥 − 1

4𝛼𝑏2 − 𝑎 𝑡 𝑠𝑒𝑐ℎ 𝑎

𝛼(𝑥 − 𝑏𝑡) (1.4)

Where 𝑎 is an arbitrary constant which ties together the amplitude, width and frequency

of the packet. The NLS equation does not have an infinite number of conservation laws

7

thus the solitons produced here are not as robust as the solitons of the KdV equation. This

equation describes envelop solitons and is often used to study the modulational instability

of these envelop solitons.

1.1.4 Kadomtsev Petviashvili (KP) equation

The two dimensional generalization of KdV equation is the KP equation

𝜕𝑥(𝜕𝑡𝜙 + 𝑎 𝜙 𝜕𝑥𝜙 + 𝑏 𝜕𝑥3𝜙) + 𝑐 𝜕𝑦2𝜙 = 0 (1.5)

Where 𝑥 and 𝑦 represents space and 𝑡 is the time variable. The solution of KP equation is

𝜙 = 3(𝑣−𝑐)2𝑎

sech2[𝑘𝑥𝑥+𝑘𝑦𝑦−𝑣𝑡∆

] (1.6)

Where ∆ is the width of the soliton having the relation ∆= 4𝑏𝑣−𝑐

and 𝑣 is the constant

speed of the wave.

It may be noted here that the propagation is always in the 𝑥 direction, however 𝑦

direction contributes the two dimensional nature of the perturbation of the wave. This

equation has been studied in many plasma physics problems.

1.1.5 Zakharov-Kuznetsov (ZK) Equation

Another multi-dimensional generalization of KdV equation is the Zakharov-Kuznetsov

(ZK) Equation which is given by this

𝜕𝜙𝜕𝑡

+ 𝑎 𝜙 𝜕𝜙𝜕𝑥

+ 𝑏 𝜕3𝜙𝜕𝑥3

+ 𝑐 𝜕3𝜙𝜕𝑥𝜕𝑦2

= 0 (1.7)

The solution of ZK equation is

𝜙 = 3𝑣𝑎

sech2[(𝑘𝑥𝑥 + 𝑘𝑦𝑦 − 𝑣𝑡)𝑣

4(𝑏+𝑐)] (1.8)

8

Where 𝑎 , 𝑏, 𝑐 are the coefficients. Zakharov-Kuznetsov (ZK) equation is derived in

(1974) for nonlinear waves in magnetized plasma. ZK [15] have shown that three-

dimensional solitons are present in ion electron plasma in the magnetic field presence.

The ZK equation is also useful in the study of vortex solitons in plasma [16].

1.1.6 Sine-Gordon Equation

It is a nonlinear partial differential equation and gets its name from the famous Klein

Gordon equation in particle physics, only here the nonlinearity occurs through the 𝑠𝑖𝑛𝜙

term and has the form

𝜙𝑥𝑥 − 𝜙𝑡𝑡 = 𝑚2𝑠𝑖𝑛𝜙 (1.9)

Sine-Gordon equation has the solution

𝜙 = 4 𝑡𝑎𝑛−1 𝑒𝑥𝑝[𝑚 𝛾(𝑥 − 𝑣𝑡) + 𝛿] (1.10)

Where 𝛾2 = (1 − 𝑣2)−1 , 𝛿 is an arbitrary constant, 𝑣 is the velocity and 𝑚 is the mass

of kink. Its solution is called a “kink” representing a twist in the variable 𝜙 having range

𝜙 = 0 to 𝜙 = 2𝜋. Kinks and anti-kinks can propagate without distortion. An example of

a kink solution is a Bloch wall between two magnetic domains in ferromagnet. The

magnetic spins turn from spin up in one domain to spin down in nearby domain. The

conversion section between up and down is called Bloch wall [17].

If linear approximation is used i,e 𝑠𝑖𝑛𝜙 = 𝜙 , the equation is called the Klein-Gordon

equation.

𝜙𝑥𝑥 − 𝜙𝑡𝑡 = 𝜙

The Klein-Gordon equation is dispersive but 𝑠𝑖𝑛𝜙 term contains both nonlinearity and

dispersion which is responsible for soliton solution.

9

1.2 Vortices

Vortices are an exceptional kind of solitary structures of 2+1 dimensions (2 spatial and 1

temporal) and have solutions to a particular type of nonlinear partial differential

equations with vector nonlinearities. This is explaining below in this section. Physically,

vortices become visible when the fluid motion has a velocity shear and when this velocity

becomes greater owing to nonlinear effects, it produces a bending of the wavefront,

which has a two dimensional solitary structure.

Historically, Charney derived a nonlinear partial differential equation for Rossby waves

in a rotating shallow fluid [18], later Hasegawa and Mima developed a similar equation

in plasma physics. It contains a nonlinearity in the form of two dimensional vector

product (Jacobian) represented by the canonical Poisson bracket for the functions 𝐹 and

𝐺 which are functions of (𝑥,𝑦). The Poisson bracket is defined as:

[𝐹,𝐺] =𝜕𝐹𝜕𝑥

𝜕𝐺𝜕𝑦

−𝜕𝐹𝜕𝑦

𝜕𝐺𝜕𝑥

Larichev and Reznik [19] developed a new method to solve the Charney equation and

established the existence of dipolar vortices as stationary solutions. There have been a

number of numerical experiments [20,21] which support the existence of vortex motion

in magnetized plasmas

The Hasegawa-Mima equation describes the low frequency propagation of vortices in a

non-uniform plasma. Owing to its simple form the Hasegawa-Mima equation has been

calculated numerically [22] and analytically [23]. Both linear and nonlinear wave’s

vortex structures can be found in HM-equation. HM-equation incorporate drift waves

because of density inhomogeneity. The HM-equation has the following form

𝜕𝜕𝑡

(∇2𝜙 − 𝜙) − [(∇𝜙 × ẑ).∇] ∇2𝜙 − ln 𝑛0𝜔𝑐𝑖 = 0 (1.11)

10

The Hasegawa-Mima equation is a generalization of the Charney equation written

originally for fluid dynamics. The general method used to solve such equations is called

“piecewise linear solution” method [24].

Different types of equations are used to give solitary solutions.

1.3 Sagdeev Potential

An alternative way of investigating solitary wave solutions is the Sagdeev potential

technique. We illustrate this method by considering the following example. Consider an

ion acoustic wave which has a linear dispersion relation 𝜔 = 𝑘𝑐𝑠/1 + 𝜆𝐷2 𝑘2 ≈ 𝑘𝑐𝑠 −12𝜆𝐷2 𝑘3𝑐𝑠. Where second term is the dispersive term. However by considering the

electrons to be inertialess and Boltzmannian and shifting to a co-moving frame of

reference moving with speed 𝑢0 , from the ion equation of continuity and motion we have

for the ion velocity and number density the following two expressions

𝑢𝑖 = 𝑢01 − 2𝑒𝜙𝑚𝑖𝑢02

(1.12)

𝑛𝑖 = 𝑛0 1 − 2𝑒𝜙𝑚𝑖𝑢02

−1/2

(1.13)

Where 𝑛0 is the equilibrium density, 𝑚𝑖 is the mass of ion, 𝑒 is the electronic charge and

𝜙 is the potential. In arriving these two expressions above we have taken that all

perturbed quantities vanish at infinity [25].

By using ion and electron densities in Poisson’s equation we get

𝜀0𝑑2𝜙𝑑𝑥2

= 𝑒(𝑛𝑒 − 𝑛𝑖) = 𝑒𝑛0 𝑒𝑥𝑝 𝑒𝜙𝐾𝑇𝑒 − 1 − 2𝑒𝜙

𝑚𝑖𝑢02−1/2

(1.14)

The above equation is a nonlinear evolution equation where so far no approximations

have been used and 𝜙 has an arbitrary nonlinearity. Traditionally, nonlinear wave

equations are expressed in dimensionless form. Using the following normalizations

11

Φ ≡𝑒𝜙𝐾𝑇𝑒

𝜉 ≡𝑥𝜆𝐷

𝑀 ≡𝑢0

(𝐾𝑇𝑒/𝑚𝑖)1/2

The above equation can be written as

𝑑2Φ𝑑𝜉2

= 𝑒Φ−1 − 2Φ𝑀2

−1/2≡ −𝑑𝑉(Φ)

𝑑Φ (1.15)

The above mentioned equation is termed as particles equation of motion moving in a

potential (Φ) , with potential Φ showing position and the position 𝜉 showing time

respectively. 𝑉(Φ) is thus a Pseudo potential most often referred to as the Sagdeev

potential. 𝑀 is the Mach number and 𝜆𝐷 = 𝜀0𝐾𝑇𝑒/𝑒2 𝑛0 is the Debye length. The

potential 𝑉(Φ) can be obtained by integrating above equation with the boundary

condition 𝑉(Φ) = 0 𝑎𝑡 Φ = 0.

𝑉(Φ) = 1 − 𝑒Φ + 𝑀2 1−1 − 2Φ𝑀2

1/2 (1.16)

Figure 1.1: Sagdeev potential V(Φ) vs potential Φ.

Sagdeev pseudo-potential is nonlinear. To study the time independent solitary structures,

we create a single variable on which all the dependent variables depend 𝜉 = 𝑥 −𝑀𝑡

(where 𝜉 is normalized by 𝜆𝐷 and 𝑀 is the Mach number) and using the boundary

conditions as Φ → 0, 𝑑Φ𝑑𝜉→ 0 at 𝜉 → ±∞. The energy equation of an oscillating particle

of unit mass with velocity 𝑑Φ𝑑𝜉

and position Φ in a potential 𝑉(Φ) can be written as

12𝑑Φ𝑑𝜉2

+ 𝑉(Φ) = 0 (1.17)

12

In order to obtain solitary waves, following conditions must be fulfilled: if 𝑑2𝑉

𝑑Φ2Φ=0< 0

so that fixed point at origin is unstable, 𝑉(Φ) < 0 when 0 > Φ > Φ𝑚𝑖𝑛 for negative-

potential solitary waves and for positive potential solitary waves 𝑉(Φ) < 0 when

0 < Φ < Φ𝑚𝑎𝑥 , where where Φ𝑚𝑖𝑛 and Φ𝑚𝑎𝑥 is the negative and positive root of the

electrostatic potential for which 𝑉(Φ) = 0 [26].

In Fig. 1, the upper arrow shows the way of soliton, it reflects back from the right. The

lower arrows show the movement of that particle who loses its energy and gets trapped in

potential well.

We would like to note here that if we make a small amplitude expansion in Φ and retain

upto Φ2 terms then an exact solution can be which is the same as the solution of the KdV

equation given above for ion acoustic wave with Mach number less than 1. Thus the

Sagdeev potential method allows us to consider arbitrary nonlinearities while

investigating the formation of soliton or solitary structures.

1.4 Adiabatic Capture of Electron

In this section we describe briefly the phenomenon of adiabatic capture or trapping of

electrons and will show how trapping changes the nonlinear properties in plasmas.

The trapping occurs due to interaction of plasma particles with when it was considered

nonlinearly [27]. Bernstein, Greene and Kruskal [28] showed that trapped particles have

considerable effect on the nonlinear dynamics of plasmas, where trapping was considered

through the wave itself. Adiabatic trapping was introduced at the microscopic level by

Gurevich [29], and it was observed that adiabatic trapping introduced a 3/2 power

nonlinearity instead of the typical quadratic one when trapping was not present. We are

dealing with this type of trapping, described in detail below.

In slowly applied electric field, consider the electron plasma distribution in a slowly

applied electric filed. Let 𝐿 be the size of the applied field and 𝜏 is the variation time of

the electric field than

13

𝜏 ≫ 𝐿𝑣𝑒

(1.18)

Here 𝑣𝑒 is the velocity of electron and also 𝜏 is relatively small compared to electron’s

mean free time, hence the plasma is collisionless. When the electron passes through the

field, the field is taken to be stationary [30]. Since the solution of collisionless transport

equation has the dependence on the particle’s integral of motion, hence for the stationary

distribution of particles amplitude only time independent solutions are taken into account.

Consider 1D case for which the potential 𝜑 depends on only one coordinate 𝑥. The

motion in 𝑦- and 𝑧- directions is not important. The distribution function is considered

according to momentum 𝑝𝑥 and the 𝑥 coordinate. Thus the integral of motion depends

upon the total energy which consists of the kinetic and potential energies and does not

depend explicitly on time and is given by

𝜖 = 𝑝𝑥2

2𝑚+ 𝑈(𝑥) (1.19)

Where 𝑈(𝑥) = −𝑒𝜑(𝑥)

The stationary distribution function is given as

𝑓 = 𝑓[𝜖(𝑥,𝑝𝑥)] (1.20)

The function 𝑓(𝜖) depends upon boundary conditions.

The field 𝑈(𝑥) can create either a potential barrier or a potential well (Fig. 2). We shall

only consider the case of a shallow potential well and the function 𝑓(𝜖) can be calculated

by electron’s distribution which are coming towards the potential well from infinity. Thus

the electrons far from the barrier have equilibrium distribution in each direction. Then the

electron’s distribution is specified by Boltzmann distribution function. i.e.

𝑓 = 𝑁0(2𝜋𝑚𝑇𝑒)1/2 exp (−𝜖/𝑇𝑒) (1.21)

The electron’s gas number density is given by

14

𝑁𝑒(𝑥) = 𝑁0exp [−𝑈(𝑥)/𝑇𝑒] (1.22)

𝑁0 is the equilibrium density which is far away from the barrier.

Figure 1.2: Potential barrier and potential well

The figure (b) is exhibited the electric field as a potential well. The electron’s distribution

with positive value of energy 𝜖 is then owing to the particles that comes from infinity.

Therefore for all space the electron distribution with 𝜖 > 0 have a Boltzmann

distribution. But there are few electrons with negative energy 𝜖 < 0 have a finite motion

in the potential well, are called “Trapped Electrons”. There is no particle at infinity with

𝜖 < 0 hence as we considered before, energy is a conserved quantity. However this is not

enough to illustrate the trapped particle distribution and we must consider that the

variation of energy in the slowly varying field is not stationary.

15

From Eq. (1.18) the field changes slightly due to the motion of trapped particles. In such

case, the motion integral is a conserved adiabatic invariant.

𝐼(𝑡, 𝜖) = 12𝜋

. 2∫ 2𝑚𝜖 − 𝑈(𝑡, 𝑥)12𝑑𝑥𝑥2

𝑥1 (1.23)

This relation describes the trapped particle distribution function. Which is shown as

𝑓𝑡𝑟 = 𝑓𝑡𝑟[𝐼(𝑡, 𝜖)] (1.24)

This above mentioned function in a slowly applied field is a continuous of energy. The

function 𝑓𝑡𝑟(𝐼) is the trapped particle distribution function which has infinite motion

above the well, for the limiting value of trapped particle energy. In the case of potential

well, for applied field the limiting energy has zero value. Then from the boundary value

this function becomes a constant:

𝑓𝑡𝑟 = 𝑓(0) (1.25)

Where the particles above the well have distribution function𝑓(𝜖). The distribution of

electrons is illustrated by the Boltzmann function.

Adding the number of electrons with 𝜖 > 0 and 𝜖 < 0 gives

𝑁𝑒 = 2∫ 𝑓(𝜖) 𝑑𝑝𝑥∞𝑝1

+ 2∫ 𝑓(0) 𝑑𝑝𝑥𝑝10 (1.26)

Here 𝑝1 = 2𝑚|𝑈| , factor 2 comes from the particles with 𝑝𝑥 > 0 and 𝑝𝑥 < 0.

Using the value of 𝑓(𝜖) from eq. (1.21), we get

𝑁𝑒(𝑡, 𝑥) = 𝑁𝑜 𝑒|𝑈|𝑇𝑒1 − 𝛷(𝜉) + 2 |𝑈|

𝜋𝑇𝑒 (1.27)

Where 𝜉 = |𝑈|𝑇𝑒

, and

16

𝛷(𝜉) = 2√𝜋∫ 𝑒−𝑢2𝑑𝑢𝜉0 (1.28)

When 𝜉 ≪ 1 , the expansion of above integral in powers of u gives

𝛷(𝜉) ≈ 2√𝜋

(𝜉 − 13𝜉3)

Hence the distribution of trapped electrons in a shallow well |𝑈| ≪ 𝑇𝑒 is

𝑁𝑒 = 𝑁𝑜 1 + |𝑈|𝑇𝑒− 4

3√𝜋|𝑈|𝑇𝑒3/2 (1.29)

The first two terms are the same as could be obtained by expanded the Boltzmann

distribution (eq. (1.22))

𝑁𝑒 = 𝑁𝑜 1 + |𝑈|𝑇𝑒

+ |𝑈|𝑇𝑒2

+ ⋯ (1.30)

but the 3/2 power term in eq. (1.29) represents nonlinearity that is different and stronger

nonlinearity from the Boltzmann distribution. This 3/2 power nonlinearity is due to the

adiabatic trapping of electrons.

When 𝜉 ≫ 1 , only the second term in eq. (1.27) is essential and the distribution is given

as

𝑁𝑒(𝑡, 𝑥) = 2𝑁𝑜|𝑈|𝜋𝑇𝑒

(1.31)

The trapped electrons density increases slowly as |𝑈| increases.

1.6 Quantum plasmas

The quantum effects become important when the de Broglie wavelength is analogous to

or it is larger compared to the average distance between the particles in the system. It is

characterized by high number density and low temperature as compared to classical

plasma which has high temperature and low number density, in a relative sense. Quantum

17

plasmas are found in different regions of the cosmos for example in astrophysical objects

[31-33] (white dwarf star, neutron star, magnetars), in micro-electronic devices [34],

nano-wires [35], ultra-cold plasmas [36,37] and microscopic plasmas [38].

The inclusion of quantum terms in the fluid equations such as the Fermi pressure and the

Bohm potential [39-41], spin effects [42-46] leads to a new physical phenomenon.

1.6.1 Basic Properties of Quantum Plasmas

In quantum plasmas, the Fermi-Dirac statistics is used instead of Maxwell-Boltzmann

distribution used in classical plasmas. When the temperature is less as compared to Fermi

temperature 𝑇𝐹 = 𝐸𝐹/𝑘𝐵 the quantum effects play a significant role. The Fermi energy is

defined as

𝐸𝐹 = ℏ2

2𝑚(3𝜋2)

23𝑛

23 (1.32)

For quantum plasma the relevant degeneracy parameter is

𝑇𝐹𝑇

= 12

(3𝜋2)23(𝑛𝜆𝐵2 )

23 ≥ 1 (1.33)

Here ℏ is the Plank’s constant and 𝑛 is the number density.

When the Fermi temperature 𝑇𝐹 becomes less than the usual temperature, then the Fermi-

Dirac distribution function reduces to a Maxwell-Boltzmann distribution function. In

quantum plasmas the scale length 𝜆𝐹 = 𝑣𝐹/𝜔𝑃 is the Debye length’s quantum analog.

Like Debye length, 𝜆𝐹 describes the scale length of electrostatic screening in a quantum

plasma. Here 𝜔𝑃 is the plasma frequency and 𝑣𝐹 is the Fermi velocity defined as

𝑣𝐹 = 2𝐸𝐹𝑚12 = ℏ

𝑚(3𝜋2𝑛)1 3 (1.34)

This is the quantum analog of the electron thermal velocity in classical plasma.

18

At large number density in quantum plasmas the Fermi length becomes considerably

different from the Debye length in classical plasmas.

One can define the quantum coupling parameter as interaction energy 𝐸𝑖𝑛𝑡 ratio to the

average kinetic energy 𝐸𝑘𝑖𝑛 . The interaction energy is similar as used in classical case

but the kinetic energy is replaced by Fermi energy 𝐸𝐹 . The coupling parameter is written

as

𝑔𝑄 ≡𝐸𝑖𝑛𝑡𝐸𝐹

= 2

(3𝜋2)23

𝑒2𝑚ℏ2𝜀0𝑛1/3

~ 1𝑛𝜆𝑇𝐹

3 23 ~ ℏ𝜔𝑃

𝐸𝐹2 (1.35)

Here 𝑔𝑄 looks like the classical coupling parameter if 𝜆𝐹 is replaced by 𝜆𝐷 . Thus the

coupling parameter is defined as the Plasmon energy ratio to the Fermi energy. For the

quantum collisionless regions the quantum coupling parameter is small. The quantum

plasma is found to be more collective at large densities as compared to classical plasma.

In a fully degenerate Fermi gas, all the low energy states are full. If more particles are

added to the gas then this would be in the high energy state. Consequently, the increase in

density results in an increase in the average kinetic energy, which decreases the value of

𝑔𝑄.

For the description of the quantum effects various models have been introduced. The

basic model for 𝑁-boby problem is the Schrodinger equation for the wave functions of 𝑁

particles. By neglecting correlations (higher order), many-particle wave function can be

written as product of one-particle functions

𝜓(𝑥1, 𝑥2, … … , 𝑥𝑁, 𝑡) = 𝜓1(𝑥1, 𝑡),𝜓2(𝑥2, 𝑡) … … .𝜓𝑁(𝑥𝑁, 𝑡) (1.36)

1.6.2 Wigner Function

The Wigner [47] representation describes the quantum mechanics in phase space

formulation. Wigner formulation has received significant attention in different branches

of physics and also encouraged the efforts to create variant quantum hydrodynamics due

to analogy with classical systems. The Wigner function is a function of the phase space

19

variables (𝑥, 𝑣) and time. The quantum mixture of single-particle wave functions is

represented by the density matrix

𝜌(𝑥,𝑦, 𝑡) = ∑ 𝑝𝑖 ∫ 𝜓𝑖∗(𝑦, 𝑡) 𝜓𝑖∞−∞

𝑁𝑖=1 (𝑥, 𝑡) (1.37)

Where sum extends over all the states contributing to mixture.

Then the Wigner function can be written as

𝑓(𝑥, 𝑣, 𝑡) = 𝑚2𝜋 ћ

∑ 𝑝𝑖 ∫ 𝑑𝑠 𝜓𝑖∗ 𝑥 + 𝑠2

, 𝑡𝜓𝑖∞−∞

𝑁𝑖=1 𝑥 − 𝑠

2, 𝑡 𝑒𝑖𝑚𝑣𝑠

ћ (1.38)

It has many important properties but it has negative values also. Though it can be useful

to calculate averages like as in classical statistical mechanics. Such as the expectation

value is

⟨𝐴⟩ = ∬ 𝑓(𝑥,𝑣)𝐴(𝑥,𝑣)𝑑𝑥𝑑𝑣∬ 𝑓(𝑥,𝑣)𝑑𝑥𝑑𝑣

(1.39)

The Wigner function regenerates the exact quantum mechanics marginal distributions, for

example the spatial density is

n(𝑥, 𝑡) = ∫ 𝑓(𝑥, 𝑣, 𝑡)𝑑𝑣 =∑ 𝑝𝑖|𝜓𝑖|2𝑁𝑖=1 (1.40)

the Wigner equation can be developed up to order 𝑂(ћ4) as

𝜕𝑓𝜕𝑡

+ 𝑣 𝜕𝑓𝜕𝑥− 𝑒

𝑚𝜕𝜙𝜕𝑥

𝜕𝑓𝜕𝑣

= 𝑒ћ2

24𝑚3𝜕3𝜙𝜕𝑥3

𝜕3

𝜕𝑣3+ 𝑂(ћ4) (1.41)

This equation have to coupled with Poisson’s equation

𝜕2𝜙𝜕𝑥2

= 𝑒𝜀0

(∫𝑓𝑑𝑣 − 𝑛0) (1.42)

1.7.3 Schrodinger-Poisson Function

Another approach equivalent to the Wigner-Poisson system is obtained by using 𝑁 wave

functions 𝜓𝑖(𝑥, 𝑡), coupled through Poisson’s equation

20

𝑖ћ 𝜕𝜓𝑖𝜕𝑡

= − ћ2

2𝑚𝜕2𝜓𝑖𝜕𝑥2

− 𝑒𝜙𝜓𝑖 , 𝑖 = 1 …𝑁 (1.43)

𝜕2𝜙𝜕𝑥2

= 𝑒𝜀0

(∑ 𝑝𝑖|𝜓𝑖|2𝑁𝑖=1 − 𝑛0) (1.44)

This model was derived by Hartree. Later on Fock introduced a correction for 𝑁 particle

wave function, therefore now this model is called Hartree-Fock model.

A quantum multistream model can be derived by introducing the real amplitude 𝐴𝑖(𝑥, 𝑡)

and the real phase 𝑆𝑖(𝑥, 𝑡) associated to the pure state 𝜓𝑖 according to

𝜓𝑖 = 𝐴𝑖exp (𝑖𝑆𝑖/ћ) (1.45)

The velocity and density of each stream is given by

𝑢𝑖 = 1𝑚𝜕𝑆𝑖𝜕𝑥

,𝑛𝑖 = |𝜓𝑖|2 = 𝐴𝑖2 (1.46)

Using equations (1.45)-(1.46) into equations (1.43)-(1.44), by separating the real and

imaginary parts we get

𝜕𝑛𝑖𝜕𝑡

+ 𝜕(𝑛𝑖,𝑢𝑖)𝜕𝑥

= 0 (1.47)

𝜕𝑢𝑖𝜕𝑡

+ 𝑢𝑖𝜕𝑢𝑖𝜕𝑥

= 𝑒𝑚𝜕𝜙𝜕𝑥

+ ћ2

2𝑚2𝜕𝜕𝑥𝜕

2(𝑛𝑖/𝜕𝑥2)

𝑛𝑖 (1.48)

Quantum effects are enclosed in the last term (called Bohm potential). If we put ћ = 0,

we retain the classical model.

1.7.4 Quantum Hydrodynamic (QHD) Fluid Description

However later Tsintsadze [48] pointed out that Wigner equation is not suitable for the

construction of fluid equations even if applied to a macroscopical body. The integration

of the Wigner function over momenta gives the configuraional probability

density∫𝑓𝑤𝑑𝒑 = |𝜓(𝒓, 𝑡)|2. Whereas the integration over spatial coordinate gives the

momentum probability ∫𝑓𝑤𝑑𝑟 = |𝜑(𝜌, 𝑡)|2. Wigner function 𝑓𝑤(𝒓,𝒑) cannot be regarded

21

as the probability for finding the particle at the point 𝒓 and with momentum 𝒑, because

𝑓𝑤(𝒓,𝒑) can become negative for some values of 𝒓 and 𝒑. Thus the Wigner distribution

function has no physical meaning. Tsintsadze derived a new quantum kinetic equation

from which the following set of QHD equations were derived and these are given below

𝜕𝑛𝛼𝜕𝑡

+ 𝛁. (𝑛𝛼𝒖𝜶) = 0 (1.49)

𝜕⟨𝒑𝛼⟩𝜕𝑡

+ (𝒖𝜶.𝛁)⟨𝒑𝛼⟩ = 𝑒𝛼 𝑬 + 𝒖𝜶×𝑩𝑐 − 1

𝑛𝛼𝛁𝑃𝛼 + ћ2

2𝑚𝛼𝛁 1𝑛𝛼

∆𝑛𝛼 + 1𝑛𝛼∫ 2𝑑𝑝

(2𝜋ћ)3𝒑𝛼 𝐶(𝑓𝛼)

(1.50)

Here last term is the collision term, 𝒖𝜶 is the velocity of the plasma, 𝒑𝛼 is the quasi-

classical distribution function and pressure term 𝑃𝛼 is given by

𝑃𝛼 = 13𝑚𝛼

∫ 𝑑3𝑝(2𝜋ћ)3

𝑝𝛼2

𝑒𝑥𝑝𝜀𝛼−𝜇𝛼𝑇𝛼+1

1.7.5 Applications of Quantum Plasmas

There is a vast variety of plasmas in space and laboratories. The interstellar medium

behaves classically because its density is low. Quantum effects are estimated to occur in

dense astrophysical plasmas [49] due to the high density. Quantum plasmas are found in

white dwarf stars and neutron stars. The plasma in neutron star is denser than in dwarf

stars [50,51]. Quantum plasma also occurs in metals where the density of electrons is

high. Numerous investigations have been achieved in laboratory to study new states of

matter and to understand the behavior of astrophysical objects.

1.8 Alfven Waves

A brief introduction to Alfven waves is given here as these waves are used in later

chapters of this thesis. In 1942, Hannes Alfven [52] investigated a low frequency wave

mode propagating along the magnetic field. The experimental verification of this wave

was given by Lundquist [53]. Later in the compressible plasma case, slow and fast

22

magnetoacoustic waves were discovered. This treatment was given by Herlofsen in 1950.

Alfven and magnetoacoustic waves are the low-frequency modes and have important role

in the transport of energy in space, laboratory and astrophysical plasmas.

Alfven wave is a low frequency electromagnetic wave. The magnetic field provides the

restoring force and inertia is provided by ion mass density. Alfven waves are of two

types. Pure Alfven waves propagate along the externally applied magnetic field. Their

group velocity and phase velocity are the same. These are non dispersive in nature. The

other type of Alfven wave is a shear Alfven wave which makes a small angle with the

externally applied magnetic field. Theoretically these waves were studied by Alfven [52].

Experimentally these waves were detected by Lundquist [53] and Lehnert [54]. Wilcox et

al [55], gave a series of experiments of the Alfven waves. The dispersion relation of pure

Alfven wave is

𝜔2 = 𝑘𝑧2 𝑣𝐴2 (1.51)

Where 𝑣𝐴 is the Alfven speed and 𝑘𝑧 is the propagation vector. Alfven waves have found

several applications in space, laboratory and astrophysical plasmas [56].

1.8.1 Kinetic and Inertial Alfven wave

When the perpendicular wavelength becomes comparable to the ion Larmor radius, ions

can no longer follow the magnetic lines of force, while electrons due to small Larmor

radius follow the magnetic field lines. Consequently charge separation occurs which is

followed by a coupling of the electrostatic longitudinal mode to the Alfven wave which

in turn produces a wave, known as the Kinetic Alfven Wave (KAW). Charge separation

can also occur if electron inertia is considered and if the electron Larmor radius is

comparable to the perpendicular wavelength, then such waves are called Inertial Alfven

Waves (IAWs) [56].

Plasmas are often classified into low and high beta plasmas where beta is defined as the

ratio of thermal to magnetic pressures. In quantum plasmas the usual temperature is

replaced by the Fermi temperature and beta is defined as (𝛽 = 2𝑛𝑘𝐵𝑇𝐹𝜇0/𝐵02). In 1965

23

Kadomtsev introduced the novel idea of using two potential theory for Alfven waves in a

low beta plasma [57]. This idea was new as Alfven waves are essentially electromagnetic

waves and using electrostatic potentials was certainly a new idea. Kadomtsev two

potential theory has been extensively used in the investigation of low frequency

hydromagnetic waves in different types of low beta plasmas. For example linear and

nonlinear Alfven waves in an electron-ion plasmas by taking into account the finite

Larmor radius effect have been investigated by Hasegawa and coauthors [58,59]. Finite

amplitude solitary Alfven waves for small but finite 𝛽 effects were later studied by Yu

and Shukla, where the nonlinear evolution equation led to the formation of solitons with

density humps with an upper limit on the amplitude [60]. Dust kinetic Alfven waves have

been investigated by Yinhua et al. [61] this confirm the existence of solitary waves in

dusty plasmas. For a comprehensive review we refer to Cramer [56].

1.8.2 Coupled kinetic Alfven-acoustic wave

For small 𝛽, the electric fields are expressed as 𝐸𝑥 = −𝜕𝜑/𝜕𝑥, 𝐸𝑧 = −𝜕𝜓/𝜕𝑧 .

The parallel component of equation of motion for electron gives

𝑝1𝑒 − 𝑖 𝑛0𝑚𝑒𝑘𝑧

𝜕𝑣1𝑒𝑧𝜕𝑡

= 𝑛0𝑒𝜓 (1.52)

If wave speed is taken less than the electron thermal speed, the temperature will be

constant parallel to the filed

𝑝1𝑒 = 𝑛1𝑒𝑘𝐵𝑇𝑒 = 𝑛1𝑒𝑚𝑖𝑣𝑠2 (1.53)

Where 𝑛1𝑒 is the perturbed number density of electron, by neglecting electron inertia we

get

𝑛1𝑒 = 𝑛0𝑒𝜓𝑚𝑖𝑣𝑠2

(1.54)

The continuity equation for ions is

24

𝑒 𝜕𝜓𝜕𝑡

+ 𝑚𝑖𝒗𝒔𝟐(𝑖𝑘𝑥𝑣1𝑖𝑥 + 𝑖𝑘𝑧𝑣1𝑖𝑧) = 0 (1.55)

From the electron continuity equation

𝑣1𝑒𝑧 = 𝑘𝑥𝑘𝑧𝑣1𝑖𝑥 + 𝑣1𝑖𝑧 (1.56)

Parallel component of current density is

𝐽1𝑧 = −𝑛0𝑒𝑘𝑥𝑘𝑧𝑣1𝑖𝑥 (1.57)

By neglecting displacement current, we get from Ampere’s law

𝜇0𝜕𝐽1𝑧𝜕𝑡

= −𝑖𝑘𝑥2𝑘𝑧(𝜑 − 𝜓) (1.58)

The 𝑥, 𝑦 and 𝑧 component of equation of motions are:

𝜕𝑣1𝑖𝑥𝜕𝑡

+ 𝑣𝐴2𝑘𝑧2

𝛺𝑖2𝑖𝑘𝑥𝑒𝜓

𝑚𝑖− 𝛺𝑖𝑣1𝑖𝑦 = 0 (1.59)

𝜕𝑣1𝑖𝑥𝜕𝑡

+ 𝛺𝑖𝑣1𝑖𝑥 = 0 (1.60)

𝜕𝑣1𝑖𝑧𝜕𝑡

+ 𝑖𝑘𝑧𝑒𝜓𝑚𝑖

= 0 (1.70)

From these equations the dispersion relation is obtained as

1 − 𝑣𝐴2𝑘𝑧2

𝜔2 1 − 𝑣𝑠2𝑘𝑧2

𝜔2 = 𝑣𝐴2𝑘𝑧2

𝜔2 𝜆𝑠 (1.71)

This is the dispersion relation for coupled kinetic Alfven-acoustic wave with the coupling

factor 𝜆𝑠 on the right hand side of above equation. By setting 2nd term on left hand side

unity, we get the dispersion relation of kinetic Alfven wave.

25

1.9 Fluid Models

There are following types of fluid models:

1.9.1 Multi-Fluid Model

In this model we consider the multi fluid in which each species of particles is indicated

by the index 𝛼, with mass 𝑚𝛼 and charge 𝑍𝛼𝑒. Each collection of particles of a specific

type is supposed to act like a fluid with its own velocity, mass density, number density

and pressure. The equation of motion is

𝑑𝒗𝛼𝑑𝑡

= 𝑍𝛼𝑒𝑚𝛼

(𝑬 + 𝒗𝛼 × 𝑩) − 1𝜌𝛼∇𝑝𝐹𝛼 − ∑ 𝜈𝛼𝛼′(𝒗𝛼 − 𝒗𝛼′)𝛼′ + ћ2

2𝑚𝛼2𝛁 1𝑛𝛼

∆𝑛𝛼 (1.72)

Where 𝜈𝛼𝛼′ is the collision frequency and 𝜌𝛼 = 𝑚𝛼𝑛𝛼 . The set of equations is completed

along with the Maxwell equations and equation of state.

1.9.2 Single Fluid Model (Ideal Fluid Model)

Single fluid models are used to describe various low frequency phenomena in

astrophysical plasmas. The multi-fluid model is assumed to act like a single fluid model

by neglecting the electron inertia.

In this model the resistivity is very small so that we can ignore it. The electrons and ions

are strongly magnetized to the magnetic field lines and called the single fluid model. The

whole plasma acts like a single fluid with same temperature and pressure. The equations

of continuity and motion for single fluid model are

𝜕𝜌𝜕𝑡

+ ∇. (𝜌 𝒗) = 0 (1.73)

𝜌 𝑑𝒗𝑑𝑡

= 𝑱 × 𝑩 − ∇𝑝𝐹 + 𝑛ћ2

2 𝑚𝛁 1√𝑛∆√𝑛 (1.74)

26

1.9.2 Hall-MHD Model

In which electron fluid and ion fluid are considered separately. The electron cyclotron

period is shorter than other time scales in the system while the ion cyclotron period is

comparable with other time scales. In this case the Hall term is included on the right hand

side of Ohm’s law

𝑬 + 𝒗 × 𝑩 =1𝑛𝑖 𝑒

𝑱 × 𝑩

1.9.3 Kinetic Model

Here we briefly describe the kinetic model used in quantum plasmas. The kinetic

equation is in some ways similar to the Vlasov model but with the additional Bohm

potential term. This kinetic equation has been discussed in detail and derived rigorously

by Tsintsadze [48]. This new quantum kinetic equation is given below

𝜕𝑓𝛼𝜕𝑡

+ 𝒗.𝛁𝑓𝛼 + 𝑒𝛼(𝑬 + 𝒗 × 𝑩) 𝜕𝑓𝛼𝜕𝒑

+ ћ2

2𝑚𝛼𝛁 1𝑛𝛼

∆𝑛𝛼𝜕𝑓𝛼𝜕𝒑

= 𝐶(𝑓𝛼) (1.75)

We note here that the distribution function 𝑓𝛼 will be the Fermi-Dirac distribution

function.

1.10 Dusty Plasmas

Dusty plasmas are found everywhere in nature and are discernible in many astrophysical

and space plasma situations such as planetary rings [62], asteroid zones [63] and the

Earth’s atmosphere [64] for example. The existence of dust has also been observed in

laboratory produced plasma for example in plasma processing and in fusion devices [65-

68]. Rao et al. [69] interpreted theoretically the existence of dust-acoustic wave (DAW)

for the first time. Shukla and Silin [70] studied the existence of dust-ion-acoustic wave

(DIAW). The experimental confirmation of these new modes has also been done [71-72].

27

The charged dust particles in a usual electron ion plasma is called a dusty plasma or

complex plasma. During the past years dusty plasma has been growing field of research

owing to its very important consequences in laboratory, space and astrophysical dusty

plasmas for example interstellar medium, nebulae, cometary tails, Earth’s environment

[73-76]. The size of the dust grain is ranging from 1 𝜇𝑚 to 1 𝑐𝑚. The mass of the dust

particle is very large. The dust grain is interacts with plasma particles because of its

charge on it. The dust grain gets charge by many processes, for instance, collision with

electrons and ions, secondary emission, thermionic emission etc. The existence of dust

grain in plasma generates new modes. Dusty plasma studies have become an attractive

field of research in the last couple of decades and continue to contribute in the

development of new findings in astrophysics and space plasma etc. We would like to

mention here that in most cases dust particles become negatively charged, due to the

electron’s small mass, however in certain instances positively charged is also observed

[77].

1.11 Layout of the Thesis

Introduction to quantum plasma physics is given in Chapter 1. In this chapter models

used in quantum plasma are also discussed. Some nonlinear properties to describe

plasmas are also given in this chapter.

In the second chapter, we use Landau quantization and investigate the effect of trapping

for ion-acoustic waves when inhomogeneities are present. KdV equation is derived from

modified Hasegawa Mima (HM) equation. Sagdeev potential approach is used to study

the solitary vortices.

In Chapter 3, we investigate the coupled Kinetic Alfven-acoustic waves in the presence

of trapped electrons in a low beta degenerate quantum plasma. Two potential theory has

been used and dispersion relation is derived. The solitary structures are discussed by

using Sagdeev potential approach. We have also seen that the nature of nonlinearity for

quantum plasma is different from classical plasma.

28

In Chapter 4, the coupled Kinetic Alfven-acoustic waves are examined in a dusty plasma.

We have taken dust both positively and negatively. The dispersion relations are derived

for both dust charges separately and their limiting cases are discussed. The solitary

structures are also discussed for both cases.

The summary of the thesis is given in chapter five.

29

Chapter 2

Effect of adiabatic trapping on vortices and

solitons in a degenerate plasma in the

presence of a quantizing magnetic field

2.1 Introduction

In the present Chapter the Effect of adiabatic trapping as a microscopic phenomenon in

an inhomogeneous degenerate plasma is investigated in the presence of a quantizing

magnetic field and a modified Hasegawa Mima equation for the drift ion-acoustic wave is

obtained. The linear dispersion relation in the presence of the quantizing magnetic field is

derived. The modified Hasegawa Mima equation is also derived and the bounce

frequencies of the trapped particles are investigated. A two dimension KdV equation is

also derived from the modified HM equation and finally the Sagdeev potential approach

is used to show the effects of different parameters on the formation of the solitary

structures.

Quantum or degenerate plasmas have become an active area of plasma research for

scientists working in both theoretical and experimental fields [78]. Several investigations

have been carried out for dense plasmas in astrophysical environments [79,80], where

electrons are dense enough to exhibit quantum behavior [81,82]. Such behavior is also

expected to be present in microelectronic devices [83]; thus it is important to understand

the quantum effects on the behavior of linear and nonlinear wave properties of these

systems. Linear theory, using quantum fluid model has been extensively used to study

different wave modes in degenerate plasmas [84]. Based on hydrodynamic version of

quantum mechanics, Manfredi and Haas [85,86] formulated the quantum multi-stream

model and fluid model for plasmas. Later this fluid model of plasma was extended to

quantum magnetohydrodynamics (QHD) by Haas [87]. An excellent survey was also

30

presented by Manfredi [88] on the modeling of quantum plasmas. Haque and Mahmood

[89] investigated the linear and nonlinear drift waves in inhomogeneous quantum

plasmas with neutrals in the background. They found that the properties of drift solitons

and shocks are modified by quantum corrections in dense magnetoplasmas. Shukla and

Eliasson [90] presented the numerical study of the dark solitons and vortices in quantum

electron plasmas.

The presence of trapping as a microscopic phenomenon has been confirmed by computer

simulations and theory [91,92] as well as by experimental work [93]. More recently

propagation characteristics of ion-acoustic solitary waves have been investigated with

trapping effect using non-Maxwellian distribution functions [94] and it has been found

that the solitary dynamics were modified and spiky solitons are obtained instead of usual

solitons. The trapping effect on the formation of vortices has also been studied in

classical plasma, and a modified Hasegawa Mima (HM) equation was derived by

considering shallow and deep potential wells, respectively [95]. Characteristics of solitary

structures have been investigated for both fully and partially degenerate plasmas [96] and

in a subsequent work fully relativistic effects were included in the investigations [97].

It is well known that in the presence of a magnetic field, the electron gas magnetization

has two independent parts; (i) Paramagnetic, and (ii) the Diamagnetic part.

Paramagnetism is caused by intrinsic or spin magnetic moment of electrons, however the

diamagnetic part is due to the fact that the orbital motion of electrons becomes quantized.

This is also known as Landau diamagnetism or Landau quantization [98]. Landau

quantization is a quantum mechanical effect as the cyclotron orbits of the electrons are

quantized in the magnetic field and affects the motion of electrons in the direction

parallel to the magnetic field itself. Therefore discrete energy levels called Landau levels

are occupied by the charged particles. The Landau levels are degenerate and the strength

of the magnetic field determines the number of electrons per level. When energy level

separation is greater than the mean thermal energy the Landau level effect is observable

for strong magnetic field and low temperature. The field is called quantizing if the

Landau quantization of electron motion in a magnetic field is taken into account [99]. By

considering Zeeman splitting each Landau level splits into pair of levels one for spin up

31

and one for spin down for electrons. The Zeeman splitting will affect the Landau levels

due to having the same energy scales 2𝜇𝐵𝐵0 = ℏ𝜔𝑐𝑒 where 𝜇𝐵 is the Bohr’s magneton,

𝐵0 is the magnetic field, ℏ is the Plank’s constant normalized by 2𝜋 and ωce = 𝑒𝐵0 𝑚𝑒

is the electron cyclotron frequency. On the other hand, the ground state energy and the

Fermi energies remain similar because when summed, the pairs of energy levels cancel

out each other. Adiabatic trapping in the presence of a quantizing magnetic field has

recently been investigated and the effect of this field was studied both theoretically and

numerically [100]. Electron holes and their coupling with Langmuir waves have also

been investigated in quantum regime by considering Wigner-Poisson model [101,102].

In the present chapter, we continue with our investigations of the influence of trapping

for ion-acoustic waves in the presence of Landau quantization for degenerate plasmas

when inhomogeneities are present in the number density. Thus in the present work we

shall consider the formation of ion-acoustic vortices under the conditions mentioned

above. We derive a modified HM equation and present its investigations. In particular

situations, we obtain Korteweg-de Vries (KdV) equation from our modified HM equation

and later use the Sagdeev potential approach to study the formation of solitary vortices.

The layout of the present work is as follows: In Sec. 2.1, we have given a general over

view of the problem. In Sec. 2.2, we have discussed some mathematical preliminaries and

have given the formulation using the Fermi-Dirac distribution function and finally have

derived a modified HM equation. In Sec. 2.3, the linear dispersion relation is derived and

has calculated the bounce frequency of the electrons trapped in the potential well. In Sec.

2.4, KdV equation is derived and in Sec. 2.5, solitary structures have been investigated by

deriving the Sagdeev potential. Finally, numerical results are discussed in Sec. 2.6.

2.2 Basic set of Equations

In this section we set up the fundamental equations needed for the investigation of ion

acoustic waves in quantum plasma. Our aim is to investigate the formation of solitary

vortices associated with these waves, and for this the background number density n0(x)

of the charged particles is taken to be inhomogeneous, which is considered to be weak i.e.

32

− 1n0 dn0

dx = κ ˂ 1 (which means that higher derivatives of n0(x) and κ 2 type of

terms are not taken into account), and the x-direction is chosen perpendicular to the

ambient magnetic field. Further for ion-acoustic waves, electrons are taken to be massless

and we need only consider the total electron density through the distribution function and

not the electron dynamics. We write here that a similar treatment was used for the

derivation of the Hasegawa Mima equation in a classical plasma [103].

Thus following the method elucidated in Landau and Lifshitz [98], we can obtain the

distribution of electrons which includes the effects of adiabatic trapping. The occupation

number for the Fermi-Dirac distribution which takes into account the effect of the

magnetic field via Landau splitting [100], is

ne = pFe2 η

2π2ℏ3 me2∑ ∫ ε−1/2

expε−UT +1dε∞

0∞ℓ=0 (2.1)

where U = eφ + µ − ℓℏωce, µ is the chemical potential, 𝜑 is the trapping potential from

the ions, ℓ represents Landau levels and pFe = 2𝑚𝑒 𝜀𝐹𝑒 is the electron Fermi

momentum. Here ℓ = 0 represents the case without quantizing magnetic field. In order to

obtain the expression for number density we separate the case ℓ = 0 from summation and

replace the summation in above equation by integration(∑ → ∫ 𝑑ℓℓmax1

ℓmax1 ). By

following the method of integration used in [100], the expression for the number density

of trapped electrons for fully degenerate plasma is given by

ne = n0(𝑥) 32η(1 + Φ)

12 + (1 + Φ− η)3/2 (2.2)

Here the effect of quantizing magnetic field appears through normalized parameter

η = ℏωce/εFe, Φ = eφεFe

is the normalized electrostatic potential and

εFe = ℏ2

2me(3π2n0)3 2 is the electron Fermi energy. In Eq. (2.2), the terms containing T2

not taken into account as cold plasma limit is assumed which is valid for fully degenerate

plasma.

33

The ions on the other hand are considered to be cold and non-degenerate due to their

heavy mass. The magnetic field is taken in the 𝑧-direction and density gradient in the 𝑥-

direction. The ion equations of continuity and motion are respectively:

∂ni∂t

+ 𝛁 ∙ (ni𝐯i) = 0 (2.3)

∂∂t

+ 𝐯i ∙ 𝛁 𝐯i = emi

(𝐄 + 𝐯i × 𝐁) − 1mini

𝛁pi (2.4)

Here 𝑛𝑖 , 𝐯i, 𝑚𝑖, pi are the ion density, velocity, mass, and pressure, respectively. Plasma

is assumed to be quasineutral and only ions are magnetized by ambient magnetic field.

By taking the curl of above equation, and using the Maxwell equation

𝛁 × 𝑬 = −𝜕𝑩𝜕𝑡

(2.5)

𝛁 × ∂∂t

+ 𝐯i ∙ 𝛁 𝐯i = − emi

𝜕𝑩𝜕𝑡

+ emi𝛁 × (𝐯i × 𝐁) − 1

mi𝛁 × ( 1

ni𝛁pi) (2.6)

Let Ω𝑖 is the vorticity which is defined as Ωi = 𝛁 × 𝐯i . By using the relation

(𝐯i ∙ 𝛁𝐯i) =12𝛁𝐯i𝟐 − 𝐯i × (𝛁 × 𝐯i)

Ωci = e𝐁mi

is the ion gyrofrequency. We are left with

𝜕Ωi𝜕𝑡− 𝛁 × (𝐯i × Ωi) = −∂Ωci

∂t+ 𝛁 × (𝐯i × Ωci) (2.7)

Here we use the vector relation

𝛁 × (𝐯i × Ω) = (Ω.𝛁)𝐯i + (𝛁.Ω)𝐯i − (𝐯i.𝛁)𝛀 − (𝛁. 𝐯i)𝛀

Since 𝛁.Ω = 0, we have

𝛁 × (𝐯i × Ω) = −(𝐯i.𝛁)𝛀 − (𝛁. 𝐯i)𝛀

Where Ω represents Ωci or Ωi. In order to express 𝛁. 𝐯i we use the ion continuity equation

34

𝛁. 𝐯i = − 𝑑𝑑𝑡

ln ni (2.8)

Using ddt

= ∂∂t

+ 𝐯i.𝛁 is the total derivative; where 𝐯i in the drift approximation [103] is

taken as 𝐯i = 𝐯e + 𝐯g, 𝐯e and 𝐯g are the E× 𝐁 and gravitational drifts, respectively. We

obtain in the absence of baroclinic pressure:

d dt

(Ωi + Ωci) − (Ωi + Ωci)𝑑𝑑𝑡

ln ni = 0 (2.9)

We have dropped both space and time derivatives of Ωci so above equation becomes

d dtln Ωi+Ωci

ni = 0 (2.10)

We consider propagation in the 𝑥 and 𝑦 directions and the vorticity only in the 𝑧-

direction which is given by

Ωi = (𝛁 × 𝐯i). ẑ (2.11)

By considering Ωi ≪ Ωci, we can write Eq. (2.10) as

d dtln Ωci

ni+ Ωi

Ωci− δn

n0+ 1

2δnn02 = 0 (2.12)

By assuming the plasma to be neutral, i.e. ni = ne = n0(x)(1 + δnn0

) we obtain, upon

using Eq. (2.2), where δn = ne − n0 which is the perturbed number density and from

Eq. (2.2), we have

δnn0

= 32η(1 + Φ)

12 + (1 + Φ− η)3/2 − 1 (2.13)

Using Eq. (2.13) in Eq. (2.12), we obtain

35

𝜕𝑡 + 𝑣𝑔𝜕𝑦(𝜌2∇2Φ) − 𝑣𝑒∗𝜕𝑦 Φ + 𝜕𝑡 + 𝑣𝑔𝜕𝑦 −3𝜂(1 + Φ)12 − 2(1 + Φ−

𝜂)32 + 1

2(1 + Φ− 𝜂)3 + 3

2𝜂(1 + Φ)

12(1 + Φ− 𝜂)

32 = 𝜌2

B0

TFe𝛁Φ ×

∧

z .𝛁(∇2Φ)

(2.14)

where 𝑣𝑒∗ = 𝜅 𝑇𝐹𝑒B0

is the electron diamagnetic drift, 𝜂 = ℏωce/𝜀𝐹𝑒 is a normalized

parameter which gives the strength of quantizing magnetic field, 𝜅 is the inverse scale

length of the number density inhomogeneity (defined earliest), 𝜌 = 𝑐𝑠Ωci

is the ion larmor

radius and 𝑐𝑠 = 𝜀𝐹𝑒mi

is the ion sound velocity.

In Eq. (2.14), we can note that the presence of trapped particles produces a modified HM

equation as the nonlinear term (last term on the LHS of the Eq. (2.14)) differs here from

the classical HM equation [95] where the additional terms containing 𝜂 occur due to the

effect of Landau quantization. The fractional power nonlinear terms will make a larger

contribution than the quadratic nonlinearity occurring in the original HM equation and

thus the higher order nonlinearities have subsequently been dropped.

𝜕𝑡 + 𝑣𝑔𝜕𝑦(𝜌2∇2Φ) − 𝑣𝑒∗𝜕𝑦 Φ + 𝜕𝑡 + 𝑣𝑔𝜕𝑦 −3𝜂(1 + Φ)12 − 2(1 + Φ−

𝜂)32 + 1

2(1 + Φ− 𝜂)3 + 3

2𝜂(1 + Φ)

12(1 + Φ− 𝜂)

32 = 0

(2.15)

We note here that this is a complicated equation and in general analytically exact

solutions are not possible to obtain. Thus in the following sections, we investigate the

equation in different limits.

2.3 Linear Dispersion Relation and Bounce Frequency

In the present section we begin by linearizing Eq. (2.15) and by using plane wave

solution we obtain the linear dispersion relation for drift ion-acoustic waves in the

presence of Landau quantization, as

36

𝜔 = 23𝑘𝑦 𝑣𝑒∗ 1 − 2

3𝜌2 𝑘2 + 5

2𝜂 + 𝑘𝑦𝑣𝑔 (2.16)

In the absence of Landau quantization 𝜂 = 0, we obtain the same linear dispersion

relation as derived in reference [104] except the last term on RHS. This equation has been

analyzed graphically in section 2.6.

We now consider that the trapped particles in the potential well which can move to and

fro in the well itself, these particles remain trapped if their energy is less than the

potential energy of the well. We expand Φ around fixed minimum value Φ0 of the

potential well by taking Φ = Φ0 + Φ1 and linearizing Eq. (2.15), we obtain

𝜔𝑏 = 𝑘𝑦 𝑣𝑒∗

𝜌2 𝑘2+32𝜂(1+Φ0)−12+3(1+Φ0)

12−32(1+Φ0)2−3𝜂(1+Φ0)−94𝜂

(2.17)

where 𝜔𝑏 = 𝜔 − 𝑘𝑦 𝑣𝑔𝑦 is the bounce frequency as the particle is reflected off the walls

of the potential well.

2.4 KdV Type of Solution

In this section, we use the reductive perturbation technique for long wavelength solution

of Eq. (2.15) and derive a KdV type equation which is valid for large scale motions

[105]. Stretched variables are introduced in the following manner:𝜉 = 𝜀12(𝑦 − 𝑢𝑡),

𝜏 = 𝜀32𝑡, 𝑥 = 𝑥 and the perturbations in potential are Φ = εΦ1 + ε2Φ2. Here 𝑢 is the

speed of perturbation in the co-moving frame of reference. By using these perturbations

in Eq. (2.15), and collecting the lowest order terms in 𝜀 (i.e. 𝜀32 ), we obtain

𝑣𝑔 − 𝑢 (𝜌2𝜕𝑥𝑥)𝜕𝜉Φ1 − 𝑣𝑒∗𝜕𝜉Φ1 = 32𝑣𝑔 − 𝑢𝜕𝜉Φ1 (2.18)

Equation (2.18) corresponds to the linear regime and integration will yield the linear

dispersion relation derived in the preceding section. By collecting the terms in the next

order, i.e. 𝜀52, we get

37

𝜕𝜏(𝜌2𝜕𝑥𝑥)Φ1+𝑣𝑔 − 𝑢𝜌2 𝜕𝜉𝜉𝜉Φ1 + 𝑣𝑔 − 𝑢 (𝜌2𝜕𝑥𝑥)𝜕𝜉Φ2 − 𝑣𝑒∗𝜕𝜉Φ2 = 32𝑣𝑔 −

𝑢)𝜕𝜉Φ2 −34𝑣𝑔 − 𝑢𝜕𝜉Φ1

2 + 32

𝜕𝜏Φ1 −158

𝜂𝑣𝑔 − 𝑢𝜕𝜉Φ12

(2.19)

Following the method used in Dodd et al. [106], we do a separation of variables as:

Φ1 = 𝐴(𝜉, 𝜏) 𝑌(𝑥) and by using Eq. (2.18) in Eq. (2.19), we obtain

𝜕𝜏(𝜌2𝑌′′)𝐴 −32𝑌𝜕𝜏𝐴 + 𝜌2𝑣𝑔 − 𝑢𝑌𝜕𝜉𝜉𝜉𝐴 +

34𝑣𝑔 − 𝑢𝑌2𝜕𝜉𝐴2

+158

𝜂𝑣𝑔 − 𝑢𝑌2𝜕𝜉𝐴2 = 0

Here 𝑌∙∙ denotes differentiation with respect to the variable 𝑥. Multiplying by ‘𝑦’ on both

sides and integrating by taking boundary conditions such that when 𝑥 → ± ∞, 𝑦 𝑑𝑦𝑑𝑥→ 0,

we get

𝜕𝜏𝐴 + 𝑎𝜕𝜉𝜉𝜉𝐴 + 𝑏 𝜕𝜉𝐴2 = 0 (2.20)

where 𝑎 and 𝑏 are the coefficients given by

𝑎 =∫ 𝜌2(𝑣𝑔 − 𝑢)𝑦2𝑑𝑥+∞−∞

∫ (𝜌2𝑌∙∙ − 32𝑦)𝑦𝑑𝑥+∞

−∞

𝑏 =34𝑣𝑔 − 𝑢

∫ 𝑦3𝑑𝑥+∞−∞

∫ 𝜌2𝑌∙∙ − 32𝑦 𝑦𝑑𝑥

+∞−∞

+158𝜂𝑣𝑔 − 𝑢

∫ 𝑦3𝑑𝑥+∞−∞

∫ 𝜌2𝑌∙∙ − 32𝑦 𝑦𝑑𝑥

+∞−∞

The solution of Eq. (2.20) is

𝐴 = 32𝜆𝑏𝑠𝑒𝑐ℎ2 1

2𝜆𝑎

(𝜉 − 𝜆𝑡) (2.21)

38

Here 32𝜆𝑏

is the amplitude and 𝜆 is the velocity of the commoving frame of reference.

The Eq. (2.21) is the standard solution of the KdV equation. However, we note that 𝑥

dependence on 𝑦 can be evaluated if a specific form of the 𝑥 dependence is given. We

here, however, left the result general.


In order to proceed further in the analysis of our modified Hasegawa Mima equation i.e.

Eq. (2.15), we derive the Sagdeev potential to investigate the formation of solitary waves

and to this end we shift to co-moving frame of reference i.e. 𝜉 = 𝛼𝑥 + 𝛽𝑦 − 𝑢𝑡. Here 𝛼

and 𝛽 are the direction cosines.

Thus Eq. (2.15) can now be reset into the following form

𝑑2Φ𝑑𝜉2

= − 𝑣𝑒∗

𝑢 − −3 𝜂(1 + Φ)

12 − 2(1 + Φ− 𝜂)

32 + 3

2𝜂(1 + Φ)

12(1 + Φ− 𝜂)

32 +

12

(1 + Φ− 𝜂)3 + −3 𝜂 − 2(1 − 𝜂)32 + 3

2𝜂(1 − 𝜂)

32 + 1

2(1 − 𝜂)3

(2.22)

By standard manipulation [107] Eq. (2.22) can be expressed in the form of an energy

integral

12𝑑Φ𝑑𝜉2

+ 𝑉(Φ) = 0 (2.23)

where 𝑉(Φ) is the Sagdeev or pseudo-potential and is given below

𝑉(Φ) = 𝑣𝑒∗

𝑢 Φ

2

2+ 3 𝜂Φ + 2(1 − 𝜂)

32Φ − 3

2𝜂(1 − 𝜂)

32Φ − 1

2(1 − 𝜂)3Φ − 2𝜂 (1 + Φ)

32 +

2𝜂 − 45

(1 + Φ− 𝜂)52 + 4

5(1 − 𝜂)

52 + 1

8(1 + Φ− 𝜂)4 − 1

8(1 − 𝜂)4 + 1

16𝜂 (1 + Φ)

12(1 +

Φ− 𝜂)128 (1 + Φ)2 − 14𝜂 (1 + Φ) − 1

16𝜂 (1 − 𝜂)

12(8 − 14𝜂)

(2.24)

39

In order to obtain solitary waves, the conditions which must be fulfilled are (i) 𝑉(Φ) < 0

when 0 > Φ > Φ𝑚𝑖𝑛 for negative-potential solitary waves and for positive-potential

solitary waves 𝑉(Φ) < 0 when 0 < Φ < Φ𝑚𝑎𝑥, where Φ𝑚𝑖𝑛 and Φ𝑚𝑎𝑥 is the negative

and positive root of the electrostatic potential for which 𝑉(Φmin) = 0 or 𝑉(Φmax) = 0

respectively (ii) 𝑉(Φ)|Φ=0 = 𝑉′(Φ)|Φ=0 = 0 and (iii)

𝑉′(Φ)|Φ=Φ𝑚𝑖𝑛 = 𝑉′(Φ)|Φ=Φ𝑚𝑎𝑥 > 0 [108]. However, as we can see that the Sagdeev

potential given by Eq. (2.24) has a complicated dependence on the different parameters

i.e. 𝜂 and 𝑣𝑒∗ as well as having whole power nonlinearities and fractional power

nonlinearities. As an exact solution of Eq. (2.24) is not possible, the behavior of the

Sagdeev potential is investigated numerically in the next section.

2.6 Results and Discussion

In this section, we present the numerical solution of our theoretical results. For numerical

values, we use the parameters of dense astrophysical objects like white dwarf star where

the values of number density and magnetic field are of the order of 1032m-3

109

and 106 T,

respectively [ ,110].

Figure 2.1 depicts the relation between the bounce frequency and the potential using Eq.

(2.17). We can note that as the potential reaches a value of 0.425, the bounce frequency

becomes infinity which means that no particle remains trapped in the potential well. If we

compare our result of bounce frequency with the results given in reference [104] in which

bounce frequency is derived for a classical plasma without taking into account the

Landau quantization 𝜂, the bounce frequency in the present case approaches infinity at a

much lower value of potential, i.e. range of trapping potential reduces.

Graphical investigations of the Sagdeev potential (Eq. (2.24)) and corresponding solitary

structures are numerically obtained which are shown in Figs. 2.2-2.5. In these plots the

dependence of the Sagdeev potential on the magnetic field, number density (through 𝜂)

and inverse of scale length 𝜅 of number density inhomogeneity are investigated. From

Fig. 2.2, it is observed that the depth and width of Sagdeev potential increases with an

increase in the magnetic field, when density is taken 𝑛0 = 1032m-3. Corresponding

40

solitary structures are shown in Fig. 2.3, where we can see that by increasing magnetic

field (as 𝜂 increases) the amplitude of solitons increases but width decreases slightly.

In Fig. 2.4, the Sagdeev potential is plotted for different values of number densities by

keeping the magnetic field fixed. In this figure, we notice that by increasing the number

density there is an enhancement in the width and depth of the Sagdeev potential. The

corresponding solitary structures are plotted in Fig. 2.5. It is found that width of soliton

decreases but amplitude increases with the increase of number density. Similarly by

increasing the strength of the inhomogeneity κ in Fig. 2.6, the Sagdeev potential becomes

deeper and the value of the potential increases as well. The same trend can be noted in the

solitary structures as well.

2.7 Conclusion

We have investigated the formation of solitary structures in an inhomogeneous

degenerate quantum plasma in the presence of a quantizing magnetic field. We have

derived the modified Hasegawa Mima equation for an ion-acoustic wave. Large scale

structures have been investigated via the KdV equation. We have investigated our

theoretical results numerically for different parameters such as magnetic field, density

and inverse of inhomogeneity scale length. These results have been presented graphically

showing the formation of solitary structures. The present study can be useful

understanding the propagation characteristics of nonlinear drift waves in dense

astrophysical plasmas such as white dwarf stars where quantum effects are expected to

play an important role.

41

Figure 2.1: bounce frequency versus Potential for fixed values of 𝑛0 = 1032𝑚−3, 𝐵0

= 106𝑇 and 𝜅 = 108𝑚−1 .

42

Figure 2.2: Sagdeev potential 𝑉(Φ) versus Φ for different values of 𝐵0 when 𝑛0 =

1032𝑚−3 and 𝜅 = 4 × 108𝑚−1.

43

Figure 2.3: Solitary structures corresponding to the Sagdeev potential 𝑉(Φ) shown in

Fig.-2.2.

44

Figure 2.4: Sagdeev potential 𝑉(Φ) versus Φ for different values of 𝑛0 when 𝐵0

= 0.5 × 106𝑇 and 𝜅 = 4 × 108𝑚−1.

45


Fig.-2.4.

46

Figure 2.6: Sagdeev potential 𝑉(Φ) versus Φ for different values of 𝜅 when 𝐵0 =

0.5 × 106𝑇 and 𝑛0 = 1032𝑚−3.

47


Fig.-2.6.

48

Chapter 3

Finite amplitude solitary structures of coupled

kinetic Alfven-acoustic waves in dense

plasmas

3.1 Introduction

In this Chapter the nonlinear propagating coupled Kinetic Alfven-acoustic waves in a low

beta (𝛽 = 2𝑛𝑘𝐵𝑇𝐹𝜇0/𝐵02) degenerate quantum plasma in the presence of trapped Fermi

electrons using the quantum hydrodynamic (QHD) model is discussed. By using the two

potential theory and the Sagdeev potential approach, we have investigated the formation

of solitary structures. It is seen that there are regions of propagation and non-propagation

for such solitary structures. It is found that the behavior of nonlinearity changes when the

quantum regime is considered.

The dispersion relation of low-frequency Alfven wave gets modified when the

perpendicular wavelength becomes comparable with the thermal ion Larmor radius. Then

the wave is called the kinetic Alfven wave (KAW). When the ion parallel motion is taken

into account in a classical plasma, the Alfven wave couples to the ion acoustic wave via

the parameter 𝜆𝑠 = 𝑘𝑥2𝑐𝑠2/𝛺𝑖2, where 𝛺𝑖 is the ion cyclotron frequency and 𝑐𝑠 is the ion

sound speed respectively producing coupled kinetic Alfven-acoustic waves (CKAAWs).

Kinetic Alfven waves (KAWs) are believed to play an important role in plasma heating,

particle acceleration, and anomalous transport [111,112].

Using the two-potential method, linear and nonlinear KAW in an electron-ion plasma by

including the finite Larmor radius effect was investigated in the references [113,114]. Yu

49

and Shukla [115] studied finite amplitude solitary KAWs for small but finite 𝛽effects.

This work was further extended by Kakati and Goswami [116] to the case of electron-

positron-ion plasma.

The study of numerous collective interactions in dense plasmas are relevant in the context

of intense laser-solid density plasma experiments [117-123] the cores of giant planets and

the crusts of old stars [124,125], superdense astrophysical objects [126-128] (e.g.,

interiors of white dwarfs and magnetospheres of neutron stars and magnetars); micro and

nano-scale objects (e.g., quantum diodes [129-131], quantum dots and nanowires [132],

nanophotonics [133,134], plasmonics [135], ultra-small electronic devices [136-138],

metallic nanostructures [139], microplasmas [140] and quantum X-ray free-electron

lasers [141]. Furthermore, a Fermi-degenerate dense plasma may also arise when a pellet

of hydrogen is compressed to many times the solid density in the fast ignition scenario

for inertial confinement fusion [142-144].

The rapid development of laser technology since the invention of chirped pulse

amplification [145,146] has given rise to unprecedented intensities on target to be

realized. Lasers are now routinely focused to an irradiance on target of Iλ2=1021 W cm−2

µm2 147 ([146] and [ ]) (where I is the intensity and λ is the wavelength of the laser

radiation). A number of new lasers promise many more order of magnitude increases in

intensity in the near [148,149] and medium-term future [150]. This field has engendered

a lot of interest and attention owing to the fact that it that has allowed the investigation of

many novel relativistic plasma physics issues, ranging from compact particle accelerators

[151-155] to high energy density laboratory astrophysics [156-158] and fast ignition

inertial fusion [159,160].

The study of radiative blast waves in atomic cluster media using intense laser pulses is

reported [161]. Atomic clusters have been shown to be very efficient absorbers of intense

laser radiation. They can be used to create high energy density plasmas that drive strong

shocks (>Mach 50) and radiative blast waves. Careful application of these equations and

similarities allow experiments to be scaled to astrophysical phenomena that have spatial

and temporal scales that are greater by as much as 15–20 orders of magnitude. In this

50

way, the radiative blast waves in the laboratory have been scaled those experienced in

supernova remnants and the physics governing their dynamics investigated under

controlled conditions. It is in the fitness of the situation to mention here that the in-situ

observations of waves in dense plasmas in extreme environments are very difficult.

However the rapid development of laser technology as mentioned above would hopefully

make it possible for us to compare the theory with experiments. Nevertheless it is

imperative that we develop the theory of dense plasmas owing to its applications to laser-

solid and compressed plasmas in the laboratory, in addition to the astrophysical

applications.

The most frequently employed approaches to describe the statistical and hydrodynamic

behavior of charged species at quantum scales in dense plasmas is the Wigner-Poisson

and the Schrödinger-Poisson models. These two approaches are the quantum equivalent

of kinetic and fluid treatments of classical plasmas. The two approaches have been

vividly explained in a review article by Manfredi [88]. The quantum hydrodynamic

(QHD) model is based on the Schrödinger-Poisson formulation. It has been extensively

applied to study the linear and nonlinear propagation of several waves in the quantum

plasma [39,87,162,163].

In classical plasmas, the effect of trapping on the vortex formation was investigated and

the modified Hasegawa-Mima equation was derived [95]. The effect of trapping as a

microscopic phenomenon in a degenerate plasma has also been investigated in the

presence of quantizing magnetic field [100]. In a self-gravitating dusty quantum plasma,

adiabatic trapping has also been found to play an effective role in the formation of

solitary structures [164]. Using two potential theory, the effect of adiabatic trapping on

obliquely propagating coupled Kinetic Alfven-acoustic waves in a low plasma was

investigated [165] for the first time.

In the present chapter, we investigate coupled kinetic Alfven-acoustic solitary structures

in a low β degenerate quantum plasma by including the effect of adiabatic trapping of

electrons. The primary difference is that electrons in this case are governed by the Fermi-

Dirac distribution and therefore the frame work of obtaining the expression for number

51

density as well as the nature of nonlinearity differs quite significantly here. The layout of

present work is as follows: In Sec. 3.2, we give the formulation of basic equations and

derive the linear dispersion relation of the coupled kinetic Alfven-acoustic in a

degenerate quantum plasma. In Sec. 3.3, the nonlinear Sagdeev potential is derived and

investigated. In Sec. 3.4, the results are discussed and finally in Sec. 3.5, the findings of

the current investigation are recapitulated.

3.2 Mathematical Formulation

At the outset, we would like to state that the electrons are considered degenerate and

follow the Fermi-Dirac distribution function; however the ions due to their heavy mass

are assumed to behave in a classical manner.

In this section, we follow the method illustrated in Cramer [56]. The limit 𝑚𝑒/𝑚𝑖 < β <

1 allows us to neglect the electron mass and leads to the investigation of coupled kinetic

Alfven-acoustic waves in the low frequency limit. The variation exists in the 𝑥𝑧 plane,

where 𝑧 is the direction of ambient magnetic field. The low-β assumption allows us to

use the two potential fields φ and 𝜓 to describe the electric field in the 𝑥 and 𝑧 direction

in the following manner 𝐸𝑥 = −𝜕𝜑/𝜕𝑥, 𝐸𝑧 = −𝜕𝜓/𝜕𝑧, 𝐸𝑦 = 0. As a consequence of

the two-potential theory, only shear perturbations in magnetic field are present which is

mathematically expressed as 𝐵𝑧 = 𝐵0 , 𝐵𝑥 = 0. The quasi neutrality condition for ions

and electrons densities leads to 𝑛𝑒 = 𝑛𝑖 = 𝑛.

For low frequency perturbations the phase velocity of the wave is much less than the

electron Fermi speed and electrons are assumed to follow the magnetic field lines. The

normalized occupation number for the Fermi-Dirac distribution is given by

𝑛 = 8√2 𝜋𝑚3/2

(2𝜋ℏ)3 ∫ √ε 𝑑εexp ε−µ−eψ

𝑇 +1

∞0 (3.1)

Where is the chemical potential, is the trapping potential from ions. Using U= eψ + µ and trapping condition ε − U = 0, and following the method described in Landau Lifshitz the expression for total number density 𝑛 including trapped electrons is obtained for a fully degenerate plasma [96] and is given by

52

𝑛 = 𝑛0(1 + 𝛹)3/2 (3.2)

𝛹 = 𝑒𝜓/𝜀𝐹𝑒 is the normalized electrostatic potential, where 𝜀𝐹𝑒 is the Fermi energy

given by 𝜀𝐹𝑒 = ℏ2(3𝜋2𝑛0)3 2 /2 𝑚𝑒. It is worth mentioning here that the implicit

assumption in the derivation of Equation (3.2) is that the electron Fermi energy is so high

that the electron Landau quantization effects can be ignored.

Although the equations for ions have been given before in an earlier paper, we reproduce

them here for the sake of completeness. Following the procedure out lined in references

[59,116,165], the 𝑥 component of ion velocity is given by

𝑣𝑖𝑥 = − 𝑚𝑖𝑒𝐵02

𝜕2𝜑𝜕𝑥𝜕𝑡

(3.3)

The parallel equation of motion for ions is

𝜕𝑡𝑣𝑖𝑧 + 𝑣𝑖𝑥𝜕𝑥𝑣𝑖𝑧 + 𝑣𝑖𝑧𝜕𝑧𝑣𝑖𝑧 = − 𝑒𝑚𝑖

𝜕𝜓𝜕𝑧

(3.4)

From Ampere’s law (Hasegawa and Mima 1976), we have

𝜇0𝜕𝑡𝑗𝑧 = 𝜕𝑧𝜕𝑥2(𝜑 − 𝜓) (3.5)

The ion continuity equation is

𝜕𝑡𝑛𝑖 + 𝜕𝑥(𝑛𝑖𝑣𝑖𝑥) + 𝜕𝑧(𝑛𝑖𝑣𝑖𝑧) = 0 (3.6)

and

𝜕𝑧𝑗𝑧 = 𝑒𝜕𝑡𝑛𝑒 + 𝑒𝜕𝑧(𝑛𝑖𝑣𝑖𝑧) (3.7)

The equation above is derived by using electron continuity equation. The algebraic

manipulation of Eqs. (3.2)-(3.7) yield the following linear dispersion relation

53

1 − 𝑣𝐴2𝑘𝑧2

𝜔2 32− 𝑐𝑠𝐹

2 𝑘𝑧2

𝜔2 = 𝑣𝐴2𝑘𝑧2

𝜔2 𝜆𝑠𝐹 (3.8)

Where 𝜆𝑠𝐹 = 𝑘𝑥2𝑐𝑠𝐹2 /𝛺𝑖2 is the coupling parameter, 𝑐𝑠𝐹 = 𝜀𝐹𝑒/mi is the ion sound speed

at the Fermi energy 𝜀𝐹𝑒, 𝑣𝐴 is the Alfven velocity and 𝛺𝑖 is the usual ion cyclotron

frequency. The wave numbers in the 𝑥 and 𝑧 directions are expressed through the

obliqueness angle with respect to the magnetic field and given by 𝑘𝑧 = 𝑘 𝑐𝑜𝑠𝜃, 𝑘𝑥 =

𝑘 𝑠𝑖𝑛𝜃. It is pertinent to mention here that the linear dispersion relation for coupled

kinetic Alfven-acoustic wave given by Eq. (3.8) differs from its classical counterpart [60]

quite appreciably. Note that the second bracket of left hand side contains not only the ion

sound velocity defined at Fermi energy but also a 3/2 factor which is the consequence of

the effect of adiabatic trapping on the linear dispersion relation. In a classical plasma, the

linear dispersion relation remains unaffected by adiabatic trapping as it manifests itself

only in the nonlinear regime.

Equation (3.8) shows the coupling of Alfven wave with the ion-acoustic wave through

the coupling parameter 𝜆𝑠𝐹. If we set the 2nd

59

factor on the left hand side to unity we get

the linear dispersion relation of KAW [ ].

𝜔2 = 𝑣𝐴2𝑘𝑧2(1 + 𝜆𝑠𝐹) (3.9)


In this section, we derive the Sagdeev potential to investigate the formation of solitary

structures and to this end we shift to a co-moving frame of reference in normalized

variables

𝜂 = 𝐾𝑥𝑥 + 𝐾𝑧𝑧 − 𝑀𝑡 (3.10)

The normalized variables are given by

𝑛 = 𝑛𝑒,𝑖/𝑛0, Φ = 𝑒𝜑/𝑇𝑒, 𝑀 = 𝑣/𝑐𝑠, 𝐾 = 𝐾𝑐𝑠/𝛺𝑖 , 𝑡 = 𝛺𝑖𝑡,

54

where 𝑛,Φ,𝑀,𝐾, 𝑡 are the normalized density, potential, effective Mach number, wave

number, and time respectively.

Eqs. (3.3)-(3.7) are recast in dimensionless form and are given below

−𝑀 𝜕𝑣𝑖𝑧𝜕𝜂

+ 𝑣𝑥𝐾𝑥𝜕𝑣𝑖𝑧𝜕𝜂

+ 𝑣𝑧𝐾𝑧𝜕𝑣𝑖𝑧𝜕𝜂

= −𝐾𝑧𝜕𝛹𝜕𝜂

(3.11)

𝑣𝑖𝑥 = 𝐾𝑥𝑀𝜕2Φ𝜕𝜂2

(3.12)

2𝐾𝑥2𝐾𝑧2𝜕𝜂4(Φ− 𝛹) = 𝛽(𝑀2𝜕𝜂2𝑛 − 𝑀𝐾𝑧𝜕𝜂2(𝑛𝑣𝑖𝑧)) (3.13)

−𝑀𝜕𝜂𝑛 + 𝐾𝑥𝜕𝜂(𝑛𝑣𝑖𝑥) + 𝐾𝑧𝜕𝜂(𝑛𝑣𝑖𝑧) = 0 (3.14)

Integrating Eqs. (3.13) and (3.14) and applying the boundary conditions 𝑣𝑖𝑧, 𝑣𝑖𝑥,𝜑,𝜓 →

0 ,𝑛0 = 1 𝑎𝑠 𝜂 → ∞, we obtain

2𝐾𝑥2𝐾𝑧2𝜕𝜂2(Φ− 𝛹) = 𝛽(𝑀2(𝑛 − 1) −𝑀𝐾𝑧(𝑛𝑣𝑖𝑧)) (3.15)

𝐾𝑥𝑣𝑖𝑥 + 𝐾𝑧𝑣𝑖𝑧 = 𝑀1 − 1𝑛 (3.16)

Φ is reduced in 𝛹 therefore it does not appear in further equations.

Using Eqs. (3.2), (3.11), and (3.16), we get the following expression for the parallel ion

velocity 𝑣𝑖𝑧

𝑣𝑖𝑧 = 𝐾𝑧𝑀(1+𝑇2)

(1 + 𝛹)3/2𝛹 (3.17)

The algebraic manipulation of Eqs. (3.12), (3.16), and (3.17) yield the following

expression

2𝐾𝑥2𝜕2𝛹𝜕𝜂2

= 2 (1+𝛹)3/2−1(1+𝛹)3/2 −

𝛽𝐾𝑧2

𝑀𝐴2 𝛹(1 + 𝛹)3/2 − 2 𝑀𝐴

2

𝐾𝑧2(1 + 𝛹)3/2 − 1 +

𝛽(𝛹(1 + 𝛹)3) (3.18)

55

In the above equation, we have used 𝑀2 = 2𝑀𝐴2/ 𝛽 , where 𝑀𝐴 is the ratio of wave

velocity to the Alfven velocity and is generally referred to as the Alfvenic Mach number.

Equation (3.18) can be expressed in the form of an energy integral through the Sagdeev

or pseudo-potential in the following manner

12𝑑𝛹𝑑𝜉2

+ 𝑉(𝛹) = 0 (3.19)

Where 𝑉(𝛹) is given by

𝑉(𝛹) = − 1𝐾𝑥2𝛹 + 2

√1+𝛹 − 𝛽𝐾𝑧2

2𝑀𝐴2

235

(1 + 𝛹)52(−2 + 5𝛹) − 𝑀𝐴

2

𝐾𝑧225

(1 + 𝛹)52 − 𝛹 +

𝛽2𝛹

2

2+ 𝛹3 + 3𝛹4

4+ 𝛹5

5 + 2

5𝑀𝐴2

𝐾𝑧2− 2

35𝛽𝐾𝑧2

𝑀𝐴2 − 2

(3.20)

3.4 Results and Discussions

In this section, we present our analysis in order to obtain the solitary structures. The

Mach number is found to obey the following conditions:

𝛽3𝐾𝑧 < 𝑀𝐴 < 𝑀1 (3.21a)

𝑀2 < 𝑀𝐴 < 𝐾𝑧 (3.21b)

where

𝑀1 =12

𝑎 − 𝑏

(1 + 𝛹) 1 − √1 + 𝛹 + 𝛹(2 + 𝛹)

𝑀2 =12

𝑎 + 𝑏

(1 + 𝛹) 1 − √1 + 𝛹 + 𝛹(2 + 𝛹)

56

𝑎 = 𝐾𝑧2 2−1 + √1 + 𝛹 + 𝛹√1 + 𝛹(2 + 𝛽(1 + 𝛹)4)

𝑏

= 𝐾𝑧4 8 − 8√1 + 𝛹 + 𝛹 𝛽2𝛹(1 + 𝛹)9 − 4𝛽(1 + 𝛹)4 1 − √1 + 𝛹 + 𝛹(2 + 𝛹)

+43 − 2√1 + 𝛹 + 𝛹(3 + 𝛹)

The above conditions are obtained by taking the power series of Sagdeev potential (3.20)

and setting the coefficients of quadratic term in 𝛹 equal to zero. Here 𝑀1 and 𝑀2 are the

lower and higher values of Mach number which satisfy the above conditions Eqs. (3.21a)

and (3.21b). Solitary structures are observed to formed only below 𝑀1 and above 𝑀2.

We now present the graphical analysis of our model by using the values of plasma

parameters that are typically found in the vicinity of white dwarf stars [109,110]. It has

been suggested that electrostatic structures could be excited in extreme events, such as

supernova explosions at the outer shells of the star [166]. It was also remarked that

electromagnetic waves should also be studied in future. Motivated by this suggestion, we

have investigated the coupling of electromagnetic kinetic Alfven wave with the acoustic

wave in the presence of quantum mechanically trapped electrons. It is worth mentioning

here that for the values of number density and magnetic field used here, the value of

plasma beta (𝛽 = 2𝑛𝑘𝐵𝑇𝐹𝜇0/𝐵02) turns out to be less than one and hence the use of two

potential theory is justified. Figure 3.1 shows the plot of Sagdeev potential for different

values of number density by keeping the values of magnetic field and Mach number

fixed. We see that by increasing the number density the maximum value of potential

increases but depth of the potential decreases. The corresponding solitary structures are

shown in Fig. 3.2, in which the amplitude as well as the width of the soliton increases

with the increase in the number density.

In Fig. 3.3, the variation of Sagdeev potential is explored for different values of magnetic

field by keeping the other plasma parameters fixed. It is observed that depth of the

potential increases but the maximum value of potential decreases by increasing the

magnetic field. The corresponding solitary structures are plotted in Fig. 3.4. It is found

57

that the amplitude as well as the width of the soliton decrease with the increase of

magnetic field.

Fig. 3.5 exhibits the change in Sagdeev potential by varying the Mach number for fixed

values of magnetic field strength and number density. It is observed that the depth of the

potential increases till the upper limit on Mach number (𝑀1) given in condition (21a) is

reached. With a further increase in the Mach number, Sagdeev potential does not

intersect the potential axis and consequently there will be no corresponding solitary

structure. However, when the Mach number further increases and reaches a certain value

(𝑀2) satisfying the condition (21b), the Sagdeev potential again intersects the potential

axis giving rise to the formation of the solitary structures again.

Figure 3.6 is plotted for the maximum amplitude of the soliton vs the Mach number for

different values of angle of propagation θ = 45o (thick black line), θ = 60o (thick dashed

line), θ = 75o

3.5 Conclusion

(black line). It is clear from the Fig. 3.6 that for a particular angle, the

maximum amplitude of the soliton first increases with the increase in Mach number and

then decreases for further increase in the Mach number. Note that the gaps in maximum

amplitude values for each angle indicate the absence of solitary structure formation as

they correspond to those values of Mach number that do not satisfy the conditions given

by Eq. (3.21). It can also be seen that with the increase in the obliqueness the gap and

range of Mach number decreases. The solitons corresponding to the Fig. 3.5 are plotted in

Fig. 3.7. Note that the amplitude as well as the width of the soliton enhances with the

increase in the Mach number as long as condition (given by Eq. (3.21a)) is satisfied,

however, both amplitude and width of the soliton show a decrease with the increase in

Mach number when the condition (given by Eq. (3.21b)) is satisfied.

In conclusion, we have investigated the adiabatic trapping of electrons in a low beta

quantum plasma for a coupled kinetic Alfven-acoustic wave. It has been observed that the

framework for obtaining the expression of number density of quantum mechanically

trapped electrons is quite different from its classical counterpart. Most importantly, the

58

nature of the nonlinearity is found to be different for quantum mechanically trapped

electrons by comparison with its classically trapped counterparts. By using the Sagdeev

potential approach, we have studied the finite amplitude nonlinear structures and also

mentioned the conditions which determine the existence regimes of the solitary

structures. We have also explored the variation of the structure of the solitary waves by

using different plasma parameters of interest such as obliqueness, magnetic field strength

and number density. The present work may be beneficial in enhancing our understanding

of the solitary structures in astrophysical environments with special reference to the

pulsating white dwarfs and also in laboratory experiments on chirped laser plasma

interactions where many astrophysical phenomena can be mimicked on the laboratory

scale.

59

Figure 3.1: Sagdeev potential 𝑉(𝛹) versus 𝛹 for different values of 𝑛0 when 𝐵0 =

0.7 × 107𝑇 and 𝑀𝐴 = 0.38.

60

Figure 3.2: Solitary structures corresponding to the Sagdeev potential 𝑉(𝛹) shown in

Fig.-3.1.

61

Figure 3.3: Sagdeev potential 𝑉(𝛹) versus 𝛹 for different values of 𝐵0when 𝑛0 = 3 ×

1033𝑚−3 and 𝑀𝐴 = 0.38.

62


Fig.-3.3.

63

Figure 3.5: Sagdeev potential 𝑉(𝛹) versus 𝛹 for different values of 𝑀𝐴 when 𝐵0 =

0.7 × 107𝑇 and 𝑛0 = 3 × 1033𝑚−3.

64

Figure 3.6: Maximum value of potential versus Mach number for different values of

angle.

65


Fig.-3.5.

66

Chapter 4

Coupled acoustic-kinetic Alfven waves in self

gravitating dusty plasmas in the presence of

adiabatic trapping

The linear dispersion relations are derived for both negatively and positively charged dust

for the coupled acoustic-kinetic Alfven wave in self-gravitating dusty plasma by taking

the trapping of electrons and ions into account separately in a low beta. Their limiting

cases are also discussed. By using Sagdeev potential the solitary structures are formed. It

is seen that solitary structures are formed for sub Alfvenic mode. We have applied our

results to the solar corona for negatively charged dust and in the vicinity of Io for

positively charged dust.

4.1 Introduction

As described in section 1.10 of chapter 1, we know that the dusty plasmas are present in

many natural and laboratory environments e.g in space plasmas, dusty plasmas have been

observed in planetary rings [62], asteroid zones [63] and the Earth’s atmosphere [64].

Dust in plasma has also been observed and investigated in laboratory plasmas and also in

fusion experimental machines [65-68]. Theoretically Rao et al. [69] give an explanation

for a first time of dust-acoustic wave (DAW), Shukla and Silin [70] investigated dust-ion-

acoustic waves (DIAW). The existence of these new modes was subsequently confirmed

in laboratory experiments [71,72,167 -175].

Dust investigations of nonlinear wave propagation quickly became an integral part of

plasma physics studies and a very large volume papers has been devoted to study the

different aspects of nonlinear interactions [176-188]. When an electron-ion plasma

contains dust grains new normal modes may appear [189]. In a dusty plasma the electrons

and ions can be considered to be Boltzmann-distributed while the dust particles which

67

can be many orders of magnitude more massive than the ions are always inertial.

Therefore, the restoring force is provided by the pressure of electrons and ions, but the

inertia comes from dust mass.

Modified Alfven and magnetosonic modes have also been observed in dusty plasma

[190]. The presence of charged dust grains modify the linear and nonlinear behavior of

different wave modes. When the mass of dust grains becomes significant, gravitational

effects on dust grains needs to be taken into account. Some studies [191-193] devoted to

wave propagation in self-gravitating dusty plasmas show that there may be competition

between gravitational self-attraction and electrostatic repulsion among the charged grains.

And consequently it is seen that the gravitational effect also affects nonlinear wave

propagation characteristics. Although most of the work referred to above has been

dedicated to negatively charged dust, however natural plasma environments provide

instances where dust is positively charged [194,195].

Stefant [56] investigated the case when the perpendicular wavelength becomes

comparable with the ion Larmor radius, ions do not follow the magnetic lines of force. As

a result charge separation is produced and the Alfven wave gets coupled to the

electrostatic longitudinal mode which leads to the kinetic Alfven wave (KAW). The

propagation of Alfven waves has been affected by the presence of dust and also modifies

the behavior of these low frequency waves in dusty plasma [196]. Solitary Alfven waves

were the subject of investigation in electron-positron-ion plasmas [56], where an exact

solitary wave solution exist for small but finite β. Nonlinear Dust kinetic Alfven waves

were investigated by Yinhua et al. in a low -𝛽𝑑 (𝑚𝑖/𝑚𝑑 < 𝛽𝑑 < 1/𝑧𝑑, where 𝛽𝑑 =𝜇0 𝑛𝑑0 𝑇

𝐵02) collisionless plasma and it was also found that the density humps are cusped and

narrower than the dips [197].

In this paper, we investigate the coupled acoustic-kinetic Alfven wave in a self-

gravitating dusty plasma with adiabatic trapping in a low β plasma. We will make use of

the two potential theory (discussed in the introduction). The layout of work is as follows:

In section 4.2, we give our set of equations. Linear dispersion relation for negative dust is

derived in section 4.3. In section 4.4, stability analysis is given. In section 4.5, the

68

nonlinear Sagdeev potential is derived for negative dust. In section 4.6, the linear

dispersion relation for positive dust is derived and stability check is in section 4.7. In

section 4.8, Sagdeev potential is also derived for positive dust. In section 4.9, the

graphical results are discussed numerically.

4.2 Mathematical Formulation

In this section we set up the governing equations which latter be used to derive the linear

dispersion for coupled acoustic-kinetic Alfven wave in a dusty plasma with effects of self

gravitation and adiabatic trapping of electrons. To this end we follow the method

illustrated in [56]. The limit 𝑚𝑖/𝑚𝑑 < 𝛽𝑑 < 1 allows us to neglect the mass of electron

and ion and leads to the investigation of coupled kinetic Alfven-acoustic waves in the low

frequency limit. In terms of 𝛽𝑑 this assumption can be written as 𝛽𝑑 =𝑧𝑑𝜇0𝑛𝑑0 𝑘𝐵 𝑇𝑖

𝐵02 . The

variation exists in the 𝑥𝑧 plane, where 𝑧 is the direction of ambient magnetic field. The

low-β assumption allows us to use the two potential fields ϕ and 𝜓 to describe the

electric field in the 𝑥 and 𝑧 direction in the following manner Ex = −∂ϕ∂x

, Ez = −∂ψ∂z

,

Ey = 0 . As a consequence of the two potential theory only shear perturbations in

magnetic field are present which is mathematically expressed as Bz = B0 , Bx = 0.

We consider three component plasma including electrons, ions and negative charged dust

grains. Both electrons and ions are taken massless in the presence of charged dust grains.

The equation of motion and continuity for dust grains can be written as

mdnd ∂∂t

+ 𝐯d ∙ 𝛁 𝐯d = − zdend(𝐄 + 𝐯d × 𝐁) − mdnd𝛁φ (4.1)

∂nd∂t

+ 𝛁 ∙ (nd𝐯d) = 0 (4.2)

Where md , nd, zd and φ are the mass, number density, charge number and gravitational

potential of dust respectively.

The electrostatic and gravitational Poisson equations are

69

𝛁2ψ = 4πe(zdnd + ne − ni) (4.3)

𝛁2φ = 4πGmdnd (4.4)

Where ni and ne are the number densities of ion and electron, 𝜓 is the electrostatic

potential and G is the gravitational constant.

The electrons are considered as classical and assume that the electrons follow the

Boltzmann distribution, which is given by

ne = ne0ee𝜓

Te (4.5)

The density for ions is

ni = ni0 1 − e𝜓Ti

+ α e𝜓Ti32 (4.6)

Where α = 43√π

∂zjz = e∂tne − e ∂tni − e ∂z(zdndvdz) (4.7)

Equations (4.1) - (4.7) are the governing equations of our model.

4.3 Linear dispersion relation with negative dust charge

For coupled acoustic-kinetic Alfven wave in a dusty plasma we ignore the inertia of

electrons and ions in the comparison of dust mass. Ions are trapped in negative dust

charge. Linearizing Eqs. (4.1)- (4.4) we obtain

(σid+γσed) 1 − kz2

ω2 csd2 1 −kz2 vA

2

ω2 −kz4 vA

2ωjd2 βdzd

ω4 1 − kx2 ω2

kz2 Ωd2 −

ω2

kz2 vA2

ω2

kz2

σedβd zd vA

2(1+γ)−kx2

vA2

Ωd2+kz

2 vA2

ω2−1+ω

2

kz2γcsd2

kx21−ωjd2

Ωd2 +kz

21+ωjd2

ω2−

ωjd2

vA2

= λskz2 vA

2

ω2 (4.8)

70

The above relation shows the dispersion relation for coupled acoustic-kinetic Alfven

wave in a dusty plasma with effects of self gravitation. Where 𝜎𝑒𝑑 is the ratio of

unperturbed electron density and unperturbed dust density, 𝜔𝑗𝑑 is the Jean frequency, 𝑐𝑠𝑑

is the dust acoustic speed and 𝜆𝑠 is the coupling parameter. It is worth mentioning here

that the trapping is a nonlinear phenomenon and therefore cannot be seen in the

dispersion relation.

By ignoring Jean term the above dispersion relation in dimensionless form can be written

as

𝑀𝐴𝑑2 =

𝛽𝑑+𝜎𝑖𝑑+γ 𝜎𝑒𝑑+𝑧𝑑𝐾𝑥2±(𝛽𝑑+𝜎𝑖𝑑+γ 𝜎𝑒𝑑+𝑧𝑑𝐾𝑥2)2−4 𝛽𝑑(𝜎𝑖𝑑+γ 𝜎𝑒𝑑)2

2(𝜎𝑖𝑑+γ 𝜎𝑒𝑑) (4.9)

Where 𝛽𝑑 = csd2 /vA2 , λs = 𝑧𝑑csd2

Ωd2 kx2 = 𝑧𝑑𝐾𝑥2 ,𝑀𝐴𝑑 = 𝜔/𝑘𝑧𝑣𝐴. This dispersion relation

has two modes. The mode with positive sign shows the super Alfvenic mode, while the

negative sign shows the sub Alfvenic mode.

In the following sub sections we consider a few limiting cases of the results obtained

above for the coupled dust acoustic-Kinetic Alfven wave to check that in the relevant

limits our equations reduce to standard results obtained elsewhere.

4.3.1 Limiting cases

Case I. (𝜔𝑗𝑑=0)

This corresponds to the case when the Jeans term is absent. We get the expression for

coupled dust acoustic-kinetic Alfven wave.

Case II. (𝜔𝑗𝑑≠ 0)

71

This corresponds to the case when the Jeans term is present. By setting the third (ion-

acoustic) factor on the left hand side of Eq. (4.8) to unity yields the dust kinetic Alfven

wave dispersion relation.

(σid + γσed)ω2 −

kz4 vA2ωjd

2 βdzdω2 1 − kx2 ω2

kz2 Ωd2 −

ω2

kz2 vA2

ω2

kz2

σedβd zd vA

2(1+γ)−kx2

vA2

Ωd2+kz

2 vA2

ω2−1+ω

2

kz21csd2

kx21−ωjd2

Ωd2 +kz

21+ωjd2

ω2−

ωjd2

vA2

=

[λs + (σid + γσed)]kz2 vA2

(4.10)

Case III. (𝜔𝑗𝑑= 0 , 𝑚𝑑 → 𝑚𝑖 ,𝑛𝑑0 → ni0)

This corresponds to the case when Jeans term is absent. We get the same expression as

obtained for kinetic Alfven wave [56].

4.4 Stability analysis with the jeans term

In this section, we will check the stability of the wave which is of the form written below

can be obtained from dispersion relation (Eq. (4.8))

𝜔6 + 𝑎1 𝜔4 + 𝑎2 𝜔2 + 𝑎3 = 0 (4.11)

Where

𝑎1 = λ kz2 ωjd2 − λ kz4 vA2 − λ kx2 kz2 vA2 1 −

ωjd2

Ωd2 − 1 −

ωjd2

Ωd2 (1 + βd)C vA2 kx2 kz2

+ ωjd2 kz2C − (1 + βd) kz4 vA2C − kz2 ωjd

2 A − zdcsd2 kz2 ωjd2 B

− B zdcsd2 vA2 kz2 kx2ωjd

2

Ωd2+ kz2 ωjd

2 C(1 + βd)

/ 1 −ωjd2

Ωd2C kx2 − C

ωjd2

vA2+ AB ωjd

2 + C kz2

72

a2 = C1 −ωjd2

Ωd2 kx2csd2 vA2 kz4 − λvA2ωjd

2 kz4 − (1 + βd)C ωjd2 kz4vA2 + C vA2csd2 kz6

− csd2 ωjd2 kz4 C + zdcsd2 ωjd

2 kz4 +zdcsd2 vA2ωjd

2 kx2 kz4

Ωd2+ Bzdcsd2 vA2ωjd

2 kz4

/ 1 −ωjd2

Ωd2C kx2 − C

ωjd2

vA2+ AB ωjd

2 + C kz2

𝑎3 = vA2csd2 ωjd2 kz6C − zdcsd2 vA2ωjd

2 kz6/ 1 −ωjd2

Ωd2C kx2 − C

ωjd2

vA2+ AB ωjd

2 + C kz2

A = σed(1 + γ) + γzd, B =1

vA2+

kx2

Ωd2, C = (𝜎𝑖𝑑 + 𝛾 𝜎𝑒𝑑)

Where 𝑎1 ,𝑎2,𝑎3 are the coefficients. If 𝑎1 𝑎2 − 𝑎3 > 0, then the system is Jeans stable

[198]. Conversely, the system will become Jeans unstable if 𝑎1 𝑎2 − 𝑎3 < 0.


In this section we proceed to derive the Sagdeev potential in the absence of Jeans term

and investigate the formation of solitary structures. Consider a wave solution which is

obliquely propagating. Shifting to a co-moving frame of reference in normalized

variables [197]

𝜂 = 𝐾𝑥𝑥 + 𝐾𝑧𝑧 − 𝑀𝑡 (4.12)

The normalized variables are given by

𝑛𝑒,𝑖,𝑑 =𝑛𝑒,𝑖,𝑑

𝑛0,𝛹 =

𝑒𝜓𝑇𝑖

,Φ = 𝑒𝜙𝑇𝑖

,𝜑 =𝜑𝑐𝑠2

,𝑀 =𝑣𝑣𝐴

,𝐾 = k𝑐𝑠𝛺𝑖

, 𝑡 = 𝛺𝑖𝑡

Where 𝑛,Ψ,Φ,𝑀,𝐾, 𝑡 are the normalized density, potentials in the 𝑥 and 𝑧 direction,

effective Mach number and time respectively. From Ampere’s law we have

𝐾𝑥2𝐾𝑧2𝜕𝜂4(Φ− 𝛹) = 𝛽𝑑𝑧𝑑

(−𝑀2𝜕𝜂2(zdnd) + 𝑀𝐾𝑧𝜕𝜂2(zdnd𝑣𝑑𝑧)) (4.13)

73

By taking parallel component of equation of motion

−𝑀 𝜕𝑣𝑑𝑧𝜕𝜂

+ 𝑣𝑥𝐾𝑥𝜕𝑣𝑑𝑧𝜕𝜂

+ 𝑣𝑧𝐾𝑧𝜕𝑣𝑑𝑧𝜕𝜂

= 𝐾𝑧𝜕𝛹𝜕𝜂− 𝐾𝑧

𝜕𝜑𝜕𝜂

(4.14)

Integrating Eqs. (4.13) and (4.14) and applying the boundary conditions 𝑣𝑑𝑧, 𝑣𝑑𝑥, 𝜙,

𝜓 → 0 𝑎𝑠 𝜂 → ∞ , we obtain

𝐾𝑥2𝐾𝑧2𝜕𝜂2(Φ− 𝛹) = 𝛽𝑑𝑧𝑑

(𝑀2zd(1 − nd) + 𝑀𝐾𝑧(zdnd𝑣𝑑𝑧)) (4.15)

Mvdz = −KzzdΨ(σid − σed) − 1

2(σid + γσed)Ψ2 − 2

5α σedγ3/2Ψ5/2 + Kz ∫nd

∂φ∂η

dη

(4.16)

Where 𝜎𝑖𝑑 = ni0nd0

, 𝜎𝑒𝑑 = ne0nd0

and 𝛾 = 𝑇𝑖𝑇𝑒

. Φ is reduced in 𝛹 therefore it does not appear

in further equations.

By using Eq. (4.16) into Eq. (4.15) we obtain

− βdKz2

zd MA2 [(σid − σed)2Ψ] + 1

zd(σid + γ σed)Ψ + α σed γ3/2 Ψ3/2 − MAd

2

zd Kz2(σid −

σed) (σid + γσed)Ψ + α σedγ3/2(σid − σed)Ψ3/2] + βdzd2 [(σid − σed)3Ψ] =

(σid − σed) Kx2 ∂η∂ηΨ (4.17)

Here, we have used 𝑀𝐴𝑑2 = 𝑀2βd. After integrating and using the boundary conditions

we get the Sagdeev potential in the following manner

12dΨdξ2

+ V(Ψ) = 0 (4.18)

where 𝑉(𝛹) is given by

74

𝑉(𝛹) = − 1𝐾𝑥2(𝜎𝑖𝑑−𝜎𝑒𝑑)

− 𝛽𝑑𝐾𝑧2

2 𝑧𝑑 𝑀𝐴𝑑2 (𝜎𝑖𝑑 − 𝜎𝑒𝑑)2𝛹2 + 1

2(𝜎𝑖𝑑 + 𝛾 𝜎𝑒𝑑)𝛹2 − 𝑀𝐴𝑑

2

2 𝑧𝑑 𝐾𝑧2(σid −

σed)(𝜎𝑖𝑑 + 𝛾 𝜎𝑒𝑑)𝛹2 + 𝛽𝑑2 𝑧𝑑

2 (𝜎𝑖𝑑 − 𝜎𝑒𝑑)3𝛹2 + 25α σed γ3/2Ψ5/2 −

25𝑀𝐴𝑑2

𝑧𝑑 𝐾𝑧2α σed γ3/2(𝜎𝑖𝑑 − 𝜎𝑒𝑑)Ψ5/2 (4.19)

𝑉(𝛹) =

− 1𝐾𝑥2𝛽𝑑21 − 𝐾𝑧2

𝑀𝐴𝑑2

2𝛹2 + 1

2 𝑧𝑑(𝜎𝑖𝑑 + 𝛾 𝜎𝑒𝑑) 1 − 𝑀𝐴𝑑

2

𝐾𝑧2𝛹2 + 2

5 𝑧𝑑α σed γ

32Ψ

52 1 − 𝑀𝐴𝑑

2

𝐾𝑧2

(4.20)

Eq. (4.20) is the Sagdeev potential when the Jean term is absent.

4.6 Linear dispersion relation with positive dust charge

We consider three component plasma including electrons, ions and positive charged dust

grains. Electrons are trapped in positive dust charge. The equation of motion and

continuity for dust grains can be written as

mdnd ∂∂t

+ 𝐯d ∙ 𝛁 𝐯d = zdend(𝐄 + 𝐯d × 𝐁) − mdnd𝛁φ (4.21)

∂nd∂t

+ 𝛁 ∙ (nd𝐯d) = 0 (4.22)

Where md , nd, zd and φ are the mass, number density, charge number and gravitational

potential of dust respectively.

The electrostatic and gravitational Poisson equations are

𝛁2ψ = 4πe(ne − ni − zdnd) (4.23)

𝛁2φ = 4πGmdnd (4.24)

Where ni and ne are the number densities of ion and electron, 𝜓 is the electrostatic

potential and G is the gravitational constant.

75

The ions are considered as classical and assume that the ions follow the Boltzmann

distribution, which is given by

ni = ni0e−e𝜓

Ti (4.25)

The density for electrons including the effect of adiabatic trapping is

ne = ne0 1 + e𝜓Te

+ α e𝜓Te32 (4.26)

Where α = − 43√π

∂zjz = e∂tne − e ∂tni + e ∂z(zdndvdz) (4.27)

Equations (4.21) - (4.27) are the governing equations of our model.

For coupled acoustic-kinetic Alfven wave in a dusty plasma we ignore the inertia of

electrons and ions in the comparison of dust mass. Linearizing Eqs. (4.21)- (4.24) we

obtain

(σid+γσed) 1 − kz2

ω2 csd2 1 −kz2 vA

2

ω2 +kz4 vA

2ωjd2 βdzd

ω4 1 − kx2 ω2

kz2 Ωd2 −

ω2

kz2 vA2

ω2

kz2

σidβd zd vA

2(1+γ)+kx2

vA2

Ωd2−kz

2 vA2

ω2−1+ω

2

kz2γcsd2

kx21−ωjd2

Ωd2 +kz

21+ωjd2

ω2+

ωjd2

vA2

= λskz2 vA

2

ω2 (4.28)

4.6.1 Limiting cases

In this section we are looking for coupled Alfven wave

Case I. (𝜔𝑗𝑑=0)

This is the case when Jeans term is not included. We get the expression for coupled dust

acoustic-kinetic Alfven wave.

76

Case II. (𝜔𝑗𝑑≠ 0)

This corresponds to the case when Jeans term is present. By setting the third (ion-

acoustic) factor on the left hand side of Eq. (4.28) to unity yields the dust kinetic Alfven

wave dispersion relation.

(σid + γσed)ω2 +

kz4 vA2ωjd

2 βdzdω2 1 − kx2 ω2

kz2 Ωd2 −

ω2

kz2 vA2

ω2

kz2

σidβd zd vA

2(1+γ)+kx2

vA2

Ωd2−kz

2 vA2

ω2−1+ω

2

kz2γcsd2

kx21−ωjd2

Ωd2 +kz

21+ωjd2

ω2+

ωjd2

vA2

=

[λs + (σid + γσed)]kz2 vA2 (4.29)

Case III. (𝜔𝑗𝑑= 0 , 𝑚𝑑 → 𝑚𝑒 ,𝑛𝑑0 → ne0)

We get the same expression as obtained for kinetic Alfven wave [56].

4.7 Stability analysis with the jeans term

In this section, we will check the stability of the wave obtained from dispersion relation

(Eq. (4.28))

𝜔6 + 𝑎1 𝜔4 + 𝑎2 𝜔2 + 𝑎3 = 0 (4.30)

Where

𝑎1 = −λ kz2 ωjd2 − λ kz4 vA2 − λ kx2 kz2 vA2 1 −

ωjd2

Ωd2 − 1 −

ωjd2

Ωd2 (1 + βd)C vA2 kx2 kz2

+ ωjd2 kz2C − (1 + βd) kz4 vA2C + kz2 ωjd

2 A + zdcsd2 kz2 ωjd2 B

− B zdcsd2 vA2 kz2 kx2ωjd

2

Ωd2− kz2 ωjd

2 C(1 + βd)

/ 1 −ωjd2

Ωd2C kx2 + C

ωjd2

vA2+ AB ωjd

2 + C kz2

77

a2 = C1 −ωjd2

Ωd2 kx2csd2 vA2 kz4 − λvA2ωjd

2 kz4 − (1 + βd)C ωjd2 kz4vA2 + C vA2csd2 kz6

+ csd2 ωjd2 kz4 C − zdcsd2 ωjd

2 kz4 +zdcsd2 vA2ωjd

2 kx2 kz4

Ωd2+ Bzdcsd2 vA2ωjd

2 kz4

/ 1 −ωjd2

Ωd2C kx2 + C

ωjd2

vA2+ AB ωjd

2 + C kz2

𝑎3 = vA2csd2 ωjd2 kz6C − zdcsd2 vA2ωjd

2 kz6/ 1 −ωjd2

Ωd2C kx2 + C

ωjd2

vA2+ AB ωjd

2 + C kz2

A = σid(1 + γ) + γzd, B =1

vA2+

kx2

Ωd2, C = (𝜎𝑖𝑑 + 𝛾 𝜎𝑒𝑑)

Where 𝑎1 ,𝑎2,𝑎3 are the coefficients. If 𝑎1 𝑎2 − 𝑎3 > 0, then the system is Jeans stable

[198].


By using the normalized variables in the same way as described above the Sagdeev

potential 𝑉(𝛹) for positive dust charge is given as

𝑉(𝛹) =

− 1𝐾𝑥2− 𝛽𝑑

21 + 𝐾𝑧2

𝑀𝐴𝑑2

2𝛹2 + 1

2 𝑧𝑑(𝜎𝑖𝑑 + 𝛾 𝜎𝑒𝑑) 1 − 𝑀𝐴𝑑

2

𝐾𝑧2𝛹2 + 2

5 𝑧𝑑α σed γ

32Ψ

52 1 −

𝑀𝐴𝑑2

𝐾𝑧2 (4.31)

4.9 Results and discussions

In this section we analyze the results of previous sections. The numerical value of the

parameters are chosen from observations of solar corona [56] and in the vicinity of Io

[199] for negatively and positively charged dust respectively.

78

The linear dispersion relation given by Eq. (4.8) is investigated using the parameters of

solar corona for negative dust where the number density and magnetic field have values

of the order of 1013𝑚−3 and 10−4𝑇. Fig. 4.1 is the plot for frequency vs wave number

for different values of magnetic field. It is seen that higher frequencies are observed as

the strength of the magnetic field increases. Fig. 4.2 is the plot for frequency vs wave

number for different values of number density. It is seen that frequency decreases as the

dust number density increases. The increase of dust number density depicts the effect of

an increasing Jeans frequency.

We also investigate numerically the dependence of Sagdeev potential 𝑉(𝛹) vs the

potential 𝛹 for different plasma parameters. Fig. 4.3 shows the plot of Sagdeev potential

for different values of Mach number for negatively charged dust. It is observed that the

width and depth of Sagdeev potential increases by increasing the value of Mach number.

Corresponding solitary structures are shown in Fig. 4.4, in which the amplitude of the

soliton increases but width decreases slightly with the increase in the Mach number.

The linear dispersion relation given by Eq. (4.29) is investigated using the parameters of

Io for positive dust where the number density and magnetic field have values of the order

of 109𝑚−3 and 10−4𝑇. Fig. 4.5 is the plot for frequency vs wave number for different

values of magnetic field. It is seen that an increase in the magnetic field enhances the

frequency. Fig. 4.6 is the plot for frequency vs wave number for different values of

magnetic field with positively charged dust. It is seen that as the dust number density

increases the frequency decreases.

Fig. 4.7 shows the plot of Sagdeev potential for different values of Mach number for

positively charged dust. It is observed that increasing the value of Mach number enhances

the width as well as the depth of Sagdeev potential. Fig. 4.8 shows the corresponding

solitary structures, the amplitude of soliton is found to increase with the increase in the

Mach number.

Fig. 4.9 shows the plot of Sagdeev potential for different values of angle of propagation

for positive dust. It is seen that the increase in the value of angle of propagation, the

width and depth of Sagdeev potential increase. Corresponding solitary structures are

79

shown in Fig. 4.10, in which the amplitude of the soliton increases with the increase in

the angle of propagation.

From the numerical parameters used above we can estimate the approximate extent of the

solitary waves. The size of the solitary structure for solar corona parameters can be

shown to have a full width around 𝜉 ≈ 1.715 × 105𝑘𝑚 and maximum potential ψ =

2.58 V with negative dust. Similarly we can estimate the size of the solitary structure with

positive dust where the full width in the vicinity of Io is 𝜉 ≈ 1.715 × 1010𝑘𝑚 and

maximum potential ψ = 43.13 V with positive dust.

4.10 Conclusion

To conclude, in this chapter, we have analyzed the coupled acoustic-kinetic Alfven wave

in self-gravitating dusty plasma and have taken the trapping of electrons and ions into

account separately. Linear dispersion relations are derived for the positively and

negatively charged dust coupled acoustic-kinetic Alfven wave separately. Their limiting

cases are also discussed. We investigate the Jeans instability in a self gravitating dusty

plasma. The Sagdeev potential approach is used for both positively and negatively dust

where gravitational effects are neglected. We have seen that the solitons are formed only

for sub Alfvenic mode in both cases. The corresponding results have been presented

graphically. We have discussed the amplitude of solitary waves for different values of

Mach number and angle of propagation. We feel our results should be applicable in

different space plasma environments which are expected to have negatively and

positively charged dust. Specifically we have applied our results in the solar corona and

Jovian magnetosphere in the vicinity of Io.

80

Figure 4.1: Dispersion relation for ω vs 𝑘 for different values of magnetic field for

negative dust.

81

Figure 4.2: Dispersion relation for ω vs 𝑘 for different values of number density for

negative dust.

82

Figure 4.3: Sagdeev potential 𝑉(𝛹) versus 𝛹 for different values of 𝑀𝐴 when 𝐵0 =

10−4𝑇 and 𝑛𝑑0 = 1012𝑚−3.

83


Fig. 4.3.

84

Figure 4.5: Dispersion relation for ω vs 𝑘 for different values of magnetic field for

positive dust.

85

Figure 4.6: Dispersion relation for ω vs 𝑘 for different values of number density for positive dust.

86

Figure 4.7: Sagdeev potential 𝑉(𝛹) versus 𝛹 for different values of 𝑀𝐴 when θ = 𝜋/3

and 𝑛𝑑0 = 107𝑚−3.

87


Fig. 4.7.

88

Figure 4.9: Sagdeev potential 𝑉(𝛹) versus 𝛹 for different values of θ when 𝑀𝐴 =

0.0000278 and 𝑛𝑑0 = 107𝑚−3.

89


Fig. 4.9.

90

Chapter 5

Summary

In chapter 1, we have introduced the basic concepts which are needed in the investigation

of nonlinear properties of wave propagation in degenerate plasmas. We have given brief

overview of the different types of nonlinear evolution equations and their solutions. We

have introduced the essential qualities of degenerate quantum plasmas. We have also

given a brief introduction to dusty plasmas as these are these are studied in chapter 4. In

this chapter we have given a fairly detailed introduction to the concept of adiabatic

trapping as this nonlinear effect is studied throughout the thesis. We have briefly

described Alfven waves, kinetic Alfven waves and coupled acoustic kinetic Alfven waves

via the use of the two potential theory which itself has been explained. Thus in the

introductory chapter we have attempted to provide a basis for further work carried out in

the thesis. At the end the layout of the thesis is given.

In Chapter 2, we study the formation of solitary structures in an inhomogeneous

degenerate quantum plasma in the presence of a quantizing magnetic field with the effect

of adiabatic trapping of electrons. We have derived the modified Hasegawa Mima

equation for an ion-acoustic wave. Large scale structures have been investigated via the

KdV equation. We have investigated our theoretical results numerically for different

parameters such as magnetic field, density and inverse of inhomogeneity scale length.

These results have been presented graphically showing the formation of solitary

structures. The present study can be useful understanding the propagation characteristics

of nonlinear drift waves in dense astrophysical plasmas such as white dwarf stars where

quantum effects are expected to play an important role.

In Chapter 3, the adiabatic trapping of electrons in a low beta quantum plasma for a

coupled kinetic Alfven-acoustic wave are explored. It has been observed that the

91

framework for obtaining the expression of number density of quantum mechanically

trapped electrons is quite different from its classical counterpart. Most importantly, the

nature of the nonlinearity is found to be different for quantum mechanically trapped

electrons by comparison with its classically trapped counterparts. By using the Sagdeev

potential approach, we have studied the finite amplitude nonlinear structures and also

mentioned the conditions which determine the existence regimes of the solitary

structures. The solitary structures are analyzed by using different plasma parameters such

as obliqueness, magnetic field strength and number density. The present work may be

beneficial in enhancing our understanding of the solitary structures in astrophysical

environments with special reference to the pulsating white dwarfs and also in laboratory

experiments on chirped laser plasma interactions where many astrophysical phenomena

can be mimicked on the laboratory scale.

In Chapter 4, we have investigated coupled acoustic-kinetic Alfven wave in self-

gravitating dusty plasma and have taken the trapping of electrons and ions into account

separately. Linear dispersion relations are derived for the positively and negatively

charged dust coupled acoustic-kinetic Alfven wave separately. The limiting cases of these

dispersion relations are also discussed. We have investigated the effect of Jeans term on

the linear properties of coupled acoustic-kinetic Alfven wave in a self gravitating dusty

plasma. Further nonlinear evolution equations are derived to study the formation of

solitary structures for these waves. The Sagdeev potential approach is used for this

purpose for both positively and negatively charged dust where gravitational effects are

neglected. We have seen that the solitons are formed only for sub Alfvenic mode in both

cases. The corresponding results have been analyzed numerically to the solar corona for

negatively charged dust and in the vicinity of Io for positively charged dust.

.

92

References

1. J. S. Russell. Report on Waves. In Rep. 14th

2. D. J. Korteweg and G. de Vries, Phil. Mag. 39, 422-443 (1895).

Meeting of the British Assoc. for the

Advancement of Science, London (1844).

3. N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15, 240 (1965).

4. Sir Michael Atiyah et al., editors New Developments in the Theory and

Application of Solitons, Royal Society of London, University Press Cambridge, 6.

Carlton House Terrace, London SW145AG, Second edition (1985).

5. R. K. Dodd et al., Solitons and Nonlinear Wave Equations, Academic Press

(1982).

6. T. H. R. Skyrme Proc. Roy. Soc. Lond., A 247, 260 (1958).

7. B. B. Kadomtsev and V. I. Petviashvili, Dokl. Akad. Nauk SSR, 192, 753 (1970).

8. Y. S. Kivshar, Phys. Rev. A 42, 1757 (1990).

9. A. D. Verga and C. F. Fontan, Plasma Phys. and Controlled Fusion 27, 19 (1985).

10. A. M. Lomonosov, P. Hess, and A. P. Mayer, Phys. Rev. Lett. 88, 076104 (2002).

11. K. Brauser, The Korteweg-ge Vries Equation, (University of Osnabruck,

Germany 1, May 2000).

12. D. J. Korteweg and G. de Vries, Phil. Mag. 39, 422 (1895).

13. M. D. Kruska1 and N. J. Zabusky, Annu. Rep. MATTQ-21. Plasma Phys. Lab.,

Princeton Univ., Princeton, New Jersey, (1963).

14. N. J. Zabusky, Nonlinear Partial Differential Equations, (W. Ames. ed.).

Academic Press, New York, (1967).

15. V. E. Zakharov and E. A. Kuznetsov, Sov. Phys. JETP 39, 285-286 (1974).

16. K. Nozaki, Phys. Rev. Lett. 46, 184 (1981).

17. R. H. Enns and G. C. McGuire, Nonlinear Physics with Mathematica for

Scientists and Engineers, Birkhauser Boston, (2001).

18. J. G. Charney, Geophys. Public Vosjones Nurs. Videnschcp. Akad. Oslo 17, 3

(1984).

19. V. D. Larichev and G. M. Reznik, Dokl. Akad. Nauk SSSR 231, 1077 (1976).

20. W. Horton and A. Hasegawa, Chaos 4, 227 (1994).

93

21. N. Iremadze and T. Jangveladze, Dokl. Samin. Inst. Prikl. Mat. IN Vekua 22, 80

(1994).

22. A. Hasegawa, K. Mima, Phys. Fluids 21, 87 (1978).

23. A. Hasegawa, S. G. Maclennan and Y. Kodama, Phys. Fluids 22, 2122 (1979).

24. W. Horton and Y. H. Ichikawa, Chaos and Structures in Nonlinear plasmas,

World scientific, 223-277 (1996).

25. F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, volume 1,

Plenum Press, Second edition, (1984).

26. E. Witt and W. Lotko, Phys. Fluids 26, 2176-2185 (1983).

27. A. Luque, H. Schamel, Physics Reports, 415, 261-359 (2005).

28. I. B. Bernstein, J. M. Greene, and M. D. Kruskal, Phys. Rev. 108, 546 (1957).

29. A. V. Gurevich, Sov. Phys. JETP 26, 575 (1968) (Zh. Eksp. Teor. Fiz., 53, 953

(1967)).

30. L. D. Landau and E. M. Lifshitz, Physical Kinetics, Pergamon, new York, (1981).

31. M. Marklund, B. Eliasson and P. K. Shukla, Phys. Rev. E 76, 067401 (2007).

32. G. Chabrier, F. Douchin and A. Y. Potekhin, J. Phys. Condens Matter 14, 9133

(2002).

33. D. Lai, Rev. Mod. Phys. 73, 629 (2001).

34. P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations

(Springer-Verlag, New York, 1990).

35. G. V. Shpatakovskaya, JETP 102, 466 (2006).

36. P. A. Markowich et al, Semiconductor Equations (Berlin: 1990, Springer).

37. M. Bonitz et al, Phys. Rev. E 49, 5535 (1994).

38. K. H. Becker, K. H. Schoenbach and J. G. Eden, J. Phys. D: Appl. Phys. 39, R55

(2006).

39. F. Hass, L. G. Garcia, J. Goedert and G. Manfredi, Phys. Plasmas 10, 3858

(2003).

40. L. G. Garcia, F. Hass, L. P. L. De Olivier and J. Goedert, Phys. Plasmas 12,

012302 (2005).

41. P. K. Shuk;a, Phys. Lett. A 352, 242 (2006).

42. M. Marklund and G. Brodin, Phys. Rev. Lett. 98, 025001 (2007).

94

43. G. Brodin and M. Marklund, New J. Phys. 9, 277 (2007).

44. G. Brodin and M. Marklund, Phys. Plasmas 14, 112107 (2007).

45. G. Brodin and M. Marklund, Phys. Rev. E 76, 055403 (R) (2007).

46. S. C. Cowley, R. M. Kulsrud and E. J. Valeo, Phys. Fluids 29, 430 (1986).

47. E. P. Wigner, Phys. Rev. 40, 749 (1932).

48. N. L. Tsintsadze, arXiv: 1010.0073v1 (2010).

49. H. M. Van Horn, Science 252, 384 (1991).

50. G. Chabrier, Astron. J. 414, 695 (1993).

51. A. Y. Potekhin, G. Chabrier, D. G. Yakovlev, Contrib. Plasma Phys. 41, 231

(2001).

52. H. Alfven, Nature (London) 150, 405 (1942).

53. S. Lundquist, Phys. Rev. 83, 307 (1951).

54. B. Lahnert, Phys. Rev. 94, 815 (1954).

55. J. M. Wilcox, A. W. De Silva and Cooper, ibid. 4, 1906 (1961).

56. N. F. Cramer, The Physics of Alfven Waves (Wiley-VCH, Berlin, 2001).

57. B. B. Kadomtsev, Plasma Turbulence (Academic, New York, 1965), p. 82.

58. A. Hasegawa and L. Chen, Phys. Rev. Lett. 35, 370 (1975).

59. A. Hasegawa and K. Mima, Phys. Rev. Lett. 37, 690 (1976).

60. M. Y. Yu and P. K. Shukla, Phys. Fluids 21, 1457 (1978).

61. C. Yinhua, L. Wei, and M. Y. Yu, Phys. Rev. E 61, 809 (2000).

62. U. De Angelis, V. Formisano and M. Giordano, J. Plasma Phys. 40, 339 (1988).

63. U. De Angelis, Phys. Scr. 45, 465 (1992).

64. C. K. Goertz, Rev. Geophys. 27, 271 (1989).

65. D. A. Mendis, Astrophys. Space Sci. 176, 163 (1991).

66. T. Nakano, Astrophys, J. 494, 587 (1998).

67. J. Winter, Plasma Phys. Control. Fusion 40, 1201 (1998).

68. E. Zweibel, Phys. Plasmas 6, 1725 (1999).

69. N. N. Rao, P. K. Shukla, M. Y. Yu, Planet Space Sci. 38, 543 (1990).

70. P. K. Shukla, V. P. Silin, Phys. Scr. 45, 508 (1992).

71. A. Barkan, R. L. Merlino and N. D’ Angelo, Phys. Plasmas 2, 2563 (1995).

72. N. Hayashi, Phys. Plasmas 8, 3051 (2001).

95

73. T. G. Northrop, Phys. Scr. 45, 475 (1992).

74. C. K. Goertz, Geophys. Rev. 27, 271 (1989).

75. F. Verheest, Sace Sci. Rev. 77, 267 (1996).

76. D. A. Mendis and M. Rosenberg, Annu. Rev. Astron. Astrophys. 32, 419 (1994).

77. H. Saleem, W. M. Moslem and P. K. Shukla, J. Geophys. Res., 117, A08220

(2012).

78. M. Bonitz, D. Semkat, A. Filinov, V. Golubnychyi, D. Kremp, D. O. Gericke, M.

S. Murillo, V. Filinov, V. Fortov and W. Hoyer, J. Phys. A 36, 5921 (2003).

79. Y. D. Jung, Phys. Plasmas 8, 3842 (2001).

80. M. Opher, L. O. Silva, D. E. Dauger, V. K. Decyk and J. M. Dawson, Phys.

Plasmas 8, 2454 (2001).

81. P. K. Shukla and S. Ali, Phys. Plasmas 12, 114502 (2005).

82. W. Masood, A. Mushtaq and R. Khan, Phys. Plasmas 14, 123702 (2007).

83. P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor Equations,

Springer, Vienna (1990).

84. B. Shokri and A. A. Rukhadze, Phys. Plasmas 6, 4467 (1999).

85. F. Haas, G. Manfredi and M. Feix, Phys. Rev. E 62, 2763 (2000) .

86. G. Manfredi and F. Haas, Phys. Rev. B 64, 075316 (2001).

87. F. Haas, Phys. Plasmas 12, 062117 (2005).

88. G. Manfredi, Fields Inst. Commun 46, 263 (2005).

89. Q. Haque and S. Mahmood, Phys. Plasmas 15, 034501 (2008).

90. P. K. Shukla and B. Eliasson, Phys. Rev. Lett. 96, 245001 (2006).

91. N. S. Erokhin, N. N. Zolnikova, and L. A. Mikhailovskaya, Fiz. Plazmy. 22, 137

(1996).

92. R. Z. Sagdeev, Review of Plasma Physics (Consultants Bureau, New York, 1996),

Vol. 4.

93. C. Hansen, A. B. Reimann, and J. Fajans, Phys. Plasmas 3, 1820 (1996).

94. A. Mushtaq and H. A. Shah, Phys. Plasmas 13, 012303 (2006).

95. H. Siddiqui, H. A. Shah, and N. L. Tsintsadze, J. Fusion Energy 27, 216 (2008).

96. H. A. Shah, M. N. S. Qureshi, and N. L. Tsintsadze, Phys. Plasmas 17, 032312

(2010).

96

97. H. A. Shah, W. Masood , M. N. S. Qureshi, and N. L. Tsintsadze, Phys. Plasmas

18, 102306 (2011).

98. L. D. Landau and E. M. Lifshitz, Statistical Physics I (Pergamon, New York,

1980).

99. A. Y. Potekhin, Astron. Astrophys. 306, 999 (1996).

100. H. A. Shah, M. J. Iqbal, N. Tsintsadze, W. Masood and M. N. S. Qureshi, Phys.

Plasmas 19, 092304 (2012).

101. A. Luque, H. Schamel and R. Fedele, Phys. Lett. A 324, 185 (2004).

102. D. Jovanovic and R. Fedele, Phys. Lett. A 364, 304 (2007).

103. J. Weiland, Collective Modes in Inhomogeneous Plasma (IOP Publishing, Bristol,

2000).

104. Q. Haque, Phys. Plasmas 15, 094502 (2008).

105. T. D. Kaladze, M. Shad and H. A. Shah, Phys. Plasmas 16, 024502 (2009).

106. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, Solitons and Nonlinear

Wave Equations (Academic Press, London, 1982).

107. E. Witt and W. Lotko, Phys. Fluids 26, 2176 (1983).

108. A. A. Mamun, Phys. Rev. E 55, 1852 (1997).

109. D. Koester and G. Chanmugam, Rep. Prog. Phys. 53, 837 (1990).

110. S. S. Ghosh and G. S. Lakhina, Nonlinear Processes Geophys. 11, 219 (2004).

111. K. Stasiewicz, P. Bellan, C. Chaston, C. Kletzing, R. Lysak, J. Maggs, O.

Pokhotelov, C. Seyler, P. Shukla, L. Stenflo, A. Streltsov and J. E. Wahlund,

Space Sci. Rev. 92, 423 (2000).

112. D. J. Wu, Prog. Phys. 30, 101(2010).

113. A. Hasegawa and L. Chen, Phys. Rev. Lett. 35, 370 (1975).

114. A. Hasegawa, and K. Mima, Phys. Rev. Lett. 37, 690 (1976).

115. M. Y. Yu and P. K. Shukla, Phys. Fluids 21, 1457 (1978).

116. H. Kakati and K. S. Goswami, Phys. Plasmas 5, 4229 (1998).

117. C. Yinhua, L. Wei and M. Y. Yu, Phys. Rev. E 61, 809 (2000).

118. H. Azechi, et al, Plasma Phys. Control. Fusion 48, B267 (2006).

119. S.X. Hu and C. H. Keitel, Phys. Rev. Lett. 83, 4709 (1999).

120. M. Marklund and P. K. Shukla, Rev. Mod. Phys. 78, 591 (2006).

97

121. V.M. Malkin, N. J. Fisch and J. S. Wurtele, Phys. Rev. E 75, 026404 (2007).

122. H.J. Lee, et al, Phys. Rev. Lett. 102, 115001 (2009).

123. P.A. Norreys et al, Phys. Plasmas 16, 041002 (2009).

124. T. Guillot, Science 286, 72 (1999).

125. J.J. Fortney et al, Phys. Plasmas 16, 041003 (2009).

126. H.G. Craighead, , Science 290, 1532 (2000).

127. S.L. Shapiro and S. Teukolsky, A Black Holes, White Dwarfs, and Neutron Stars,

The Physics of Compact Objects (2004).

128. G. Chabrier, F. Douchin, A.Y.J. Potekhin, Phys. Condens. Matter 14, 9133

(2002).

129. Y.Y. Lau, et al, Phys. Rev. Lett. 66, 1446 (1991).

130. L.K. Ang, et al, Phys. Plasmas 13, 056701 (2006).

131. P.K. Shukla and B. Eliasson, Phys. Rev. Lett. 100, 036801 (2008).

132. G.V. Shpatakovskaya, Zh. Eksp. Teor. Fiz. 129, 533 (2006).

133. W.L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).

134. D.E. Chang et al, Phys. Rev. Lett. 97, 053002 (2006).

135. M. Marklund, et al, Europhys. Lett. 84, 17006 (2008).

136. P.A. Markovich et al, Semiconductor Equation (Wien: Springer, 1990).

137. E. Abrahams, S.V. Kravchenko, and M.P. Sarachik, Rev. Mod. Phys. 73, 251

(2001).

138. W.C.J. Magnus, W.J. Schoenmaker, Quantum Transport in Submicron Devices

(Berlin: Springer, 2002).

139. N. Crouseilles, P.A. Hervieux and G. Manfredi, Phys. Rev. B 78, 155412 (2008).

140. K.H. Becker, K.H. Schoenbach and J. G. Eden, J. Phys. D 39, R55 (2006).

141. A. Serbeto et al, Phys. Plasmas 15, 013110 (2008).

142. N. Piovella et al, Phys. Rev. Lett. 100, 044801 (2008).

143. J. Lindl, Phys. Plasmas 2, 3933 (1995).

144. M. Tabak et al, Phys. Plasmas 12, 057305 (2005).

145. D. Strickland and G. Mourou, Opt. Commun. 56, 219 (1985).

98

146. M. D. Perry, D. Pennington, B. C. Stuart, G. Tietbohl, J. A. Britten, C. Brown, S.

Herman, B. Golick, M. Kartz, J. Miller, H. T. Powell, M. Vergino and V.

Yanovsky, Opt. Lett. 24, 160 (1999).

147. C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, B. Fell, A. Frackiewicz,

S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson,

A. Kidd, W.J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D.

A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt and B. E.

Wyborn, Nucl. Fusion 44, S239 (2004).

148. C. J. Hooker, J. L. Collier, O. Chekhlov, R. Clarke, E. Divall, K. Ertel, B. Fell, P.

Foster, S. Hancock, A. Langley, D. Neely, J. Smith and B. Wyborn, J. Phys. IV

133, 673 (2006).

149. V. Yanovsky, V. Chvykov, G. Kalinchenko, P. Rousseau, T. Planchon, T.

Matsuoka, A. Maksimchuk, J. Nees, G. Cheriaux, G. Mourou and K. Krushelnick,

Opt. Express 16, 2109 (2008).

150. E. Gerstner, Nature (London) 446, 16 (2007).

151. T. Tajima, Dawson, J. M.: Phys. Rev. Lett. 43, 267 (1979).

152. A. Pukhov and J. Meyer-ter-Vehn, Appl. Phys. B: Lasers Opt. 74, 355 (2002).

153. S. P. D. Mangles, C. D. Murphy, Z. Najmudin, A. G. R. Thomas, J. L. Collier, A.

E. Dangor, E. J. Divall, P. S. Foster, J. G. Gallacher, C. J. Hooker, D. A.

Jaroszynski, A. J. Langley, W. B. Mori, P. A. Norreys, F. S. Tsung, R. Viskup, B.

R. Walton and K. Krushelnick, Nature (London) 431, 535 (2004).

154. C. G. R. Geddes, Cs. Toth, J. Van Tilborg, E. Esarey, C. B. Schroeder, D.

Bruhwiler, C. Nieter, J. Cary and W. P. Leemans, Nature (London) 431, 538

(2004).

155. J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko, E. Lefebvre, J. P.

Rousseau, F. Burgy and V. Malka, Nature (London) 431, 542 (2004).

156. D. D. Ryutov, R. P. Drake, J. O. Kane, E. Liang, B. A. Remington and W. M.

Wood-Vasey, Astrophys. J. 518, 821 (1999).

157. D. D. Ryutov, B. A. Remington, H. F. Robey and R. P. Drake, Phys. Plasmas 8,

1804 (2001).

158. D. D. Ryutov and B. A. Remington, Phys. Plasmas 10, 2629 (2003).

99

159. M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Wood-worth,

E. M. Campbell, M. D. Perry and R. J. Mason, Phys. Plasmas 1, 1626 (1994).

160. R. Kodama et al, Nature (London) 412, 798 (2001).

161. R. A. Smith, Bull. Am. Phys. Soc. 53, 221 (2008).

162. M. Marklund, L. Stenflo, P. K. Shukla and B. Brodin, Phys. Plasmas 12, 072111

(2005).

163. W. Masood and A. Mushtaq, Phys. Plasmas 15, 022306 (2008).

164. M. Ayub, H. A. Shah and M. N. S. Qureshi, Phys. Scr. 84, 045505 (2011).

165. H. A. Shah, W. Masood and Z. Ali, Phys. Plasmas 20, 032301 (2013).

166. B. Eliasson and P. K. Shukla, Europhys. Lett. 97, 15001 (2012).

167. H. R. Prabhakara and V. L. Tanna, Phys. Plasmas 3, 3176 (1996).

168. J. B. Pieper and J. Goree, Phys. Rev. Lett. 77, 3137 (1996).

169. R. L. Merlino, A. Barkan and N. D’ Angelo, Phys. Plasmas 5, 1607 (1998).

170. V. E. Fortov et al., Phys. Plasmas 7, 1374 (2000).

171. D. A. Mendis and M. Rosenberg, Annu. Rev. Astron. Astrophys. 32, 419 (1994).

172. P. K. Shukla, Phys. Plasmas 10, 1619 (2003).

173. P. K. Shukla, Phys. Plasmas 8, 1791 (2001).

174. O. Ishihara, J. Phys. D 40, R121 (2007).

175. G. E. Morfill and A. Ivlev, Rev. Mod. Phys. 81, 1353 (2009).

176. V.E. Zakharov, Sov. Phys. JETP 35, 908 (1972).

177. R. Z. Sagdeev, Review of Plasma Physics 4, (Ed. M. A. Leontovich), Consultants

Bureau, New York, (1996).

178. P. K. Shukla, N. N. Rao, M. Y. Yu and N. L. Tsintsadze, Phys. Reports 138, 1

(1986).

179. P. K. Shukla, N. L. Tsintsadze and L. N. Tsintsadze, Phys. Fluids B 5, 233 (1993).

180. N. L. Tsintsadze, Astrophys. Space Sci. 191, 151 (1992).

181. V. I. Berezhiani, N. L. Tsintsadze and P. K. Shukla, J. Plasma Phys. 48, 139

(1992).

182. N. L. Tsintsadze and P. K. Shukla, Phys. Lett. A 187, 67 (1994).

183. L. N. Tsintsadze and V. I. Berezhiani, Plasma Phys. Rep. 19, 132 (1993).

184. N. L. Tsintsadze, Phys. Scripta 50, 413 (1994).

100

185. L. N. Tsintsadze and V. I. Berezhiani, Phys. Rev. E 53, 4228 (1996).

186. M. Lontano, S. Bulanov, J. Koga and M. Passoni, Phys. Plasmas 9, 2562 (2002).

187. N. L. Tsintsadze, Phys. Plasmas 2, 4462 (1995).

188. N. L. Tsintsadze, K. Nishikawa, T. Tajima and J. T. Mendonca, Phys. Rev. E 60,

7435 (1999).

189. J. Vranjes, H. Saleem and S. Poedts, Planetry and Space Sci. 50, 807 (2002).

190. N. N. Rao, Planet Space Sci. 41, 21 (1993).

191. N. N. Rao, J. Plasma Phys. 53, 317 (1995).

192. F. Verheest, P. K. Shukla, N. N. Rao and P. Meuris, J. Plasma Phys. 58, 163

(1997).

193. F. Verheest, Space Sci. Rev. 77, 267 (1996).

194. T. K. Baluku, M. A. Hellberg and R. L. Mace, Phys. Plasmas 15, 033701 (2008).

195. F. Verheest, Phys. Plasmas 16, 013704 (2009).

196. N. Rubab, N. V. Erkaev, D. Langmayr and H. K. Biernat, Phys. Plasmas 17,

103704 (2010).

197. C. Yinhua, L. Wei and M. Y. Yu, Phys. Rev. E 61, 809 (2000).

198. L. A. Pipes and L. R. Harvill, Applied Mathematics for Engineers and Physicists,

McGraw-Hill, Japan, (1971).

199. M. Horanyi, Phys. Plasmas 7, 3847 (2000).

Contentsprr.hec.gov.pk/jspui/bitstream/123456789/9518/1/Sabeen_Arshad_Ph… · plasma and introduce...

Documents

Transcript of Contentsprr.hec.gov.pk/jspui/bitstream/123456789/9518/1/Sabeen_Arshad_Ph… · plasma and introduce...