Nonlinear electrostatic shock waves in inhomogeneous plasmaswith nonthermal electrons

W. Masood,1,2 H. Rizvi,1,a) and H. Hasnain3,4

1Theoretical Plasma Physics Division (TPPD), PINSTECH, P. O. Nilore, Islamabad, Pakistan2National Center for Physics (NCP), Islamabad 44000, Pakistan3NILOP, P. O. Nilore, Islamabad, Pakistan4PIEAS, P. O. Nilore, Islamabad, Pakistan

(Received 27 September 2011; accepted 23 January 2012; published online 29 March 2012)

Density inhomogeneity driven linear and nonlinear ion drift waves are investigated in a plasma

consisting of heavy ions and non-thermal electrons. The dissipation is introduced in the system by

the ion-neutral collision frequency. The nonlinear Korteweg de Vries Burgers (KdVB) and Burgers

like equations are derived in the small amplitude limit, and the solution is obtained using the

tangent hyperbolic method. It is found that the system under consideration admits rarefactive shock

structures. It is observed that the ion-neutral collision frequency, nonthermal electron population,

inverse density inhomogeneity scalelength, and the ambient magnetic field affect the propagation

characteristics of the drift shock waves. The present study may be applicable in regions of space

where nonthermal electrons and heavy ions have been observed. VC 2012 American Institute ofPhysics. [http://dx.doi.org/10.1063/1.3688869]

I. INTRODUCTION

The nonuniform magnetized plasmas are known to sup-

port a variety of drift oscillations. The interest in these insta-

bilities arose owing to their importance in the anomalous

transport of a plasma transverse to a magnetic field. The

electrostatic drift waves involve a two-dimensional ion

motion in a plane perpendicular to the external magnetic

field B0z, along with the Boltzmann distributed inertialess

electrons. Thus, the low-frequency (by comparison with the

ion gyrofrequency Xci ¼ eB0=mic) waves having parallel

(meaning along B0) phase velocity much smaller than the

electron thermal velocity arise due to a balance between the

time derivative density fluctuations and the E� B0 convec-

tion of the unperturbed density. The dispersion of the drift

waves is provided by the ion polarization drift.

There has been a great deal of interest in the study of

shock wave-like and soliton-like solutions of drift waves in

electron-ion plasmas. Many years ago, Buchelnikova1 showed

that under the condition of Q-machine, the development of the

drift instabilities results in regular drift structures of a saw-

shaped form like cnoidal waves, i.e., a train of solitons. This

led to the development of a theory to explain these nonlinear

structures. The possibility of the existence of regular struc-

tures was originally discussed theoretically by Tasso2 and

Oraevskii.3 The ideology was also carried out in the papers by

Refs. 4 and 5. It followed, from the above mentioned papers,

that the necessary condition for the existence of such struc-

tures is the presence of an electron temperature gradient.

However, in the Buchelnikova1 experiment conditions, the

temperature gradient disappeared in the region of localization

of the drift structures. The possibility of the existence of such

structures in the absence of the temperature gradient followed

also from the experiments of Hendel et al.6 Lakhin et al.7

showed that for the correct analysis of the problem of drift sol-

itons, the vector nonlinearity must be taken into account. The

authors showed the existence of the drift solitons in the ab-

sence of the temperature gradient. It was also shown that for

the existence of the drift solitons, a sufficiently high level of

the drift activity is necessary, which is characterized by the

qualitative relation ~n=n0 � 1=k?Ln, where n0 and ~n are the

equilibrium and perturbed parts of the plasma number density,

k? is the characteristic transverse wave number of the soliton,

and Ln is the characteristic scale of inhomogeneity of the

plasma number density.

It is a well established idea that in a non-linear dispersive

media, shock waves are formed owing to the interplay

between the non-linearity (causing wave steepening) and dis-

sipation (e.g., caused by viscosity, collisions, wave particle

interaction, etc.). When the wave breaking due to non-

linearity is balanced by the combined effect of dispersion and

dissipation, a monotonic or oscillatory dispersive shock wave

is generated in a plasma.8 A number of papers have explored

the effects of various dissipative processes on the propagation

of ion inertia driven waves both in homogeneous and inhomo-

geneous plasmas. Ostrikov et al.9 studied the current driven

ion acoustic instability in a collisional dusty plasma and found

that the threshold for the excitation of the dust ion-acoustic

waves could become higher on account of the large dissipa-

tion rate induced by the dust particles. The effects of the elec-

trons, ions, and neutrals as well as the dust charge fluctuation

on the ion acoustic waves have also been investigated.10,11

The nonlinear propagation of drift waves in multicomponent

plasmas has also been studied in recent years with regard to

space, astrophysical, and laboratory plasmas.12–15

Over the last two decades, the observations of space

plasmas have indicated the ubiquitous presence of ion and

electron populations which are far away from their respec-

tive thermodynamic equilibria.16–19 Cairns et al.20 showed

that the presence of a nonthermal distribution of electronsa)Electronic mail: hdrrizvi@gmail.com. Tel.: þ92-321-4244674.

1070-664X/2012/19(3)/032314/5/$30.00 VC 2012 American Institute of Physics19, 032314-1

PHYSICS OF PLASMAS 19, 032314 (2012)

could change the nature of ion acoustic solitary structures to

allow for the existence of structures observed by the Freja

and Viking satellites.21,22 Nonthermal distributions are com-

mon feature of the auroral zone.23,24

In this paper, we investigate the non-linear propagation

of drift waves in a nonuniform magnetized plasma compris-

ing of oxygen ions and electrons (both thermal and nonther-

mal) in a one dimensional (1-D) planar geometry. In this

regard, we derive the one dimensional nonlinear Korteweg

deVries-Burgers (KdVB) and Burgers like equations in the

small amplitude limit. The paper is organized in the follow-

ing manner. In Sec. II, the basic set of equations for the sys-

tem under consideration are given. In Secs. III, we derive the

linear dispersion relation for the system under consideration.

In Secs. IV and V, we derive the 1-D nonlinear equations for

the KdVB and Burgers like equations and obtain the solution

using the tangent hyperbolic method. In Sec. VI, we show

the results obtained numerically and discuss them. Finally, in

Sec. V, we recapitulate the main findings of the paper.

II. MATHEMATICAL MODEL

Consider a plasma consisting of heavy ions (oxygen in

our case) and electrons of mass Mþ and me, respectively,

where Mþ ¼ ZmiðMþ ¼ 16mp in our caseÞ. Such a considera-

tion has been motivated by observations of heavy ions and

nonthermal electrons in different regions of space. The gov-

erning equations for the system under consideration are

@Nþ@tþ $ � ðNþvþÞ ¼ 0: (1)

The equation of motion for heavy ions is

MþNþdvþdt¼ qNþðEþ

1

cvþ � B0Þ �MþNþ�nþvþ; (2)

where we have assumed cold ions and Nþ; vþ;Mþ, q, and �nþdenote the number density, fluid velocity, ion mass, charge,

and neutral-ion collisions, respectively. The system of equa-

tions is closed with the help of quasineutrality condition.

~ne ’ ~Nþ; (3)

where ~Nþ is the perturbed ion number density. Since the ox-

ygen atoms are singly charged, therefore, Z¼ 1 is assumed

here. The perpendicular velocity component of ion becomes

vþ? ’ �c

B0

ru� z� c

B0Xþ

d

dtr?u� c

B0

�nþXþr?u

¼ vE þ vp þ vc; (4)

where vE ¼ �c=B0ru � z; vp ¼ �c=B0Xþ ddtr?u; vc

¼ �c�þðr?uÞ=B0Xþ, and ddt ¼ ð@=@t þ vE:rÞ. We use

drift approximation @=@tj j � Xþ . Ignoring the parallel

motion, the ion continuity equation can be written as

d

dtNþ þ

c

B0

z�ru � rNþ þcNþ

B0Xþ

d

dtr2?u

þ c

B0

1

XþNþ�nþ r2

?u ¼ 0; (5)

where Nþ is the total ion number density that contains both

the unperturbed and the perturbed contributions. The non-

thermal electron density is given by20

~ne ¼ ne0ð1� bUþ bU2Þ exp Uð Þ; (6)

where ~ne is the perturbed number density of electrons and

the quantity with subscript zero represents its equilibrium

counterpart. b is given by 4C=ð1þ 3CÞ, where C is a param-

eter that determines the population of the nonthermal elec-

trons. Using Eq. (6), the quasineutrality condition in the

normalized form becomes

~Nþ ¼ ð1� bÞUþ 1

2U2; (7)

where

U ¼ eu=Te:

III. LINEAR ANALYSIS

Assuming that the perturbations are proportional to

exp½ik?y� xt� in the linear case, the algebraic manipulation

of Eqs. (5) and (7) yields the following dispersion relation

for the dispersive ion drift waves.

x ¼ x�þ � i�nþq2þk2?

ð1� bþ q2þk2?Þ; (8)

qþ ¼ cs=Xþ is the ion Larmor radius that moves with the

ion acoustic speed cs;x�þ ¼ v�þky is the drift frequency, v�þ is

the diamagnetic drift velocity and is given by cTe=eB0ð Þjni,

where jni ¼ �ð1=Nþ0Þ dNþ0=dxð Þ, where Nþ0 is the back-

ground density inhomogeneity.

IV. NONLINEAR ANALYSIS I (KDVB LIKE EQUATION)

In order to obtain the nonlinear structure, we proceed as

follows. Using Eq. (7), the ion continuity equation. (i.e.,

Eq. (5)) becomes

(1 - b)@

@tUþ U

@

@tUþ v�þð1� bÞU @U

@yþ v�þ

@U@y

� q2þ@

@t

@2U@y2� q2

þ�nþ@2U@y2¼ 0: (9)

To find the stationary solution, we transform by introducing

the variable n ¼ k y� utð Þ and obtain the following equation

for the nonlinear dispersive ion drift waves

� u�v�þ

ð1� bÞ

� �@U@nþ v�þ �

u

1� b

� �U@U@n

þuq2þk2

ð1� bÞ@3U

@n3�

q2þ�nþk

ð1� bÞ@2U

@n2¼ 0; (10)

where second term second, third and fourth terms in the

above equation represent the nonlinear, dispersive, and dissi-

pative terms, respectively. Using the tangent hyperbolic

method,25,26 Eq. (10) in the comoving frame of the nonlinear

structure admits the following shock type solution

032314-2 Masood, Rizvi, and Hasnain Phys. Plasmas 19, 032314 (2012)

U n; sð Þ ¼ 9

25

q2þ�

2nþ

u u� v�þð1� bÞð Þ �6

25

q2þ�

2nþ

u u� v�þð1� bÞð Þ tanhðnÞ

� 3

25

q2þ�

2nþ

u u� v�þð1� bÞð Þ tanh2ðnÞ; (11)

where the nonlinear velocity u is

u ¼v�þ

2ð1� bÞ 1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 24

25ð1� bÞ q

2þ�

2nþ

v�þð Þ2

s" #:

The above equation shows that the nonlinear dispersive drift

wave potential depends upon the temperatures of electrons

and ions, magnetic field strength, density inhomogeneity,

and the propagation velocity of the nonlinear structure.

V. NONLINEAR ANALYSIS II (BURGERS LIKEEQUATION)

In order to obtain the nonlinear structure in a dispersion-

less plasma, we proceed as follows. Using Eq. (7), the ion

continuity (i.e., Eq. (5)) in the absence of the polarization

drift (note that the polarization drift gives us the wave dis-

persion) reads as follows:

ð1� bÞ @@t

Uþ U@

@tUþ v�þð1� bÞU @U

@yþ v�þ

@U@y

� q2þ�nþ

@2U@y2¼ 0: (12)

To find the stationary solution, we transform by introducing

the variable n ¼ k y� utð Þ and obtain the following equation

for the nonlinear ion drift shock waves

� u�v�þð1�bÞ

� �@U@nþ v�þ �

u

1�b

� �U@U@n�

q2þ�nþk

ð1�bÞ@2U

@n2¼ 0:

(13)

Using the tangent hyperbolic method,25,26 Eq. (10) in the

comoving frame of the nonlinear structure admits the follow-

ing shock type solution

Uðn; sÞ ¼2q2þ�nþ

1� bð Þv�þ � u1� tanh nð Þ½ �; (14)

where the nonlinear velocity u is

u ¼2q2þ�nþ þ v�þ1� bð Þ :

FIG. 1. Variation of the electrostatic drift potential U with decreasing

Oþ � neutral collision frequency �nþ i.e., �nþ ¼ 0:2 Hz (small dashed),

�nþ ¼ 0:1 Hz (long dashed), and �nþ ¼ 0:05 Hz (thick). Other parameters

are Te ¼ 1000 K;C ¼ 0:3, and B0 ¼ 0:3 G.

FIG. 2. Variation of the electrostatic drift potential U and perturbed number density ~Nþ with nonthermal electron population C i.e., C ¼ 0:1 (small dashed),

C ¼ 0:2 (long dashed), and C ¼ 0:3 (thick). Other parameters are Te ¼ 1000 K; �nþ ¼ 0:1 Hz, and B0 ¼ 0:3 G.

FIG. 3. Variation of the electrostatic shock potential with magnetic field B0

i.e., B0 ¼ 0:1 (small dashed), B0 ¼ 0:2 (long dashed), and B0 ¼ 0:3 (thick).

Other parameters are Te ¼ 1000 K; �nþ ¼ 0:1 Hz, and C ¼ 0:3:

032314-3 Masood, Rizvi, and Hasnain Phys. Plasmas 19, 032314 (2012)

Note that the advantage of using tangent hyperbolic method

is that it gives us the velocity of the nonlinear structure in

terms of the plasma parameters of the system under

consideration.

VI. RESULTS

In this section, we numerically explore the dependence

of one-dimensional nonlinear drift shock waves on various

plasma parameters. For illustration, we have chosen the pa-

rameters that are typically found in the ionosphere.27,28 The

value of the ion Larmor radius that moves with the ion

acoustic speed, qþ, for the chosen parameters turns out to be

400 cm. Using the condition k? < 1=qþ and jni < k?, the

values of k? and jni have been calculated. It is observed that

the rarefactive shock potential is obtained for the system

under consideration. Fig. 1 investigates the effect of increas-

ing collision frequency of oxygen ions with neutrals affects

the drift shock wave propagation. It is observed that the

increase in ion-neutral collision frequency enhances, in terms

of magnitude, the drift shock wave potential owing to the

fact that the increase in collision frequency increases the dis-

sipation in the system. Fig. 2 explores the effect of the

increasing nonthermal electron population on the nonlinear

drift shock wave potential. It is found that the increase in the

nonthermal electron population mitigates the shock strength

in terms of magnitude. Note that we have also plotted a

graph besides the rarefactive shock potential to show the cor-

responding effect on the number density which shows that

the increasing nonthermal population decreases the depletion

of heavy (Oþ in our case) ions.

Fig. 3 explores the effect of the increasing magnetic

field on the propagation characteristics of drift shock poten-

tial. It is observed that the increasing magnetic field brings

about an increase (in terms of magnitude) in the drift shock

wave potential. Fig. 4 explores the effect of inverse scale-

length of density inhomogeneity, jni, on the propagation

characteristics of the drift shock wave potential. It is

observed that, in terms of magnitude, the increasing jni

decreases the shock potential.

In Fig. 5, we investigate the effect of increasing collision

frequency, electron thermal population, and magnetic field on

the coefficients of nonlinearity, dispersion, and dissipation. It

is observed that the increase in collision frequency and non-

thermal population enhances the nonlinearity and dissipation

coefficients, whereas they decrease the dispersion. On the

other hand, the increase in magnetic field decreases the nonli-

nearity and dissipation coefficients, whereas they enhance the

dispersion. Fig. 5 enables us to understand the overall behav-

ior of the drift shock potential by varying different plasma

parameters.

Finally, Fig. 6 explores the effect of the increasing non-

thermal electron population on the nonlinear drift shock

wave potential. It is found that the increase in the nonthermal

electron population enervates the Burger shock strength in

terms of magnitude; however, it should be noted that the

Burgers shock is stronger (two orders of magnitude) than the

KdVB shock due to the fact that the competing dispersive

term is absent in the Burgers shock. Furthermore, it is noted

that the Burgers equation does not get affected appreciably

by the observed values of other plasma parameters in the

ionosphere. However, if the magnetic field gets increased or

the ion-neutral collisional frequency gets decreased in some

other region of space, the effect will then be significant on

the Burgers shock.

It is worth mentioning here that since the variable n is

normalized by qþ, therefore, the width of the shock turns out

to be of the order of one tenth of a kilometer, whereas the

width of the F2 layer is 80 – 100 km. Hence, we arrive at the

conclusion that the shock formation in the F2 layer is

possible.

FIG. 4. Variation of the electrostatic shock potential with inverse density

inhomogeneity scalelength, jni. Other parameters are Te ¼ 1000 K,

�nþ ¼ 0:1 Hz, and C ¼ 0:3.

FIG. 5. Variation of the coefficients of nonlinear (solid), dispersive (thin dashed), and dissipative (thick dashed) terms corresponding to Oþ � neutral collision

frequency �nþ (blue, 1), nonthermal electron population C (red, 2), and magnetic field B0 (green, 3).

032314-4 Masood, Rizvi, and Hasnain Phys. Plasmas 19, 032314 (2012)

VII. CONCLUSION

In this paper, the linear and nonlinear propagation of

small amplitude drift shock waves are investigated in a

plasma consisting of heavy ions (oxygen in our Case) and

electrons (both thermal and nonthermal). In this regard, non-

linear KdVB and Burgers like equations are derived in an in-

homogeneous plasma using the drift approximation, and the

solutions are obtained using the tangent hyperbolic method.

It is found that the system under consideration admits rare-

factive shocks. The effects of varying ion-neutral collision

frequency, nonthermal electron thermal population, inverse

density inhomogeneity scalelength, and the magnetic field

strength on the nonlinear drift shock potential are numeri-

cally illustrated in Figs. 1–5. Moreover, the drift Burgers

shock is also investigated, and it is found that the nonthermal

electron population affects the Burgers shock, and it is also

observed that the Burgers shock is stronger by comparison

with the drift KdVB shock. It is observed that the increase in

ion-neutral collision frequency and magnetic field strength

increases the rarefactive drift shock potential, whereas the

increasing nonthermal population mitigates it. The shock

width turns out to be of the order of one tenth of a km, and it

is shown that the shock formation is possible in the F2 layer.

The results presented here may have relevance in regions of

space where the satellite observations have indicated the

presence of nonthermal electrons and heavy ions.

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FIG. 6. Variation of the electrostatic drift potential U for dispersionless

plasma with nonthermal electron population C i.e., C ¼ 0:3 (small dashed),

C ¼ 0:325 (long dashed), and C ¼ 0:35 (thick). Other parameters are

Te ¼ 1000 K; �nþ ¼ 0:1 Hz, and B0 ¼ 0:3 G.

032314-5 Masood, Rizvi, and Hasnain Phys. Plasmas 19, 032314 (2012)