Jyotirmoy Goswami Department of Physics, Jadavpur University, Kolkata-700032

ISCA YS Lecture

Highlights of the talkIntroduction MotivationIon acoustic wavee-p-i PlasmaHydrodynamic ModelLinear Dispersion CharacteristicsKorteweg de Vries equation & Solitary StructuresResultsReferences

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IntroductionProperties:Quasineutral collection of charged and neutral particalsCollective behaviourLong-range electromagnetic intractionsCriteria for ionized gas to be a plasma: Again,

Occurrences: Natural Plasmas: Lightening, aurora borealis etc. Man-made Plasmas: Fluorescent lamp. *

Dense Space Plasma*

MotivationNonlinear propagation of intense electrostatic waves in electron-positron plasmas has received large amount of theoretical interest mainly because such plasma are naturally produced under certain astrophysical conditions. Since electron-positron are thought to have been present in the early Universe ,plasma possesses are expected to have played an important role in the early history, as well as the evolution of the Universe. But to the best of our knowledge no investigation has been made of the solitary structure of electrostatic waves in electron-positron-ion plasmas by formulating and solving KdV equation. The present work is to investigate the linear and nonlinear properties of plasma consisting of electrons , positrons and ions.*

Ion acoustic waveIn ,Plasma Physics, an ion acoustic wave is one type of longitudinal oscillation of the ions and electrons in a plasma, much like acoustic waves travelling in natural gas. However, because the waves propagates through positively charged ions, ion acoustic waves can interact with their electromagnetic fields, as well as simple collisions.For a single ion species plasma and in the long wavelength limit, the waves are dispersionless.*

Electron-positron-ion PlasmaElectron-positron plasmas are believed to exist in early universe, in active galactic nuclei and in pulsar magnetosphere. Most of the astrophysical plasmas usually contains ion as well in addition to electrons and positrons. *

Hydrodynamic ModelHydrodynamic is the study of motion of liquids, and in particular ,water.The basis of computational hydrodynamic model is the set of equations the motion of fluids; the Navier-Stokes equations.When the density of plasma is high it becomes an impossible task to follow the trajectory of each particle and to predict plasma behavior. Fortunately when collisions between plasma particles become very frequent each species can be treated as fluid described by local density, temperature, velocity. *

BASIC EQUATIONS Continuity eqn.

Momentum eqn.

Poission`s eqn.

Pressure eqn.

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The positron density is given by,

And electron density is given by

Where

And

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Using the following normalization conditions

We get the normalized equations

Linear Dispersion CharacteristicsThe perturbation expansion for the field quantities :

Liner dispersion relation is given by:

Therefore here we get fast mode and slow mode.*

Linear Dispersion CharacteristicsFig. 1(a): Dispersion curves for different values of .

Fig. 1(b): Dispersion curves for different values v0.*

Linear Dispersion CharacteristicsFig. 1(c): Dispersion curves for different values of

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Linear Dispersion CharacteristicsResults:The wave frequency is shown to increase with the increase in the values of k, One mode vanishes if the electrons are considered inertialess.This proves the fact that as more number of particles come in the picture, the more the system behaves like a fluid.We can see from the equation that there are two modes of frequency fast and slow mode.

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Korteweg de Vries (KdV) EquationFollowing the standard reductive perturbation technique we use the usual stretching of the space and time variables: and solving for the lowest order equation with the boundary condition that

as , the following solutions are obtained:

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Going for the next higher order terms in and following the usual method we obtain the desired KdV equation:

Where

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KdV Equation Solution To find the solution of KdV equation we transform the the independent variables and into one variable = - M where M is the normalized constant speed of the wave frame. Applying the boundary conditions that as ; the possible stationary solution of Eq. (20) is obtained as:

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Solitary wave profileFig. 2(a): Solitary profiles for different values of v0.

Fig. 2(b): Solitary wave profile for different values of *

Solitary wave profileFig. 2(c): Solitary profiles for different values of M.

Fig. 2(d): Solitary wave profile for different values of *

Solitary wave profileFig. 2(e): Solitary profiles for different values of .

Fig. 2(f): Solitary wave profile for different values of *

Solitary Profile CharacteristicsResults: The amplitude of electron-acoustic solitary structure increases with increase in M and , but increasing is broad for M and steep for . On the other hand the soliton decreases with increase in v0 , , and . Here the decreasing is steep for , , .

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References[1] F.Hass, L.G.Garcia, J.Goedert, G.Manfredi, Physics of Plasmas, 10, 3858(2003) [2] B.Ghosh, S.Chandra and S.N.Paul., Physics of Plasma, 18, 012106 (2011) [3] B.Ghosh, S.Chandra and S.N.Paul., Pramana-journal of Physics, 78, 779-790 (2012) [4] S.Chandra, S.N Paul and B.Ghosh, Indian Journal of Pure and Appl. Phys, 50, 314(2012)[5] M. Akbari-Moghanjoughi, Astrophysics and Space Sciences 332, 187 (2011) [6] Chandrasekhar, S.: An Introduction to the Study of Stellar Structure. The University of Chicago Press, Chicago, (1939). p. 360*

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