Non-linear electrostatic waves in pair-ion plasmas - National Centre ...

Nonlinear electrostatic structures inunmagnetized pair-ion (fullerene) plasmasS. MahmoodTheoretical Plasma Physics Division, PINSTECH IslamabadCollaborators:H. SaleemNational Center for Physics, QAU Campus IslamabadandW. Masood, H. Ur-Rehman and N. ImtiazTheoretical Plasma Physics Division, PINSTECH Islamabad2 nd International Scientific Spring Conference, March 01-06 2010, NCP Islamabad1

Outline of presentation1. Introduction of soliton2. Brief introduction of pair-ion (fullerene)plasmas3. Model for electrostatic waves in pair-ionplasmas4. KdV solitons5. Double layer structures in pair-ion plasmas6. Kadomstev-Petviashvili-Burgers (KPB)equation is obtained for solitons andshocks in PI plasmas7. Conclusion2

1. Soliton‣ Soliton is a nonlinear structure (bell shaped) which conserves itsshape and interacts with surroundings or other solitons as anindependent particle.‣ Solitons are robust against perturbations.‣ Soliton was first noticed in the month of August 1834 by the Scottishengineer John Scott Russell (1808-1882), while conductingexperiments to determine the most efficient design for canal boatsalong the Union canal linking Edinburg with Glasgow.‣ In 1885, Korteweg and de Vries derived a model equation (knownas KdV equation) for the dynamics of nonlinear wave as follows,3∂u∂u∂ u+ au + b3∂t∂x∂x= 0Here a and b are non zero constants, which depends on fluid properties.The KdV equation posses a soliton (or solitray wave) solution.3

Soliton (contd.)Fig.1.2: Soliton solutions of stationary KdV equation. The speed of solitondepends on its amplitude.4

Soliton in Plasmas (contd.)Linear waves in v rplasmas are studied by assuming a harmonic wavei( k .−ωt)solution ~ ein the small amplitude limit.When the wave amplitude is sufficiently large the nonlinearities cannotbe ignored.Nonlinearities in plasmas enter through‣ Harmonic generation (involving fluid convection),‣ Nonlinear Lorentz force,‣ Ponderomotive force, trapping of the particles in the wavepotential etc.Importance in PlasmasThe study of soliton is important to understand the particle orenergy transport mechanisms in plasmas.5

Nonlinear Plasma DynamicsWhen wave amplitude grows to such high values that linear perturbationtheory cannot be applied to describe the interactions in plasmas. Theninstead of applying Fourier analysis, we transform the original set ofnonlinear equations, that describe the wave dynamics, into lowest ordernonlinear equations (such as KdV equation, Burgers equation and NLSequation) whose properties are well known.For arbitrary amplitude waves, one has to solve the nonlinear differentialequations with full nonlinear glory and difficulties. This method haslimitations and cannot be used in general and applicable to only somespecific problems.Reductive perturbation method is used when one cannot deal theequations with full nonlinearity and weak nonlinearity are assumed in thesystem. The stretched variables in space and time are defined and slowtime variations are induced by the nonlinearity of the system. For example,the KdV equation is obtained for low amplitude nonlinear wave with6appropriate scaling of space and time variables.

2. Brief Introduction of pair-ion Plasmas‣In usual electron-ion plasmas, asymmetry in collective phenomenonoccur due to large difference of mass between electrons and ions.‣However, pair plasmas (e-p) consists of positively and negativelycharged particles of same mass to keep the space, time symmetrybecause the mobility of equal mass particles is same in theelectromagnetic field.‣Pair plasma consists of positrons and electrons have been producedin laboratory experiments. But the identification of collective modes isvery difficult because the annihilation time is short compared with theplasma period.‣Therefore, attention is concentrated on the stable generation of apair-ion plasma in laboratory consisting of positive and negative ionswith an equal mass for collective modes identification.‣It has been found that fullerene (C 60) can be adopted as an ionsource for pair-ion plasma production, based on the fact thatinteraction of electrons with fullerenes lead to the production of bothnegative and positive ions.7

A fullerene is any molecule composed entirely of carbon, in the form of ahollow sphere, ellipsoid, or tube. Spherical fullerenes are also calledbuckyballs, and cylindrical ones are called carbon nano-tubes or buckytubes.The discovery of fullerenes greatly expanded the number of known carbonallotropes, which until recently were limited to graphite, diamond, and amorphous8carbon such as soot and charcoal.

Production of pair-ion (fullerene) PlasmasChamberLength=260 cm,diameter=15.7 cm,B=0.3 T (solenoidcoils), gas pressure=2x10 -4 PaE e=0 - 150 eV(electron beamenergy)The production process of pair-ion plasma is divided into three regions:Region I: Consists of electron beam region and fullerene ion production region.+Positive ions C 60 are produced by electron impact ionization and low energyelectrons are produced in connection with this process.−Negative ions C 60 are produced by attachment of these low energy electronsproduced during the charging process of positive ions.C 260− + −+ e → C60+ e,+−242( C60)= 25×10 cm− −C60 + e → C ,−−24260σ ( C60)= 100 × 10 cmσat 100 eVat 5eV9

Production of pair-ion (fullerene) Plasmas (contd.)Region II: Electrons and ions are rapidly separated by a magnetic filtering+−effect and only and are expected to exist in midmost of the cylinder.C 60C 60Charge particle gyroradii ~(mass) 1/2 /charge+C 60−C 60(for same thermal velocity in theperpendicular direction of B)Region III: Electron free pair-ion ( , ) plasma generated is attained hereand plasma parameters are measured by Langmuir probes.The typical values of pair-ion plasma density ~ 2x10 8 cm -3 at E e=100eVThe temperature of the positively and negatively charged fullerene ions lies inthe range 0.3-0.5 eV.[W. Oohara, Y. Kuwabara and R. Hatakeyama, PRE 75, 056403 (2007)]The temperature of positive ions is found to be different from negativefullerene ions because the charging processes are quite different for boththe plasma species.10

Electrostatic Waves in pair-ion PlasmasW. Ohara, D. Date and H. Hatakeyama [PRL 95,175003 (2005)], firstly reportedthe electrostatic waves in paired fullerene-ion plasmas.Three types of electrostatic waves can propagate parallel to the magnetic fieldin pair-ion plasmas i.e.,2 2 2 2 γT1. Ion acoustic wave (IAW) ω = C sk , C s= (acoustic speed)m2. Intermediate frequency wave (IFW)2 2 2 23. Ion plasma wave (IPW)ω = C + ωThe IFW has the features that group velocity is negative but the phasevelocity is positive i.e., the mode resembles as a backward wave.11sk2 p

3. Nonlinear Electrostatic Waves in pair-ion PlasmasModel and nonlinear set of equationsConsider homogeneous, unmagnetized fullerene (C 60) plasma consists of positivelyand negatively charged ions.The positive and negative ions are considered to be isothermally heated but havedifferent temperatures.Using two fluid theory, the Continuity equations of positive and negativeions are given as∂n∂t+∂+ ( nv+ + ) =∂x0(1)∂n∂t−∂+ ( nv− −)=∂x0(2)12

Nonlinear Electrostatic Waves in Pair-ion Plasmas (contd.)Momentum equations for positively and negatively charged ions are describedas,(where φ is electrostatic potential) i.e.,2∂ φPoisson equation gives, = 4π e n− −2 +∂xEquilibrium is defined as,∂ v+ ∂v+e ∂φT+1 ∂n++ v+= − −∂t∂xm ∂xm n ∂x∂v∂t∂v+∂φ−∂x− − e− 1+ v−=∂xm−m−n−( n )( )n = n = n say0+ 0−0T++∂n−∂x∂φE = −∂x(3)(4)(5)(6)Dispersion relation⎡⎢⎢⎣1 1⎤1+ ⎥ − =− + ⎦2 2 2 2 2 2 2( ω −VTk ) ( ω −VTk ) ⎥ ωp0(7)whereVTα⎛Tα⎞= ⎜ ⎟⎝m⎠12andωp12 2⎛4πne ⎞0= ⎜ ⎟⎝ m ⎠m +=m -(=m say)13

Nonlinear Electrostatic Waves in pair-ion Plasmas (contd.)Linear analysis of the modes in pair-ion plasmasCase (i)If T +=T -(=T say), then Eq.(7) givesω= 2ω+ V k2 2 2 2p Twhich is the usual Langmuir waves in pair-ion unmagnetized plasmas.(8)Case (ii) If T +≠T -, then Eq.(7) gives quartic equation as followswhere( ) ( )ω − V k + V k + 2ω ω + ω V + V k + k V V = 04 2 2 2 2 2 2 2 2 2 2 4 2 2T T p p T T T Tω+ − + − + −1= ( V k T+ VT k + 2ωp)± Δ+ −2{ }2 2 2 2 2 22( 2ω) 4 ω ( ){ }Δ= V k + V k + − V + V k + k V V2 2 2 2 2 2 2 2 2 4 2 2T T p p T T T T+ − + − + −(9)(10)here (+ve) and (-ve) sign gives higher and lower frequency Langmuir wavesin pair-ion plasmas.14

Nonlinear Electrostatic Waves in pair-ion Plasmas (contd.)Under the approximation of small k (long wave length), the +ve root i.e., the fastLangmuir wave gives the relation,Under the approximation of small k (long wave length) , the -ve root i.e., theslow Langmuir wave gives the relation,which is low frequency ion acoustic mode in pair-ion plasmas.15

Nonlinear Electrostatic Waves in Pair-ion Plasmas (contd.)Nonlinear set of equations in normalized form are written as follows:∂n∂t+∂+ ( nv+ + ) =∂x0∂n∂t−∂+ ( nv− −)=∂x0∂v⎛ ∂ ⎞ ∂Φ 1 ∂n+ ⎜v+ ⎟v+= − −∂t ⎝ ∂x ⎠ ∂x n ∂x+ +∂v−⎛ ∂ ⎞ ∂Φ μ ∂n+ ⎜v−⎟v−= −∂t ⎝ ∂x ⎠ ∂x n ∂x2∂Φ( n− n2 = −+ )∂x⎛eφ⎞T−where Φ=⎜ ⎟⎝Tand μ = have been defined.+ ⎠TNormalization of time, space and velocityhas been done byVT++−⎛T+⎞= ⎜ ⎟⎝m⎠+12λD−+⎛ T ⎞+= ⎜ 2 ⎟⎝4πne0 ⎠12ωp12 2⎛4πne ⎞0= ⎜ ⎟⎝ m ⎠16

Nonlinear Electrostatic Waves in pair-ion Plasmas (contd.)Using reductive perturbation method, the perturbed quantities can beexpanded in the power series of ε as follows:( 1) 2 ( 2)n = 1 + εn + ε n + .....j j j( 1) 2 ( 2)v = εv + ε v + .....j j j( 1) 2 ( 2)Φ= εΦ + ε Φ + .....The stretching of independent variables are introduced in the following fashion:1ξ = ε 2( x −λt),τ = ε3 2twhere 0

Nonlinear Electrostatic Waves in Pair-ion Plasmas (contd.)Now collecting the next higher order terms of continuity and momentum equationsof ions (~ε 5/2 ) and Poisson equation (~ ε 2 ) and using relation of parameter λ givenin Eq.(11) after some simplification, we obtain the KdV equation as follows:( 1) ( 1) ( 1) 3 ( 1)∂Φ + AΦ ∂Φ + B∂Φ =τ ξ ξ(12)⎡ 2 232 23( 3λ −1)( λ − μ) −( 3λ − μ) ( λ −1)⎤⎢⎥A =⎣⎦2 2λ( λ − )( λ − μ) ⎡ 22222 1 ( λ − μ) + ( λ −1)⎤⎢⎣⎥⎦2222( λ −1) ( λ − μ)B =λ ⎡ 22222 ( λ − μ) + ( λ −1)⎤⎢⎣⎥⎦Note: The soliton solution will be formed only when T +≠ T -0(coefficient of nonlinear term)(coefficient of dispersive term)holds.18

Nonlinear Electrostatic Waves in Pair-ion Plasmas (contd.)If T +≠T -, then soliton solution of Eq.(12) is given bywhere( 1)2 ⎛ η ⎞Φ = φ msec h ⎜ ⎟⎝ W ⎠φm= 3u0/ A(13)(amplitude of the soliton)W 4B/u=0(width of the soliton)Transformed variableη = ξ − u 0τhas been defined.The velocity of the transformed frame u 0has been normalized with thermalvelocity of positive ion.19

Numerical solutions in pair-ion plasmasFig.1: The electrostatic potential dips areshown for μT -for μ=0.9(dotted curve), μ= 0.8(broken curve) andμ=0.7 (solid curve) at u 0=0.8.The potential dip is increased with theincrease in positive ion temperature.F0- 0.1- 0.2- 0.3- 0.4- 1 - 0.5 0 0.5 1hFig.2: The electrostatic potential humps areshown for μ>1 i.e., the T +

Double Layer in Plasmas5. Double Layers in Pair-ion PlasmasA double layer is a structure in a plasma consists of two parallel layers withopposite electrical charge. The sheets of charge cause a strong electric field anda correspondingly sharp change in voltage (electrical potential) across the doublelayer. Ions and electrons, which enter the double layer are accelerated,decelerated or reflected by the electric field.In general, double layers (which may be curved rather than flat) separate regionsof plasma with quite different characteristics.Mathematical Description of Double Layer in Pair-ion PlasmasFor double layers structures, using the stretching for independent variables asfollows,3ξ = ε ( x − λt),..............................τ = ε tThe perturbed quantities are expanded in the powers of ε as defined earlier .Now collecting the higher order terms of continuity and momentum equations ofions (~ε 3 ) and Poisson equation (~ ε 4 ) and using relation of parameter λ given inEq.(11) after some simplification, we obtain the mKdV equation as follows:21

Double layer in Pair-ion Plasmas (contd.)The mKdV equation is described asA ∂τϕ23 31+ Q ∂ξϕ1+B ∂ξϕ1+ ∂ξϕ1=0(14)where the coefficients are defined asB= ⎡ 1 1 ⎤A 2λ⎢+2 2 2 ⎥⎣(λ − μ)( λ −1)2⎦2⎡⎛ 2 1 ⎞⎛ 2 μ ⎞⎢⎜ 2 λ − ⎟⎜ 2 λ −2 22 2⎟⎢ (3λ− 1) ⎝ 3 ⎠ (3λ− 1)= −+−⎝ 3 ⎠2 52 42524⎢ 2 ( λ − 1) ( λ − 1) 2 ( λ − μ ) ( λ − μ )⎢⎣22(3λ− μ)(3λ−1)Q = −2 3 2 32( λ − μ)2( λ −1)2⎤⎥⎥⎥⎥⎦Now using the transformationη = ξ −Uτthe above mKdV equation22

Double layer in Pair-ion Plasmas (contd.)can be described in the form of energy integral equation as follows,1 ⎛ ∂ϕ1⎞⎜ ⎟2 ⎝ ∂η⎠2+ V ( ϕ ) =10(15)where the Sagdeev potential is defined asUA 2 Q 3 BV ( ϕ1)= − ϕ1+ϕ1+ϕ2 3 4For the formation of DLs, the Sagdeev potential must satisfy the following conditions,41(16)VddVϕ1)= 0, = 0 at ϕ1= 0 and ϕ = ϕm.dϕ(12V2dϕQ1=1< 0 at ϕ1= 0 and ϕ1= ϕm−32B ϕmU= −12BϕA2m(17)The above conditions are satisfied only if , coefficients ‘Q’ and ‘U’ described as,23

Double layer in Pair-ion Plasmas (contd.)The Sagdeev potential satisfying the DLs boundary conditions as follows,V ( ϕB ϕ4221) = ϕ1( ϕm−1)(18)Using Eq.(18), the solution of energy integral equation (15) can be written as,ϕ ⎡ ⎧⎫⎤m- Bϕ1 = ⎢1− tan h⎨ϕm( η −Uτ) ⎬⎥2 ⎢⎣⎩ 8⎭⎥⎦(19)where amplitude and width of the DLs are defined as,ϕm=−2 Q3 B- B28Width = where B 0, compressive DLs and for Q

Double layer in Pair-ion Plasmas (contd.)Numerical solutions of rarefactive DLs (μ>1) in pair-ion plasmas0.20.1V0-0.1-0.2- 0.4 -0.2 0 0.2 0.4F0-0.1F-0.2-0.3-0.4-3 -2 -1 0 1 2 3x(solid curve) and(dashed curve)25

Double layer in Pair-ion Plasmas (contd.)Numerical solutions of compressive DLs (μ

6. Dissipative shocks and solitons in Pair-ion PlasmasKadomstev-Petviashvili-Burgers (KPB) Equation in PI PlasmasElectrostatic structures are studied in unmagnetized, weakly dissipative PI plasmas inthe presence of weak transverse perturbations.The dissipation in the system is incorporated by taken into account the kinematicviscosity of both the species.Normalized set of dynamic equations are written as(Continuity equation for +veions)(Momentum equations for+ve ions)(Continuity equation for -veions)(Momentum equations for-ve ions)u∂n+∂t∂ ∂+ ( n+u+) + ( n+v+) = 0∂x∂y∂u∂uϕ 1 ∂n2∂+ ++ ∂+++ u++ v+= − − + η+ 2∂t∂x∂y∂xn+∂x∂x2∂ v+ ∂v+∂v+∂ϕ1 ∂n+∂ u++ u++ v+= − − + η+ 2∂t∂x∂y∂yn+∂y∂y∂u−∂t∂v∂t−∂n−∂t+ u+ u−−∂ ∂+ ( n−u−) + ( n−v−) = 0∂x∂y∂u−∂x∂v∂x−+ v+ v−−∂u−∂y∂v∂y−2∂ϕβ ∂n−∂ u−= − + η− 2∂xn−∂x∂x2∂ϕβ ∂n−∂ v= − + η−2∂yn ∂y∂y−∂u−27

Dissipative shocks and solitons in Pair-ion PlasmasThe Poisson equation is written as follows,2 2∂ ϕ ∂ ϕ+ = n − −n2 2+∂x∂ywhere+β = T−/ T+μμ, η+ -= −and ηλ( λ )have been defined.v D+sIn fully ionized gas, the coefficient of viscosity in the absence of magnetic field isgiven by5 / 2 1/ 2−15T±Aiμ ±= 2.21×10gm / cm sec Braginskii (1962)4Z lnΛUsing reductive perturbation method, the perturbed quantities can be expandedabout their equilibrium values in powers of ε such thatv D−snvα(1)(2)23= 1+εnα+ ε nα+ ε nα(3)+ ....(1) 2 (2) 3 (3)α= εuα+ ε uα+ ε uα+ ....3/ 2 (1) 5 / 2 (2) 7 / 2 (3)α= ε vα+ ε vα+ ε vαuϕFor weak dissipation,(1) 2 (2) 3 (3)= εϕ + ε ϕ + ε ϕ+ ......η = ε 1/ 2η± 0±where η 0±is+ ....0 (1)28

Dissipative shocks and solitons in Pair-ion PlasmasThe stretching of the independent variables for 2-D electrostatic waves in PIplasmas is defined as1/ 2ξ = ε ( x − λt),χ = εy,τ = εNow collecting the terms of different powers of ε, we finally obtained the KPB equationas follows∂ ⎛ ∂ϕ⎜ +∂ξ⎝ ∂τ3∂ϕ∂ ϕAϕ+ B − C3∂ξ∂ξwhere the coefficients as defined as,3 / 222∂ ϕ ⎞ ∂= 02⎟ + ϕD2∂ξ⎠ ∂χt(20)A =B =2 2[3λ−1)(λ − β )2 222λ(λ −1)(λ − β )[( λ −1)2 2 2 2( λ −1)( λ − β )22λ[(λ −1)232+ ( λ − β )22− (3λ− β )( λ −1)2]232+ ( λ − β )]2](Nonlinear coefficient)(Dispersive coefficient)Cη=0+2( λ − β )22[( λ −1)D = λ / 222−η0−2( λ −1)2+ ( λ − β )2]2(Dissipative coefficient)(weakly dispersive coefficient)29

Dissipative shocks and solitons in Pair-ion PlasmasUsing transformationζ = k( ξ + χ −Uτ ),the solution of KPB equation is given by,hereϕ(ζ , χ,τ )k=C10B=Soliton solution625+3252C ⎡⎢1− tan hAB ⎣2C ⎡⎢sechAB ⎣2C10BC10B2⎧ 6C× ⎨ζ+ χ − (⎩ 25B2⎧ 6C× ⎨ζ+ χ − (⎩ 25B⎫⎤+ D)τ ⎬⎥⎭⎦⎫⎤+ D)τ ⎬⎥⎭⎦(wave number of nonlinear wave) has been defined.(21)The KP equation is obtained by putting C=0 in KPB equation, which givessoliton solution such aswhere12Bϕ(ζ , χ,τ ) = sechAU = −( 4B+ D)2, λ =[ ζ + χ −Uτ]+ β2andTβ =T1 -+(22)have been defined.30

Dissipative shocks and solitons in Pair-ion PlasmasTβ = -T+For β=1.4 (left)and β=1.6 (right)T −> T +For β=0.8 (left)and β=0.6 (right)T −> T +η0 += 0.15/2η = 0 −η0+(β )31

Dissipative shocks and solitons in Pair-ion PlasmasFor β=1.4 (left)and β=1.6 (right)T −> T +For β=0.8 (left)and β=0.6 (right)T −> T +32

7. Conclusion• Linear and nonlinear electrostatic waves are studied inunmagnetized pair-ion plasmas.• Korteweg-de Vries (KdV) equation is obtained using reductiveperturbation method.• Both electrostatic potential hump and dip solitons are obtainedin unmagnetized pair-ion plasmas.• The electrostatic potential humps are formed, when thetemperature of negative ion species in greater than the positiveions, while in reverse temperature conditions potential dips areobtained.• Both compressive and rarefactive potential double layers areobtained depending on the ratio between difference in positiveand negative ion species.33

7. Conclusion• Similarly, both compressive and rarefactive shocksand solitons are obtained from KPB equationdepending on the temperature ratio between positiveand negative ion species.• No electrostatic structures are obtained, when thetemperature of both positive and negative ions aresame in unmagnetized pair-ion plasmas.34

Thank You35