13.1 Ion Acoustic Soliton and Shock Wave

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1 13 Nonlinear Waves In linear theory, the wave amplitude is assumed to be sufficiently small to ignore contributions of terms of second order and higher (ie, nonlinear terms) in wave amplitude In such a case, it is sufficient to consider only one Fourier component at a time, and a general solution can be expressed by appropriate superposition of Fourier components However, when the wave amplitude becomes larger, the linear approximation breaks down and nonlinear effects must be taken into account Linear theory predicts exponential growth of unstable waves, but nonlinear effects cause saturation and limit the wave amplitude at a finite level 131 Ion Acoustic Soliton and Shock Wave Consider the idealized potential profile of an ion acoustic shock wave The wave is travelling to the left with velocity u 0 In the frame moving with the wave, the potential profile ϕ(x) is constant in time, and the plasma is impinging on the wave from the left with a velocity u 0 For simplicity, assume T i = 0 and let the electrons be Maxwellian From energy conservation, the velocity of the ions in the shock wave is u i = u 0 1 2eϕ Mu 2 0 Taking the density on the left to be, the ion density in the shock is Using the normalized parameters n i = n ( 0u 0 = 1 2eϕ ) 1/2 u i Mu 2 0 n i = n, u 0 c s = u, x λ D = ξ, ω pi t = τ, eϕ T e = φ equations are expressed as follows continuity equation n τ + ξ (nu) = 0 ion equation of motion u τ + u u ξ + φ ξ = 0 electron equation of motion leads to Boltzmann distribution n e = e φ Poisson equation 2 φ ξ 2 = eφ n 57

2 Rewriting 2 φ ξ 2 ( = eφ 1 2φ ) 1/2 dv (φ) u 2 = dφ This equation can be regarded as the equation of motion for a particle moving in the potential V (φ), with the potential φ playing the role of position and the position ξ playing the role of time, respectively The quasi-potential V (φ) is called the Sagdeev potential Fig 1 The Sagdeev potential The function V (φ) can be found by integrating with the boundary condition V (φ = 0) = 0, ) V (φ) = 1 e φ + u ( φ u 2 A particle will make a single excursion to positive φ and return to φ = 0 Such a potential structure is called a soliton It is a potential and density disturbance propagating to the left with velocity u 0 If there is dissipation, the potential will oscillate in space about some positive value of φ, and never return to φ = 0 Reflection of ions from the shock front has the same effect as dissipation Expanding n = 1+n, and transforming to a coordinate system moving with velocity c s, y = ξ τ, n τ + y (u n + n u) = 0 u τ + u u y + (φ u) = 0 y 2 φ y 2 = n φ φ2 2 58

3 Fig 2 The potential of a soliton moving to the left In the wave frame, ions stream into the shock front from the left Fig 3 Typical potential structure of an ion-acoustic shock moving to the left 59

4 The 1st order solution, with the boundary condition that φ, u, n 0 as y, is u = n = φ The 2nd order equation is u τ + u u y u 2 y 3 = 0 This equation is called the Korteweg-de Vries (K-dV) equation The second term is the convective term v v which leads to wave steepening The third term arises from wave dispersion, ie, k dependence of the phase velocity The K-dV equation has a localized solution (soliton) ( ) x ct u = u + Bsech 2 δ where c = u + B 3, δ = 6 B The nonlinearity produces harmonic components (2ω, 2k), (3ω, 3k), Low harmonics grow because they satisfy the linear dispersion relation, and cause wave steepening High harmonics are suppressed because they deviate from the linear dispersion relation ω = kc s A stable soliton solution is obtained by balancing the steepening due to nonlinearity and the dispersion effect of higher order terms Fig 4 A large amplitude ion acoustic wave steepens so that the leading edge has a steeper slope than the trailing edge 132 Ponderomotive Force Equation of motion is m d v dt = q [ E( r) + v B( r) ] 60

5 Take E( r) = E s ( r) cos ωt, and expand r = r 0 + δ r 1 +, E( r) = E( r 0 ) + (δ r 1 ) E( r 0 ) + The first order equation can be solved to give m d v 1 dt v 1 = The second order equation is From E = B/ t, = q E( r 0 ) q E mω s ( r 0 ) sin ωt = dδ r 1 dt δ r 1 = q mω 2 E s ( r 0 ) cos ωt, m d v 2 dt = q [ (δ r 1 ) E( r 0 ) + v 1 B 1 ] B 1 = 1 ω E s ( r 0 ) sin ωt which can be substituted in the equation of motion and time averaged to give d v2 m = q2 1 [ dt mω 2 ( E 2 s ) E s + E s ( E ] s ) q 2 = 1 4 mω 2 E2 s The nonlinear force is given by F NL = 1 q 2 4 mω 2 E2 s = ω2 p ω 2 ϵ0 E 2 2 where E 2 = Es 2 /2 This force is called the ponderomotive force The ponderomotive force works in the direction to expel particles from a region of strong electric field, and is stronger for particles with lighter mass and for lower frequencies This is a nonlinear force because it depends on the square of the electric field strength A laser beam of finite diameter travelling through a plasma causes a radially outward ponderomotive force This force moves plasma out of the beam so that ω p is lower and the dielectric constant ϵ and the index of refraction n are higher inside the beam than outside, since n = ϵ = 1 ω2 pe ω 2 The plasma acts as a convex lens, and focuses the beam to a smaller diameter 133 Caviton and Envelope Soliton In case of electron plasma wave, higher harmonic components do not satisfy the linear dispersion relation, so generation of harmonics is not important If the linear wave is represented by A cos(kx ωt), the second order components are A 2 cos 2 (kx ωt) = 1 2 A2 [cos(2kx 2ωt) + 1] 61

6 Fig 5 Self-focusing of a laser beam is caused by the ponderomotive force The first term in the bracket describes second harmonic generation, which in this case is unimportant The second term is the k = 0, ω = 0 component and describes the slow evolution of nearly uniform modulation The wave equation for electron plasma wave can be expressed in linear approximation as 2 2 t 2 u ex + ωpeu 2 ex 3vte 2 x 2 u ex = 0 Note that ωpe 2 is proportional to the electron density and vte 2 is proportional to the electron temperature, and that the third term can be ignored in the limit k 0 Assume that the plasma density is modulated from to + δn e (x, t), and that the time variation of δn e is much slower than plasma oscillation In such a case 2 t 2 u ex 3vte 2 2 [ x 2 u ex + ωpe δn ] e(x, t) u ex = 0 Writing the solution as u ex = R[u(x, t) exp( iω pe t)] and ignoring 2 u/ t 2 compared to ω 2 peu, the following equation is obtained i t u(x, t) + 3 vte ω pe x 2 u(x, t) ω pe 2 δn e (x, t) u(x, t) = 0 This equation has the form of Schrödinger equation with the third term as the nonlinear potential arising from the ponderomotive force As will be shown later in this sub-section, δn e / is proportional to u(x, t) 2 This equation is called the nonlinear Schödinger equation The solution that corresponds to a particle trapped in the potential well represents a standing wave trapped in the density well Because the soliton solution of this equation is oscillating at high frequency, it is called the envelope soliton The slow evolution of the density modulation δn e is described by the following equation, obtained by averaging the electron equation of motion over the fast oscillation (ω pe ), m e u ex t + m e u ex x u ex = 1 n e p e x + e ϕ x The first term on the left hand side is the electron inertia term, and can be ignored when the time evolution is slow The second term is the nonlinear term 62

7 Fig 6 An envelope soliton which can be expressed as x ( me ) 2 u2 ex = ( me u(x, t) 2) x and represents the ponderomotive force The first term on the right hand side can be rewritten as (T e / ) δn e / x, so the equation above can be expressed as ( me u(x, t) 2) = ( ) Te δn e eϕ x x This equation can be solved with the boundary condition δn e, ϕ 0 as u(x, t) 0 to give δn e = eϕ m e u(x, t) 2 T e Finally, the ion equation of motion must be solved to obtain ϕ Since ions cannot respond to high frequency electron plasma oscillation, linearization can be used, u i m i t = e ϕ x 1 x (n it i ) If the time evolution is slow enough, the ion inertia term can be ignored, and eϕ = T i δn i Therefore, δn e = m e u(x, t) 2 T e + T i Substituting this relationship in the potential term of the Schrödinger equation yields i t u(x, t) + 3 vte ω pe x 2 u(x, t) + ω pe m e u(x, t) 2 u(x, t) = 0 2 T e + T i Since the potential depends on the wave intensity, this equation is called the nonlinear Schrödinger equation 63