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This velocity may be low enough to lie within the iondistribution function. There can then be an energy exchange with the resonantions. The potential the ions see is the effective potential due to the ponderomotiveforce (Fig. 8.24), and Landau damping or growth can occur. This damping providesan effective way to heat ions with high-frequency waves, which do not ordinarilyinteract with ions.
If the ion distribution is double-humped, it can excite the electronwaves. Such an instability is called a modulational instability.Problems8.15 Make a graph to show clearly the degree of agreement between the echo dataof Fig. 8.21 and Eq. (8.87).8.16 Calculate the bounce frequency of a deeply trapped electron in a plasma wavewith 10-V rms amplitude and 1-cm wavelength (Fig. 8.23).8.8 Equations of Nonlinear Plasma Physics307Fig. 8.24 The ponderomotive force caused by the envelope of a modulated wave can trapparticles and cause wave-particle resonances at-the group velocity8.8Equations of Nonlinear Plasma PhysicsThere are two nonlinear equations that have been treated extensively in connectionwith nonlinear plasma waves: The Korteweg–de Vries equation and the nonlinearSchr€odinger equation.
Each concerns a different type of nonlinearity. When an ionacoustic wave gains large amplitude, the main nonlinear effect is wave steepening,whose physical explanation was given in Sect. 8.3.3. This effect arises from thev · ∇v term in the ion equation of motion and is handled mathematically by theKorteweg–de Vries equation. The wave-train and soliton solutions of Figs. 8.5 and8.7 are also predicted by this equation.When an electron plasma wave goes nonlinear, the dominant new effect is thatthe ponderomotive force of the plasma waves causes the background plasma tomove away, causing a local depression in density called a caviton. Plasma wavestrapped in this cavity then form an isolated structure called an envelope soliton orenvelope solitary wave. Such solutions are described by the nonlinear Schr€odingerequation.
Considering the difference in both the physical model and the mathematical form of the governing equations, it is surprising that solitons and envelopesolitons have almost the same shape.8.8.1The Korteweg–de Vries EquationThis equation occurs in many physical situations including that of a weaklynonlinear ion wave:3∂U∂U 1 ∂ UþUþ¼0∂τ∂ξ 2 ∂ξ3ð8:92Þwhere U is amplitude, and τ and ξ are timelike and spacelike variables, respectively.Although several transformations of variables will be necessary before this form isobtained, two physical features can already be seen. The second term in Eq. (8.92)is easily recognized as the convective term v · ∇v leading to wave steepening.3088 Nonlinear EffectsThe third term arises from wave dispersion; that is, the k dependence of the phasevelocity.
For Ti ¼ 0, ion waves obey the relation (Eq. (4.48))1ω2 ¼ k2 c2s 1 þ k2 λ2Dð8:93ÞThe dispersive term k2λ2D arises from the deviation from exact neutrality. ByTaylor-series expansion, one finds12ω ¼ kcs k3 cs λ2Dð8:94Þshowing that the dispersive term is proportional to k3. This is the reason for the thirdderivative term in Eq. (8.92). Dispersion must be kept in the theory to prevent verysteep wavefronts (corresponding to very large k) from spuriously dominating thenonlinear behavior.The Korteweg–de Vries equation admits of a solution in the form of a soliton;that is, a single pulse which retains its shape as it propagates with some velocityc (not the velocity of light!).
This means that U depends only on the variable ξ cτrather than ξ or τ separately. Defining ζ ξ cτ, so that ∂/∂τ ¼ cd/dζ and∂/∂ξ ¼ d/dζ, we can write Eq. (8.92) ascdUdU 1 d3 UþUþ¼0dζdζ 2 dζ 3ð8:95ÞThis can be integrated with the dummy variable ζ 0 :cð1ζdU 0 10 dζ þ2dζð1ζdU2 0 10 dζ þ2dζð1ζ!d d2 U0dζ ¼ 0:002dζ dζð8:96ÞIf U(ζ) and its derivatives vanish at large distances from the soliton (|ζ| ! 1), theresult is12cU U2 21d U2 dζ 2¼0ð8:97ÞMultiplying each term by dU/dζ, we can integrate once more, obtaining1 2cU2or 11 dU 2¼0 U3 64 dζð8:98Þ8.8 Equations of Nonlinear Plasma Physics 2dU2¼ U2 ð3c U Þdζ3309ð8:99ÞThis equation is satisfied by the soliton solutionhiUðζ Þ ¼ 3c sech2 ðc=2Þ1=2 ζð8:100Þas one can verify by direct substitution, making use of the identitiesdðsech xÞ ¼ sech x tanhxdxð8:101Þsech2 x þ tanh2 x ¼ 1ð8:102ÞandEquation (8.100) describes a structure that looks like Fig. 8.7, reaching a peak atζ ¼ 0 and vanishing at ζ ! 1.
The soliton has speed c, amplitude 3c, and halfwidth (2/c)1/2. All are related, so that c specifies the energy of the soliton. The largerthe energy, the larger the speed and amplitude, and the narrower the width. Theoccurrence of solitons depends on the initial conditions. If the initial disturbancehas enough energy and the phases are right, a soliton can be generated; otherwise, alarge-amplitude wave will appear.
If the initial disturbance has the energy of severalsolitons and the phases are right, an N-soliton solution can be generated. Since thespeed of the solitons increases with their size, after a time the solitons will dispersethemselves into an ordered array, as shown in Fig. 8.25.Fig. 8.25 A train of solitons, generated at the left, arrayed according to the relation among speed,height, and width3108 Nonlinear EffectsWe next wish to show that the Korteweg–de Vries equation describes largeamplitude ion waves. Consider the simple case of one-dimensional waves with coldions.
The fluid equations of motion and continuity are∂vi∂vie ∂ϕþ vi¼m ∂x∂t∂xð8:103Þ∂ni ∂þ ð ni v i Þ ¼ 0∂t ∂xð8:104ÞAssume Boltzmann electrons (Eq. (3.73)); Poisson’s equation is then2ε0∂ ϕeϕ=KT e¼enen0i∂x2ð8:105ÞThe following dimensionless variables will make all the coefficients unity:0x0tχ0v1=2¼ x=λD ¼ xðn0 e2 =ε0 KT e Þ1=2¼ Ω p t ¼ tðn0 e2 =ε0 MÞ0¼ eϕ=KT en ¼ ni =n0¼ v=vs ¼ vðM=KT e Þ1=2ð8:106ÞOur set of equations becomes00∂v∂χ0 ∂v0 þ v0 ¼∂x0∂t∂xð8:107Þ0∂n∂ 0 0nv ¼00 þ∂x0∂t2∂ χ∂x02¼ eχ n0ð8:108Þð8:109ÞIf we were to transform to a frame moving with velocity v0 ¼ M, we would recoverEq.
(8.27). As shown following Eq. (8.27), this set of equations admits of solitonsolutions for a range of Mach numbers M.Problem8.17 Reduce Eqs. (8.107)–(8.109) to Eq. (8.27) by assuming that χ, n0 , and v0depend only on the variable ξ0 x0 Mt0 . Integrate twice as inEqs. (8.96)–(8.98) to obtain12hi1=20 2dχ=dξ¼ eχ 1 þ M M 2 2χMShow that soliton solutions can exist only for 1 < M < 1.6 and 0 < χ max < 1.3.8.8 Equations of Nonlinear Plasma Physics311To recover the K–dV equation, we must expand in the wave amplitude and keepone order higher than in the linear theory.
Since for solitons the amplitude andspeed are related, we can choose the expansion parameter to be the Mach numberexcess δ, defined to beδM1ð8:110ÞWe thus write0n ¼ 1 þ δn1 þ δ2 n2 þ χ ¼ δχ 1 þ δ2 χ 2 þ 0v ¼ δv1 þ δ2 v2 þ ð8:111ÞWe must also transform to the scaled variables1so that 00ξ ¼ δ1=2 x t0τ ¼ δ3=2 tð8:112Þ∂∂∂ δ1=2¼ δ3=20∂τ∂ξ∂t∂∂¼ δ1=2∂x0∂ξSubstituting Eqs. (8.111) and (8.113) into Eq. (8.109), we find that the lowest-orderterms are proportional to δ, and these giveχ 1 ¼ n1ð8:114ÞDoing the same in Eqs.
(8.107) and (8.108), we find that the lowest-order terms areproportional to δ3/2, and these give∂v1 ∂χ 1 ∂n1¼¼∂ξ∂ξ∂ξð8:115ÞSince all vanish as ξ ! 1, integration givesn1 ¼ χ 1 ¼ v 1 Uð8:116ÞThus our normalization is such that all the linear perturbations are equal and can becalled U. We next collect the terms proportional to δ2 in Eq.
(8.109) and to δ5/2 inEqs. (8.107) and (8.108). This yields the set1It is not necessary to explain why; the end will justify the means.3128 Nonlinear Effects2∂ χ11¼ χ 2 n2 þ χ 212∂ξ2ð8:117Þ∂v1 ∂v2∂v1∂χþ v1¼ 2∂ξ∂τ∂ξ∂ξð8:118Þ∂n1 ∂n2 ∂þ ðv2 þ n1 v1 Þ∂τ∂ξ ∂ξð8:119ÞSolving for n2 in Eq. (8.117) and for ∂v2/∂ξ in Eq.
(8.118), we substitute intoEq. (8.119):3∂n1 ∂ χ 1 ∂χ 2 1 ∂χ 21 ∂v1∂v1 ∂χ 2 ∂þ ð n1 v 1 Þ ¼ 0þþþ v1þ∂ξ 2 ∂ξ∂ξ ∂ξ∂τ∂τ∂ξ∂ξ3ð8:120ÞFortunately, χ 2 cancels out, and replacing all first-order quantities by U results in3∂U∂U 1 ∂ UþUþ¼0∂τ∂ξ 2 ∂ξ3ð8:121Þwhich is the same as Eq. (8.92). Thus, ion waves of amplitude one order higher thanlinear are described by the Korteweg–de Vries equation.Problem8.18 A soliton with peak amplitude 12 V is excited in a hydrogen plasma withKTe ¼ 10 eV and n0 ¼ 1016 m3. Assuming that the Korteweg–de Vriesequation describes the soliton, calculate its velocity (in m/s) and its fullwidth at half maximum (in mm).
(Hint: First show that the soliton velocityc is equal to unity in the normalized units used to derive the K–dV equation.)8.8.2The Nonlinear Schr€odinger EquationThis equation has the standard dimensionless form2i∂ψ∂ ψþ p 2 þ qj ψ j 2 ψ ¼ 0∂t∂xð8:122Þwhere ψ is the wave amplitude, i ¼ (1)1/2, and p and q are coefficients whosephysical significance will be explained shortly. Equation (8.122) differs from theusual Schr€odinger equation8.8 Equations of Nonlinear Plasma Physics313Fig. 8.26 The ponderomotive force of a plasma wave with nonuniform intensity causes ions toflow toward the intensity minima.
The resulting density ripple traps waves in its troughs, thusenhancing the modulation of the envelope2ih∂ψ h2 ∂ ψþ V ðx; tÞψ ¼ 0∂t 2m ∂x2ð8:123Þin that the potential V(x, t) depends on ψ itself, making the last term nonlinear.Note, however, that V depends only on the magnitude jψj2 and not on the phase ofψ. This is to be expected, as far as electron plasma waves are concerned, becausethe nonlinearity comes from the ponderomotive force, which depends on thegradient of the wave intensity.Plane wave solutions of Eq.
(8.122) are modulationally unstable if pq > 0; thatis, a ripple on the envelope of the wave will tend to grow. The picture is the same asthat of Fig. 8.24 even though we are considering here fluid, rather than discreteparticle, effects. For plasma waves, it is easy to see how the ponderomotive forcecan cause a modulational instability. Figure 8.26 shows a plasma wave with arippled envelope. The gradient in wave intensity causes a ponderomotive forcewhich moves both electrons and ions toward the intensity minima, forming a ripplein the plasma density.
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