ebook img

Velocity-Space Diffusion in a Perpendicularly Propagating Electrostatic Wave PDF

0.17 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Velocity-Space Diffusion in a Perpendicularly Propagating Electrostatic Wave

Velocity-Space Diffusion in a Perpendicularly 5 Propagating Electrostatic Wave 0 0 2 Charles F. F. Karney n Plasma Physics Laboratory, Princeton University a J Princeton, New Jersey 08544, U.S.A. 7 ] h p - m Abstract s a The motionofionsin the fieldsB=B0zˆ andE=E0yˆcos(k⊥y−ωt)is con- l sidered. When ω ≫Ω andv >ω/k , the equations ofmotionmaybe reduced p i ⊥ ∼ ⊥ . toasetofdifferenceequations. Theseequationsexhibitstochasticbehaviorwhen s c E0 exceeds a threshold. The diffusion coefficient above the threshold is deter- i mined. Far above the threshold, ion Landau damping is recovered. Extension of s y the method to include parallel propagationis outlined. h p [ 1 v 5 3 0 1 0 5 0 / s c i s y h p : v i X r a Presented at the International Workshop on Intrinsic Stochasticity in Plasmas, Carg`ese, Corsica, France, June 18–23, 1979. Published in Intrinsic Stochas- ticity in Plasmas, edited by G. Laval and D. Gr´esillon (Editions de Physique Courtaboeuf, Orsay, 1979), pp. 159–168. –1– –2– Equations of Motion Consideranioninauniformmagneticfieldandaperpendicularlypropagat- ing electrostatic wave, B=B0zˆ, E=E0yˆcos(k⊥y−ωt). (1) Normalizing lengths to k⊥−1 and times to Ω−i 1 (Ωi =qiB0/mi), the Lorentz force law for the ion becomes y¨+y =αcos(y−νt), x˙ =y, (2) where ν =ω/Ω , (3a) i E0/B0 α= . (3b) Ω /k i ⊥ We solve (2) by approximating the force due to the wave by impulses at those points where the phase is slowly varying, i.e., at y˙ = ν. The trajectory of the ion is given in Fig. 1. Expanding the trajectory about the resonance point, we find that the magnitude of the impulses is given by ∞ B = αcos(φ − 1y t′2)dt′ Z c 2 c −∞ =α(2π/|y |)1/2cos[φ −sign(y )π/4], (4) c c c where φ =y −νt and t and y are the time and position of the wave-particle c c c c c “collision.” WemaydeterminetheLarmorradiusandphaseoftheionattheend of the jth orbit[the beginning of the (j+1)th orbit] in terms of these quantities at the beginning of the jth orbit. (Details are given in Ref. 1.) The resulting difference equations are u=θ−ρ, v =θ+ρ, (5a) θ = 1(v+u), ρ = 1(v−u), (5b) 2 2 uj+1−uj =2πδ−2πAcosvj, (5c) vj+1−vj =2πδ+2πAcosuj+1, (5d) where θ =νt (mod 2π), (6a) ρ=(r2−ν2)1/2−νcos−1(ν/r)+νπ−π/4, (6b) 2 1/2 αν(r2−ν2)1/4 A= , (7a) (cid:18)π(cid:19) r2 δ =ν−n. (7b) –3– HereristhenormalizedLarmorradius,k v /Ω ,andnisaninteger. Thelimits ⊥ ⊥ i of validity of (5) are ν ≫1, r−ν ≫(1ν)1/3, A≪(r2−ν2)3/2/r2. (8) 2 In Fig. 2 we compare the trajectories obtained using the exact equations of motion, (2), with those obtained from the difference equations, (5). We see that the agreement is very good indicating that (5) is an excellent approximation of (2). There are three advantages to using the difference equations in preference to the Lorentz force law. Firstly, they are much quicker to solve numerically. Secondly, because of the way the equations were derived, the results are easier to interpret. Lastly, the equations have separated out two velocity-space scales, the ρ scale (∼ Ω /k ) and the r scale (∼ ω/k ). We therefore treat A which i ⊥ ⊥ is a function of r as a constant when iterating the equations. This means that the diffusion coefficient is independent of ρ and so is much easier to determine numerically. Examples of Trajectories When A is infinitesimal, (5) may be solved by integrating (summing?) over unperturbed orbits. Substituting uj = u0 +2πδj and vj = v0 +2πδj into the right hand sides of (5c) and (5d) gives ρN −ρ0 =2πAcos(ρ0−πδ) (−1)δcosθ0N, for δ =integer, ×sin(2πδN +θ0)−sinθ0, for δ 6=integer. (9) 2sin(πδ) Note that the trajectory is secular or not depending on whether or not δ is an integer (ω is a cyclotron harmonic). Formally, we may compute a diffusion coefficient using h(ρN −ρ0)2i D= Lim . (10) N→∞ 2N Substituting (9) into (10) gives ∞ D=π2A2cos2ρ0 δˆ(δ−m), (11) mX=−∞ where δˆis the Dirac delta function. Converting back to r and t and undoing the normalizations gives πq2E2 ω 2 ∞ k v D = i 0 J2 ⊥ ⊥ δˆ(ω−mΩ ), (12) 2 m2 (cid:18)k v (cid:19) m(cid:18) Ω (cid:19) i i ⊥ ⊥ mX=−∞ i which is the usual quasi-linear diffusion coefficient. IfweconsiderfinitebutsmallA,thenallthetrajectoriesarebounded. There arethreedistinctcases,δ =0(whichisthecaseconsideredbyFukuyamaetal.2), δ = 1, and δ 6=0 or 1. The trajectories for these cases are shown in Fig. 3. 2 2 –4– When A is increased, the system undergoes a stochastic transition, an ex- ampleofwhichisshowninFig.4forδ =0.23. Belowthestochasticitythreshold, nearly all the trajectoriesare integrable [Fig. 4(a)]or, if there are stochastic tra- jectories, they are bounded in ρ [Fig. 4(b)]. Above the threshold, nearly all the trajectories are stochastic and unbounded. The value of the threshold may be numerically determined and is found to be A= A = 1. Above this value of A, s 4 the kick received by the ion during one transit through resonance is sufficient to change the phase of the kick received when next in resonance by π/2. Diffusion Coefficient When computing the diffusion coefficient numerically, it is convenient to work with the correlation function, C , where k Ck =hajaj+ki, (13) aj istheparticleacceleration,aj =ρj+1−ρj,andtheaverageisoveranensemble of particles and over the length of a given trajectory (i.e., over j). Then the diffusion coefficient, D, is given by ∞ 1 D= C0+ Ck. (14) 2 kX=1 [This definition is equivalent to (10).] The advantages of defining D in this way aretwofold. Firstly,thestatisticalfluctuationsinthecomputationareminimized. Secondly,itiseasytointroducetheeffectsofcollisionsonthediffusioncoefficient. This is accomplished as follows: If k0 is the mean number of cyclotron periods between decorrelating collisions (such collisions need only result in deflection by 1/ν, a small angle), then the probability of such a collision taking place in k periods is 1 − exp(−k/k0) since collisions are independent events. Collisions may then be included in the computations of D by replacing C in (14) by k Ckexp(−k/k0). Inthe limit A≫A , the kicks the ionreceivesareuncorrelatedsothatonly s C0 is nonzero. Assuming that the trajectory is ergodic, we obtain D = 21π2A2. When A is not large, we account for the correlations between the kicks received by the ion by writing D= 1π2A2g2(A), (15) 2 Numerically determining g(A) we find that approximately g(A)=max(1−A2/A2,0), (16) s with A = 1. s 4 Converting (15) back into usual variables we obtain 1q2E2 ω 2 g2(A) D = i 0 . (17) 2 m2 (cid:18)k v (cid:19) (k2v2 −ω2)1/2 i ⊥ ⊥ ⊥ ⊥ In the limit A ≫ A , when g(A) ≈ 1, this is just the zero-magnetic-field result, s (π/2)(q /m )2E2δˆ(ω−k v ), averaged over Larmor phase. Thus, in this limit, i i 0 ⊥ y we recover ion Landau damping. –5– Extension to Parallel Propagation The diffusion coefficient was so easily calculated above because of the simp- lification obtained by reducing the problem to difference equations, (5). This reductionmay be achievedin similarproblems. We consider here the case where the wave has some component of parallel propagation so that (1) becomes B=B0zˆ, E=E0(yˆ+kkzˆ/k⊥)cos(k⊥y+kkz−ωt). (18) Adopting the same normalization as before, we obtain y¨+y =αcos(y+ζz−νt), (19a) z¨=αζcos(y+ζz−νt), (19b) whereζ =k /k . ThedifferenceequationsforaparticlewithnormalizedLarmor k ⊥ radius, r, and normalized parallel velocity, w=k v /Ω , are ⊥ k i u=θ−ρ, v =θ+ρ, (20a) uj+1−uj =2πγj+1/2−2πAcosvj, (20b) vj+1−vj =2πγj+1/2+2πAcosuj+1, (20c) γ =δ−β(ρ +πAcosv ), (20d) j+1/2 j j where the variables θ and ρ are given by θ =νt−ζz (mod 2π), (21a) ρ=(r2−µ2)1/2−µcos−1(µ/r)+µπ−π/4−2πm. (21b) The definitions of the parameters A, β, and ρ are 1/2 2 αµQ A= , (22a) (cid:18)π(cid:19) r(r2−µ2)1/4 β =ζ2r/(µQ), (22b) δ =µ+βρ−n. (22c) Here, m and n are integers and µ=ν−ζw, (23a) (r2−µ2)1/2 µ ζ2r Q= − π−cos−1 . (23b) r r µ h (cid:16) (cid:17)i Despite appearancesδ is a parameterindependent of ρ since the quantity µ+βρ is a constant. (This follows from energy conservation in the wave frame.) The restrictions on the validity of (20) are µ≫1, r−µ≫(1µ)1/3. (24) 2 –6– Acomparisonbetweentheexactequationsofmotion,(19),andthedifference equations, (20), is shown in Fig. 5 for ζ = 1 (propagation at 45◦). Again, there is excellent agreement. The results of Smith and Kaufman3 may be obtained in the limit β → ∞ and A → 0 (Q → 0). In that case, the change in ρ is negligible so that it is necessary to rescale the velocity variable by defining σ =4πβρ −2πδ. (25) j j Equation (20) then becomes θj+1−θj =−σj − 21Kcos(θj +ρ), (26a) σj+1−σj = 21K[cos(θj+1−ρ)+cos(θj +ρ)], (26b) where K =4π2Aβ. In (26) ρ is a constant. Setting ψ =σ+ 1Kcos(θ+ρ) (27) 2 gives θj+1−θj =−ψj, (28a) ψj+1−ψj =Kcosρcosθj+1. (28b) This is the “standard mapping” studied by Chirikov.4 The island overlap condi- tion for this mapping is |Kcosρ|>π2/4 or |α|> 16ζ2J (r) −1, (29) µ (cid:12) (cid:12) whichisthestochasticitythresholdo(cid:12)btainedby(cid:12) SmithandKaufman. Theisland overlapcriterionisasignificantoverestimateofthestochasticitythresholdforthe standard mapping.4 Greene5 has calculated that the true threshold is a factor (π2/4)/0.971635≈21 smaller than the result given above. 2 Acknowledgments The author wishes to thank N. J. Fisch, J. M. Greene, J. A. Krommes, and A. B. Rechester for useful discussions. ThisworkwassupportedbytheU.S.DepartmentofEnergyunderContract No. EY–76–C–02–3073. References 1C. F. F. Karney, Phys. Fluids 21, 1584 (1978); Phys. Fluids 22, 2188 (1979). 2A. Fukuyama, H. Momota, R. Itatani, and T. Takizuka, Phys. Rev. Lett. 38, 701 (1977). 3G.R.SmithandA.N.Kaufman,Phys.Rev.Lett.34,1613(1975);Phys.Fluids 21, 2230 (1978). 4B. V. Chirikov, Phys. Repts. 52, 265 (1979). 5J. M. Greene, J. Math. Phys. 20, 1183 (1979). –7– Fig. 1. Motion of an ion in velocity space, showing the kicks it receives when passing through wave-particle resonance. –8– Fig. 2. Comparison of the difference equations with the Lorentz force law. (a) Trajectories computed using (2) with ν = 30.23 and α = 2.2. (b) Trajectories computed using (5) with δ = 0.23 and A = 0.1424, which are given by (7) with ν =30.23,α=2.2, andr =47.5. In eachcasethe trajectoriesof 24particles are followed for 300 orbits. –9– Fig. 3. Trajectories for small A and (a) δ =0, (b) δ = 1, and (c) δ 6=0 or 1. 2 2 –10– Fig. 4. The trajectoriesofionswithδ =0.23and(a)A=0.05,(b)A=0.2,(c) A=0.35. The initial positions are shown by crosses.

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.