Contour Dynamics for One-Dimensional Vlasov-Poisson Plasma with the Periodic Boundary
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b Contour Dynamics for One-DimensionalVlasov-Poisson Plasma with the PeriodicBoundary
Hiroki Sato ∗ , T.-H. Watanabe and Shinya MaeyamaDepartment of Physics, Nagoya University, Furo-cho, Nagoya, 4646-8602, Japan Abstract
We revisit the contour dynamics (CD) simulation method which isapplicable to large deformation of distribution function in the Vlasov-Poisson plasma with the periodic boundary, where contours of distributionfunction are traced without using spatial grids. Novelty of this studylies in application of CD to the one-dimensional Vlasov-Poisson plasmawith the periodic boundary condition. A major difficulty in applicationof the periodic boundary is how to deal with contours when they crossthe boundaries. It has been overcome by virtue of a periodic Green’sfunction, which effectively introduces the periodic boundary conditionwithout cutting nor reallocating the contours. The simulation resultsare confirmed by comparing with an analytical solution for the piece-wiseconstant distribution function in the linear regime and a linear analysisof the Landau damping. Also, particle trapping by Langmuir wave issuccessfully reproduced in the nonlinear regime.
Kinetic equations for plasma dynamics describe many interesting physical phe-nomena, but are generally difficult to be solved analytically or numerically.For example, a long term nonlinear evolution of the distribution function isnot yet fully understood even in the one-dimensional Vlasov-Poisson system.Three types of simulation methods for kinetic plasma are widely known, suchas Lagrangian, semi-Lagrangian, and Eulerian methods. The Particle-In-Cell(Lagrangian) method has a problem of numerical noise, while resolution of theVlasov method (Eulerian) is limited by the grid size. Indeed, it is shown that,in the Vlasov simulation of the nonlinear Landau damping, fine structures ofthe distribution function continue to grow in phase space and are stretchedexponentially in time, increasing numerical errors (Ref. [1]).The water-bag model, which assumes a piece-wise constant distribution func-tion ( f ), has been studied for the Vlasov-Poisson plasma (Refs .[2, 3, 4]) since1960s, and successfully resolved stretching and strong deformation of f in thephase space ( x, v ). In 1979, as a generalization of the water-bag model, con-tour dynamics (CD) method is introduced by Zabusky, Hughes, and Roberts(Ref. [5]) for solving inviscid and incompressible fluid motions in the two-dimensional configuration space ( x, y ). The CD method employs nodes on each1ontour of which motion is given by calculating line integrals of the Green’s func-tion along contours. Because CD employs no spatial grid but nodes on contours(Lagrangian), the numerical resolution is not limited by spatial grids, whichmakes the CD method tough against large deformation of vorticity (Ref .[6]).In this paper, we revisit the CD method and apply it to the Vlasov-Poissonplasma with the periodic boundary condition. Although the basic idea of theCD method stems from the water-bag model, there has been a few applicationsto the Vlasov-Poisson plasma. The CD method differs from the water-bag modelas no spatial grid is used in the former in solving Poisson equation (Ref. [2]).Although a modern implementation of the water-bag method by Colombi &Touma (Ref. [7]) did not use spatial grids, the application is limited to a systemwith no spatial boundary. Novelty of the present paper lies in application of theCD method to the Vlasov-Poisson plasma with the periodic boundary, wherewe consider time development of contours of the distribution function withoutusing spatial grids. The simulation results are confirmed by comparing withan analytical solution for the piece-wise constant distribution function in thelinear regime and a linear analysis of the Landau damping. Furthermore, theparticle trapping by Langmuir waves is successfully reproduced in the nonlinearregime. Here, it should be remarked that no contour surgery (Ref. [8]) nor noderedistributions (Ref. [9]) is employed in numerical simulations in this paper,because we focus on validity of our implementation for the periodic boundarycondition.This article is organized as follows. After a brief introduction to the CD inSection 2, application to the Vlasov-Poisson system with the periodic boundaryis described in Section 3. Validity of the CD method is confirmed by comparingthe simulation results with the analytical solution for the piece-wise constantdistribution function in Section 4. A bench mark test for the linear Landaudamping is described in Section 5.1. Application to the nonlinear Landau damp-ing is shown in Section 5.2. Finally, we summarize the results in Section 6.2 Contour Dynamics
Zabusky, Hughes, and Roberts have proposed contour dynamics algorithm forthe Euler equation of fluid dynamics in two dimensions (Ref. [5]). The governingequations are
DωDt = ∂ω∂t + u ∂ω∂x + v ∂ω∂y = 0 , (1) ∇ ψ = ∂ ψ∂x + ∂ ψ∂y = − ω, (2) u = ∂ψ∂y , v = − ∂ψ∂x ,ω = − ∂u∂y + ∂v∂x , (3)where ψ is the stream function and ω means the vorticity. Time developmentof the vorticity is calculated by tracing motions of contours of the piece-wiseconstant vorticity distribution.The flow velocity is given by the line integrals ofthe Green’s function on the contours, such that( u, v ) = (cid:18) ∂ψ∂y , − ∂ψ∂x (cid:19) = X m (∆ ω m ) I C m G ( ξ, η ; x, y ) dr ′ m , (4)where m is a label of contours, C m is the contour labeled by m , and ∆ ω m is ajump of vorticity when crossing the contour C m inward. The two-dimensionalGreen’s function, G ( x, y ; ξ, η ), is given as G ( x, y ; ξ, η ) = − π log p ( x − ξ ) + ( y − η ) . (5)The contours are discretized by nodes with label n , and motion of the contours isdetermined by solving the Hamilton equations of the node points (Fig. 1). Eachcontour represents a constant vorticity line, and is advected by an incompressibleflow as given in Eq. (1). The incompressibility also guarantees conservation ofvolumes surrounded by each contour.Figure 1: In contour dynamics, Motion of the contours are determined by solvingthe Hamilton equations of the node points3 Appication to Vlasov-Poisson system
Here, we consider application of the CD method to the Vlasov-Poisson systemwith the periodic boundary. The normalized Vlasov-Poisson equations are ∂f∂τ + v ∂f∂x + a ∂f∂v = 0 , (6) a = ∂φ∂x , (7) − ∇ φ = 1 − Z ∞−∞ f ( x, v ) dv =: F ( x ) , (8)where f is the distribution function of electrons, while stationary backgroundions are assumed. The particle density is normalized so that R L/ − L/ dx R ∞−∞ dvf ( x.v ) = L , where L denotes the system length. The periodic boundary conditions at x = ± L/ ǫ →− φ (cid:18) L ǫ (cid:19) = lim ǫ ′ → +0 φ (cid:18) − L ǫ ′ (cid:19) , (9)lim ǫ →− φ ′ (cid:18) L ǫ (cid:19) = lim ǫ ′ → +0 φ ′ (cid:18) − L ǫ ′ (cid:19) (quasi neutrality) , (10)The Liouville’s theorem and Eq. (6) guarantee the volume conservation and dfdτ = 0, which are required for contour dynamics method. In order to implementthe CD, we employ the Green’s function, G , that satisfies ∇ G ( x ; ξ ) = 1 L − δ ( x − ξ ) , (11)lim ǫ →− G (cid:18) L ǫ (cid:19) = lim ǫ ′ → +0 G (cid:18) − L ǫ ′ (cid:19) , (12)lim ǫ →− G ′ (cid:18) L ǫ (cid:19) = lim ǫ ′ → +0 G ′ (cid:18) − L ǫ ′ (cid:19) . (13)Solving Eqs. (11), (12), and (13) to obtain G (Ref. [10]), one finds G ( x ; ξ ) = 12 L (cid:18) | x − ξ | − L (cid:19) . (14)Therefore, φ ( x ) = Z L − L G ( ξ ; x ) F ( ξ ) dξ + const for x ∈ (cid:18) − L , L (cid:19) . (15)The acceleration of each particle at x is given by the CD representation, a ( x ) = N m X m ∆ f m I c m G ( ξ ; x ) dv, (16)4here N m is the number of contours, C m is a contour labeled by m and ∆ f m is the jump of distribution function when crossing the contour C m inward. Wediscretize the contours with nodes labeled counterclockwise by n and connectnodes with straight line segments. Then, Eq. (16) breaks down into a ( x ) = X m ∆ f m X n v n +1 − v n (cid:18) w n ( x ) L + δ n L − I n ( x ) (cid:19) , (17)where I n ( x ) = ( | w n ( x ) | for | w n ( x ) | ≥ δ n w n ( x ) δ n + δ n for | w n ( x ) | < δ n with w n ( x ) := x − x n +1 + x n δ n := | x n +1 − x n | . (18)Although n is a function of the label of contours ( m ), we use the notation of n = n ( m ) for simplicity. Equation of motion of each node point labeled by i isgiven by d x i dτ = a ( x = x i ) = X m ∆ f m X n v n +1 − v n (cid:18) w n ( x ) L + δ n L − I n ( x ) (cid:19) . (19)For the time integration, we use the leap-frog scheme with the time step size∆ τ = 0.01 in the all simulations shown below. A difficulty of implementation arises in the CD method with the periodic bound-ary, when a node( x n , v n ) moves across the boundaries and comes into the sim-ulation box from the another side. Straightforwardly, we may cut the contourat the boundary and reallocate a node point ( x n , v n ) as x n / ∈ ( − L/ , L/ ⇒ ( x n x n − L/ L/ < x n x n x n + L/ x n < − L/ x s , ˜ v s ) and ( ˜ x t , ˜ v t ) on the boundary between( x n +1 , v n +1 ) / ∈ (cid:0) − L , L (cid:1) and ( x n , v n ) ∈ (cid:0) − L , L (cid:1) , ( ˜ x s , ˜ v s ) = (cid:16) L , v n +1 − v n x n +1 − x n (cid:0) L − x n +1 (cid:1) + v n +1 (cid:17) ; if L/ ≤ x n +1 · · · A (cid:16) − L , v n +1 − v n x n +1 − x n (cid:0) − L − x n +1 (cid:1) + v n +1 (cid:17) ; if x n +1 ≤ − L/ · · · B (21)( ˜ x t , ˜ v t ) = (cid:16) − L , v n +1 − v n x n +1 − x n (cid:0) L − x n +1 (cid:1) + v n +1 (cid:17) ; if L/ ≤ x n +1 · · · A ′ (cid:16) L , v n +1 − v n x n +1 − x n (cid:0) − L − x n +1 (cid:1) + v n +1 (cid:17) ; if x n +1 ≤ − L/ · · · B ′ (22)for calculation of the line integrals (see fig. 2). here, we call this method thereallocation scheme. since we must know the sequence of the nodes for the cdmethod, the reallocated nodes complicate the computational algorithm. actu-ally, we need to count how many times each node moved across the boundaries5 x L/2-L/2 AA’ B’B AA’ B’B Figure 2: Conventional implementation of the periodic boundary. Contourswhich get out of simulation box are cut and reallocated (
A, B, A ′ and B ′ aredefined on Eqs. (21) and (22)).and to use the counts every time in calculation of v n +1 − v n (cid:16) w n l + δ n l − i n (cid:17) .however, it increases numerical costs and makes the code implementation com-plicated.In the following, we propose a novel scheme to implement the periodicboundary in CD, which is named a periodic Green’s function method. Be-cause we consider the periodic problem; f ( a ) = f ( a + L ) , φ ( a ) = φ ( a + L ) , and φ ′ ( a ) = φ ′ ( a + L ). Thus, it may be possible to eliminate the boundariesat x = ± L , while extending the simulation box to ( −∞ , ∞ ) and imposing theperiodicity to the Green’s function, that is, ∀ x ∈ ( −∞ , ∞ ) , a ( x ) = N m X m =1 ∆ f m I c m G ′ ( ξ ; x ) dv ′ , (23)with G ′ ( ξ ; x ) = G ( M od ( ξ − ( x − L/ , L ) + ( x − L/
2) ; x ) , (24)where M od ( a, b ) = M in { c ( ≥ |∃ r ∈ Z , a = br + c } . (25)In this way, we can avoid cutting or reallocation of the contours, but equivalentlythe contours feel the periodicity of the system through G ′ instead of G (SeeFig. 3). If ∀ n, | x n − x n − | < L is satisfied, then (17) becomes a ( x ) = X m ∆ f m X n v n +1 − v n (cid:18) w ′ n L + δ n L − I ′ n (cid:19) , (26)6here w ′ n ( x ) := x + r ( x n , x ) L − x n +1 + x n , (27) r ( x n , x ) := x n − (cid:0) x − L (cid:1) − M od (cid:0) x n − (cid:0) x − L (cid:1) , L (cid:1) L = ⌊ x n − (cid:0) x − L (cid:1) L ⌋ , and δ n := | x n +1 − x n | , I ′ n ( x ) = ( | w ′ n ( x ) | for | w ′ n ( x ) | ≥ δ n w ′ n ( x ) δ n + δ n for | w ′ n ( x ) | < δ . (28)Here, the periodicity is introduced not in C m , but in G ′ . Therefore we do notneed to count how many times each node moves across the boundaries, whichmakes a faster and simpler implementation. Our new implementation with theperiodic Green’s function G ′ accelerate the computation speed faster than thatof the reallocation scheme. It owes to no cutting nor reallocation, which makesthe code avoid many ”if” branches. v x x+L/2x-L/2 Figure 3: Implementation of the periodic boundary using G ′ = G ( M od ( ξ − ( x − L/ , L ) + ( x − L/ x ). The periodic boundary is effectively introducedwithout cutting nor reallocating the contours.7 Benchmark Test for Piece-Wise Constant Dis-tribution Function
In order to check validity of our application, we consider the initial distributionfunction given by f ( x, v, τ = 0) = N m X m = − N m b m U ( v − v m ( x, , (29) U ( x ) = ( < x x ≤ , (30)where b m = − b − m < v m ( x,
0) = v m + v m ( x ). Also v m = m ∆ v and v m ( x ) = αe ikx with ∆ v ∈ R + and kL/ π ∈ N , where α ≪ | v m | ≪ | v m | . This function Eq. (29) was also used to study the water-bag model (Ref.[11]).As shown in Appendix, the linear dispersion relation is derived as D ( ω ) = 1 + X m> b m v m ω − ( kv m ) = 0 . (31)Solving the initial value problem analytically, the acceleration a ( x, τ ) is deter-mined by means of ω l which satisfies D ( ω l ) = 0 with kv l < ω l < kv l +1 , a ( x, τ ) = Re X j> X m> n ( − b m ) αe i ( kx − π ) Π n ( = m ) > (cid:16) ( ω j ) − (cid:0) kv n (cid:1) (cid:17)o k Π n ( = j ) > (cid:0) ω j − ω n (cid:1) ω j τ . (32)We make comparison of a simulation result with Eqs. (31) and (32).For N m = 2 , b = b = − / , ∆ v = 1 . , k = 1 . α = 0 . a ( x = − . π, τ ) = A cos( ω τ ) + A cos( ω τ ) with A = 0 . , A = 0 . , ω = 1 .
13 and ω = 2 . . The numerical resultof a ( x = − . π, τ ) shown in Fig. 4 agrees well with the analytical prediction,Eq. (32). It shows validity of our implementation of the CD method for theVlasov-Poisson system with the periodic boundary condition.8 a τ numerical result predicted behavior Figure 4: Benchmark test1: Time evolution of a = ∂φ/∂x . Red points aresimulation results and green line is predicted by the Eq. (32). N m = 2 , b = b = − / , ∆ v = 1 . , k = 1 . α = 0 .
01 are given for initial distributionfunction (29). 9
Benchmark test for the Landau Damping
We also verify our code for the linear Landau damping. We set the initialcontour distribution as follows. The initial (continuous) distribution function f is given by f ( x, v, τ = 0) = 1 √ π exp (cid:18) − v (cid:19) (1 + α cos( kx )) . (33)By means of a sequence { ∆ f m } N Max m =1 : ∆ f m ∈ R + and N Max X m =1 ∆ f m < M ax x,v ∈ R { f ( x, v, τ = 0) } , we define ˜ f , a piece-wise constant approximation of f ,˜ f ( x, v, τ = 0) := N m X m =1 ∆ f m I [Σ mm ′ =1 ∆ f m ′ < f ( x, v, τ = 0)] , (34)where I [ P ( x, v, τ )] = ( P ( x, v, τ ) is true0 otherwise (35)with a propositional function P ( x, v, τ ). However, Eq. (34) does not satisfy R L − L dx R ∞−∞ ˜ f ( x, v, t = 0) dv = L because ˜ f is a piece-wise constant functiondefined by means of contours of f . Therefore instead of ˜ f , we use ˜ f ′ normalizedby (1 + ǫ ),˜ f ′ = (1 + ǫ ) ˜ f , where ǫ = L R L − L dx R ∞−∞ ˜ f ( x, v, t = 0) dv − , (36)and thus ∆ f ′ m = (1 + ǫ )∆ f m .The simplest way of giving { ∆ f m } is ∆ f m =constant. However, contourdynamics method does not require constant ∆ f m , and non-uniform contourintervals have an advantage over the constant ∆ f m in approximation of f . Incase with ∆ f m = constant, the contours are densely distributed where thevelocity space gradient of f is steep around v ∼ ± v th ( v th means the thermalvelocity), while no contour is found for | v | > v th when we use 40 contours. Itmeans that there is no particle in | v | > v th , while the super thermal particlescan be included in the case of ∆ f m = constant. From the linear theory, real andimaginary parts of the eigenfrequency for k = 0 . ω r = 1 . γ = − . ω r /k ∼ . v th . This is the reasonwhy the high speed particle should be included in this application. Otherwise,many contours are necessary in the constant ∆ f m case in order to introducecontributions of the super thermal particles. Thus, we employ the non-uniformcontour intervals of ∆ f m which is defined as∆ f m := ( f ( x = x , v = V m ) m = 1 f ( x = x , v = V m ) − f ( x = x , v = V m − ) 2 ≤ m ≤ N Max , (37)10ith dv ∈ R + , N Max ∈ N and V m := ( N Max + 1 − m ) dv . Since Eq. (33) hasthe maximum at x = 0, we set x = 0. In this application, we used 40 contourswith dv = 0 .
1, where the contour spacing in v is nearly constant covering thevelocity space of | v | < v th .A simulation result for the linear Landau damping is presented in Fig. 5,where the initial distribution function is given by Eq. (37) with α = 0 .
01 and k = 0 . . The simulation box size is L = 4 π . We also used 2000 nodes/contour.One clearly finds the linear damping rate of γ = − .
153 successfully reproducedby the present CD method. ∫ φ d x τ ∝ exp(-2 γτ ) .Figure 5: Time history of the quadratic integral of the electrostatic potential φ obtained from the simulation of the linear Landau damping. The damping rate γ = -0.153 is successfully confirmed by the contour dynamics method.11 .2 Nonlinear Landau damping Next, we consider a benchmark test for the nonlinear Landau damping (Ref. [1])with the initial distribution function in Eq. (33) where α = 0 . k = 0 .
5. Weemploy 40 contours and 8000 nodes/contour. It is noteworthy that intersectionsof contour lines are not observed till τ = 30 in the present simulation. It is,thus, appropriate not to use the node redistribution nor the contour surgery inthe current test case. Fig. 6 shows a snapshot of contour distribution in thephase space at τ = 30, where the particle trapping by the Langmuir wave issuccessfully reproduced by the CD method. -6-4-2 0 2 4 6 -6 -4 -2 0 2 4 6 v x tau = 30.000000 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure 6: Phase space structure of nonlinear Landau damping at τ = 30. Par-ticles trapped by waves are reproduced. The color bar represent the magnitudeof f .Soundness of our implementation is also confirmed by conservation of energy,as shown in Figs (7) and (8). Total energy, E t = RR v f dxdv + R | ∂φ∂x | dx ,is conserved with an error, ǫ E ( τ ) = | E t ( τ ) − E t (0) | /E (0), less than 2 . × − for τ ≤ ǫ N := | N ( τ ) − N (0) | /N (0), for the case with 8000 nodes/contour, where N ( τ ) is atotal integral of the particle density defined as N ( τ ) := Z Z ˜ f ′ ( x, v, τ ) dxdv, (38)where ˜ f ′ ( x, v, τ ) := N m X m =1 ∆ f ′ m I [( x, v ) ∈ S m ( τ )] (39)with S m ( τ ) denoting the closed polygonal region determined by nodes on the m th contour. Errors in the particle conservation, ǫ N , is less than 10 − whileincreasing in time. Generally speaking, the CD method tends to fail in followingthe strong deformation of contours with large curvature, because the contoursconsist of finite straight segments connected by nodes.12 E n e r gy τ Total energyKinetic energyEnergy of electric field
Figure 7: Conservation of energy. Blue line represents energy of electric field, E φ = R | ∂φ∂x | dx , Green kinetic energy, E k = RR v f dxdv , and Red Totalenergy, E t = E φ + E k . ε E ( τ ) τ Figure 8: The error found in total energy, ǫ E ( τ ) = | E t ( τ ) − E t (0) | /E (0), isplotted. Total energy is conserved with the error, ǫ E ( τ ) < . × − until τ = 30. 13 .000E-081.000E-071.000E-061.000E-051.000E-041.000E-03 0 5 10 15 20 25 30 | N ( τ )- N ( ) | / N ( ) τ Figure 9: Time evolutions of an error found in the particle conservation, | N ( τ ) − N (0) | /N (0). 8000 nodes/contour are employed, where N ( τ ) means the integralof the particle density defined in Eq. (38)14 Summary and Conclusion
We have newly implemented contour dynamics method for the Vlasov-Poissonsystem with the periodic boundary. The major difficulty in application of theperiodic boundary is how to deal with contours when they cross the boundaries.It has been overcome by introducing periodic Green’s function defined on theinfinite phase space, instead of the Green’s function derived for the boundedsystem with the periodic boundary condition. The new scheme enables imple-mentation without cutting nor reallocating the contours and node points, andaccelerates the computational speed.Validity of the CD method for the Vlasov-Poisson system with the periodicboundary is confirmed by comparing the simulation results with the analyticalsolution for the piece-wise constant distribution function in the linear regime,and by the bench mark test for the linear Landau damping. Nonlinear Landaudamping simulation using the CD method successfully reproduces the electrontrapping by the Langmuir wave. Soundness of our method is also demonstratedby the energy and particle conservation with errors less than 2 . × − and10 − , respectively. Improvement of the CD method to reduce the conservationerrors remains for future works.Because this paper focused on the verification of our basic CD scheme forthe periodic system, detailed analyses of the physics problem by means of theCD method are remained for future studies. Application of the CD method toa variety of issues in kinetic plasma physics is currently in progress, and will bereported elsewhere. 15 ppendix Here, we calculate the analytical solution of a ( x, τ ) = ∂φ/∂x for the initialdistribution function in Eqs (29) and (30). For a node with the index m , v m satisfies the equation a ( x, τ ) = dv m dτ = ∂v m ∂τ + v m ∂v m ∂x . (40)Eqs. (8) and (29) lead to − ∂a∂x = 1 − N m X − N m ( − b m ) v m . (41)For the zeroth order, the electron density is assumed to be the same as thatof the uniform background ions, N m X − N m ( − b m ) v m = 1 (namely, we choose ∆ v tosatisfy this relation). Therefore, Eq.(40) is linearized as a = ∂v m ∂τ + v m ∂v m ∂x , (42)and Eq. (41) reads ∂a∂x = N m X − N m ( − b m ) v m . (43)Assuming a, v m ∝ e ikx , The Laplace transform of Eqs (42) and (43) give L ( a ) = − v m (0) + sL (cid:0) v m (cid:1) + v m ikL (cid:0) v m (cid:1) , (44) − ikL ( a ) = N m X − N m b m L (cid:0) v m (cid:1) , (45)where L ( f ( τ )) := R ∞ f ( τ ) e − sτ dτ . Thus, L ( a ) = 1 D ( is ) N m X − N m (cid:8) ( − b m ) v m (0)Π l = m (cid:0) is − kv l (cid:1)(cid:9) , (46)with D ( is ) = k Π N m m = − N m (cid:0) is − kv m (cid:1) + N m X m = − N m (cid:8) b m Π l = m (cid:0) is − kv l (cid:1)(cid:9) . (47)It is known that for a choice of b n < ∀ n ≥ , the solutions of D ( is ) = 0 arepurely real (see Refs. [11] and [12]). We define ω := is and ω m : D ( ω m ) = 016ith kv m < ω m < kv m +1 so that D ( ω m ) is written as D = k Π N m m = − N m ( ω − ω m ) . The inverse Laplace transform of Eq. (46) is a = N m X j = − N m Res (cid:0) L ( a ) ( s ) e st , − iω j (cid:1) (48)= N m X j = − N m lim s →− iω j ( s + iω j ) 1 k N m X − N m (cid:8) ( − b m ) v m (0)Π l = m (cid:0) is − kv l (cid:1)(cid:9) Π N m m = − N m ( is − ω m ) e st (49)Since ω m = − ω − m , b m = − b − m and v m = − v − m , one finds a ( x, τ ) = Re X j> X m> n ( − b m ) αe i ( kx − π ) Π n ( = m ) > (cid:16) ( ω j ) − (cid:0) kv n (cid:1) (cid:17)o k Π n ( = j ) > (cid:0) ω j − ω n (cid:1) ω j τ , (50)where ω j satisfies the dispersion relation of1 + X m> b m v m ω − ( kv m ) = 0 . (51) References [1] T-H Watanabe and Hideo Sugama. Vlasov and drift kinetic simulationmethods based on the symplectic integrator.
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