Semi-Lagangian simulations of the diocoton instability Eic Madaule, Seve Adian Histoaga, Michel Mehenbege, Jéôme Péti To cite this vesion: Eic Madaule, Seve Adian Histoaga, Michel Mehenbege, Jéôme Péti. Semi-Lagangian simulations of the diocoton instability. [Reseach Repot] 213. <hal-84154> HAL Id: hal-84154 https://hal.inia.f/hal-84154 Submitted on 5 Jul 213 HAL is a multi-disciplinay open access achive fo the deposit and dissemination of scientific eseach documents, whethe they ae published o not. The documents may come fom teaching and eseach institutions in Fance o aboad, o fom public o pivate eseach centes. L achive ouvete pluidisciplinaie HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau echeche, publiés ou non, émanant des établissements d enseignement et de echeche fançais ou étanges, des laboatoies publics ou pivés.
Semi-Lagangian simulations of the diocoton instability Seve Histoaga Eic Madaule Michel Mehenbege Jéôme Péti July 3, 213 Abstact We conside a guiding cente simulation on an annulus, following [2, 3]. Thee, the PIC method was used. We popose hee to evisit this test case by using a classical semi-lagangian appoach [4]. Fist, we obtain the consevation of the electic enegy and mass fo some adapted bounday conditions. Then we ecall the dispesion elation fom [1] and discussions on diffeent bounday conditions ae detailed. Finally, the semi-lagangian code is validated in the linea phase against analytical gowth ates given by the dispesion elation. Also we have validated numeically the consevation of electic enegy and mass. Numeical issues/difficulties due to the change of geomety can be tackled in such a test case which thus may be viewed as a fist intemediate step between a classical guiding cente simulation in a 2D catesian mesh and a slab 4D dift kinetic simulation. 1 The guiding cente model We conside the guiding cente model in pola coodinates fo ρ = ρ(t,, θ) t ρ θφ ρ + Φ θρ =, (1.1) coupled with the Poisson equation fo the potential Φ = Φ(t,, θ) 2 Φ 1 Φ 1 2 2 θφ = γρ, whee γ is a numbe which is fixed to eithe 1 o 1. Geneally, we take γ = 1 (see [1]); taking γ = 1 can be intepeted as looking fo solutions backwad in time. The domain is (, θ) = [ min, max ] [, 2π]. The initial and bounday conditions will be discussed late on. We assume fo the moment that we have peiodic bounday conditions in θ. Poposition 1.1. We define the electic enegy E(t) = Φ 2 + 1 θφ 2 ddθ, (1.2) 1
and the mass We have M(t) = ρddθ. (1.3) (i) (ii) (iii) t E(t) = 2 E(t) = [Φ Φ] =max dθ + γ ρφddθ. [Φ t Φ] =max dθ + 2γ t M(t) = Poof. Note that the Poisson equation wites [Φ θ ρ] =max dθ [Φρ θ Φ] =max dθ. ( Φ) 1 2 θφ = γρ. (1.4) We have, by integating by pats the electic enegy (1.2), in fo the fist tem and in θ fo the second tem, 1 E(t) = [Φ Φ] =max dθ Φ ( Φ)ddθ Φ 2 θφ ddθ, and get (i), by using the Poisson equation (1.4). We then have t E(t) = I 1 + γ(i 2 + I 3 ), (1.5) with I 1 = [ t (Φ Φ)] =max dθ, I 2 = Φ t ρddθ, I 3 = ρ t Φddθ. By using the guiding cente equation (1.1) and integating as above (in fo the fist tem and in θ fo the second tem), we compute I 2 = Φ( θ Φ ρ Φ θ ρ)ddθ = [Φρ θ Φ] =max dθ (Φ θ Φ)ρddθ + θ (Φ Φ)ρddθ = [Φρ θ Φ] =max dθ, since we have the elation (Φ θ Φ) = θ (Φ Φ). Using the Poisson equation (1.4), and integating as usually, we have γi 3 = t Φ ( ( Φ) + 1 2 θφ ) ddθ = [ Φ t Φ] =max dθ + I 4, 2
whee I 4 = ( Φ t Φ ) ddθ + t θ Φ 1 θφddθ. Integating as usually and using then the Poisson equation (1.4), we get ( I 4 = [Φ t Φ ) ( ] = =max min dθ Φ t Φ ) ddθ Φ t θφ 2 1 ddθ = [Φ t ( Φ ) ] =max dθ + γi 2. Replacing the fomulae fo I 1, I 2, I 3 in (1.5) gives t E(t) = [ t (Φ Φ)] =max dθ + 2γ [ Φ t Φ] =max dθ + [Φρ θ Φ] =max dθ [Φ t ( Φ ) ] =max dθ which is (ii), using the elation t (Φ Φ) = t Φ Φ + Φ t Φ. It emains to pove the fomula (iii) fo the mass. We use the guiding cente equation (1.1) and integate as usually t M(t) = t ρddθ = = = ( θ Φ ρ + Φ θ ρ) ddθ [Φ θ ρ] =max dθ + (Φ θ ρ Φ θ ρ) ddθ [Φ θ ρ] =max dθ. Poposition 1.2. We suppose Diichlet bounday conditions at min and at max : Φ(t, min, θ) = Φ(t, max, θ) =. Then the electic enegy and the mass ae constant in time: t E(t) =, t M(t) =. Poof. We get immediately the esult fom Poposition 1.1. Poposition 1.3. We suppose the following bounday conditions: (i) Diichlet bounday condition at max Φ(t, max, θ) = (ii) Inhomogeneous Neumann bounday condition at min fo the Fouie mode in θ: whee Q is a given constant. Φ(t, min, θ)dθ = Q, 3
(iii) Diichlet bounday condition at min fo the othe modes, which eads Φ(t, min, θ) = 1 Φ(t, min, θ )dθ 2π Then, the electic enegy and the mass ae constant in time: t E(t) =, t M(t) =. Poof. Fom Poposition 1.2 and using fist the Diichlet bounday condition at max (i), we get t E(t) = 2 Φ(t, min, θ) ( min t Φ(t, min, θ) + γρ(t, min, θ) θ Φ(t, min, θ)) dθ. Note that, by (iii), Φ(t, min, θ) does not depend on θ and thus, we get Φ(t, min, θ) = Φ(t, min, ) and θ Φ(t, min, θ) = which leads to We thus get, fom (ii) t E(t) = 2Φ(t, min, ) min t Φ(t, min, θ)dθ. t E(t) = 2Φ(t, min, ) min t Φ(t, min, θ)dθ = 2Φ(t, min, ) min t Q =. Fo the mass, we get similaly t M(t) = Φ(t, min, ) θ ρ(t, min, θ)dθ =. Remak 1. We do not know whethe the electic enegy and the mass emain constant in time when we conside the following bounday conditions: (i) Diichlet at max : Φ(t, max, θ) =. (ii) Neumann at min : Φ(t, min, θ) =. If we impose that ρ(t, min, ) =, we can check that we get also consevation of mass and electic enegy. In the semi-lagangian code, we have not imposed this condition (we have tied such a condition but got wose consevation fo such bounday condition on Φ); if the foot of the chaacteistic is out of the domain at a point (, θ ), we take the value at ( min, θ ), if < min o at ( max, θ ), if > max. 4
2 Study of the electostatic eigenvalue equation and discetization The eigenvalue equation which pemits to obtain the gowth ates of instability, has been deived in Davidson [1] (equation (6.28), p 296). We ecall this deivation, in ou notations. We begin with the following poblem t ρ θφ ρ + Φ θρ = 2 2 Φ Φ 2 θ 2 Φ 2 = γρ (2.6) We conside at fist an equilibium n () that does not depend on θ and coesponding potential Φ (), which satisfies ( Φ ()) = n (). As an equilibium, the couple (ρ, Φ) = (n, γφ ) satisfies (2.6). Then, we petub this equilibium and seach an appoximate solution of (2.6) in the fom We set also ρ(t,, θ) n () + εn 1 (t,, θ) Φ(t,, θ) γφ () + εγφ 1 (t,, θ) We inject these appoximations in (2.6) to get γ t n 1 θφ 1 n + Φ θ n 1 = O(ε) 2 θ Φ 2 1 = n 2 1 + O(ε) 2 2 Φ 1 Φ 1 We now look fo a paticula solution of the lineaized poblem in the fom γ t n 1 θφ 1 n + Φ θ n 1 = 2 2 Φ 1 Φ 1 2 θ 2 Φ 1 2 = n 1 n 1 (t,, θ) = n 1,l () exp(ilθ) exp( iωt/γ), Φ 1 (t,, θ) = Φ 1,l () exp(ilθ) exp( iωt/γ). This type of solution can be obtained though Laplace tansfom in time and Fouie tansfom in θ. We look fo unstable equilibium, that means that we want to have a solution with I(ω)/γ >. We obtain iω n 1,l il n 2 2 Φ 1,l Φ 1,l Φ 1,l + il Φ n 1,l = + l2 2 Φ 1,l = n 1,l 5
We expess n 1,l in tems of Φ 1,l to get an equation on Φ 1,l : ( iω + il Φ )( 2 Φ 2 1,l Φ 1,l Φ 2 1,l ) = il n Φ 1,l, (2.7) whee i = min + i and = ( max min )/N and N N. We can poceed in the numeical esolution of the poblem by making the following discetization: Φ 1,l ( i ) = φ i, Φ1,l ( i ) = φ i+1 φ i 1, 2 2 Φ 2 1,l ( i ) = φ i+1 2φ i + φ i 1 (2.8) 2 The poblem can then be witten as follows Aφ = cbφ. We then have to find the eigenvalues c of the poblem B 1 Aφ = cφ, whee c = ω. l We look fo an eigenvalue c such that c/γ has the geatest imaginay pat which should be stictly positive. 3 Diocoton instability fo an annula electon laye + l2 We conside the following initial data, min <, ρ(,, θ) = 1 + ε cos(lθ), +,, + < max, whee ε is a small paamete. In that specific case, we can do analytical computations. The linea analysis is pefomed in [1]. We thus can find an explicit solution of (2.7): Φ 1,I (), min <, Φ 1,l () = Φ 1,II (), < +, Φ 1,III (), + < max, whee Φ 1,I () = ( /) l (B( ) l + C( ) l ) (/ min) 2l + ɛ ( / min ) 2l + ɛ, Φ 1,II () = B l + C l, Φ 1,III () = ( + /) l (B( + ) l + C( + ) l ) 1 (/ max) 2l 1 ( + / max ) 2l. Hee, we have consideed Diichlet bounday condition at max. At min, we have Diichlet bounday condition, if ɛ = 1 and Neumann bounday condition if ɛ = 1. The constants B and C ae chosen so that (see [1]) and ( Φ 1,II ( ) Φ 1,I ( Φ 1,I ( ) )) + l ω lω m ( ) =, 6
with and + ( Φ 1,III ( + ) Φ 1,II ( + Φ 1,II ( + ) )) + l ω lω m ( + ) =, ( ω m () = 1 ( ) ) 2 ( ) 2 1 + ω q, 2 ω q = 2 Q ( ). 2 This leads to a 2 2 linea system fo (B, C). We look fo a solution with (B, C) (, ). This implies that the deteminant of the coesponding matix is zeo. This is the dispesion elation. In the case of bounday conditions given in Poposition 1.3, the dispesion elation can be explicitely given (see [1]) ω 2 b l ω + c l =, whee and and b l = l(1 ( / + ) 2 + ω q (1 + ( / + ) 2 ))(1 ( min / max ) 2l ) + (1 ( / + ) 2l ) (+ / max ) 2l ( min / ) 2l 1 ( min / max ) 2l (3.9) c l (1 ( min / max ) 2l ) = l 2 ω q (1 ( / + ) 2 + (ω q )( / + ) 2 )(1 ( min / max ) 2l ) lω q (1 ( min / + ) 2l )(1 ( + / max ) 2l ) + l(1 ( / + ) 2 + ω q ( / + ) 2 )(1 ( / max ) 2l )(1 ( min / ) 2l ) (1 ( + / max ) 2l )(1 ( min / ) 2l )(1 ( / + ) 2l ) (3.1) ω q = 4 Q ( ). 2 The case of the bounday conditions of Poposition 1.2 is obtained by taking in the dispesion elation. Q = ( ) 2 ( + ) 2 + 2( ) 2 ln( max / ) + 2( + ) 2 ln( + / max ), 8 ln( min / max ) Remak 2. In [1], bounday conditions of Poposition 1.3 wee consideed. Hee, we can also conside bounday conditions of Poposition 1.2 and Neumann bounday conditions (using ɛ = 1). The computations have been done using Maple. Remak 3. The system (2.6) fo γ = 1 o γ = 1 leads to the same gowth ate in the linea phase, since the dispesion elation has eal coefficients. 7
Remak 4. By using the linea appoximation ρ = n + εn 1 and Φ = γ(φ + εφ 1 ), which is supposed to be valid in the linea phase, we obtain fo bounday conditions of Remak 1 (Neumann fo all the modes at min ) t E(t) 2γ 3 Φ ( min )ε 2 n 1 (t, min, θ) θ Φ 1 (t, min, θ)dθ + O(ε 3 ), which leads a pioi to a loss of enegy consevation fo this linea appoximation, which may even gow exponentially in time, with a gowth ate dictated by the dispesion elation! On the othe hand, we can check that the enegy emains constant fo the linea appoximation when using bounday conditions of Popositions 1.2 and 1.3, as we have poven it fo the initial non linea poblem. This may explain the linea gowth ate obtained by numeical appoximations fo the electic enegy in [3]. 4 Numeical esults 4.1 Analytical and discetized dispesion elation We obtain the following esults in Tables 1, 2, 3 fo the analytical dispesion elation. We take min = 1, max = 1, γ = 1. The esults ae coheent with those given in [2]. Remak 5. We have taken the same modes as in [2]. Fo mode 5 of Table 3, we had to change + in ode to get an unstable mode. We have also tested a discetization of the dispesion elation (2.8) with Diichlet bounday conditions. The esults ae shown on Table 4. The esults ae coheent with espect to the analytical case. 4.2 Semi-Lagangian method We use a classical backwad semi-lagangian method [4]. A pedicto coecto scheme is used as time scheme, togethe with a symplectic Velet algoithm fo the computation of the chaacteistics, The intepolation is done with cubic splines. Fo the Poisson equation, finite diffeence of ode two ae used in the adial diection and Fouie in the θ diection. We denote by N (esp. N θ ) the numbe of intevals in the adial (esp. θ) diection. The time step is t. We give the gowth ate obtained by the semi-lagangian method in Tables 5, 6, 7. We obtain esults in ageement with those obtained with the dispesion elation, except fo the mode 5, when Neumann bounday conditions ae used fo the mode (o fo all the modes). Fo this specific case, the obtained value is stongly dependent on the discetization paametes. By taking N = 256, N θ = 512 and t =.25, we get I(ω) =.51 on the inteval [123, 241], which is close to the theoetical value. An example of gowth ate and of density function is given on Figue 1. The time evolution of electic enegy and of elative mass eo is given on Figues 2,3,4,5, fo diffeent bounday conditions at min (we suppose Diichlet bounday condition at max ). 8
We take hee ε =.5 and l = 3. We obseve convegence of enegy and mass consevation fo Diichlet and Neumann mode bounday conditions at min. On the othe hand, fo Neumann bounday conditions at min, these quantities ae no moe conseved in long time. We had to take a quite lage amplitude of ε to see this phenomenon. 5 Conclusion We have studied the consevation of electic enegy and mass of the continuous model; we have enlightened adhoc bounday conditions fo having such consevations. Then, we have detailed the dispesion elations fo such bounday conditions. Finally, we have given some numeical esults with a classical semi-lagangian method and validated the linea phase and the consevation of mass and electic enegy. A compaison on numeical methods (PIC vs. semi-lagangian method) with egads to consevative quantities fo example, as well as highe dimensional poblem with velocity discetization ae envisaged. Refeences [1] R. C. Davidson, Physics of non neutal plasmas, 199 [2] J. Péti, The diocoton instability in a pulsa cylindical electosphee, Astonomy & Astophysics, Febuay 5, 28. [3] J. Péti, Non-linea evolution of the diocoton instability in a pulsa electosphee: 2D PIC simulations, Astonomy & Astophysics, May 7, 29. [4] E. Sonnendücke, J. Roche, P. Betand, A. Ghizzo, The Semi-Lagangian Method fo the Numeical Resolution of the Vlasov Equation, J. Comput. Phys. 149, 21-22 (1999). 9
l + ω 2 4 5.3772986486486 +.71764354531424i 3 4 5.545674435361935 +.22672491815516i 4 4 5.7216199562166 +.298891133128847i 5 7 9 1.1478697769771 +.57879684756166i 7 6 7.931577231955 +.33764942142169i Table 1: Bounday conditions of Poposition 1.3 with Q = (Neumann fo mode and Diichlet fo othe modes) l + ω 2 4 5.37961164883512 +.6733857665534i 3 4 5.545854564457936 +.22666519518819i 4 4 5.721631794378213 +.298891598934172i 5 7 9 1.147869781983 +.57879699378142i 7 6 7.9315772312837 +.3376494214854i Table 2: Bounday conditions of Remak 1 (Neumann fo all the modes). l + ω 2 4 5.5821252541581 +.28873922755427i 3 4 5.6745316194978 +.36731589514246i 4 4 5.8344693673229 +.38481542249742i 5 7 8.4571134448886 +.323578172647371i 7 6 7.5223326524272 +.33757342425866i Table 3: Bounday conditions of Poposition 1.2 (Diichlet fo all the modes). 1
l + I(ω), fo N = 256, N = 512, N = 124 2 4 5.29293,.2993,.28985 3 4 5.3743,.36899,.36817 4 4 5.38587,.38498,.38453 5 7 8.3276,.32539,.3245 7 6 7.33852,.33783,.33765 Table 4: Numeical esolution of the dispesion elation with bounday conditions of Poposition 1.2 (Diichlet fo all the modes). l + I(ω) time inteval of validity 2 4 5.2875 [26, 55] 3 4 5.3667 [26, 72] 4 4 5.3852 [26, 51] 5 7 8.3222 [22, 62] 7 6 7.3424 [26, 61] Table 5: Gowth ate fo the semi-lagangian method with bounday conditions of Poposition 1.2 (Diichlet fo all the modes). N = N θ = 128, t =.5 l + I(ω) time inteval of validity 2 4 5.719 [93, 138] 3 4 5.2264 [33, 87] 4 4 5.2991 [34, 69] 5 7 9.12 [83, 165] 7 6 7.3338 [35, 53] Table 6: Gowth ate fo the semi-lagangian method with bounday conditions of Poposition 1.3 (Neumann fo mode and Diichlet fo the othe modes). N = N θ = 128, t =.5 l + I(ω) time inteval of validity 2 4 5.672 [89, 151] 3 4 5.2263 [32, 89] 4 4 5.2991 [34, 69] 7 6 7.3338 [35, 53] Table 7: Gowth ate fo the semi-lagangian method with bounday conditions of Poposition 1.3 (Neumann fo all the modes). N = N θ = 128, t =.5 11
Figue 1: (Left) Squae modulus of the 7 th Fouie mode of max min Φ(t,, θ)d vs time t fo bounday condition of Poposition 1.3 with Q = (Neumann fo mode and Diichlet fo othe modes). (Right) Density ρ at t = 95. Discetization paametes ae N = 512, N θ = 256 and t =.5. Figue 2: Time evolution of electic enegy (left) and elative mass eo (ight) fo Neumann and Neumann mode bounday conditions, with diffeent discetizations (N N θ t on legend). 12
Figue 3: Long time evolution of electic enegy (left) and elative mass eo (ight) fo Neumann bounday conditions, with diffeent discetizations (N N θ t on legend). Figue 4: Long time evolution of electic enegy (left) and elative mass eo (ight) fo Neumann mode bounday conditions, with diffeent discetizations (N N θ t on legend). Figue 5: Long time evolution of electic enegy (left) and elative mass eo (ight) fo Diichlet bounday conditions, with diffeent discetizations (N N θ t on legend). 13