Minimization formulation of a bi-kinetic sheath

Transcription

1 Minimization formulation of a bi-kinetic sheath Mehdi Badsi (PHD), B. Després and Martin Campos-Pinto Laboratoire Jacques-Louis Lions UPMC-Paris VI, CNRS UMR 7598, Paris, France Sponsors: ANR Chrome, FRFCM 29 septembre 2014 Minimization formulation of a bi-kinetic sheath p. 1 / 25

2 Section 1 Minimization formulation of a bi-kinetic sheath p. 2 / 25

3 What is a sheath? Sheath = boundary layer for a quasi-neutral plasma (i +, e ) facing a ( isolated, metallic) wall. Important parameter ε λ D << 1. Physics : due to the high mobility ratio (m i /me 2000 >> 1), the local quasi-neutrality in the core is highly violated at the wall, and the plasma continuously flows towards the wall. This is kind of a paradox since there is no exterior forces, and the trivial solution n i n e = φ = 0 is not the physical one. Terminology : core, Debye length, pre-sheath, sheath, Boltzmanian e (n e = n 0 e φ ), Bohm criterion, floating potential, ambipolar hypothesis,... From Stangeby : The Plasma Boundary of Magnetic Fusion Devices (2011). Fundamental for applications : Tokamaks (ITER), ionic engine (satellites),... Minimization formulation of a bi-kinetic sheath p. 2 / 25

4 From Chen The physical solution is n i = n i2 > n e. Here E = φ 0 Condition on the Mach number : M > 1. Kinetic Bohm criterion : 0 f given i v 2 dv < 1. Minimization formulation of a bi-kinetic sheath p. 3 / 25

5 Some references Physics : Chen ( to Plasma Physics 74 ) ; Review by Riemann (The Bohm criterion and sheath formation 91 ), Baalrud-Hegna (Kinetic theory of the presheath and the Bohm criterion, Manfredi-Devaux, Magnetized plasma-wall transition. Consequences for wall sputtering and erosion, Stangeby The Plasma Boundary of Magnetic Fusion Devices, Math : Raviart- Greengard, (A Boundary-Value problem for the stationary Vlasov-Poisson equations : The Plane Diode, 1990) P-H. Maire, Fluid models with a plasma and b.c., PHD thesis 96 (with Sentis and Golse) H. Guillard, M 1 in fluid models at the SOL (2012). Additional references : the Milne problem Marschak 47 : variational equation for the mean field. Modern treatment in Golse and al,... Minimization formulation of a bi-kinetic sheath p. 4 / 25

6 Our angle of attack Do not use Maxwellian e which ends up in a compatibility issue with the ambipolar hypothesis, and usually with different models in different parts of the domain. = Following Stangeby (see below), use truncated Maxwellians for e Do not use fluid equations for ions : = start from bi-kinetic (Vlasov) + Poisson equation Study exact stationary solutions with convenient B.C., eliminate everything f i, f e, n i, n e in function of the electric potential φ Solve the equation for φ Minimization formulation of a bi-kinetic sheath p. 5 / 25

7 Section 2 Minimization formulation of a bi-kinetic sheath p. 6 / 25

8 Equations Domain is (x, v) [0, 1] R v x f i φ (x) v f i = 0, Vlasov i + v x f e + m i φ m (x) v f e = 0, e Vlasov e ε 2 φ (x) f i dv + f edv = 0, Poisson, (ε λ D ) vfi dv vf edv = 0, Ambipolarity, (J = te = 0) f i (0, v) = f given i (v), v > 0, f i (1, v) = 0, v < 0, f e(0, v) = n 0 me m i e me v 2 2m i, v > 0, Scaled Maxwellian at entrance f e(1, v) = αf e(1, v), v < 0, Material dependent reemission of e with parameter 0 α = α given < 1, φ(0) = 0, φ(1) = φ w, Just a convention, Floating potential, unknown at this stage. Additional condition is E = φ 0 Warning : n 0 and φ w are unknowns at this level. Minimization formulation of a bi-kinetic sheath p. 6 / 25

9 Trajectories Assume the physical condition E = φ > 0 e i + Char. : 1 2 v 2 m i φ(x) = cst m e Char. : 1 2 v 2 + φ(x) = cst Sep. : 1 2 v 2 m i φ(x) = m i φ m e m w e Sep. : 1 2 v 2 + φ(x) = 0 Minimization formulation of a bi-kinetic sheath p. 7 / 25

10 ( v Notice for example that me 2 2 m ) i me φ m i R e me 2m i dv = 2πe φ. One gets after more computations n e(x) = R fedv = n 0 ( 2πe φ(x) (1 α) γ e = R vfedv = (1 α)n 0 mi m e e φw, 2φ w e v 2 2 Formulas ) v v 2 +2φ(x) dv, n i (x) = R f i dv = 0 f given v i (v) dv, v 2 2φ(x) γ i = R vf i dv = 0 f given i (v)vdv. One still has two degrees of freedom. In order to simplify the mathematical analysis, we consider two equations { γi = γ e Ambipolarity, n i (0) = n e(0) Local neutrality at entrance. Elimination of n 0 yields the resulting non linear equation with φ w only γ e n = γ i e(0) n i (0) Minimization formulation of a bi-kinetic sheath p. 8 / 25

11 Equation for φ w The equation writes W(φ w ) = b, < φ w 0, with ( mi ) W(ψ) = f given i (v)dv e ψ + m e 0 0 f given i (v)vdv e v 2 2ψ 2 dv and b = 2π 1 α 0 f given i (v)vdv. A first result : The function W being monotone (increasing) there exists a unique φ w if and only 0 f given i (v)vdv 0 f given i (v)dv 2m i 1 α πm e 1 + α. Corollary : No solutions for α = 1 (that would imply Boltzmanian e ). Corollary : If the is a =, then φ w = 0 and the solution is trivial. Corollary : One has the bound [ ln ( 2π ) ln ( 2 + mi m e 0 0 f given (v)dv i f given (v)vdv i )] ln(1 α) φ w 0. Minimization formulation of a bi-kinetic sheath p. 9 / 25

12 Non linear Poisson equation Now that φ w and n 0 are uniquely determined, one can solves { ε 2 φ (x) = n i (φ) n e(φ) := q (φ), φ(0) = 0 and φ(1) = φ w, where the potential is q(φ) = f given i (v)v v 2 2φdv 0 ( 2πe +n φ 2 0 (1 α) 2 v ) v 2 + 2φdv 0. 2φw e v Set with the space 1 J ε(φ) = ε 2 φ (x) q(φ(x))dx, φ V V = { v H 1 (0, L), 0 = v(0) v v(l) = φ w }. Proposition (evident) : There exists a minimum of the problem J ε(φ ε) J ε(φ), φ ε V, φ V. Minimization formulation of a bi-kinetic sheath p. 10 / 25

13 Preliminary properties of the minimizer Since 0 q by construction, the inequality J ε(φ ε) J(xφ w ) yields a bound on the H 1 norm : φ ε H 1 (0,1) = O ( ε 1). Minimizers are non increasing 0 x y 1 = 0 φ ε(x) φ ε(y) φ w. This is proved by comparison methods. If 0 > x > 1 = 0 > φ ε(x) > φ w, the minimizer is C 2 solution of the non linear Poisson equation. Since now ε 2 φ ε (x) = n i n e, it yields n i n e H 1 (0,1) = O(ε) which shows quasi-neutrality with a boundary layer (equal to the physical sheath) of size ε in a weak norm. Minimization formulation of a bi-kinetic sheath p. 11 / 25

14 Bohm criterion The interesting physical situation is when n i n e. At entrance (x = 0) (n i n e) (φ) = q (φ) = q (0)φ + O(φ 2 ) Since φ 0, a necessary condition is q (0) 0 (Bohm criterion) Complete Bohm crit. : 0 f given i (v) dv 2π + (1 α) 2φw v 2 0 f given i (v)dv e v2 2 dv v 2. 2π (1 α) 2φw e v2 2 dv Proposition : If this inequality is true, then n i n e everywhere in [0, 1]. Notice the simplified Bohm criterion 0 f given i (v) dv v 2 0 f given i (v)dv < 1. Minimization formulation of a bi-kinetic sheath p. 12 / 25

15 Compatiblity issue We need at the same time two inequalities. First one says : low velocities 0 f given i (v)vdv φ w : 0 f given i (v)dv < 2m i 1 α πm e 1 + α. Second one says : high velocities Simplified Bohm criterion : Lemma : There exists non trivial f given i conditions if 1 < 2mi 1 α πm e 1+α. 0 f given i (v) dv v 2 0 f given i (v)dv < 1. L 1 (R + ) L (R + ) that satisfy all Proof of the necessary condition : based on double Cauchy-Schwarz inequality ( 4 ( f given i (v)dv) f given i (v) dv ) 2 ( ) 2 f given i (v)vdv 0 ( 0 ) ( f given i (v)dv 0 0 v 0 f given i (v) dv ) ( 2 v 2 f given i (v)vdv). 0 Minimization formulation of a bi-kinetic sheath p. 13 / 25

16 Mobility ratio Quite fortunately the high mass ratio 1 << m i 3000 (for Z = 1.5) yields m e 2m i 43.7 πm e Proposition : There exists a important interval in which sheath solutions exist 0 α α c 0.93 These are also exact stationary solutions of bi-kinetic+maxwell equations. Minimization formulation of a bi-kinetic sheath p. 14 / 25

17 Summary of the claim Claim : Assume 0 α α c, ρ 0 = n i (0) n 0 (0) = 0, f given i I ad (ρ 0, α) and ε > 0. Assume the Bohm criterion. Then the Vlasov-Poisson-Ampere formulation of the sheath is well-posed in the following sense : There exists a unique strictly concave and decreasing φ ε C 2 [0, 1] solution of the non linear Poisson equation with b.c. φ(0) = 0 and φ(1) = φ w < 0, where the equation on φ w does not depend on ε. The densities f i, f e ( L 1 L ) ( (0, 1) R +) are weak solutions : note that f e is not a strong solution since it is a truncated Maxwellian. There is a sheath (i.e. a boundary layer) : φ ε H 1 (0,1) = O ( ε 1) and n i n e H 1 (0,1) = O(ε). The local charge is positive : ni(x) > n e(x) for x (0, 1]. Minimization formulation of a bi-kinetic sheath p. 15 / 25

18 Section 3 Minimization formulation of a bi-kinetic sheath p. 16 / 25

19 Algorithm Numerical algorithm developed by Mehdi. Choose a f given i that satisfies all conditions and inequalities. Use high order quadratures to compute q. Solve the equation on φ w Use a gradient algorithm to compute φ. Converge until φ n+1 φ n ɛ. Minimization formulation of a bi-kinetic sheath p. 16 / 25

20 The potential q for the chosen data Here φ w 2.4. Seems q is convex in this case : characterization still an open problem. Minimization formulation of a bi-kinetic sheath p. 17 / 25

21 Ion distribution : ε = 1, 0.1, 0.01 and zoom Note ε is a scaling factor : y = x ε Minimization formulation of a bi-kinetic sheath p. 18 / 25

22 Electron distribution : ε = 1, 0.1, 0.01 and zoom Minimization formulation of a bi-kinetic sheath p. 19 / 25

23 n i and n e ε = 1 ε = 0.1 ε = 0.01 ε = 0.01 (zoom) Minimization formulation of a bi-kinetic sheath p. 20 / 25

24 φ and u = fi vdv fi dv φ u ε = 1 ε = 0.1 ε = 0.01 Minimization formulation of a bi-kinetic sheath p. 21 / 25

25 Influence of the reemission factor Seems important in view of engineering and applications α (n i, n e) is increasing function. Minimization formulation of a bi-kinetic sheath p. 22 / 25

26 Section 4 Minimization formulation of a bi-kinetic sheath p. 23 / 25

27 What is a RF sheath? It is an new problem discussed in plasma physics community : Heuraux (U. Nancy), Colas (CEA-IRFM), Myra-D ippolito,.... Heating of a fusion plasma can be described by a radio-frequency wave φ rf cos ωt. PDE model is linear Maxwell s equations by with the cold plasma dielectric tensor, but......sheath modify strongly the boundary conditions. The boundary condition might be non linear due to the sheath. Minimization formulation of a bi-kinetic sheath p. 23 / 25

28 A fundamental simple idea Boltzmannian electrons are n e = n 0 e φ. A RF wave modifies the electric potential φ tot = φ + φ rf cos(ωt) so that averaging in time modify the apparent electronic density since 1 T e φ+φ rf cos(ωt) dt = e φ 1 T e φ rf cos(ωt) dt = e φ I 0 (φ rf ), T = 2π T 0 T 0 ω where the rectification factor I 0 (φ rf ) 1 is the modified Bessel function of the first kind. Finally n e = n apparent 0 e φ, n apparent 0 = n 0 I 0 (φ rf ). = It is possible to add the multiplicative rectification factor in the previous equations. = At first order of approximation : φ w and φ are the same. I do not know if this conclusion is physically correct for RF sheath. Minimization formulation of a bi-kinetic sheath p. 24 / 25

29 A sheath solution has been constructed, starting from stationary solutions of two species Vlasov + Poisson. A well-posed non linear minimization equation shows up after reduction for the mean field (Poisson). It is reminiscent of the seminal Marshak s proposition of a variational formulation of the Milne s problem. is very robust. More to be done to compare with physical solutions. This can be a basis for further mathematical developments on kinetic sheath solutions. Preprint soon online. Minimization formulation of a bi-kinetic sheath p. 25 / 25