Full-wave simulation of tokamak plasma heating

Transcription

1 Full-wave simulation of tokamak plasma heating Institut Elie Cartan de Nancy, UMR 7502, Université de Lorraine, CNRS & INRIA (project-team CALVI). B.P , Vandœuvre les Nancy Cedex, France. Université Pierre et Marie Curie, 21 juin 2012

2 Joint work with: Jean R. Roche (IECN) Takashi Hattori (IECN) Christoph Kirsch (ex-iecn) Pierre Bertrand (Institut Jean Lamour, U. Lorraine) Jean-Hugues Chatenet (Institut Jean Lamour, U. Lorraine) Yves Peysson (CEA Cadarache)

3 Outline

4 Outline 1 Physical and mathematical modelling 1 Physical assumptions 2 Mixed and augmented formulations 3 Spectral properties of the dielectric tensor 4 Well-posedness of the variational formulations 5 Discretisation (Fourier Finite element) 6 A first numerical test

5 Outline 1 Physical and mathematical modelling 1 Physical assumptions 2 Mixed and augmented formulations 3 Spectral properties of the dielectric tensor 4 Well-posedness of the variational formulations 5 Discretisation (Fourier Finite element) 6 A first numerical test 2 The issue of accessibility 1 Electromagnetic dispersion relation 2 Accessibility criteria 3 More physical test cases

6 Outline 1 Physical and mathematical modelling 1 Physical assumptions 2 Mixed and augmented formulations 3 Spectral properties of the dielectric tensor 4 Well-posedness of the variational formulations 5 Discretisation (Fourier Finite element) 6 A first numerical test 2 The issue of accessibility 1 Electromagnetic dispersion relation 2 Accessibility criteria 3 More physical test cases 3 In progress and to do

7 Part One: Physical and mathematical modelling

8 The name of the game

9 The name of the game (simplified)

10 Physical assumptions Strong external, time-invariant magnetic field B ext. Harmonic time dependence of self-consistent electromagnetic fields (of the form e iωt, with ω > 0).

11 Physical assumptions Strong external, time-invariant magnetic field B ext. Harmonic time dependence of self-consistent electromagnetic fields (of the form e iωt, with ω > 0). Plasma response described by the matrix: S id 0 K = id S 0 + i γ e ε 0 0 P 0 ω }{{}}{{} ε r Landau term (1) Stix frame: third coordinate B ext.

12 Physical assumptions Strong external, time-invariant magnetic field B ext. Harmonic time dependence of self-consistent electromagnetic fields (of the form e iωt, with ω > 0). Plasma response described by the matrix: S id 0 K = id S 0 + i γ e ε 0 0 P 0 ω }{{}}{{} ε r Landau term (1) Stix frame: third coordinate B ext. Cold plasma approximation of ε r is used, valid in the lower hybrid (LH) regime ω ci ω ω ce. (More on this later.)

13 Physical assumptions Strong external, time-invariant magnetic field B ext. Harmonic time dependence of self-consistent electromagnetic fields (of the form e iωt, with ω > 0). Plasma response described by the matrix: S id 0 K = id S 0 + i γ e ε 0 0 P 0 ω }{{}}{{} ε r Landau term (1) Stix frame: third coordinate B ext. Cold plasma approximation of ε r is used, valid in the lower hybrid (LH) regime ω ci ω ω ce. (More on this later.) Both Landau and collisional damping present.

14 Electromagnetic waves: model problem The domain Ω is a torus (tokamak plasma volume). curl curl E (ω/c) 2 KE = 0 in Ω, (2) div(ke) = 0 in Ω. (3)

15 Electromagnetic waves: model problem The domain Ω is a torus (tokamak plasma volume). curl curl E (ω/c) 2 KE = 0 in Ω, (2) div(ke) = 0 in Ω. (3) Boundary condition: Ω = Γ A Γ C = antenna + the rest E n = 0 on Γ C. (Phoney?) (4)

16 Electromagnetic waves: model problem The domain Ω is a torus (tokamak plasma volume). curl curl E (ω/c) 2 KE = 0 in Ω, (2) div(ke) = 0 in Ω. (3) Boundary condition: Ω = Γ A Γ C = antenna + the rest E n = 0 on Γ C. (Phoney?) (4) On Γ A several possible models... Neumann: curl E n = iωµ 0 J s (5) Dirichlet: E n = E A n (6) Silver Müller: (E + i (c/ω) curl E n) n = E n (7) where J s = J s (0, 0, 1) is a current source equivalent to the electric field E A excited at the antenna.

17 Variational Formulations (1) Neumann BC on Γ A (fss) variational space for (2): H C 0 (curl; Ω) = {v H(curl; Ω) : v n = 0 on Γ C} Eq. (3): constraint mixed (unaugmented) formulation: Find (E, p) H C 0 (curl; Ω) H1 0 (Ω) or H1 0,C (Ω) such that a pl (E, F ) + b pl (F, p) = L(F ), F H C 0 (curl; Ω); (8) b pl (E, q) = 0, q H0,(C) 1 (Ω), (9) where H0,C 1 (Ω) = {w H1 (Ω) : w = 0 on Γ C } and a pl (E, F ) := (curl E curl F ) (ω/c) 2 (KE F ), b pl (F, p) := (KF grad p), L(F ) := iωµ 0 J s, F H 1/2 (Γ A ),H 1/2 (Γ A ). Above, ( ) = scalar product in L 2 (Ω).

18 Variational Formulations (2) Other approaches: consider the space X C N (K; Ω) := HC 0 (curl; Ω) H(div K; Ω) Mixed augmented formulation: Find (E, p) X C N (K; Ω) L2 (Ω) such that a s (E, F ) + b(f, p) = L(F ), F X C N (K; Ω); (10) b(e, q) = 0, q L 2 (Ω), (11) where a s (E, F ) := (curl E curl F ) + s (div KE div KF ) (ω/c) 2 (KE F ), b(f, p) := (div KF p).

19 Variational Formulations (2) Other approaches: consider the space X C N (K; Ω) := HC 0 (curl; Ω) H(div K; Ω) Mixed augmented formulation: Find (E, p) X C N (K; Ω) L2 (Ω) such that a s (E, F ) + b(f, p) = L(F ), F X C N (K; Ω); (10) b(e, q) = 0, q L 2 (Ω), (11) where a s (E, F ) := (curl E curl F ) + s (div KE div KF ) (ω/c) 2 (KE F ), b(f, p) := (div KF p). Augmented formulation: Find E X C N (K; Ω) such that a s (E, F ) = L(F ), F X C N (K; Ω). (12)

20 More on plasma response (1) Experimentalists / coupling with other codes maps of external magnetic field B ext (x), density n s (x) and temperature T s (x) of each species s (electrons e, various ions).

21 More on plasma response (1) Experimentalists / coupling with other codes maps of external magnetic field B ext (x), density n s (x) and temperature T s (x) of each species s (electrons e, various ions). Compute maps of plasma and cyclotron frequencies: n s qs 2 ω ps =, ω cs = q s B ext, ε 0 m s m s then plasma parameter and collision frequency Λ = 4π 3 n e ( ) ε0 k B T 3/2 e n e qe 2 ; ν c = ω pe ln Λ. Λ

22 More on plasma response (2) Setting α(x) = ω + iν c (x), the entries of ε r are: S = 1 α ωps 2 ω α s 2 ωcs 2 (13) D = 1 ω cs ωps 2 ω α s 2 ωcs 2 (14) P = 1 1 ωps 2 (15) ωα s

23 More on plasma response (2) Setting α(x) = ω + iν c (x), the entries of ε r are: S = 1 α ωps 2 ω α s 2 ωcs 2 (13) D = 1 ω cs ωps 2 ω α s 2 ωcs 2 (14) P = 1 1 ωps 2 (15) ωα Landau damping rate (phenomenological): ( ) π ω 2 n e qe 2 m 1/2 e γ e = 4(k B T e ) 3/2 k 3 exp ω2 m e 4k B T e k 2. (16) Relies on plane wave analysis: k = k B ext / B ext. s

24 More on plasma response (2) Setting α(x) = ω + iν c (x), the entries of ε r are: S = 1 α ωps 2 ω α s 2 ωcs 2 (13) D = 1 ω cs ωps 2 ω α s 2 ωcs 2 (14) P = 1 1 ωps 2 (15) ωα Landau damping rate (phenomenological): ( ) π ω 2 n e qe 2 m 1/2 e γ e = 4(k B T e ) 3/2 k 3 exp ω2 m e 4k B T e k 2. (16) Relies on plane wave analysis: k = k B ext / B ext. If ν c > 0 or γ e > 0, K is not hermitian. s

25 More on plasma response (2) Setting α(x) = ω + iν c (x), the entries of ε r are: S = 1 α ωps 2 ω α s 2 ωcs 2 (13) D = 1 ω cs ωps 2 ω α s 2 ωcs 2 (14) P = 1 1 ωps 2 (15) ωα Landau damping rate (phenomenological): ( ) π ω 2 n e qe 2 m 1/2 e γ e = 4(k B T e ) 3/2 k 3 exp ω2 m e 4k B T e k 2. (16) Relies on plane wave analysis: k = k B ext / B ext. If ν c > 0 or γ e > 0, K is not hermitian. Actually, a relief! s

26 Spectral properties of K (1) Eigenvalues and uniform (in x), unitary eigenvector basis λ 1 = S + D, λ 2 = S D, λ 3 = P + i γ e ε 0 ω := P L (17) i u 1 = 1 2, u 2 = i, u 3 = 0. (18)

27 Spectral properties of K (1) Eigenvalues and uniform (in x), unitary eigenvector basis λ 1 = S + D, λ 2 = S D, λ 3 = P + i γ e ε 0 ω := P L (17) i u 1 = 1 2, u 2 = i, u 3 = 0. (18) There always holds: Im λ i (x) > 0, hence: Im(v K(x)v) ζ(x) v v, v C 3. Suitable bounds on n s, T s, B ext : v C 3, x Ω, v K(x)v Im(v K(x)v) ζ v 2. (19)

28 Spectral properties of K (2) On the contrary, in the LH frequency range: Re λ 1 (x) > 0, Re λ 2,3 (x) < 0. Without the imaginary parts (damping), the well-posedness proof breaks down!!! (Still OK if ν c = 0 or γ e = 0, but not both.)

29 Well-posedness of the variational formulations (1) Lemma (Sébelin, Bénilan 97) Let V H two Hilbert spaces, with continuous embedding. Let a(, ) be a sesquilinear form on V satisfying the inequalities: Re a(v, v) α v 2 V λ v 2 H, Im a(v, v) γ v 2 H, (20) for some strictly positive constants α, γ, λ and all v V. Then a(, ) is V -elliptic, i.e., a(v, v) α v 2 V.

30 Well-posedness of the variational formulations (1) Lemma (Sébelin, Bénilan 97) Let V H two Hilbert spaces, with continuous embedding. Let a(, ) be a sesquilinear form on V satisfying the inequalities: Re a(v, v) α v 2 V λ v 2 H, Im a(v, v) γ v 2 H, (20) for some strictly positive constants α, γ, λ and all v V. Then a(, ) is V -elliptic, i.e., a(v, v) α v 2 V. With the bound (19): Im(Kv v) ζ v L 2, v L 2 (Ω). a pl (u, v) = (curl u curl v) (ω/c) 2 (Ku v) satisfies (20) on V = H(curl; Ω), with H = L 2 (Ω);

31 Well-posedness of the variational formulations (1) Lemma (Sébelin, Bénilan 97) Let V H two Hilbert spaces, with continuous embedding. Let a(, ) be a sesquilinear form on V satisfying the inequalities: Re a(v, v) α v 2 V λ v 2 H, Im a(v, v) γ v 2 H, (20) for some strictly positive constants α, γ, λ and all v V. Then a(, ) is V -elliptic, i.e., a(v, v) α v 2 V. With the bound (19): Im(Kv v) ζ v L 2, v L 2 (Ω). a pl (u, v) = (curl u curl v) (ω/c) 2 (Ku v) satisfies (20) on V = H(curl; Ω), with H = L 2 (Ω); a s (u, v) = a pl (u, v) + s (div Ku div Kv) satisfies (20) on V = H(curl, div K; Ω) if Re s > 0 and Im s 0.

32 Well-posedness of the variational formulations (1) Lemma (Sébelin, Bénilan 97) Let V H two Hilbert spaces, with continuous embedding. Let a(, ) be a sesquilinear form on V satisfying the inequalities: Re a(v, v) α v 2 V λ v 2 H, Im a(v, v) γ v 2 H, (20) for some strictly positive constants α, γ, λ and all v V. Then a(, ) is V -elliptic, i.e., a(v, v) α v 2 V. With the bound (19): Notice: Im(Kv v) ζ v L 2, v L 2 (Ω). a pl (u, v) = (curl u curl v) (ω/c) 2 (Ku v) satisfies (20) on V = H(curl; Ω), with H = L 2 (Ω); a s (u, v) = a pl (u, v) + s (div Ku div Kv) satisfies (20) on V = H(curl, div K; Ω) if Re s > 0 and Im s 0. coercivity ( Fredholm alternative); no BC required.

33 Well-posedness of the variational formulations (2) Augmented formulation (12) well-posed on the closed subspace X C N (K; Ω) by Lax Milgram.

34 Well-posedness of the variational formulations (2) Augmented formulation (12) well-posed on the closed subspace X C N (K; Ω) by Lax Milgram. (K grad u grad v) elliptic on H0 1 (Ω) well-posedness of K u := div(k grad u) = f H 1 (Ω).

35 Well-posedness of the variational formulations (2) Augmented formulation (12) well-posed on the closed subspace X C N (K; Ω) by Lax Milgram. (K grad u grad v) elliptic on H0 1 (Ω) well-posedness of K u := div(k grad u) = f H 1 (Ω). "Warped" Hodge decomposition for all u L 2 (Ω): u = grad ϕ + u T, with: ϕ H 1 0 (Ω), div Ku T = 0.

36 Well-posedness of the variational formulations (2) Augmented formulation (12) well-posed on the closed subspace X C N (K; Ω) by Lax Milgram. (K grad u grad v) elliptic on H0 1 (Ω) well-posedness of K u := div(k grad u) = f H 1 (Ω). "Warped" Hodge decomposition for all u L 2 (Ω): u = grad ϕ + u T, with: ϕ H 1 0 (Ω), div Ku T = 0. inf-sup condition for b pl (v, p) = (Kv grad p) on H 0 (curl; Ω) H 1 0 (Ω)

37 Well-posedness of the variational formulations (2) Augmented formulation (12) well-posed on the closed subspace X C N (K; Ω) by Lax Milgram. (K grad u grad v) elliptic on H0 1 (Ω) well-posedness of K u := div(k grad u) = f H 1 (Ω). "Warped" Hodge decomposition for all u L 2 (Ω): u = grad ϕ + u T, with: ϕ H 1 0 (Ω), div Ku T = 0. inf-sup condition for b pl (v, p) = (Kv grad p) on H 0 (curl; Ω) H 1 0 (Ω) and HC 0 (curl; Ω) H1 0,(C) (Ω)

38 Well-posedness of the variational formulations (2) Augmented formulation (12) well-posed on the closed subspace X C N (K; Ω) by Lax Milgram. (K grad u grad v) elliptic on H0 1 (Ω) well-posedness of K u := div(k grad u) = f H 1 (Ω). "Warped" Hodge decomposition for all u L 2 (Ω): u = grad ϕ + u T, with: ϕ H 1 0 (Ω), div Ku T = 0. inf-sup condition for b pl (v, p) = (Kv grad p) on H 0 (curl; Ω) H 1 0 (Ω) and HC 0 (curl; Ω) H1 0,(C) (Ω) inf-sup condition for b(v, p) = (div Kv p) on X N (K; Ω) L 2 (Ω) and X C N (K; Ω) L2 (Ω)

39 Well-posedness of the variational formulations (2) Augmented formulation (12) well-posed on the closed subspace X C N (K; Ω) by Lax Milgram. (K grad u grad v) elliptic on H0 1 (Ω) well-posedness of K u := div(k grad u) = f H 1 (Ω). "Warped" Hodge decomposition for all u L 2 (Ω): u = grad ϕ + u T, with: ϕ H 1 0 (Ω), div Ku T = 0. inf-sup condition for b pl (v, p) = (Kv grad p) on H 0 (curl; Ω) H 1 0 (Ω) and HC 0 (curl; Ω) H1 0,(C) (Ω) inf-sup condition for b(v, p) = (div Kv p) on X N (K; Ω) L 2 (Ω) and X C N (K; Ω) L2 (Ω) Mixed unaugmented (8) (9) and mixed augmented (10) (11) formulations well-posed

40 Well-posedness of the variational formulations (2) Augmented formulation (12) well-posed on the closed subspace X C N (K; Ω) by Lax Milgram. (K grad u grad v) elliptic on H0 1 (Ω) well-posedness of K u := div(k grad u) = f H 1 (Ω). "Warped" Hodge decomposition for all u L 2 (Ω): u = grad ϕ + u T, with: ϕ H 1 0 (Ω), div Ku T = 0. inf-sup condition for b pl (v, p) = (Kv grad p) on H 0 (curl; Ω) H 1 0 (Ω) and HC 0 (curl; Ω) H1 0,(C) (Ω) inf-sup condition for b(v, p) = (div Kv p) on X N (K; Ω) L 2 (Ω) and X C N (K; Ω) L2 (Ω) Mixed unaugmented (8) (9) and mixed augmented (10) (11) formulations well-posed Equivalence of various formulations

41 Dimension reduction (1)

42 Dimension reduction (2) The domain Ω is described by cylindrical coordinates (R, Z, φ), which are related to the cartesian ones (x, y, z) by x = R cos φ, y = R sin φ, z = Z. The 3D problem is reduced to a series of 2D ones by expanding all fields u(r, Z, φ) as Fourier series in φ: u(r, Z, φ) = ν Z u ν (R, Z ) e iνφ, (21) where the coefficients u ν (R, Z ) are defined on a meridian section Ω := {(R, Z ) R 2 (R, Z, π/2) Ω}. Furthermore: curl u = ν Z (curl ν u ν ) e iνφ, div u = ν Z(div ν u ν ) e iνφ, (22) where curl ν, div ν are obtained by setting φ iν in the expressions of curl ν, div ν in cylindrical coordinates.

43 Dimension reduction (3) Simplifying assumptions: External magnetic field B ext (x) = B ext (R, Z ) e φ. Particles constrained to move and diffuse freely along field lines of B ext

44 Dimension reduction (3) Simplifying assumptions: External magnetic field B ext (x) = B ext (R, Z ) e φ. Particles constrained to move and diffuse freely along field lines of B ext plasma properties averaged in φ n s, T s, hence S, D, P, are independent of φ.

45 Dimension reduction (3) Simplifying assumptions: External magnetic field B ext (x) = B ext (R, Z ) e φ. Particles constrained to move and diffuse freely along field lines of B ext plasma properties averaged in φ n s, T s, hence S, D, P, are independent of φ. The r.h.s. J s has a single Fourier mode: J s = J ν 0 s assign a definite value to k (x) = ν 0 /R in (16) the whole matrix K is independent of φ. As a consequence: e iν 0φ div Ku = ν Z(div ν Ku ν ) e iνφ, and, by orthogonality: a s (u, v) = ν Z a s,(ν) (u ν, v ν ), b(v, p) = ν Z b (ν) (v ν, p ν ).

46 Discretisation: Fourier Finite Element Method A 2D problem is obtained for each pair of Fourier coefficients (E ν, p ν ), ν Z, set in the spaces X C N (K; Ω) L 2 ( Ω), traces on Ω of the 3D spaces: a s,(ν) (E ν, F ) + b (ν) (F, p ν ) = L ν (F ), F X C N (K; Ω); (23) b (ν) (E ν, q) = 0, q L 2 ( Ω). (24) The r.h.s. and the solution are zero except when ν = ν 0.

47 Discretisation: Fourier Finite Element Method A 2D problem is obtained for each pair of Fourier coefficients (E ν, p ν ), ν Z, set in the spaces X C N (K; Ω) L 2 ( Ω), traces on Ω of the 3D spaces: a s,(ν) (E ν, F ) + b (ν) (F, p ν ) = L ν (F ), F X C N (K; Ω); (23) b (ν) (E ν, q) = 0, q L 2 ( Ω). (24) The r.h.s. and the solution are zero except when ν = ν 0. We use a Taylor-Hood P 2 -iso-p 1 finite element (continuous fields and multipliers) to discretize the MAVF (23) (24). ( As,(ν) B ) ( ) ( ) (ν) E h Lν ν B (ν) 0 p ν 2h =, (25) 0 where E ν h and p ν 2h nodal values of E ν and p ν. Notice that the matrix of the system (25) is complex-valued and non-hermitian.

48 Essential boundary conditions In a local basis {n, τ, e φ = n τ }, we have: E 1 B 1 0 E n = EB 2 0 = E 3 EB 3 B. 0 EB 2 The boundary condition E n = 0 on Γ C becomes: (EB 2, E B 3 ) = (0, 0). (26) Denoting P = transformation matrix between {e R, e Z, e φ } and {n, τ, n τ } on Γ C nodes, identity elsewhere: v M u = v f v B (P MP) u B = v B (P f ). Thus, one sets (E 2 B, E 3 B ) = (0, 0) in P MP and P f. Generalisation to E n = E A n on Γ A obvious.

49 Numerical test We consider a 2D test problem analogous to (23) (24), with K = I (and s = 1 in the augmented formulation): curl ν curl ν E η 2 E = f in Ω, (27) div ν E = g in Ω, (28) E n = 0 on Γ C, (29) curl ν E n = h on Γ A. (30) The r.h.s. f, g, h are chosen so that the exact solution is E(R, Z ) = (R R 0, Z, 0) H 0 (curl, Ω) H(div, Ω) and ν = 220. This gives rise to six independent tests, for the real and imaginary parts of the three coordinates of E.

50 Figure: The real part of the test solution E.

51 Figure: The imaginary part of the test solution E.

52 Part Two: The issue of accessibility

53 Dispersion relation (1) We look for plane wave solutions of the form Ee i(k x ωt) to Maxwell s equations in the plasma. we set E = Ee ik x in the propagation equation (2): curl curl E (ω/c) 2 KE = 0. No BC is assumed (propagation in an infinite medium).

54 Dispersion relation (1) We look for plane wave solutions of the form Ee i(k x ωt) to Maxwell s equations in the plasma. we set E = Ee ik x in the propagation equation (2): curl curl E (ω/c) 2 KE = 0. No BC is assumed (propagation in an infinite medium). Therefore: ik (ik E) (ω/c) 2 K E = 0. Multiplying by (c/ω) 2, we get the dispersion relation n (n E) + KE = 0. (31) where n = c ω k is the vector index of refraction.

55 Dispersion relation (2) Take a (local) Stix frame (e 1, e 2, e 3 ), with e 3 in the direction of B ext and k e 2 = 0. Thus: n = (n, 0, n ).

56 Dispersion relation (2) Take a (local) Stix frame (e 1, e 2, e 3 ), with e 3 in the direction of B ext and k e 2 = 0. Thus: n = (n, 0, n ). The dispersion relation (31) is rewritten as: ME = 0, (32) where: M = S n 2 id n n id S n 2 n2 0 n n 0 P L n 2.

57 Dispersion relation (2) Take a (local) Stix frame (e 1, e 2, e 3 ), with e 3 in the direction of B ext and k e 2 = 0. Thus: n = (n, 0, n ). The dispersion relation (31) is rewritten as: ME = 0, (32) where: M = S n 2 id n n id S n 2 n2 0 n n 0 P L n 2. To have a non-trivial solution, there must hold det M = 0: Sn 4 [(S + P L)(S n 2 ) D2 ]n 2 + P L[(S n 2 )2 D 2 ] = 0. (33)

58 Accessibility (1) To simplify the discussion, we assume there is no damping (ν c = γ e = 0). Effect of actual damping on this analysis: A propagating wave will still propagate (with a little attenuation). An evanescent wave will not become propagating. Therefore, the entries S, D, P L = P of K are real.

59 Accessibility (1) To simplify the discussion, we assume there is no damping (ν c = γ e = 0). Effect of actual damping on this analysis: A propagating wave will still propagate (with a little attenuation). An evanescent wave will not become propagating. Therefore, the entries S, D, P L = P of K are real. Under this assumption, we say there is strict accessibility if n R 3 solution to (33); vague accessibility if n C 3 solution to (33), with Im n Re n.

60 Accessibility (1) To simplify the discussion, we assume there is no damping (ν c = γ e = 0). Effect of actual damping on this analysis: A propagating wave will still propagate (with a little attenuation). An evanescent wave will not become propagating. Therefore, the entries S, D, P L = P of K are real. Under this assumption, we say there is Recall: strict accessibility if n R 3 solution to (33); vague accessibility if n C 3 solution to (33), with Im n Re n. in the LH regime, there holds: P < 0 < S < D and S, D P. (34)

61 Accessibility (2) We fix n R (Phenomenology...) and write r = S n 2. Then, det M is a second degree polynomial in n 2 : det M = Q(n 2 ) = a r n 4 b r n 2 + c r, where: a r = S, b r = (S + P)r D 2, c r = P(r 2 D 2 ).

62 Accessibility (2) We fix n R (Phenomenology...) and write r = S n 2. Then, det M is a second degree polynomial in n 2 : det M = Q(n 2 ) = a r n 4 b r n 2 + c r, where: a r = S, b r = (S + P)r D 2, c r = P(r 2 D 2 ). Let r := b 2 r 4 a r c r be the discriminant of Q. 1 If r 0 and Q has a non-negative root, there is strict accessibility. 2 If r 0 and both roots are negative, there is strict inaccessibility (all waves are evanescent). 3 If r < 0, there cannot be strict accessibility, but there may be vague accessibility.

63 Strict Accessibility One calculates: r = (S P) 2 r 2 2D 2 (S + P) r + D 2 (D 2 + 4SP). 1 r 0 r / (r, r + ), with: r ± := D2 (S + P) ± δ (S P) 2, δ := 4D 2 SP(D 2 (S P) 2 ). Actually, in the LH regime (34), δ > 0.

64 Strict Accessibility One calculates: r = (S P) 2 r 2 2D 2 (S + P) r + D 2 (D 2 + 4SP). 1 r 0 r / (r, r + ), with: r ± := D2 (S + P) ± δ (S P) 2, δ := 4D 2 SP(D 2 (S P) 2 ). Actually, in the LH regime (34), δ > 0. 2 Both roots negative (strict inaccessibility) iff b r a r < 0 and c r a r > 0 r > r := D2 S + P and r D.

65 The case r < 0 We have vague accessibility iff n C s.t. det M = 0 and Re(n ) Im(n ).

66 The case r < 0 We have vague accessibility iff n C s.t. det M = 0 and Re(n ) Im(n ). We calculate: n 2 = (b r ± i r )/(2a r ) = Re(n ) 2 = b r + br 2 r, Im(n ) 2 = b r + b 2 r r. 4a r 4a r Necessary condition: b r 0.

67 The case r < 0 We have vague accessibility iff n C s.t. det M = 0 and Re(n ) Im(n ). We calculate: n 2 = (b r ± i r )/(2a r ) = Re(n ) 2 = b r + br 2 r, Im(n ) 2 = b r + b 2 r r. 4a r 4a r Necessary condition: b r 0. In this case: Re(n ) Im(n ) = κ κ, where: κ := b2 r r. Thus: vague accessibility b r 0 and κ large enough

68 The case r < 0 We have vague accessibility iff n C s.t. det M = 0 and Re(n ) Im(n ). We calculate: n 2 = (b r ± i r )/(2a r ) = Re(n ) 2 = b r + br 2 r, Im(n ) 2 = b r + b 2 r r. 4a r 4a r Necessary condition: b r 0. In this case: Re(n ) Im(n ) = κ κ, where: κ := b2 r r. Thus: vague accessibility b r 0 and κ large enough r r and b2 r r large enough.

69 The case r < 0 (continued) We investigate the condition ɛ 2 b 2 r + r 0, for ɛ > 0 given. The l.h.s. is a second-degree polynomial in r, it is positive outside its roots: ρ ɛ ± = (ɛ 2 + 1)(S + P)D 2 ± δ ɛ (ɛ 2 + 1)(S 2 + P 2 ) 2(ɛ 2 1)SP, with: δ ɛ = 4D 2 SP[(ɛ 2 + 1)(D 2 S 2 P 2 ) 2(ɛ 2 1)SP]. In the LH regime (34), δ ɛ > 0 and ρ ɛ r ρ ɛ +.

70 Accessibility: conclusions Proposition 1 Assume ω ci ω ω ce and ν c = γ e = 0. Then for ɛ small enough. r < ρ ɛ < r < ρ ɛ + < r + < S, 2 Let λ = D2 P. One has strict accessibility iff n > S r S + 2 Sλ + λ. One has vague accessibility ( Re(n ) / Im(n ) > K ) iff S ρ ɛ < n < S r, with ɛ = 2K /(K 2 1).

71 "Physical" test case We consider an electron-proton plasma with densities n e = n i = m 3 Plasma and cyclotron frequencies (rad/s): (constant). ω pe , ω pi , ω ce , ω ci , for an external magnetic field of intensity B ext = 3.2 T. Lower hybrid frequency ω LH = ω ci ω ce rad/s. Electron temperature: k B T e = 1 kev ν c, γ e.

72 "Physical" test case We consider an electron-proton plasma with densities n e = n i = m 3 Plasma and cyclotron frequencies (rad/s): (constant). ω pe , ω pi , ω ce , ω ci , for an external magnetic field of intensity B ext = 3.2 T. Lower hybrid frequency ω LH = ω ci ω ce rad/s. Electron temperature: k B T e = 1 kev ν c, γ e. We present a series of tests on the physical problem (2) (5) and its 2D VF (23) (24), for the values ω = ω LH, 2ω LH or ω LH /2 of the frequency, and several values of the Fourier index ν.

73 Influence of parameters Parallel wave vector: k (x) = ν R or n (x) = νc Rω.

74 Influence of parameters Parallel wave vector: k (x) = ν R or n (x) = νc Rω. ω = const., ν = { more accessible γ e 0 for ν large enough,

75 Influence of parameters Parallel wave vector: k (x) = ν R or n (x) = νc Rω. ω = const., ν = ν = const., ω = { more accessible γ e 0 for ν large enough, { less accessible γ e 0 for ω large enough. Note: k ν/r 0 in γ e = K = constant.

76 Influence of parameters Parallel wave vector: k (x) = ν R or n (x) = νc Rω. ω = const., ν = ν = const., ω = { more accessible γ e 0 for ν large enough, { less accessible γ e 0 for ω large enough. Note: k ν/r 0 in γ e = K = constant. On the other hand we should have an a priori estimate E ν C ν α rhs (in the suitable spaces). So, for a fixed ω, there is an optimal ν where the wave emitted by the antenna penetrates most the plasma.

77 Figure: The real part of E for ω = ω LH and ν = 200

78 Figure: The real part of E for ω = ω LH and ν = 300

79 Figure: The real part of E for ω = ω LH and ν = 400

80 Figure: The real part of E for ω = ω LH /2 and ν = 100

81 Figure: The real part of E for ω = ω LH /2 and ν = 200

82 Figure: The real part of E for ω = ω LH /2 and ν = 300

83 Figure: The real part of E for ω = 2 ω LH and ν = 400

84 Figure: The real part of E for ω = 2 ω LH and ν = 600

85 Figure: The real part of E for ω = 2 ω LH and ν = 700

86 Figure: The real part of E for ω = 2 ω LH and ν = 800

87 Provisional conclusions

88 In progress and to do Feasibility of simulations incorporating most of the physics

89 In progress and to do Feasibility of simulations incorporating most of the physics (to do: variable plasma properties) Simulations confirm accessibility analysis

90 In progress and to do Feasibility of simulations incorporating most of the physics (to do: variable plasma properties) Simulations confirm accessibility analysis Non-hermitianness crucial for well-posedness, but precludes the use of many numerical methods

91 In progress and to do Feasibility of simulations incorporating most of the physics (to do: variable plasma properties) Simulations confirm accessibility analysis Non-hermitianness crucial for well-posedness, but precludes the use of many numerical methods To do: comparison of antenna boundary conditions

92 In progress and to do Feasibility of simulations incorporating most of the physics (to do: variable plasma properties) Simulations confirm accessibility analysis Non-hermitianness crucial for well-posedness, but precludes the use of many numerical methods To do: comparison of antenna boundary conditions More realistic test-cases: transversal wave length domain size Domain decomposition

93 In progress and to do Feasibility of simulations incorporating most of the physics (to do: variable plasma properties) Simulations confirm accessibility analysis Non-hermitianness crucial for well-posedness, but precludes the use of many numerical methods To do: comparison of antenna boundary conditions More realistic test-cases: transversal wave length domain size Domain decomposition Non-definite and non-hermitian operator make subdomains small enough to use a direct method