Recent issues in the propagation and absorption of high-frequency wave beams in magnetized plasmas

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1 Recent issues in the propagation and absorption of high-frequency wave beams in magnetized plasmas Omar Maj, Katlenburg-Lindau, Germany. Plasmaphysik, Garching bei München, Germany. 14th European Fusion Theory Conference - Frascati Thanks to: G. V. Pereverzev, A. A. Balakin, E. Poli, N. Bertelli, M. Bornatici, R. Bilato and M. Brambilla. In memory of Grigory Pereverzev

2 Resent (and not so recent) issues Schematic overview of open problems in linear wave theory Refraction Plasma Fluctuations Fluctuations Reflectometry Antenna Resonance Layer Linear scattering from fluctuations Reflectometry diagnostics: Doppler reflectometry requires fast calculations of the wave field. Reflection from the cut-off must be accurately described. The effects of scattering from random fluctuations should be assessed. Stabilization of MHD modes in ITER: Fluctuations at the plasma edge could lead to a shift in the power deposition profile.

3 Resent (and not so recent) issues Schematic overview of open problems in linear wave theory Diffraction Normalized particle distribution Normalized transmitted spectrum Normalized launched spectrum Diffraction Refraction Resonant particles Spectral gap for LH waves The spectral gap problem for lower-hybrid waves: diffraction broadens the launched power spectrum, thus, increasing the overlap with the electron distribution function.

4 Resent (and not so recent) issues Schematic overview of open problems in linear wave theory Absorption Incident beam Resonance layer N toroidal r Resonance layer Spatial inhomogeneity Spectral inhomogeneity The shape (position and width) of power deposition profile is important for the stabilization of MHD modes via electron cyclotron current drive: Spatial inhomogeneity of the absorption coefficient leads to asymmetric absorption of the beam. Spectral inhomogeneity (dispersive absorption) can broaden the power deposition profile.

5 Outline Wave equation in the high-frequency limit ( GHz) Standard assumptions Overview of semiclassical solutions (in spatially dispersive media) Geometrical optics and ray tracing Complex geometrical optics Beam tracing: the case of Gaussian beams Modeling of applications Reflectometry Absorption at the EC resonance layer Gaussian beams of LH waves Conclusions

6 Wave equation for beams in a hot magnetized plasma. Maxwell s equations for harmonic (e iωt ) fields are equivalent to [ E(ω, x) ] κ 2ˆεE(ω, x) = 0, κ = ωl c, ˆεE(ω, x) = ˆε(ω, x, x )E(ω, x )dx, where ω is the frequency, c the speed of light in free space, while x = (x 1,..., x d ) are normalized coordinates, d = 2, 3, L is the normalization scale length. The plasma dielectric operator ˆε is assumed to be known (!): Cold plasma model: local response, ˆε(ω, x, x ) = ε(ω, x)δ(x x ), ε = (ε ij ). Finite temperature plasma: non-local response and spatial dispersion, ˆε(ω, x, x ) decays with increasing x x.

7 High-frequency asymptotic expansion A preparatory calculation on the wave equation We consider the high-frequency regime κ = ωl/c 1, Electron Cyclotron (EC) waves: Lower Hybrid (LH) waves: f = 170GHz, k 0 = ω/c 35.7cm 1 f = 5GHz, k 0 = ω/c 1cm 1 Ansatz for the wave field, (allowing for fast variations of the amplitude) E(ω, x) = e iκs(x) a(ω, x), ( ) m a = O(κ mr ), κ +, 0 r < 1; for r = 0 this is just the eikonal ansatz, but we keep r free. The substitution into the (properly written) wave equation gives i D 0 κ p i a x = 1 2 D 0 i 2κ 2 p i p j 2 a [ x i x + D j 0 i ( [ D0 ] +2iD 2κ x i 1 )]a p i + O(κ 2+r + κ 3+3r ), where p = S(x) and [ ] denotes the total derivative, and D(ω, x, p) = D 0(x, p)+κ 1 D 1(x, p)+o(κ 2 ), is the local dispersion tensor.

8 Standard assumptions There are three standard assumptions on the dispersion tensor, D(ω, x, p) = D 0(x, p) + 1 κ D1(x, p) + O( 1 κ 2 ). (WA) Weak Absorption: the leading-order term is Hermitian, D 0(x, p) = D 0(x, p), which implies that D a (ω, x, p) = D(ω, x, p) D (ω, x, p) = O(1/κ). (AC) Absence of linear mode Conversion: the real eigenvalues λ j(x, p) of D 0(x, p) are well separated, namely, λ i(x, p) λ j(x, p) C i,j > 0, uniformly in (x, p) for i j. (WI) Weakly Inhomogeneous medium: the scale length L represents the scale of spatial variations of the medium.

9 PART 1: Overview of semiclassical solutions (with generalizations to spatially dispersive media) 1. Geometrical optics (as in [McDonald 1988, Littlejohn & Flynn 1991]) 2. Complex geometrical optics (generalized) 3. Beam tracing (generalized) - Here, we consider Gaussian beams only NOT addressed here: Parabolic wave equation [Fock 1965; Permitin & Smirnov 1996], and the quasi-optical method [Balakin et al. 2008]. Mode conversion theory of Kaufman and co-workers, based on the variational principle for Hermitian operators. Code: RAYCON [Jaun, Tracy & Kaufman 2009]. Wave kinetic equation [McDonald 1988]. Many others are not explicitly addressed [Keller 1978; Littlejohn 1985; Saveliev 2009; Cairns & Fuchs 2010;...], cf., also, [Richardson, Bonoli & Wright 2010].

10 Geometrical optics Standard eikonal theory and ray tracing Let us recall the asymptotic expansion, i D 0 κ p i a x = 1 2 D 0 i 2κ 2 p i p j The wave electric field is 2 a [ x i x + D j 0 i ( [ D0 ] +2iD 2κ x i 1 )]a p i + O(κ 2+r + κ 3+3r ). E(ω, x) = e iκs(x) a(ω, x), a a 0 + κ 1 a 1 +, r = 0, and, separating the powers of κ one gets, to leading orders, D 0(x, S)a 0(x) = 0, D 0(x, S)a 1(x) = i D0 a 0 p i x + i ( [ D0 ] + 2iD i 2 x i 1 )a 0. p i

11 Geometrical optics Standard eikonal theory and ray tracing - continued Standard analysis of the lowest order, D 0(x, S)a 0(x) = 0: With both assumptions (WA) and (AC), one finds a solution for each eigenvalue λ j = H of the dispersion tensor, { H(x, S) = 0, (eikonal equation), ( I π(x, S) ) a0 = 0, (polarization condition). where π = π j is the projection onto the eigenspace corresponding to the eigenvalue H = λ j. Standard analysis of the first order, D 0(x, S)a 1(x) = r.h.s.: A solution a 1 exists only if π(r.h.s.) = 0, namely, V a ( divv + 2iη ) a0 = 0, (amplitude transport), V (x) = ph(x, S) and η is a matrix (absorption, polarization rotation, Berry phase shift and curvature phase shift [Emmrich & Weinstein 1996]). What if one drops an assumption...

12 Geometrical optics An analytical example from reflectometry Let us consider the well-known linear layer model for reflectometry: κ =20 p y = dimension d = 2, [ + κ 2 (1 x) ] E(x, y) = 0, E(x, y) = ẑe iκpyy u(x, p y), u(x, p y) = u 0Ai ( κ 2 3 (x β) ), β = 1 p 2 y. The geometrical optics solution is y Caustic x u GO (x, p y) = E GO (x, y) = ẑe iκpyy u GO (x, p y), a ( ) in e iκs +(x) + e iκs (x)+i π (β x) 1/4 2, S ±(x) = 2 3 (β x)3/2, which agrees with the asymptotic expansion of Ai ( κ 2 3 (x β) ) for x < β. Caustic: the geometrical optics solution blows up at the turning points.

13 Diffraction from ray tracing Construction of Lagrangian solutions Local parametrization of Lagrangian manifolds: find a phase ϕ(x, α) such that, [Guillemin & Sternberg 1977; Duistermaat 1996] Λ : x β + [ xϕ(x, α)] 2 = 0, (x, α) C ϕ, C ϕ : αϕ(x, α) = Λ = { x β +p 2 x =0 } Since Λ has two branches, p x 0.0 αϕ = v(x) + α 2, ϕ(x, α) = v(x)α α3, v(x) = x β x β At last, after solving the amplitude transport equation, the wave field is u Λ(x, p y) = e iκϕ(x,α) a 0(x, α)dα = 2π [ a 0(0)Ai ( κ 2/3 (x β) ) + b 0(0)Ai ( κ 2/3 (x β) ) /κ 1 2/3],!! where a 0(x, α) is constant in x and b 0(x, α) = αa 0(x, α).

14 Complex geometrical optics Complex eikonal and basic estimates Complex eikonal ansatz, E(ω, x) = e iκψ(x) a(ω, x), ψ(x) = S(x) + iφ(x), φ(x) 0. The condition Imψ(x) = φ(x) 0 implies that, the wave field is exponentially localized around the zero-level set R = {x; φ(x) = 0}, and φ R = 0, φ(x) = 1 2 v2 (x), v R = 0. Physically this is the far field diffraction scaling for the beam width W, L κ W = 2 v, W 2 λ 0L, λ 0 = 2πc/ω. A simple, but crucial estimate follows, [Maslov 1996, Pereverzev 1996] v n e κφ = κ n 2 ξ n e 1 2 ξ2 Cκ n 2, which implies φe κφ = O(κ 1 2 ), φ φe κφ = O(κ 1 ),...

15 Complex geometrical optics Asymptotics of the wave equation with complex phase Let us recall again the asymptotic expansion, i D 0 κ p i a x = 1 2 D 0 i 2κ 2 p i p j 2 a [ x i x + D j 0 i ( [ D0 ] +2iD 2κ x i 1 )]a p i + O(κ 2+r + κ 3+3r ). One can substitute a e κφ a with r = 1/2. Then, the expansion a a 0 + κ 1/2 a 1/2 + κ 1 a 1 + gives a 1/2 = 0, D 0(x, ψ)a 0(x) = 0, D0 a 0 D 0(x, ψ)a 1(x) = i[ p i x + 1 ( 2 D ψ 2 D ) 0 + 2iD i 2 x i p i x i x j 1 a 0 ]. p i p j Being p = p + ip the complex refractive index vector and given f(x, p), f(x, p) = f(x, p) + ip f i (x, p) 1 p i 2 p ip 2 f j (x, p). p i p j

16 Complex geometrical optics Complex rays and extended rays The lowest-order implies the complex eikonal equation, H(x, ψ) = 0. Complex rays: let s assume that there exists an analytical continuation, H(z, ψ(z)/ z ) = 0, z = (z 1,..., z d ) C d, and apply the method of characteristics in the complex space [Egorchenkov & Kravtsov 2001, Amodei et al. 2006]. Remark: Complex rays can be used for non-hermitian media [Bravo-Ortega & Glasser 1991; Cardinali 2000; Cardinali 2001]. Extended rays: let s separate the real and imaginary parts, H(x, S) 1 φ φ 2 H (x, S) = 0, 2 x i x j p i p j φ H (x, S) = 0, x i p i a constrained Hamilton-Jacobi equation, for a bundle of interacting rays [Mazzucato 1989, Nowak & Orefice 1993, Peeters 1996, Farina 2007].

17 Complex geometrical optics Extended amplitude transport equation The application of the projector π(x, ψ) to the first-order equation gives [ D0 a 0 π p i x + 1 ( 2 D ψ 2 D ) ] 0 + 2iD i 2 x i p i x i x j 1 a 0 + φ [ ] = 0, p i p j and the last term yields a contribution of O(κ 3/2 ) to the wave equation. After some calculus, and dropping terms φ, V φ a [ 2 φ 2 H ] divv φ + 2iη φ + i a 2 x i x j 0 = 0, p i p j V φ (x) = ph(x, S), η φ (x) is a matrix analogous to the η. Here S(x) depends on φ(x) due to the interaction of extended rays! Assumption (AC) ensures that this is sufficient for the first-order equation to be satisfied modulo an O(κ 3/2 ) remainder in the wave equation. Amplitude transport in complex geometrical optics is currently under consideration by Daniela Farina and co-workers.

18 Beam tracing: the case of Gaussian beams Local coordinates and paraxial expansion Let us assume that R = {x; φ(x) = 0} is a regular curve {x = x(τ)}, v 2 x { s = s(x), v α = v α (x), α = 1, 2. v 1 R An additional limitation: The local curvilinear coordinates (s, v 1, v 2 ) should be valid in the range of a beam width W from R, K = curvature of the curve R = 1/W, We note that R is a locus of minima for φ(x), i.e., φ(x) = 1 2 φ αβ(s)v α v β. Taylor expansion in v, paraxial expansion [Pereverzev 1998] H(x, ψ) = H( x, p) + v α H R + 1 v α 2 vα v β 2 H R + O( v 3 ) = 0, v α v β where p = S( x), and the remainder amounts to an O(κ 3/2 ).

19 Beam Tracing: the case of Gaussian beams Construction of the wave field The wave field now reduces to a Gaussian beam, { E GB(ω, x) = a 0(τ) exp iκ [ S(τ) + p i(τ) x i ψij(τ) xi x j]}, τ = τ ( s(x) ), x = x x(τ), τ being the parameter along the curve R. Separating different powers of v in the paraxial expansion produces ordinary differential equations for: 1. S(τ), the integral of p i (τ)dx i (τ), 2. position x(τ) and the carrier refractive index p(τ), 3. Reψ ij (τ), which is related to the curvature of the phase front, 4. Imψ ij (τ), which is related to the Gaussian envelope of the beam. For the amplitude a 0, da 0 dτ = 1 [ divv φ + 2iη φ + i 2 2 φ x i x j 2 H a 0. p i p j ]R

20 Summary of PART 1 A schematic overview of the derivation Asymptotic expansion of the wave equation Geometrical Optics Complex Geometrical Optics Beam Tracing [G. V. Pereverzev] Parabolic Wave Equation Lagrangian solutions Extended Rays Gaussian Beams Eikonal Theory Complex Rays Ray Tracing codes [Many] Complex Ray Tracing codes [Bravo-Ortega et al.] [Cardinali 2000 and 2001] [Egorchenkov & Kravtsov] [Amodei et al.] Extended Ray Tracing codes GRAY and GREY [Farina] Beam Tracing codes TORBEAM [Poli] LHBEAM [Bertelli] Quasi-Optical Code [Balakin et al.]

21 PART 2: Modeling of applications in critical cases (i.e., when applicability conditions are marginal or broken) 1. Reflectometry applications (caustics, large curvature of the beam trajectory). 2. EC heating and current drive (strong absorption). 3. LH heating and current drive (in progress).

22 Reflectometry Large curvature of the trajectory: limitation on the incidence angle Parameters: κ = 50, α = 0.3 κ, β = 0. Linear energy density u(x, y) 2 dy =45 BT EX Integration line x = =80 BT EX BT EX

23 Reflectometry Quantitative estimates In reflectometry applications, the caustic coincides with the probed region: we need its position and width (resolution of the diagnostic). Focussed beam The caustic appears as a pronounced maximum in the line wave energy density. W (x) u(x, y) 2 dy, normalized to its maximum. Case x QO(%) x BT(%) w QO(%) w BT(%) non focused focused

24 Absorption at the EC resonance layer A simple analytical model Dispersive absorbing half-plane (Doppler-shifted resonance layer), ε(x, p y) = 1 + iγ(x, p y), { 0, γ(x, p y) = γ 1 for x p 2 y, x + p 2 y, for x > p 2 y. The model is better understood in the mixed Fourier (x, p y) space: Physical space y Mixed Fourier space p y Absorption x Launched spectrum x x= x 0 x= x 0 Absorption

25 Absorption at the EC resonance layer Spatial inhomogeneity: effect of asymmetric incidence on Gaussian beam Parameters: ϑ = 70 o, f = 170GHz, w 0 = 1.14cm, and γ = Spatial dispersion is switched off, only spatial inhomogeneity is addressed. u Ex u Ex u BT y/ y/ x / x /

26 Absorption at the EC resonance layer Qualitative comparison of solutions The errors in the beam tracing solution are due to spatial dispersion. Amplitude contours y y y x x x Beam tracing Aberration-free Quasi-optics Error rel. to max. y y y x x x

27 Absorption at the EC resonance layer Extrapolation to ITER: geometry Let us consider coordinates adapted to the line of sight of the upper EC port of ITER and neglect the third dimension. Launching direction y = x Parameters: κ = 200 (limited by numerical cost), f = 170GHz, with k 0 = ω = c 35cm 1, L = κ/k 0 = 5.7cm (!!!), a = 200cm, R 0 = 600cm, n e0 = cm; T e = 15keV, B 0 = Gauss. EC resonance layer

28 Absorption at the EC resonance layer Extrapolation to ITER: absorption coefficient The absorption coefficient for ITER plasmas is computed numerically (DAMPBQ [Westerhof, Rijnhuizen Report 1989]) and fitted by { 0, for x < a 1p 2 y, (1 p y 2 ) 2 γ(x, p y) = g(p y) g(p y) = γ x + a 1p 2 y, for x a 1p 2 1 y, 1 + p y/p yref. 3 Parameters: γ 1 = 0.017, a 1 = 55, p yref = 0.22, in the range p y [ 1, +1].

29 Absorption at the EC resonance layer Extrapolation to ITER: results For the ITER-like model, the effects of dispersive absorption on deposition profiles are mitigated by the envelope function g(p y). w focus =3cm =22 o w =22 o focus =2 cm w focus =3cm =28.04 o x dep w dep x dep w dep x dep w dep BT Ex error (%)

30 Reconstruction of a LH wave beam Post-processing of the beam tracing code LHBEAM A post-processing code in fortran 90 and python is under development for the output of the lower hybrid beam tracing code LHBEAM [Bertelli]. POLOIDAL SECTIONS HORIZONTAL SECTIONS

31 Conclusions and outlook A systematic derivation of semiclassical solutions in spatially dispersive media (hot plasmas) has been presented on the basis of a preparatory asymptotic expansion of the wave equation. Beam tracing modeling of critical applications. [Maj, Pereverzev & Poli 2009; Maj, Balakin & Poli 2010] Reflectometry: there is a limitation on the incidence angle. EC resonance heating: spatially asymmetric absorption has no significant effect on the power deposition profile. EC resonance heating: spectrally asymmetric absorption can broaden the power deposition ( 10% for ITER). Outlook Beam tracing near caustics (reflectometry). Definition and calculation of the local spectrum (dispersive absorption and LH spectral gap). Effects of fluctuations on Gaussian (and non-gaussian) beams.