Propagation of Radio Frequency Waves Through Density Filaments

Transcription

1 PSFC/JA Propagation of Radio Frequency Waves Through Density Filaments A. K. Ram and K. Hizanidis a May 015 a National Technical University of Athens (part of HELLAS) School of Electrical and Computer Engineering Association EURATOM-Hellenic Republic, GR Greece. Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge, MA 0139 U.S.A. This work was supported by the U.S. Department of Energy grants DE-FG0-91ER and DE-FG0-99ER-5455-NSTX, and by the National Programme on Controlled Thermonuclear Fusion associated with the EUROfusion Consortium. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted. To be published in: Proceedings of the 1 st Topical Conference on Radio Frequency Power in Plasmas (015).

2 Propagation of radio frequency waves through density filaments Abhay K. Ram and Kyriakos Hizanidis Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA USA National Technical University of Athens, Association EURATOM, Greece Abstract. In tokamak fusion plasmas, coherent fluctuations in the form of blobs or filaments are routinely observed in the scrape-off layer. In this paper we develop an analytical formalism for the scattering of radio frequency waves by filaments which are cylindrical with their major axis aligned along the toroidal magnetic field lines. Since the magnitude of the ratio of the density inside the filaments to the background density is generally of order 1, the geometric optics approximation cannot be used to describe the scattering. A full-wave model is formulated which assumes that the plasma is cold and that the plasma in the cylindrical filament has uniform density. The background plasma, in which the filament is present, is also assumed to be cold and uniform. The theoretical framework applies to the scattering of any plasma wave. Keywords: Plasma waves, scattering, fluctuations PACS: 5.5.Os, 5.35.Hr INTRODUCTION The existence of turbulent structures such as blobs and filaments in the edge region of tokamak plasmas has been addressed theoretically [1, ] and experimentally [3, 4]. Radio frequency (RF) waves, which are routinely used for heating and current drive in fusion plasmas, have to propagate through this active region before delivering their energy and momentum to the core plasma. Since the density inside the blobs or filaments is significantly different from the background plasma density, the use of geometric optics to determine the refractive effects on RF waves is inadequate. In this paper, we formulate a full-wave analytical model which describes the scattering of plasma waves by a cylindrical plasma filament embedded in a background magnetized plasma. The model is similar to the Mie formalism for scattering of electromagnetic waves in vacuum by dielectric spheres [5]. The present description is based on the scattering of RF waves by spherical plasma blobs of arbitrary density [6, 7]. We assume that the axis of the cylindrical filament is aligned along the magnetic field. The center of the coordinate system is taken to be the center of the filament. The background plasma and the plasma inside the filament is assumed to be uniform and cold, with the ratio of the two densities being completely arbitrary. The filament is assumed to be stationary and of infinite extent in the axial direction. The electromagnetic fields inside and outside the filament are obtained by solving the vector Helmholtz equation. The formulation of the model is quite lengthy; only a brief description is provided in this paper. A full theoretical description of the scattering model, and the associated numerical results illustrating the underlying physics for different RF waves, will be discussed elsewhere. WAVE ELECTRIC FIELDS IN A PLASMA For a cold fluid plasma, described by a linearized set of continuity and momentum equations for electrons and ions, Faraday s and Ampere s equations in Maxwell s system of equations can be combined [8] to yield the following equation for the spatial variation of the electric field, E(r) ω K(r).E(r) = 0, (1) c where ω is the angular frequency, c is the speed of light, and K(r) is plasma permittivity tensor. Here we assumed that the plasma equilibrium is time independent, while the linearized perturbed electromagnetic fields and the plasma density have a time dependence of the form e iωt. In the cylindrical coordinate system where the ambient magnetic 1

3 field B 0 = B 0 ẑ is aligned along the z-axis, K(r) has the form [8] K ρ ik φ 0 K = ik φ K ρ 0, () with 0 0 K z K ρ = 1 K φ = ω ce ω K z = 1 ω pe ω ω pe ω ω ce ω pe ω ω ce + ω, ω ωci, ω ci ω ω ωci, (3) where ω pe (ω pi ) and ω ce (ω ci ) are the angular electron (ion) plasma frequency and cyclotron frequency, respectively, and the index represents all the ion species in the plasma. The plasma and cyclotron frequencies can, in general, be functions of space. The permittivity tensor of the background plasma and of the filament are expressed in terms of their respective densities and ion compositions. Vector cylinder functions The background plasma surrounding the cylindrical filament and the plasma inside the filament are assumed to have constant, but different, densities. Subsequently, the elements of K are constants in each region. The solutions to the vector Helmholtz equation (1) can be constructed from the scalar Helmholtz equation ψ (r) + k ψ (r) = 0, (4) using the procedure described in [9]. Here k = k.k is assumed to be a constant and ψ is a scalar field that is a function of the position vector r. The spatial unit vectors in cylindrical coordinates are taken to be ( ˆρ, ˆφ,ẑ ). The solution to (4) is obtained by separation of variables [9] ψ (r) = ψ m m= ( Z m kρ ρ ) e ikzz+imφ, (5) m= where Z m represents either Bessel functions or Hankel functions of order m, k z = k.ẑ is the component of k along the z-direction, k ρ = k k z ẑ, and φ is the azimuthal angle. The Bessel and Hankel functions can be of the first or second kind. The vector cylinder function are then obtained from (5) as follows [9] [( ) { } ] im l m = ψ m = ρ Z m ˆρ + ρ ˆφ + ik z ẑ Z m e ik zz+imφ [ ( ) ] im m m = (ψ m ẑ) = ρ Z m ˆρ ρ Z m ˆφ e ik zz+imφ (6) [ n m = 1 ( ) { } ] k m ik z k z m m = Z m ˆρ k ρ k ρ ˆφ k ρ k ẑ Z m e ik zz+imφ The spatial variation of a plane wave is [9] e ik.r = e ik ρ ρ cos(φ φ k ) = Dyadic representation of the plane wave i m ( J m kρ ρ ) e im(φ φk) m= i m ψ 1m e imφ k, (7) m=

4 where J m is the Bessel function of the first kind of order m, and φ k is the azimuthal angle between k and the x-axis. Furthermore, ψ 1m is the same as ψ m defined in (5) with Z m replaced by J m. From (7) it follows that, ˆρ ˆφ e ik.r = i m e imφ k ( al 1m + bm 1m + cn 1m ), (8) ẑ m= where the right hand side is a sum of dyadics with column vectors a, b, and c given by a = i k ρ cos(φ φ k ) k k ρ sin(φ φ k ),b = i sin(φ φ k) cos(φ φ k ),c = 1 k z cos(φ φ k ) k z sin(φ φ k ) k k ρ z 0 kk ρ k ρ. (9) ELECTROMAGNETIC FIELDS INSIDE THE CYLINDRICAL FILAMENT The spatial variation of the electric field inside the filament is obtained by solving (1) using the Fourier transform E(r) = d 3 k E(k) e ik.r. (10) Substituting this form for the field into (1) leads to d 3 k D(k,ω).E(k) e ik.r = 0, (11) where D(k,ω) = c ω ( kk k I ) + K (1) is the dielectric tensor. Here kk is a dyadic formed by the wave vector, and I is the unit tensor. A non-trivial solution to the electric field in (11) requires that det ( D(k,ω) ) K ρ n 4 ρ + [ Kφ ( )( K ρ + K z Kρ n )] [ (Kρ z n ρ + K z n ) ] z K φ = 0, (13) where n ρ = ck ρ /ω and n z = ck z /ω. The boundary conditions for the electromagnetic waves, that need to be satisfied at the interface of the filament and the background plasma, require that n z be the same for all waves. Consequently, it is the same as that of the incoming plane wave. Then the dispersion relation (13) is an algebraic equation for n ρ. It can be shown easily that, for n z real, n ρ is real. Consequently, n ρ is either real or imaginary. For a real root, we are constrained to choose the positive one. For an imaginary root, we choose the root with a negative imaginary part. By definition, one pair of independent roots of n ρ is negative of the other pair. Each pair corresponds to the two electromagnetic waves which can exist in a cold plasma. The polarizations of these waves follow from the requirement D.E = 0. The conditions that n z is fixed and that n ρ is determined from (13) leads to d 3 k dk ρ i=1 1 k ρi δ ( k ρ k ρi ) π 0 dφ dk z δ (k z k z0 ), (14) where δ represents the Dirac delta function. The delta function in the first integral on the right hand side corresponds to the two independent values of k ρ obtained from (13) the summation is over the two modes. The second integral over φ ensures that the other two roots, which are the negative of the first pair of roots, are included in the formulation of the electric field inside the filament. The negative sign corresponds to a rotation in φ by π. The delta function in the third integral sets the k z of all the waves to be the same as that of the incident plane wave k z0. The electric field inside the filament follows from substituting the plane wave dyadic (8) in (10) with the integrals given in (14). It is important to note that the electromagnetic fields inside the filament are due to the two primary cold plasma waves that are propagating in all directions relative to the magnetic field. 3

5 BOUNDARY CONDITIONS A critical component in the description of full-wave scattering is that the electromagnetic fields satisfy boundary conditions at the interface between the filament plasma and the background plasma. The boundary conditions follow from Maxwell s equations with the requirement that there be no free surface charges or currents at the interface. The boundary conditions at the interface, for all the plasma waves, are the continuity of the tangential components of the electric and magnetic fields, and the continuity of the normal components of the electric displacement field and the magnetic field. Of these six scalar conditions, it can be shown that only four are independent. This implies that for an incident plane wave, chosen to be a particular branch of the dispersion relation, the electromagnetic fields inside the filament, and those that are scattered by the filament, will be composed of both branches of the plasma dispersion relation, and will be propagating in all directions relative to the ambient magnetic field. CONCLUSIONS We have given an outline of the full-wave analytical model which describes the scattering of RF waves by a cylindrical plasma filament aligned along the magnetic field line. The model is for an arbitrary ratio of the density inside the filament to the density of the background plasma. It also applies to waves having any frequency; thus, it is appropriate for ion cyclotron, lower hybrid, and electron cyclotron waves. An important result, which easily follows from the full-wave description with the appropriate boundary conditions, is that, due to the scattering by a filament, an incident plane wave can couple power to the two cold plasma waves which propagate in all directions relative to the ambient magnetic field. For example, an incident ordinary wave, in the electron cyclotron range of frequencies, can couple power to the extraordinary wave; the scattered ordinary and extraordinary waves propagating in all directions, not just along the direction of the incident ordinary wave. The scattering broadens the wave vector spectrum of the RF waves which propagate into the core of the plasma. The filament can also lead to diffractive focusing which enhances the amplitude of the plasma waves near the interface separating the filament from the background plasma. The regions of enhanced fields can be a source of parametric processes. ACKNOWLEDGMENTS The research reported in this paper has been supported in part by the US Department of Energy Grant numbers DE- FG0-91ER and DE-FG0-99ER-5455-NSTX, and by National Programme on Controlled Thermonuclear Fusion associated with the EUROfusion Consortium. REFERENCES 1. S. I. Krasheninnikov, Phys. Lett. A 83, (001).. J. R. Myra, D. A. D Ippolito, D. P. Stotler, S. J. Zweben, B. P. LeBlanc, J. E. Menard, R. J. Maqueda, and J. Bodeo, Phys. Plasmas 13, (006). 3. O. Grulke, J. L. Terry, B. LaBombard, S. J. Zweben, Phys. Plasmas 13, (006). 4. S. J. Zweben, R. J. Maqueda, D. P. Stotler, A. Keesee, J. Boedo, C. E. Bush, S. M. Kaye, B. LeBlanc, J. L. Lowrance, V. J. Mastrocola, R. Maingi, N. Nishino, G. Renda, D. W. Swain, J. B. Wilgen, and the NSTX Team, Nucl. Fusion 44, (004). 5. H. C. van de Hulst, Light Scattering by Small Particles, Dover, New York, 1957, Chapter A. K. Ram, K. Hizanidis, and Y. Kominis, Phys. Plasmas 0, (013). 7. A. K. Ram and K. Hizanidis, Radiation Effects and Defects in Solids: Incorporating Plasma Science and Plasma Technology 168, (013). 8. T. H. Stix, Waves in Plasmas, Springer, New York, 199, pp J. A. Stratton, Electromagnetic Theory, McGraw Hill, New York, 1941, pp