PSFC/JA-18-38

Diffraction of radio frequency waves by spatially modulated interfaces in the plasma edge in tokamaks

A. D. Papadopoulos1, E. N. Glytsis1, S. I. Valvis1, P. Papagiannis1, A. K. Ram3, K. Hizanidis1, A. Zisis2

1School of Electrical and Computer Engineering, National Technical University of Athens, 9 Iroon Polytechniou Street, Athens 15780, GR 2Faculty of Physics, National and Kapodistrian University of Athens, University Campus, Zografou, Athens 15784, GR 3Plasma Science and Fusion Center, Massachusetts Institute of Technology, 175 Albany Street, Cambridge MA 02139, USA

September 2018

Plasma Science and Fusion Center Massachusetts Institute of Technology

Cambridge MA 02139 USA

A.K.R. is supported by the US Department of Energy Grant numbers DE-FG02-91ER- 54109, DE-FG02-99ER-54525-NSTX, and DE-FC02-01ER54648. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.

Submitted to Journal of Plasma Physics

Under consideration for publication in J. Plasma Phys. 1

Diffraction of radio frequency waves byspatially modulated interfaces in the plasma

edge in tokamaks

A. D. Papadopoulos1†, E. N. Glytsis1, S. I. Valvis1, P. Papagiannis1,A. K. Ram3, K. Hizanidis1 and A. Zisis2

1School of Electrical and Computer Engineering, National Technical University of Athens, 9Iroon Polytechniou Street, Athens 15780, GR

2Faculty of Physics, National and Kapodistrian University of Athens, University Campus,Zografou, Athens 15784, GR

3Plasma Science and Fusion Center, Massachusetts Institute of Technology, 175 Albany Street,Cambridge MA 02139, USA

(Received xx; revised xx; accepted xx)

The use of radio frequency (RF) waves in fusion plasmas for heating, for non-inductivecurrent generation, for profile control, and for diagnostics has been well established. TheRF waves, excited by antenna structures placed near the wall of a fusion device, have topropagate through density fluctuations in the plasma edge. These fluctuations can modifythe properties of the RF waves that propagate towards the core of the plasma. A full-wave electromagnetic computational code ScaRF based on the finite difference frequencydomain (FDFD) method has been developed to study the effect of density turbulence onRF waves. The anisotropic plasma permittivity used in the scattering studies is that for amagnetized, cold plasma. The code is used to study the propagation of an RF plane wavethrough a modulated, spatially periodic density interface. Such an interface could arisein the edge region due to magnetohydrodynamic instability or drift waves. The frequencyof the plane wave is taken to be in the range of the electron cyclotron frequency. Thescattering analysis is applicable to ITER-like plasmas, as well as to plasmas in mediumsized tokamaks such as TCV, ASDEX-U, and DIII-D. The effect of different densitycontrasts across the interface and of different spatial modulations are discussed. WhileScaRF is used to study a periodic density fluctuation, the code is general enough toinclude different varieties of density fluctuations in the edge region – such as blobs andfilaments, and spatially random fluctuations.

1. Introduction

In fusion devices like tokamaks, radio frequency (RF) electromagnetic waves aregenerated by antennas near the wall of the device. The launched waves propagate througha turbulent edge plasma region before coupling power to the core plasma for heating andfor generating non-inductive currents. The tenuous plasma in the edge is composed ofblobs and filamentary structures (see Grulke et al. 2006; Krasheninnikov 2001; Myra et al.2006a,b; Pigarov et al. 2012; Zweben et al. 2007), drift waves and rippling modes (seeRitz et al. 1984), as well as random fluctuations. Various theoretical and computationalstudies have shown that the propagation characteristics of the RF waves are modifieddue to their interaction with blobs and filaments (see Ram & Hizanidis 2016; Ioannidiset al. 2017). The theoretical studies have provided physical insight into the scattering

† Email address for correspondence: arpapad@mail.ntua.gr

2 A. D. Papadopoulos, et al.

process by constructing analytical solutions to the full-wave equations. However, theyare limited in their scope as the composition of the edge plasma is assumed to be eithera single spherical blob (see Ram et al. 2013) or a single cylindrical filament (see Ram& Hizanidis 2016). However, the edge plasma is a more intricate mixture of blobs andfilaments of different shapes and sizes as well as wave like magnetohydrodynamic (MHD)instabilities. For complicated representations of the edge plasma density, one has to useresort to fully computational resources.

There are two approximations that are made in all these studies. First, that theedge plasma is cold so that all thermal effects are ignored. Consequently, only the coldplasma RF waves can propagate in the edge region. Second, that the edge plasma isstationary. This is based on the fact that all time scales associated with the RF wavesare much shorter than those associated with the edge turbulence. The time scales foredge turbulence are in the MHD realm – frequencies in the kHz range and fluctuationspeeds in the ion-acoustic range. On the other hand, the RF time scales are in the kineticregime – frequencies ranging from 10s of MHz to 100s of GHz and group speeds near thespeed of light. The full-wave studies for RF propagation use the complete complement ofMaxwell’s equations in which the plasma permittivity is an anisotropic tensor which is afunction of the local density in the edge region. The ambient magnetic field is assumedto be uniform and at arbitrary direction.

Lacking detailed measurements of the plasma density in the edge region, one has tomodel the effective permittivity based on some assumptions. One possible scheme is basedon the generalization of the Maxwell-Garnet homogenization technique (see MacKay &Lakhtakia 2015; Bairaktaris et al. 2017), which assumes some distribution of filamentarystructures of random sizes and densities. But there are many different representations ofdensity fluctuations which get implemented in various studies. In addition, given somerepresentation of the fluctuations, one can use either approximate methods or full-wavemethods to determine their effect on RF waves. The approximate methods are generallybased on the eikonal WKB solution of Maxwell’s equations.

A WKB ray tracing approach was used to study changes in the wave vectors of theRF waves due to a random distribution of blobs (see Hizanidis et al. 2010). In (seeSysoeva et al. 2015), two analytical models for the ordinary mode propagation throughturbulent edge plasma layer were developed based on the eikonal perturbation methodand the weak turbulence theory approach. In (see Snicker et al. 2018), RF scatteringfrom turbulent density fluctuations was studied by a Monte-Carlo solver (WKBeam) forthe kinetic wave equation by use of the integral form of the scattering operator withingthe Born approximation. There have been several studies using full-wave computationaltools. The effect of multiple filamentary structures was studied numerically by solvingthe full-wave equation within the COMSOL framework (see Ioannidis et al. 2017). In (seeKohn et al. 2018), scattering of RF beams from edge density fluctuations was investigatedusing the WKBeam code and the full-wave code IPF-FDMC, which is the standard 2DFinite Difference Time Domain (FDTD) code for isotropic media.

While these previous studies have considered the effect of blobs and filaments, andrandom fluctuations on RF waves, we are interested in the effect of periodic densityinterfaces (plasma gratings) on RF waves. This is primarily driven by experimentalobservations which indicate the presence of drift waves and rippling modes in theedge region (see Ritz et al. 1984). The density interface is at the edge separating thevacuum region in which the antenna structure is located and the turbulent plasmain the edge region and scrape-off layer. The periodic density interface could also bedue to the presence of a MHD instability. In order to have a detailed understandingof the effect of such an interface on the scattering of a RF wave, we have developed

RF waves in the plasma edge 3

the full-wave code ScaRF. ScaRF is based on the finite difference frequency domainmethod (FDFD) (see Smith 1996). FDFD solves Maxwell’s equations in the frequencydomain. It is a full wave method and thus describes reflection, refraction and diffractioneffects. ScaRF is this work is used not only for the RF scattering analysis of a densityinterface, approximated by a single mode periodic interface (plasma grating) but moregenerally by an arbitrary profile periodic interface generated as a superposition of spatialharmonics with random weights. In addition the permittivity of the turbulent region isapproximated by the aforementioned homogenized anisotropic permittivity tensor. Bothpermittivities (incidence and turbulent regions) are approximated by the cold plasmaones. To handle these problems, ScaRF uses FDFD formulated for anisotropic media (seeRumpf et al. 2014), in conjunction with the Total-Field Scattered Field (TFSF) method(see Papadopoulos & Glytsis 2014; Rumpf et al. 2014), for inserting the RF excitationinto the computational grid, the Perfect Matching Layer (PML) absorbing boundarycondition (see Oskooi & Johnson 2011), for absorption of irrelevant boundary reflectedwaves, and Floquet-Bloch-Periodic boundary conditions (FBPBC) for the definition ofthe periodic interface. ScaRF is a 3D code , and thus can model cases of arbitrarymagnetic field orientation. In addition it considers general density fluctuations and isnot restricted to periodic interfaces. We are not aware of the existence of a similar finitedifference (FD) full wave code for RF-plasma scattering. Other FD codes for plasmascattering, are usually time domain codes (FDTD). The advantage of FDFD is that ithandles numerical dispersion, anisotropy, incident field definition and periodic boundaryconditions, more straightforward than FDTD (see Glytsis et al. 2018), in which anisotropyand periodicity could lead in some cases to inaccurate results and instability (see Taflove& Hagness 2005). Thus FDFD is ideal for the class of problems ScaRF is designed tosolve. It is important to understand that the rippling modes or modulated interfacesexist in the edge region in conjunction with random fluctuations, blobs, and filaments.While ScaRF can take any representation of the edge plasma density into account, weare primarily motivated in understanding the effect of density ripples on RF waves.Our study complements previous studies which have dealt with blobs, filaments, andrandom fluctuations. In the future, ScaRF will be used for studying the effect of variousrepresentations of the edge density on the propagation of RF waves.

The structure of the paper is organized as follows. In section 2 the geometry of theplasma structure (plasma grating) is presented and the relation of the coordinate systemfor the microwave diffraction analysis with the magnetic field and plasma coordinatesystems in the torus is explained. Next the anisotropic permittivity tensors for coldplasma in the interface regions are derived (see Stix 1992). A detailed summary of theFDFD formulation follows. In section 3, the dispersion relation in anisotropic medium ispresented which is used to define the modes of the incident plane wave used in the FDFDmethod in conjunction with the TFSF technique. In section 4 the FDFD numerical resultsare shown for a sinusoidal plasma grating of various amplitudes and periods, for O andX modes of the incident RF wave, and for various interface region contrasts. Finally therandom periodic (containing multiple spatial frequencies) plasma grating is analyzed forO and X incident modes. Finally in section 5, the main results and conclusions of thiswork are summarized.

2. Geometrical Configurations and the FDFD method

The coordinate system used in the FDFD method for the RF diffraction analysisis based on the simplistic toroidal plasma configuration shown in Fig. 1, where twodifferent coordinate systems are shown relative to the toroidal plasma configuration.

4 A. D. Papadopoulos, et al.

xp

yp

zpzB

yBxB

xp

yp

yBxB

Plasma torus

Figure 1: A tokamak plasma torus is shown with the corresponding coordinate systemsof interest. The coordinate system (xB , yB , zB) corresponds to the magnetic flux density,~B, coordinate system while the (xp, yp, zp) correspond to the plasma coordinate system.The zp-component of the magnetic flux density corresponds to the toroidal magnetic fluxdensity component, Btor, while the xpyp-component corresponds to the poloidal magneticflux density component, Bpol.

The coordinate system (xB , yB , zB) corresponds to the magnetic field density coordinatesystem where the zB direction corresponds to the direction of the magnetic field (thisis the toroidal direction). The (xp, yp, zp) coordinate system corresponds to the plasmacoordinate system. Since the toroidal magnetic flux density can have a poloidal com-ponent the two coordinate systems could be related via the Euler rotation angles as itis represented in Fig. 2. The Euler angles that connect the two coordinate systems aredenoted by φB , θB , and ψB and all are measured counter-clockwise as shown in Fig. 2. Itis straightforward to show that the unit vectors of the two coordinate systems are relatedby: x̂BŷB

ẑB

= m11 m12 m13m21 m22 m23m31 m32 m33

x̂pŷpẑp

= M̃ x̂pŷpẑp

, (2.1)where

m11 = cosψB cosφB − cos θB sinφB sinψB ,m12 = cosψB sinφB + cos θB cosφB sinψB ,

m13 = sinψB sin θB ,

m21 = − sinψB cosφB − cos θB sinφB cosψB ,m22 = − sinψB sinφB + cos θB cosφB cosψB ,m23 = cosψB sin θB ,

m31 = sin θB sinφB ,

m32 = − sin θB cosφB ,

RF waves in the plasma edge 5

zB

yB

xB

zp

yp

xp

θΒ

φΒ

ψΒ

Figure 2: The relation between the (xB , yB , zB) and the (xp, yp, zp) coordinate systems.The angles φB , θB , and ψB are the Euler angles that connect the two coordinate systems.All the angles are defined positive as counter-clockwise.

m33 = cos θB ,

where the hatted variables correspond to the unit vectors in the corresponding coordinatesystem. According to Stix (see Stix 1992) the relative permittivity of the cold plasma isgiven by the following equation in the (xB , yB , zB) coordinate system.

ε̃B =

S −iD 0+iD S 00 0 P

, (2.2)where the needed parameters are defined below:

S =1

2(R+ L) ,

D =1

2(R− L) ,

P = 1 + Pe + Pi,

R = 1 +Re +Ri,

L = 1 + Le + Li,

Re =1 + incol

1 + Ce + incolPe,

Ri =1 + incol

1 + Ci + incolPi,

Le =1 + incol

1− Ce + incolPe,

Li =1 + incol

1− Ci + incolPi,

Pe = −(ωpeω

)2 11 + incol

,

6 A. D. Papadopoulos, et al.

Pi = −(ωpiω

)2 11 + incol

,

Ce = −ωceω,

Ci = +ωciω,

ωpe = qe

√neme�0

,

ωpi = qi

√nimi�0

= Zqe

√ni

Amp�0,

ωce =qeBzBme

,

ωci =ZqeBzBAmp

,

ncol =νcolω.

The variables in the above equations are: the magnitude of the electron charge qe, theelectron rest mass me, the atomic number Z, the atomic mass number A, the proton restmass mp, the permittivity of free space �0, the frequency of the electromagnetic radiationω = 2πf (where f the frequency in Hz), the electron and ion plasma densities ne and ni,respectively (in m−3), and the electron and ion collision rate νcol. The frequencies ωpeand ωpi are the electron and ion plasma resonant frequencies and the ωce and ωci are theelectron and ion cyclotron frequencies respectively. In this work ωce = 7.915·1011 rad/secand ωci = 2.155 ·108 rad/sec. Collisional absorption is not included in the current model,but can be incorporated by modifying the relative permittivity tensor in Eq. 2.2. Therelative permittivity tensor of Eq. (2.2) can be expressed in the plasma coordinate system(xp, yp, zp) using the following transformation

ε̃p = M̃−1ε̃BM̃, (2.3)

where ε̃B is defined in Eq. (2.2) and the Euler angles transformation matrix is definedin Eq. (2.1). A possible ripple at the torus plasma surface could cause diffraction of theincident microwave radiation. In order to study this effect a periodic ripple at the plasmatorus surface is studied. The geometry is shown in Fig. 3. The ripple can be defined as

h(x) = d− d cos(

2π

Λx

), (2.4)

where h(x) is the cosinusoidally varying ripple height and Λ the spatial period of theripple. The microwave diffraction is analyzed in the (x, y, z) coordinate system that isrelated to the plasma coordinate system (xp, yp, zp) as follows: x̂ŷ

ẑ

= 0 1 00 0 1

1 0 0

x̂pŷpẑp

= Q̃ x̂pŷpẑp

, (2.5)where the hatted variables are the unit vectors along the corresponding axes, respectively.The incident microwave radiation is modeled as a plane wave with an azimuthial angleof incidence φ and a polar angle of incidence θ (as shown in Fig. 3). The regions aboveand below the periodic ripple correspond to plasma regions of different plasma densities.Their tensor relative permittivities ε̃l,p can be determined by suitable use of Eq. (2.3) inthe plasma coordinate system. Then the relative permittivities ε̃1 and ε̃2 needed for the

RF waves in the plasma edge 7

z = xp

y = zp x = yp

!

kinc~

1

~ 2

d

Plasma ripple

Figure 3: A plasma ripple at the torus boundary is considered as a periodic spatialmodulation, i.e. as a plasma grating. The microwave radiation is represented as a planewave incident from the top towards the bottom region. The incident wavevector is shownas ~kinc and the incident angles are defined as φ and θ. The scattering coordinate system(x, y, z) is related to the plasma coordinate system by x = yp, y = zp and z = xp as shownin the figure. The plasma relative permittivities of the top and of the bottom regions aredefined as ε̃1 and ε̃2 respectively. The plasma grating is assumed to have a sinusoidalprofile of periodicity Λ along the x direction, and an amplitude spatial variation of d.

diffraction analysis in the (x, y, z) coordinate system can be found from

ε̃` = Q̃ε̃`,pQ̃T , ` = 1, 2. (2.6)

Next the FDFD method is formulated in the (x, y, z) coordinate system that will be usedin the analysis of the plasma grating of Fig. 3, with FBPBC in the xy plane and PMLin xy planes parallel to the z axis. FDFD solves Maxwell’s equations in the frequencydomain. It is a rigorous and stable method of known error sources (see Taflove & Hagness2005; Sadiku 2001; Smith 1996) that can model systems of complex geometry and can beapplied in parallel for computationally demanding problems. In FDFD finite differencesare used to approximate Maxwell’s equations, leading to a large linear algebraic system,whose solution provides the electromagnetic fields in space. In particular after normalizing

the magnetic field according to, ~̃H ≡ −iZ0 ~H, where Z0 is the free space impedanceand i is

√−1, Maxwell’s equations with the wave-absorbing PML layer truncating the

computational grid, become:

∇ ~E = k0 [�̃] ~̃H (2.7)

∇ ~̃H = k0 [µ̃] ~E, (2.8)

where [�̃] ≡ J [�] JT /det (J) and [µ̃] ≡ J [µ] JT /det (J) are relative permittivity andpermeability tensors, defined so as to implement the PML for anisotropic media, (seeOskooi & Johnson 2011), where J ≡ diag

(s−1x , s

−1y , s

−1z

)and sw ≡ κw+iσwω , w = {x, y, z}

8 A. D. Papadopoulos, et al.

z

yx

ExEy

Ez

HxHy

Hz

Figure 4: 3D Yee cell and electric and magnetic fields staggered in space by half acell. Electric field components are staggered along their direction, while magnetic fieldcomponents are staggered perpendicular to their direction.

are the PML stretching factors with κw > 1 the evanescent wave absorption parameter,and σw the PML conductivity. The parameters of the stretching factors are polynomials(see Oskooi & Johnson 2011) spatially varying along the w direction. It is convenient tosimplify Maxwell’s equations (2.7)-(2.8) by normalizing the grid coordinates as w̃ = k0w,w = {x, y, z}, which leads to:

∂Ez∂ỹ− ∂Ey

∂z̃= µ̃xxH̃x + µ̃xyH̃y + µ̃xzH̃z (2.9)

∂Ex∂z̃− ∂Ez

∂x̃= µ̃yxH̃x + µ̃yyH̃y + µ̃yzH̃z (2.10)

∂Ey∂x̃− ∂Ex

∂ỹ= µ̃zxH̃x + µ̃zyH̃y + µ̃zzH̃z (2.11)

∂H̃z∂ỹ− ∂H̃y

∂z̃= �̃xxEx + �̃xyEy + �̃xzEz (2.12)

∂H̃x∂z̃− ∂H̃z

∂x̃= �̃yxEx + �̃yyEy + �̃yzEz (2.13)

∂H̃y∂x̃− ∂H̃x

∂ỹ= �̃zxEx + �̃zyEy + �̃zzEz (2.14)

In FDFD-method the discretization of Eqs. (2.9)-(2.14) is done by approximating spatialderivatives using central differences, assuming that fields are placed on the Yee cell asshown in Fig. 4, and permittivity and permeability tensors elements �̃mn, µ̃mn (m,n ={x, y, z}) are placed at the same grid point as the En and H̃n fields respectively. Inthe resulting finite difference equations each term must exist at the same point on theYee grid. In order to satisfy this rule the terms containing off diagonal tensor elements inEqs. (2.9)-(2.14) are linearly interpolated to the correct grid points by using interpolationmatrices R−w , R

+w which averages the grids points along the direction w = {x, y, z}, with

the next (+) or previous (−) grid points. Eventually the discrete form is Eqs (2.9)-(2.10)is:

Deyez −Dezey = µ̃′xxh̃x +R−x R+y µ̃′xyh̃y +R−x R+z µ̃′xzh̃z (2.15)

RF waves in the plasma edge 9

Dezex −Dexez = R−y R+x µ̃′yxh̃x + µ̃′yyh̃y +R−y R+z µ̃′yzh̃z (2.16)

Dexey −Deyex = R−z R+x µ̃′zxh̃x +R−z R+y µ̃′zyh̃y + µ̃′zzh̃z (2.17)

Dhy h̃z −Dhz h̃y = �̃′xxex +R+xR−y �̃′xyey +R+xR−z �̃′xzez (2.18)

Dhz h̃x −Dhx h̃z = R+y R−x �̃′yxex + �̃′yyey +R+y R−z �̃′yzez (2.19)

Dhx h̃y −Dhy h̃x = R+z R−x �̃′zxex +R+z R−y �̃′zyey + �̃′zzez, (2.20)

where Dew, Dhw are derivative matrices, and ew, h̃w are vectors containing electric and

magnetic field values respectively, at discrete spatial points, in the direction w = {x, y, z}and µ̃′mn, �̃

′mn are diagonal matrices containing the relevant tensor elements of the grid

along their diagonals. Equations (2.15)-(2.20) are simplified by redefining the permittivityand permeability diagonal tensors, �̃′mn, µ̃′mn, so as to include the interpolation matricesaccording to:

�̃′′mn =

{R+mR

−n �̃′mn, m 6= n,

�̃′mn, m = n,(2.21)

µ̃′′mn =

{R−mR

+n µ̃′mn, m 6= n,

µ̃′mn, m = n.(2.22)

Then Eqs. (2.15)-(2.20) are equivalent to the linear system:

A

[~e~̃h

]= 0, (2.23)

where A ≡[

Ce −[µ̃′′]

−[�̃′′]

Ch

], (2.24)

Ce ≡

0 −Dez DeyDez 0 −Dex−Dey Dex 0

, (2.25)Ch ≡

0 −Dhz DhyDhz 0 −Dhx−Dhy Dhx 0

, (2.26)~e ≡ [ex, ey, ex]T and ~̃h ≡

[h̃x, h̃y, h̃x

]T. Since the r.h.s. of Eq. (2.23) is zero no meaningful

solution exists. A nonzero r.h.s of Eq. (2.23) is generated by introducing the plane waveexcitation in the computational grid by use of the total-field/scattered-field (TFSF)technique. In the TFSF method, a boundary S is defined separating the computationaldomain into regions where only total fields exist and regions where only scattered fieldsexits. The TFSF interface for the plasma grating in Fig. 3 is parallel to the xy plane. Toapply the TFSF method, the incident source electric (~esrc ≡ [ex,src, ey,src, ey,src]T ) and

magnetic (~̃hsrc ≡

[h̃x,src, h̃y,src, h̃y,src

]T) fields, propagating in the anisotropic medium,

free of scatterers (with only the background present), should be known. This informationis available and is given by Eqs. (3.5)-(3.7), as described in detail in section 3. Then the

r.h.s of Eq. (2.23), ~b is given by:

~b = (QA−AQ)[~esrc,

~̃hsrc

]T(2.27)

Q ≡ diag(Qex, Q

ey, Q

ez, Q

hx, Q

hy , Q

hz

), (2.28)

10 A. D. Papadopoulos, et al.

where Qew or Qhw are diagonal matrices operating on the electric or magnetic field source

vector, in the direction w = {x, y, z}. These Qew, Qhw matrices have 1 or 0 in the diagonalif they act on a source field node that belongs to the scattered or to the total fieldregion respectively. Then the electric and magnetic fields are obtained everywhere in the

computational domain, from the solution of the linear system A[~e,~̃h]T

= ~b.

3. Dispersion and polarization of RF waves in plasma

In order to define the excitation plane wave source for the FDFD method, it is necessaryto derive the dispersion relation for anisotropic media. There are two possible orthogonalpolarizations in the anisotropic region and the incident plane wave is selected as oneof them. In particular with respect to Fig. 3, an incident plane wave is assumed, withwavevector, ~k = kinc (−x̂ cos θ sinφ− ŷ sin θ sinφ+ ẑ cos θ), and kinc = k0n1, with k0the free space wavenumber and n1 the effective index on the wave in the anisotropicmedium with relative permittivity tensor �̃1. The Faraday’s and Ampere’s equations forthe planewave solution become:

~k × ~E = ωµ0 ~H (3.1)~k × ~H = −ω�0�̃1 ~E (3.2)

Elimination of the field components (z) vertical to the interface of Fig. 3, in Eqs. (3.1),(3.2) leads to a 4× 4 eigenvalue-eigenvector equation for the tangential to the interfacefield components ~vT ≡ [Ex Ey Hx Hy] of the form:

M̃~v = k̃z~v (3.3)

and the matrix M̃ is:

−�zxk̃x�zz

Z0

(1− k̃

2x

�zz

)−�zyk̃x

�zz

k̃yk̃xZ0�zz

1

Z0

(−k̃2y + �xx −

�xz�zx�zz

)− k̃x�xz

�zz

1

Z0

(−k̃xk̃y + �xy −

�xz�zy�zz

)k̃y�xz�zz

−�zxk̃y�zz

− k̃yZ0�zz

−�zy�zz

k̃y Z0

(k̃2y�zz− 1

)1

Z0

(−k̃xk̃y − �yx −

�yz�zx�zz

)�yz k̃x�zz

1

Z0

(−k̃2x − �yy −

�yz�zy�zz

)−�yz k̃y

�zz

(3.4)

In general there are four solutions to Eq. (3.3), two forward (+z) and two backward (−z),which can be propagating or evanescent. In order to classify them, Poynting’s vector isused. In particular Pz ≡ Re{Sz} is calculated, where ~S is the Poynting vector. If Pz > 0or Pz < 0 the solution is a forward or backward wave respectively, and if Pz = 0 the waveshould be evanescent. Since e−ikzz is the z-dependence and kz ≡ kzr + ikzi, it holds thatkzi < 0 or kzi > 0 for forward or backward evanescent wave respectively. The generalsolution of Eq. (3.3) is written as:

~v = [~v1+ ~v2+ ~v1− ~v1−]

A1+e

−ik1+z

A2+e−ik2+z

A1−e−ik1−z

A2−e−ik2−z

e−i~kt·~rt , (3.5)

RF waves in the plasma edge 11

where ~v1+, ~v2+, ~v1−, ~v1− are the eigenpolarizations [eigenvectors of Eq. (3.3)] thatcorrespond to the two forward and backward waves with kz1+, kz2+, kz1−, kz2− thecorresponding eigenvalues, with A1+, A2+, A1−, A1− the corresponding amplitudes,

with e−~kt·~rt the transverse field dependence. The other two field components, Hz, Ez

are specified directly from Eqs. (3.1) and (3.2) by use of Eq. (3.5) as:

Hzi± =1

Z0

[k̃xEyi± − k̃yExi±

](3.6)

Ezi± =Z0�zz

[k̃yHxi± − k̃xHyi±

]− �zxExi±

�zz− �zyEyi±

�zz(3.7)

where i = 1, 2, and Z0 the free space impedance.

4. Numerical Results

It is emphasized that the purpose of the following numerical results is twofold. Onone side, it is to highlight the capabilities of the developed FDFD solver, ScaRF, andon the other side for the rigorous analysis of RF scattering by plasma-blobs, which is animportant problem, with various applications. Usage of the ScaRF solver, for the analysisof additional plasma-blobs systems is left for future work.

By use of ScaRF, the electric and magnetic field vectors e, h, where h = iZ−10 h̃,are calculated at every node of the computational domain, and consequently the time-averaged Poynting vector, S is calculated as:

S =1

2Re{e× h∗}. (4.1)

In the following figures the Poynting vector components are presented in the code’scoordinate system x, y, z. Transformation of the results to the plasma coordinate systemxp, yp, zp is straighforward by use of Eq. 2.5. The y component of the Poynting vectoris not shown since it is significantly smaller than the x, z, components. Fig. 5 is themain-simulation, and subsequent simulations are cases where a single design parameteris varied.

In Fig. 5, the normalized Poynting vector (normalized to the incident plane wavePoynting vector), is shown for a common blob configuration, where it is assumed thatan O-mode or an X-mode plane wave is incident on the interface, as seen in Fig. 3, atangle θ = 30 deg and at frequency of 170 GHz. The periodic interface region has a periodΛ = 10λ1 and amplitude d = λ1, where λ1 is the wavelength of the incident wave. Theinterface region contrast is weak , where the electron density of background and blob isset to nbg. = 3 · 1020m−3, nbl. = 3.2 · 1020m−3, respectively. The magnetic field is at a 6deg inclination, with toroidal component, of 4.5T and poloidal component of 0.4730T.

It is observed in Fig. 5(e)-(f) that the density of the energy flow lines of the normalizedPoynting vector increases in regions of high values of the Poynting vector amplitude. Inaddition these flow lines are spatially varying due to the spatial variation of the periodicinterface. In regions where the Poynting amplitude is very small, close to numerical noiselevel, the pattern of the flow lines changes irregularly (X mode).

Next the normalized Poynting vector is shown when a single design variable (theamplitude, period or interface region contrast) is varied relatively to the ScaRF simulationof Fig. 5. In particular in Fig. 6 the amplitude of the modulation is decreased to 0.2wavelength of the incident wave and a different spatial pattern of the Poynting vector isobserved compared to Fig. 5. It is observed that relative to the main simulation of Fig.

12 A. D. Papadopoulos, et al.

Re(Sz)/|Re(S

0)|

-20 -10 0 10 20

x (mm)

-20

-10

0

10

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mm

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-1

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mm

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0

1

2

3

4

5

(d)

(e) (f)

Figure 5: Main Simulation where an O-mode or an X-mode plane wave is incident onthe interface (Fig. 3) at angle θ = 30 deg, at frequency of 170 GHz, Λ = 10λ1, d = λ1(a)-(b): Components of the normalized Poynting vector, O-mode. (c)-(d): X-mode. (e)-(f) Normalized Poynting amplitude and Poynting vector flow, for the O and X moderespectively.

RF waves in the plasma edge 13

5, the power flow lines are almost straight due to the flatter periodic interface, that leadsto weaker diffraction effects.

In Fig. 7 the period of the modulation is increased to 20 wavelengths of the incidentwave, compared to the modulation period of Fig. 5. This change in period, results in quitedifferent spatial distribution of the Poynting vector compared to Fig. 5. In particularincreasing the period results in weaker diffraction effects in the sense that there are fewersignificant diffracted orders. This can also be seen from the power flow in Fig. 7(e)-(f),where the flow lines are smoother compared to Fig. 5.

With respect to the main simulation (Fig. 5), in Fig. 8 the electron density of the blobis increased from nbl. = 3.2 · 1020m−3 to nbl. = 3.6 · 1020m−3 for the O mode, and fromnbl. = 5.5 · 1020m−3 to nbl. = 6.3 · 1020m−3 for the X mode. This increased interfaceregion contrast leads to strong reflections of the incoming wave and consequently, as canbe seen in Fig. 8, the Poynting vector amplitude in the blob region is very small since forthe chosen blob-electron densities, at 170GHz, the blob-system is in the frequency cutoff.This is also quantitatively shown by calculation of the reflection, r, and transmission,t, coefficients. These are the square root of the spatial integral in the y-direction of theSx vector for the scattered fields divided by the power of the incident wave in the xdirection. The values of r, t, are r = 1.005, t = 0.013 and r = 0.98, t = 1.11 · 10−6 forthe O and X mode respectively. In addition in Fig. 8(e)-(f) the power flow lines are veryirregular in the reflection region due to interference of the incident plane wave and thestrong reflected power from the interface. This is also supported from the fact that themaximum Poynting amplitude value is higher than any other of the cases considered.The power flow lines are also irregular in the transmission regions, since the Poyntingamplitude is very small, close to the numerical noise level.

Finally in Fig. 9 the interface region is generated as a superposition of 4 spatialmodes (multimode) with periods of 2.5, 5, 10, 20 wavelengths of the incoming wave,and with randomly selected heights, 0.8147, 0.9058, 0.1270, 0.9134 respectively.The bloband background electron densities are the same as in Fig. 5 (nbl. = 3.2 · 1020m−3,nbg. = 3.0 · 1020m−3). In principle a periodic plasma grating of any shape, can berepresented by a superposition of spatial modes (Fourier decomposition). This interface,defined by an appropriate number of modes and weights, could represent a possiblerealization of an experimentally observed interface region geometry. Similar experimentalscenarios have been observed in (see Ritz et al. 1984) where at the Results and Discussionsection, it is stated that : “Dissipative drift waves and rippling modes, driven by densityand temperature gradients, are strong candidates to describe the density and potentialfluctuations...”. Again the components of spatially varying normalized Poynting vectorare shown for O and X modes of the incident wave in Fig. (a)-(d) . In Fig 9(e)-(f) atthe reflection region, the power flow lines are almost straight lines, since it appears thatthere’s weak reflection due to the weak density contrast and mainly the incident planewave is present. In the transmission region the power flow lines are irregular due to thecomplex diffraction effects from the multimode interface.

5. Conclusions

In this work the FDFD code ScaRF, is developed for the analysis of RF wave propa-gation through a turbulent edge plasma region where dissipative drift waves and ripplingmodes characterize density fluctuations (see Ritz et al. 1984). The interfaces between theregion of incidence of the RF wave and the turbulent (blob) region are approximatedas sinusoidal interfaces or more generally, as arbitrary periodic interfaces. The tworegions are anisotropic and the respective permittivities are specified by the cold plasma

14 A. D. Papadopoulos, et al.

Re(Sz)/|Re(S

0)|

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mm

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0

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(d)

(e) (f)

Figure 6: Same parameters as the main simulation of Fig. 5, except the periodic interfaceamplitude which is 20% of the incident wavelength, (a)-(b) Components of the normalizedPoynting vector, O-mode. (c)-(d): X-mode. (e)-(f) Normalized Poynting amplitude andPoynting vector flow, for the O and X mode respectively.

RF waves in the plasma edge 15

Re(Sz)/|Re(S

0)|

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x (mm)

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mm

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mm

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(d)

(e) (f)

Figure 7: Same parameters as the main simulation of Fig. 5, expect the periodic interfaceperiod which is 20 times the incident wavelength, (a)-(b) Components of the normalizedPoynting vector, O-mode. (c)-(d): X-mode. (e)-(f) Normalized Poynting amplitude andPoynting vector flow, for the O and X mode respectively.

16 A. D. Papadopoulos, et al.

Re(Sz)/|Re(S

0)|

-20 -10 0 10 20

x (mm)

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mm

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Re(Sz)/|Re(S

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(e) (f)

Figure 8: Same parameters as the main simulation of Fig. 5, expect that the electrondensity of the blon region is stronger: nbl. = 3.6 · 1020m−3, nbl. = 5.5 · 1020m−3 for theO-X mode, respectively.(a)-(b) Components of the normalized Poynting vector, O-mode.(c)-(d): X-mode. (e)-(f) Normalized Poynting amplitude and Poynting vector flow, forthe O and X mode respectively.

RF waves in the plasma edge 17

Re(Sz)/|Re(S

0)|

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0

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2

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(e) (f)

Figure 9: Same parameters as the main simulation of Fig. 5, except the background-blobregion is a multimode interface, (a)-(b) Components of the normalized Poynting vector,O-mode. (c)-(d): X-mode. (e)-(f) Normalized Poynting amplitude and Poynting vectorflow, for the O and X mode respectively.

18 A. D. Papadopoulos, et al.

permittivity tensor (see Stix 1992). For the turbulent region a homogenized permittivitytensor is utilized. The FDFD method in ScaRF is formulated for anisotropic media,using Floquet-Bloch-periodic boundary conditions to describe the periodic interface, inconjunction with PML’s to remove irrelevant reflections, and the TFSF interface forthe plane wave excitation, which is defined in the background medium and satisfies theanisotropic-media dispersion relation. A number of simulations are shown, where thePoynting vector is calculated for O and X mode excitation, and for variations of theperiod and the amplitude of the modulation, and also of the interface region contrast.Finally by calculating the Poynting vector of a general periodic interface region, defined asan arbitrary superposition of 4 sinusoidal (with freq. ωn = nω0, n = 1, . . . , 4) interfaces,the potential of the method to analyze arbitrary configurations is emphasized. It is notedthat there are no limitations in the ScaRF code as to what sort of density fluctuationsare prescribed. It can be used for more general representations of the edge fluctuations,periodic or non-periodic. In this work the ScaRF code is applied to a problem which hasnot been addressed before–nameley periodic interfaces.

In the near future, Scarf will be used for more general representations of the plasmaedge fluctuations, and in conjunction with the Polynomial Chaos Expansion method(PCE) (see Papadopoulos et al. 2018), will rigorously analyze the turbulent region-incidence region scattering for random variation of the material and geometric parame-ters, for periodic or non-periodic structures, and calculate the statistical moments of thePoynting vector.

6. Acknowledgment

This work has been carried out within the framework of the EUROfusion Consortiumand has received funding from the Euratom research and training programme 2014-2018and 2019-2020 under grant agreement No 633053 (except author A.K.R.). The views andopinions expressed herein do not necessarily reflect those of the European Commission.A.K.R. is supported by the US Department of Energy Grant numbers DE-FG02-91ER-54109, DE-FG02-99ER-54525-NSTX, and DE-FC02-01ER54648.

RF waves in the plasma edge 19

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coverA. D. Papadopoulos1, E. N. Glytsis1, S. I. Valvis1, P. Papagiannis1, A. K. Ram3, K. Hizanidis1, A. Zisis2

18ja038_fullIntroductionGeometrical Configurations and the FDFD methodDispersion and polarization of RF waves in plasmaNumerical ResultsConclusionsAcknowledgment