Spectral broadening from turbulence in multiscale lower hybrid current drive simulations Bodhi Biswas1 Paul Bonoli1 Abhay Ram1 and Anne White1

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Spectral broadening from turbulence in multiscale lower hybrid

current drive simulations

Bodhi Biswas,∗1, Paul Bonoli1, Abhay Ram1, and Anne White1

1Plasma Science and Fusion Center, Massachusetts Institute of Technology

October 26, 2022

Abstract

The scattering of lower hybrid (LH) waves due to scrape-oﬀ layer (SOL) ﬁla-

ments is investigated. It is revealed that scattering can account for the LH spectral

gap without any ad hoc modiﬁcation to the wave-spectrum. This is shown using a

multiscale simulation approach which allows, for the ﬁrst time, the inclusion of full-

wave scattering physics in ray-tracing/Fokker-Planck calculations. In this approach,

full-wave scattering probabilities are calculated for a wave interacting with a statis-

tical ensemble of ﬁlaments. These probabilities are coupled to ray-tracing equations

using radiative transfer (RT) theory. This allows the modeling of scattering along

the entire ray-trajectory, which can be important in the multi-pass regime. Sim-

ulations are conducted for lower hybrid current drive (LHCD) in Alcator C-Mod,

resulting in excellent agreement with experimental current and hard X-ray (HXR)

proﬁles. A region in ﬁlament parameter space is identiﬁed in which the impact of

scattering on LHCD is saturated. Such a state coincides with experimental LHCD

measurements, suggesting saturation indeed occurs in C-Mod, and therefore the ex-

act statistical properties of the ﬁlaments are not important.

1 Introduction

Lower hybrid waves are an eﬃcient means to non-inductively drive current in a tokamak

via electron Landau damping (ELD) [1]. It is an attractive actuator for current proﬁle

shaping, which has been successfully demonstrated in several tokamaks [2–6]. There is

also interest in targeted LHCD for neoclassical tearing mode suppression [7,8].

The condition for strong linear ELD is [9]



N||

≥N||,ELD ≡c

3vte

(1)

where N|| ≡N·B/B is the parallel refractive index (with respect to the background

magnetic ﬁeld B), cis the speed of light, and vte =p2Te/meis electron thermal velocity

(with Teand mebeing the electron temperature and mass, respectively). If the initial

parallel refractive index 

N||0

< NELD, the wave is expected to ﬁrst undergo |N|||-upshift

due to toroidicity until the condition in Eq. (1) is met [10]. For a suﬃciently large

spectral-gap between 

N||0

and NELD, the wave may require multiple passes through the

plasma before this gap is bridged. This scenario is typical of present-day tokamaks due

to low core Te.

A large spectral-gap poses diﬃculties for theory and modeling of LHCD. Ray-

tracing/Fokker-Planck modeling of LHCD reveal large discrepancies between simulations

and experiments in several tokamaks [11–14]. In high aspect ratio devices (A ≡ R0/a &5,

where R0is the major radius, and ais the minor radius), like WEST and TRIAM-1,

toroidicity is insuﬃcient to bridge the spectral-gap via 

N||

-upshift. In moderate aspect

∗Present aﬃliation: University of York, UK. E-mail: bodhi.biswas@york.ac.uk

arXiv:2210.02344v2 [physics.plasm-ph] 25 Oct 2022

Bodhi Biswas 1 INTRODUCTION

ratio devices (A ≈ 3), like Alcator C-Mod and EAST, toroidicity can accurately model

the total current drive eﬃciency, but not the radial current or HXR proﬁles [11,13]. For

example, in C-Mod, the simulated LH current proﬁles are not smooth and are peaked far

oﬀ-axis, while experimental motional Stark eﬀect (MSE) and HXR measurements suggest

smooth proﬁles robustly peaked on-axis [11]. Consequently, N||-upshift due to toroidicity

is not adequate for understanding experimental observations. There exist other contribut-

ing factors that must be taken into account in theoretical models.

Several mechanisms have been proposed as possible explanations for the spectral gap.

Ad hoc modiﬁcations to the launched LH wave-spectrum reveal that either N|| broaden-

ing [15] or angle-broadening of the perpendicular refractive index (N⊥) [16] can explain

experimental current drive results in C-Mod.

The two theorized causes of N|| broadening are parametric decay instabilities (PDI)

[17,18] and scattering from parallel density ﬂuctuations in front of the LH antenna [19].

In FTU, a combination of modeling and experiment shows PDI is likely responsible for

low LHCD eﬃciency at high densities [20]. However, there is little theoretical support for

- or experimental evidence of - strong PDI in low and moderate density C-Mod discharges

(¯ne<1020 m−3) where the spectral gap persists [21]. Likewise, there is little evidence

of density ﬂuctuations with the large parallel gradients required to induce a signiﬁcant

N||-broadening.

The most likely mechanism for N⊥angle-broadening is scattering from turbulent

scrape-oﬀ layer (SOL) ﬂuctuations. Prior models have employed either ray-tracing [22,23]

or wave-kinetic treatments [10,24,25] for scattering in drift-wave turbulence. These mod-

els have demonstrated modiﬁed current proﬁles, but cannot match experimental observa-

tions. Gas-puﬀ imaging (GPI) [26] and statistical analysis of Langmuir probe measure-

ments [27] in the SOL have motivated the modeling of LH scattering from intermittent,

ﬁeld-aligned ﬁlaments. The extent of N⊥angle-broadening from ﬁlaments is greater than

in “equivalent” drift-wave turbulence [28]. A recent hybrid full-wave/statistical model

for wave-ﬁlament interactions was developed to model the modiﬁcation to the LH wave-

spectrum in front of the antenna [29]. Multiple wave-ﬁlament interactions are accounted

for using the radiative transfer (RT) equation. This treatment allows the modeling of

realistic turbulence parameters without being restricted to the validity constraints of the

ray-tracing or random phase approximation. The study ﬁnds a large angle-broadening of

the incident wave-spectrum, enough to robustly direct a fraction of LH power to damp

on-axis on ﬁrst pass through the plasma. In turn, the LH current proﬁle is monotonic

and peaked on-axis, in much closer agreement with experiment. In addition, asymmet-

ric scattering of the LH wave is observed in angle-space. This is a full-wave eﬀect only

possible with intermittent density ﬂuctuations, and therefore is not accounted for in prior

ray-tracing or wave-kinetic treatments.

The hybrid full-wave/statistical scattering model discussed above is limited to a slab

geometry with homogeneous background and turbulence parameters. It also only treats

slow wave to slow wave (S→S) scattering and ignores the fast (F) wave. Therefore this

model can only approximately treat scattering directly in front of the LH antenna. In this

paper, the more general RT equation is solved using a multiscale full-wave/ray-tracing

solver. It allows the modeling of arbitrary geometry and both like-mode (S→S,F→F)

and unlike-mode (S→F,F→S) scatter. An arbitrary geometry allows accounting for

realistically tapered SOL turbulence proﬁles in a tokamak, and models scattering along the

entire ray-trajectory (important in the multi-pass regime). The inclusion of all scattering

modes is especially important near the mode-conversion density where like- and unlike-

mode scattering probabilities are comparable.

This multiscale model is applied to Alcator C-Mod, allowing, for the ﬁrst time, the

Bodhi Biswas 2 MULTISCALE SCATTERING MODEL

inclusion of self-consistent full-wave scattering physics in LHCD simulations. The result-

ing current and HXR proﬁles provide excellent matches to experimental measurements.

Thus, the spectral-gap in C-Mod is resolved via turbulent scattering, and without any ad

hoc modiﬁcation to the wave-spectrum.

This paper is organized as follows. Section 2 will describe the multiscale scattering

model. Section 3 discusses its application to LHCD in Alcator C-Mod discharges. Section

4 provides a discussion and summary of the ﬁndings.

2 Multiscale scattering model

From a modeling perspective, the LH wave is in an interesting and challenging range

of wavelengths. The wavelength is small enough to employ ray-tracing in the quiescent

core [30], but large enough that full-wave modeling is required for common SOL turbulence

parameters. On the other hand, the LH wavelength is small enough such that whole-

device full-wave modeling in a turbulent plasma is prohibitively expensive. Therefore, a

multiscale model that employs ray-tracing in the core and a full-wave solver in the SOL

is a promising concept. The implementation is as follows. A Mie-scattering approach

eﬃciently calculates the scattering probabilities for an incident plane wave interacting

with a ﬁlament. The probabilities are averaged over a statistical ensemble of ﬁlament

parameters. Next, the scattering of RF power in phase-space is modeled using the RT

approximation. The RT equation is solved in a ray-tracing solver, where the scattering

terms act as stochastic kicks to the ray-trajectory. Finally, the rays are used to calculate

current drive using the quasi-linear Fokker-Planck equation.

2.1 Review of single wave-ﬁlament interaction

The Mie-scattering model for a single wave-ﬁlament interaction is brieﬂy reviewed. An

incident LH plane wave, either the slow (S) or fast (F) mode, travels through a cold

magnetized plasma with a homogeneous background density n0and magnetic ﬁeld B.

The magnetic ﬁeld is aligned along ˆez, such that the parallel wave-number k|| =kz.

The wave trajectory is aligned such that vgr⊥=vgr⊥ˆex, where vgr⊥is the perpendicular

group velocity and vgr⊥>0. An inﬁnitely long, cylindrical, ﬁeld-aligned ﬁlament passes

through the origin with density nb. Given this is a poloidally symmetric system (where θ=

tan−1(y/x)), the Jacobi-Anger expansion is used to write the electric ﬁeld in cylindrical

coordinates (ρ, θ, z). Consequently, the ﬁeld everywhere is a series solution in poloidal

mode-numbers (m≡kθρ),

Ejβ =ei(k||z−ωt)

+∞

m=−∞

EjmWjβmeimθ;β=ρ, θ, z (2a)

Wjρm =ξjxJ0

m(kj⊥ρ)−iξjy

kj⊥ρJm(kj⊥ρ) (2b)

Wjθm =iξjx

kj⊥ρJm(kj⊥ρ) + ξjyJ0

m(kj⊥ρ) (2c)

Wjzm =iξjzJm(kj⊥ρ) (2d)

where j= 0, ..., 4 is the wave index, indicating (0) the incident wave, (1,2) the

slow/fast mode inside the ﬁlament, and (3,4) the slow/fast mode outside the ﬁlament.

ξj={ξjx, ξjy, ξjz}is the plane-wave polarization of wave j.Jmis the Bessel function

of the ﬁrst kind and order m.J0

mis the ﬁrst derivative of Jmwith respect to its argu-

ment. The coeﬃcients Ejm for the incident plane wave (j= 0) are known through the

Bodhi Biswas 2 MULTISCALE SCATTERING MODEL

Jacobi-Anger expansion. The coeﬃcients for the remaining waves are found by imposing

Maxwell’s boundary conditions at the ﬁlament perimeter. This is detailed in Ram et al.,

(2016) [31].

This technique can be extended to a radially varying ﬁlament density proﬁle, as de-

tailed in Biswas et al., (2021) [29]. A similar method was used to model ion-cyclotron

wave scattering [32]. For the purposes of this work, it is assumed the ﬁlament has a

Gaussian density proﬁle such that

n(ρ)−n0=n0nb

−1e−2√ln(2)ρ

ab2

(3)

where nis the density, nb/n0is the relative density of the ﬁlament, and abis the full-width

half max of the ﬁlament radial proﬁle.

Following [29,33], the diﬀerential scattering-width is calculated by evaluating the ra-

dially scattered Poynting ﬂux in the far-ﬁeld limit

σj→j0(θ) = ∓2

|ξj0y|2+|ξj0z|2+kz

kj0⊥Reξj0xξ∗

j0z

kjx(|ξjy|2+|ξjz|2)−kzReξjxξ∗

jz



+∞

m=−∞

i±mEj0meimθ



(4)

where now j=S, F is the incident mode, j0is the scattered mode, ξj` is the `-component of

the electric ﬁeld polarization of mode j, and kj` is the `-component of the wave-number

of mode j. The sign ∓depends on the scattered mode. The diﬀerential scattering-

width is analogous to a diﬀerential scattering cross-section. Likewise, σ=Rσ(θ)dθ is the

scattering-width, which is analogous to a scattering cross-section. Lastly, ˆσ(θ)≡σ(θ)/σ is

the normalized diﬀerential scattering-width and will be useful in writing the RT equation.

2.2 Scattering through statistical ensemble of ﬁlaments

To account for the statistical variation in ﬁlament relative density (nb/n0) and radial width

(ab), a joint probability distribution function p(nb/n0, ab) is deﬁned for an ensemble of

ﬁlaments. The quantities hnb/n0iand habidenote the average relative density and radial

width of ﬁlaments. The average, or “eﬀective”, diﬀerential scattering-width is

σeﬀ,j→j0(θ) = hσj→j0(θ)i=Z∞

dabZ∞

d (nb/n0)σj→j0(θ;nb/n0, ab)p(nb/n0, ab) (5)

A wave packet traveling in a straight line will encounter, on average, fp

πhabi2ﬁlaments

per unit length in the perpendicular plane. Thus, the inverse mean-free-path to scatter is

Σeﬀ,j→j0=fp

πhabi2σeﬀ,j→j0(6)

where fpis the packing-fraction, which is deﬁned as the fractional area (in the perpen-

dicular plane) inhabited by ﬁlaments. Note that Σeﬀ,j→j0and σeﬀ,j→j0(θ) are dependent

on the local plasma parameters, the incident wave’s frequency and wavenumber, and the

ﬁlament PDF. These functional dependencies are only stated explicitly when needed.

2.3 Radiative transfer approximation

Radiative transfer (RT) theory models the RF wave-spectrum as a wave-packet distribu-

tion function in phase-space, akin to the Fokker-Planck equation for a particle distribution

Bodhi Biswas 2 MULTISCALE SCATTERING MODEL

function [24]. We deﬁne Pj(r,k) as the distribution corresponding to the LH mode j. The

RT equation governing Pjis

dPj

dt r

+ 2γ(kj,r)Pj=dPj

dt sct

(7)

The (...)rterm is the convective term accounting for the trajectory for a wave-packet/ray.

The second term accounts for wave damping, where γis the damping rate. The left hand

side (LHS) is routinely solved using ray-tracing/Fokker-Planck codes. The right hand

term accounts for the added eﬀect of scattering,

dPj

dt sct

j0=S,F

−Σeﬀ,j→j0(k||,r)|vgr⊥(kj,r)|Pj

j0=S,F

Σeﬀ,j0→j(k0

||,r)|vgr⊥(kj0,r)|Zπ

−π

ˆσeﬀ,j0→j(χ−χ0;k||,r)Pj0(χ0, k||,r)dχ0

(8)

The transformation of θ→χmust ﬁrst be explained. The variable θdescribes the angle

of scatter in the frame of a single ﬁlament-wave interaction, as described in Section 2.1.

In contrast, χis deﬁned through b·(ˆ

eψ×k⊥) = k⊥sin χ, where ˆ

eψis the unit vector

normal to the ﬂux surface and directed outwards, and b≡B/B is the unit vector along

the magnetic ﬁeld. Thus, by using χ, the angular orientation of k⊥is unambiguously

deﬁned in a tokamak geometry. The functional dependence of Pj(r,k) can be mapped to

Pj(χ, k||,r) and vice-versa using the dispersion relation for the appropriate mode, and the

local magnetic geometry.

The two RHS terms in Eq. (8) account for angular (χ) rotation of the perpendicular

wave-vector k⊥due to a wave-ﬁlament interaction. The ﬁrst term accounts for out-scatter

of Pjfrom a phase-space volume element centered at χ. The second term accounts for

in-scatter into this volume element from all other angles χ0. Note that the scattering

probabilities and background plasma parameters are allowed to vary in real-space. In

addition, the summation over the slow and fast modes now ensure that both like-mode

and unlike-mode scatter (S→F, F →S) are accounted for. In contrast, the system

studied in [29] was restricted to homogeneous scattering and background parameters, and

neglected mode-conversion.

It should be clariﬁed that, in generating scattering probabilities using a full-wave

formalism, interference eﬀects during a single wave-ﬁlament interaction are accounted

for. However, in using the RT approximation, the interference eﬀects of simultaneous

multi-ﬁlament scattering are ignored. This is a reasonable approximation as long as

k⊥d1, where dis the average distance between ﬁlaments [34]. Code comparison with

a fully numeric full-wave solver ﬁnds that this multiscale method generally overestimates

the eﬀects of scattering for fp&0.15 [29]. Nevertheless, the multiscale model retains

many important full-wave eﬀects and is therefore a signiﬁcant improvement over prior

reduced RF-turbulence scattering models in the SOL.

2.4 Coupling to ray-tracing code GENRAY

The initial wave-spectrum, launched from the LH antenna, is discretized into rays. Each

ray-trajectory is evolved using the ray-tracing equations in GENRAY [35]. The quasi-

linear calculation of γ, accounting for Landau damping and collisions, is accomplished

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SpectralbroadeningfromturbulenceinmultiscalelowerhybridcurrentdrivesimulationsBodhiBiswas,*1,PaulBonoli1,AbhayRam1,andAnneWhite11PlasmaScienceandFusionCenter,MassachusettsInstituteofTechnologyOctober26,2022AbstractThescatteringoflowerhybrid(LH)wavesduetoscrape-olayer(SOL)la-mentsisinvestigated.Iti...

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Spectral broadening from turbulence in multiscale lower hybrid current drive simulations Bodhi Biswas1 Paul Bonoli1 Abhay Ram1 and Anne White1

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