The shear Alfv en continuum with a magnetic island chain in tokamak plasmas Z. S. Qu1zand M. J. Hole12

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The shear Alfv´en continuum with a magnetic island

chain in tokamak plasmas

Z. S. Qu1‡and M. J. Hole1,2

1Mathematical Sciences Institute, the Australian National University, Canberra ACT

2600, Australia

2Australian Nuclear Science and Technology Organisation, Locked Bag 2001,

Kirrawee DC NSW 2232, Australia

E-mail: zhisong.qu@ntu.edu.sg

Abstract. The shear Alfv´en continuum spectrum is studied for a tokamak with a

single island chain using the ideal Magnetohydrodynamics (MHD) theory. We have

taken into account the toroidal geometry and toroidal mode coupling with the island

considered as a highly-shaped stellarator. Various new frequency gaps open up inside

the island due to its asymmetry both poloidally and toroidally, such as the Mirror-

induced Alfv´en Eigenmode (MAE) gap and the Helicity-induced Alfv´en Eigenmode

(HAE) gap. We have shown that the MAE gap acts as the continuation of the

outside Toroidal Alfv´en Eigenmode (TAE) gap into the island. However, the combined

TAE/MAE gap is getting narrower as the island grows, leaving only half of its original

width with a moderate island size as much as 3.2% of the minor radius. In addition,

the two-dimensional eigenfunction of the continuum mode on the lower tip of the MAE

gap now has highly localised structures around the island’s long axis, contrary to the

usual oscillatory global solutions found with no or a low level of toroidal asymmetry -

an indication of the continuous spectrum becoming discrete and dense. These results

have implications for the frequency, mode structure and continuum damping of global

TAEs residing in the gap.

1. Introduction

Magnetically conﬁned fusion plasmas contain signiﬁcant fast populations originating

from fusion products and external heating such as the neutral beam injection (NBI)

and the ion cyclotron resonance heating (ICRH) [1]. These fast particles, when

slowed down, can excite a zoo of Alfv´en eigenmodes as discrete solutions of the ideal

Magnetohydrodynamics (MHD) spectrum in a process similar to the inverse Landau

damping, leading to enhanced fast ion transport and therefore worse energy output [2].

Of all the Alfv´en eigenmodes, the most experimentally proliﬁc is the Toroidicity-induced

Alfv´en Eigenmode (TAE) [3,4], which resides in the band gaps of the shear Alfv´en

‡Present Address: School of Physical and Mathematical Sciences, Nanyang Technological University,

637371 Singapore, Singapore.

arXiv:2210.15086v2 [physics.plasm-ph] 22 Nov 2022

The shear Alfv´en continuum with a magnetic island chain in tokamak plasmas 2

continuum spectrum induced by the poloidal modulation of the magnetic ﬁeld and

geometry.

The classic theory and numerical solvers of TAEs in tokamaks generally assume

nested ﬂux surfaces and perfect toroidal symmetry. However, broken symmetry is

introduced unavoidably, by the ﬁnite number of ﬁeld coils or spontaneous instabilities

such as tearing modes [5], and deliberately, through the use of resonant magnetic

perturbation (RMP) coils [6], to suppress large explosive instabilities known as edge

localised modes (ELMs) [7]. With the loss of symmetry and thus integrability, the

ﬁeld lines can tangle around a ﬁxed-point, creating so-called magnetic islands, or when

multiple islands overlap, regions of ﬁeld line chaos. The impact of symmetry-breaking

ﬁelds on Alfv´en eigenmodes is an emerging research topic. Several experiments in

NSTX [8,9] and KSTAR [10] have found an RMP ﬁeld to either reduce or enhance the

amplitude of the TAE, depending on RMP phasing and plasma conditions. Existing

works focused on the inﬂuence of RMP on the energetic particle distribution function [11]

and the change of background plasma parameters such as the rotation, taking the mode

frequency and structure to be the same as if the symmetry is not broken. Nevertheless,

islands and chaos could modify the frequency, mode structure and damping rate of the

TAEs and thus aﬀect energetic particle conﬁnement.

As an important ﬁrst step, one needs to answer the question of how a single island

chain changes the shear Alfv´en continuum spectrum, in particular the TAE gap where

global eigenmodes reside. The continuum with a magnetic island has been studied

in slab and cylindrical geometries [12,13,14,15,16,17], where the change of the ﬁeld

strength poloidally, as well as the coupling of modes with diﬀerent toroidal numbers due

to an absence of toroidal symmetry, are neglected. The idea is to separate the island

chain from the rest of the plasma volume, construct a coordinate system within the

island aligning with the magnetic surfaces inside, and then apply the same continuum

equations as the outside. In other words, one considers the island itself as a straight ﬂux

tube with its O point being the new magnetic axis with nested ﬂux surfaces surrounding

it. The main ﬁnding is that the island has its own frequency gaps, with the dominant

one being the Ellipticity-induced Alfv´en Eigenmode (EAE) gap [18], or the Magnetic-

island-induced Alfv´en Eigenmode (MiAE) gap named by Biancalani et al [12], due to

the elongation of the island. Moreover, the lowest-frequency continuum accumulation

point (CAP) is shifted up on the island separatrix, thanks to the strong poloidal mode

coupling there. A recent publication [19] extends the three-dimensional (3D) continuum

code CONTI [20] to compute the continuum in Wendelstein 7-X with islands. The

discovery of the island EAE/MiAE gap prompts a further search for discrete MiAEs,

with candidate modes being identiﬁed in TJ-II [21] and J-TEXT [22] experimentally,

and in Madison Symmetric Torus (MST) [23] using the SIESTA-Alfv´en code [15], an

extension of the SIESTA [24] 3D equilibrium code with a kinetic normalisation matrix.

Magnetic islands are also found to interact and excite Beta-induced Alfv´en Eigenmodes

(BAEs) in FTU [25,26] and later in HL-2A [27] and J-TEXT [22].

Despite these great advancements, important questions regarding the connection

The shear Alfv´en continuum with a magnetic island chain in tokamak plasmas 3

between the inside/outside continuum remain unanswered. For instance, does the TAE

gap outside extends into the island chain? If the answer is yes, does the width of the

inside gap match that of the outside? One major limitation of the aforementioned

analytical works is the absence of toroidicity and toroidal mode coupling. In tokamak

geometry, an island winds around the core of the plasma following the magnetic ﬁeld

lines, creating a complicated 3D magnetic structure with both poloidal and toroidal

asymmetry. It is no longer appropriate to be viewed as a straight tube, but rather,

as a mini-stellarator. One can therefore make use of the established methodology

and knowledge of continuum in stellarators to compute and analyse that in the island

chain. In stellarators, the toroidal mode number is no longer a good quantum number,

and, as a consequence, each eigenmode consists of multiple toroidal harmonics, while

the toroidal coupling also gives rise to new gaps such as the Mirror-induced Alfv´en

Eigenmode (MAE) gaps [28,29] and the Helicity-induced Alfv´en Eigenmode (HAE)

gaps [30,31]. We will show in the paper that within the ideal MHD theory these new

gaps are ultimately responsible for the continuation of the outside TAE gap into the

island, whose width is determined by the interaction between them. Finally, a high level

of toroidal asymmetry could also change the continuous spectra into localised discrete

ones [32], which may have a fundamental impact on their interaction with existing global

modes, or create new ones.

In this work, we aim to study the shear Alfv´en continuum in the presence of an

island chain in tokamak geometry. The current paper is organised as follows. Section

2introduces the magnetic ﬁeld, geometry and the equations for the shear Alfv´en

continuum. Section 3brieﬂy describes the numerical scheme and benchmarks it against

analytical results in the literature. With the newly developed code, we compute the

continuum both inside the island-stellarator and outside in the bulk of the plasma, as

detailed in Section 4. The width of the combined gap and the eigenfunction of the mode

are also investigated. Finally, Section 5discusses the results and draws the conclusions.

2. Theory

2.1. The magnetic ﬁeld with an island chain

We start with a magnetic ﬁeld Bgiven by

B=∇ψ× ∇ϑ− ∇ψp× ∇ζ, (1)

in which ψis the toroidal ﬂux and will be used as the radial coordinate. The two angles

ϑand ζare generalised angles in the poloidal and toroidal directions, respectively. The

physical quantities are by default in SI units. The poloidal ﬂux function ψpis the

superposition of an unperturbed axisymmetic equilibrium ﬁeld and a surface-breaking

perturbation and is given by

ψp=Zψ

dψ0

q(ψ0)+Acos(m0ϑ−n0ζ),(2)

The shear Alfv´en continuum with a magnetic island chain in tokamak plasmas 4

where qis the safety factor, and Ais the amplitude of the ﬂux perturbation, while m0and

n0label the helicity of the island chain. We have used the “constant-ψ” approximation

[33] for a nonlinear tearing mode in which Ais assumed to be a constant. The contra-

variant ﬁeld components are given by

JBψ=Am0sin(m0ϑ−n0ζ), JBϑ=1

q, JBζ= 1,(3)

in which J= (∇ψ× ∇ϑ· ∇ζ)−1is the Jacobian of the coordinate system.

The unperturbed geometry is a large aspect ratio, circular-cross-section tokamak.

The cylindrical coordinates (R, ϕ, Z) are written in terms of the toroidal coordinates

(ψ, ϑ, ζ) with the relationship given by [34]

R=R0+rcos ϑ−∆(r) + rη(r)(cos 2ϑ−1),(4)

ϕ=−ζ, (5)

Z=rsin ϑ+rη(r) sin 2ϑ, (6)

where R0is the major radius of the plasma boundary, with the metric tensor and the

Jacobian given in Appendix A. The boundary of the plasma is circular with a radius of

a. The unperturbed toroidal ﬂux is related to the radius rby

ψ=B0

2r2,(7)

where B0is the ﬁeld strength on axis. The ﬂux surfaces are approximated by circles with

radius rand their centres are shifted from the centre of the boundary by a distance ∆(r).

This shift, known as the Shafranov shift, is a consequence of a non-zero pressure gradient

and current density, and is determined by solving the Grad-Shafranov equation [35]. Its

derivative with respect to rin the zero pressure limit is given by

∆0(r) = q2

R0r3Zr

q2dr, (8)

and with a zero or moderate shear ∆0≈r/(4R0). The quantity η(r) = (r/R0+∆0)/2 and

its existence in (4) and (6) is to ensure ϑand ζare straight-ﬁeld-line angles to order O().

This makes ϑslightly diﬀerent from the geometric poloidal angle. A demonstration of

the constant ψand ϑsurfaces is given by the black lines in Figure 1. We note that

the generalised toroidal angle ζcoincides with the (negative) true toroidal angle ϕand

therefore the straight-ﬁeld-line coordinates in this paper are PEST coordinates.

If A= 0, then Bψ= 0 and ψpis a function of ψonly, meaning that the unperturbed

magnetic ﬁeld in (1) is completely integrable, i.e. the ﬁeld lines are lying on concentric,

nested surfaces known as the ﬂux surfaces labelled by ψ. Moreover, ϑand ζare straight-

ﬁeld-line angles, such that Bζ/Bϑ=q(ψ) is a constant on each ﬂux surface. When

|A|>0, the coordinate system is kept as it is, while a magnetic ﬁeld perpendicular

to the constant ψsurfaces is introduced. An island chain will develop around the ﬂux

surface where

q(ψ0) = q0=m0

,(9)

The shear Alfv´en continuum with a magnetic island chain in tokamak plasmas 5

in which ψ0is the radial location of resonance in terms of the unperturbed radial

coordinate. The integer m0gives the number of O points/X points on a toroidal cross-

section, while n0gives the ﬁeld period in the toroidal direction. A Poincar´e plot of

an m0= 5, n0= 2, A = 10−4island chain with ψ0= 0.125, a = 1, R0= 3, B0= 1,

dq/dψ = 4 and a linear rotational transform proﬁle is overplotted in Figure 1. The

Poincar´e plot is constructed by ﬁeld-line tracing, i.e. solving the ordinary diﬀerential

equation dX/dl =B(X) for a number of initial locations, where Xis the location of a

point on a ﬁeld line and lis a distance-like variable along the ﬁeld line, and recording

a point whenever a ﬁeld line penetrate the ϕ= 0 cross section. Now with the island

chain neither ψnor ψpis a good ﬂux label, as the ﬂux surfaces shown by the Poincar´e

plot are no longer aligned with the coordinate surfaces. Also, the angles ϑand ζare not

straight-ﬁeld-line angles any more. The next step is to construct a new radial coordinate

aligning with the ﬂux surfaces and new straight-ﬁeld-line angles both inside and outside

the island.

Figure 1. Poincar´e plot of a m0= 5, n0= 2, A = 10−4island chain with

dq/dψ = 4, ψ0= 0.125, a = 1, R0= 3, B0= 1 and a linear rotational transform

proﬁle. Constant ψand ϑsurfaces are indicated by black solid lines.

2.2. Straight-ﬁeld-line coordinates

Now let α=ϑ−ζ/q0be the helical angle labelling the rotation around the axis of the

island. One can rewrite (1) as

B=∇ψ× ∇α− ∇χ× ∇ζ, (10)

where the helical ﬂux χis given by

χ(α, ψ) = ψp−ψ/q0−χ0=Zψ

ψ01

q−1

q0dψ0+Acos(m0α),(11)

with χ0being an integration constant. If the rotational transform ι-=1/q is a linear

function of ψ, (11) has a simpler form given by

χ(α, ψ) = −q0

2q2

(ψ−ψ0)2+Acos(m0α).(12)

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TheshearAlfvencontinuumwithamagneticislandchainintokamakplasmasZ.S.Qu1zandM.J.Hole1;21MathematicalSciencesInstitute,theAustralianNationalUniversity,CanberraACT2600,Australia2AustralianNuclearScienceandTechnologyOrganisation,LockedBag2001,KirraweeDCNSW2232,AustraliaE-mail:zhisong.qu@ntu.edu.sgAbstra...

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The shear Alfv en continuum with a magnetic island chain in tokamak plasmas Z. S. Qu1zand M. J. Hole12

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