# Poloidal trapping of high-frequency Alfvén eigenmodes in stellarators

Yu. V. Yakovenko¤, Ya. I. Kolesnichenko¤, O. P. Fesenyuk¤, A. Weller†,

S. Zegenhagen†, A. Werner† and J. Geiger†

¤Institute for Nuclear Research, 03680 Kyiv, Ukraine

†Max-Planck-Institut für Plasmaphysik, D-17489 Greifswald, Germany

Abstract. Properties of shear Alfvén oscillations in three-dimensional toroidal magnetic configurations (stellarators) are

studied. It is shown that in the case when two continuum gaps are very close to each other, the joint effect of several

Fourier harmonics of the configuration parameters traps the wave functions of the Alfvén continuum in waveguides. This

situation is typical for the high frequency range in stellarators (the frequency range of the helicity-induced and mirror-induced

eigenmodes), when the waveguides are typically located on either inner or outer circumference of the plasma torus. The

Alfvén eigenmodes in this case are shown to be localized in waveguides of the same shape, too. Qualitative agreement with

results of experiments on the stellarator Wendelstein 7-AS is demonstrated.

Keywords: Alfvén continuum, Alfvén eigenmodes, stellarator

PACS: 52.35.Bj, 52.35.Py, 52.55.Hc

INTRODUCTION

The Alfvén waves and instabilities are frequently observed in space and laboratory plasmas. It is well known that

the ideal Alfvén waves in a non-uniform plasma possess a continuous spectrum (the Alfvén continuum, AC), which

is associated with their property to spread along the field lines. The Alfvén oscillations with the frequencies in the

AC are strongly damped through phase mixing [1]. This explains the interest paid to the discrete spectrum of the

Alfvén oscillations (Alfvén eigenmodes, AE). The AEs are weakly damped and can be destabilized by fast ions. The

resulting instabilities may be harmful, causing losses of the fast ions. On the other hand, they can be utilized for plasma

diagnostics [2].

A most common type of ideal AEs in tokamaks and stellarator are gap AEs, which reside in gaps of the AC. These

gaps arise due to asymmetry (the deviation of the shape and the magnetic field of the device from the cylindrical

symmetry). In particular, the poloidal asymmetry due to toroidicity results in toroidicity-induced AEs (TAE) [3].

The lack of toroidal symmetry in stellarators produces additional types of Alfvén eigenmodes in the high-frequency

part of the spectrum, namely, the helicity-induced and mirror-induced AEs (HAE and MAE) [4, 5, 6]. There are

evidences that such modes have been observed in experiments [7, 8, 9]. Another, more profound consequence of the

absence of symmetry is that the parameters of the magnetic configuration are not periodic along a field line. Then the

AC gaps can cross [5] and form a complicated net on the plane (y;w), where y is the flux surface label, and w is the

frequency.

In this work, the structure of the AC and AEs is studied for the case when the AC gaps are close (with the distance

between them about their widths). In this case, which is typical for the high frequency range in high-N machines (N is

the number of the field periods of the device), the wave functions of both the AC waves and the AEs are shown to be

trapped in special “waveguides.” An experiment on W7-AS is discussed, in which such trapping seems to be observed.

TRAPPED CONTINUUMWAVES

The continuous spectrum of ideal Alfvén oscillations are normalized components of the metric tensor,

ˆ s is the magnetic shear, fk is a parameter characterizing the transversal wave number. All coefficients in (4) are taken

at a single field line, q =if +a with a = const. One can show that (4) turns into (1) in the limit of f !§¥.

Again, we consider the case when two gaps are close and keep only the corresponding two harmonics in all the

coefficients. On averaging over the fast scale, d=df »˜kX , we reduce (4) to a Schrödinger equation with the potential

schematically shown in Fig. 3. We conclude that the the AEs in this case are also trapped in the same sectors of the

plasma cross section as the neighbouring continuum wave functions. In particular, this means that HAEs and MAEs

are typically localized on either inner or outer circumference of the plasma.

As an example, we consider shot # 54937 of the stellarator Wendelstein 7-AS (W7-AS). At the end of the shot,

plasma oscillations with rather high frequencies in the range of 200¡420kHz were registered by Mirnov coils. The

AC calculated with the continuum code COBRA [5] for the time instant t = 0:36s is shown in Fig. 4. We observe that

the oscillations are in the frequency range of several rather wide HAE and MAE gaps. In the region hatched in green,

between the (0;1)- and (2;1)-gaps, the calculations did not converge (a probable reason for this is the strong coupling

between the (7;0)- and (1;1)-gaps, which lie in this region). Analysis shows that all the (m;1)-gaps are produced

mainly by the (2;1)- and (3;1)-harmonics of the plasma shape (the helical ellipticity and triangularity). The AEs in

this case are to be trapped at the inner circumference (the high-field side) of the torus.

Figure 5 shows the spectra of the signals on the coils 1 and 4, which are located on outer and inner circumferences,

respectively. The comparison of the two spectra shows that some spectral bands (about 250, 290 and 410 kHz at

t = 0:36s) are much stronger at the inner circumference, which correlates with the theory predictions. The 320-kHz

frequency band has approximately equal amplitudes at both coils (slightly stronger at the outer circumference), which

may mean that the waveguides of this eigenmode has a more complicated shape, resulting from the interaction of gaps

with different values of n (e.g., the (1;1)- and (7;0)-gaps, which lie near 320 kHz).

SUMMARY

It has been shown that in the case when continuum gaps are very close in frequency (the relative width of the gaps

exceeds the characteristic relative distance between them), the continuum waves are trapped in certain “waveguides.”

The trapping arises as a result of a joint action of several (two or more) Fourier harmonics of parameters of the

magnetic configuration. The continuum in this case turns into thin threads, the widths of the threads being determined

by wave tunneling through the evanescence zone.

This situation is typical for the high frequency range in high-N stellarators, i.e., the range of the HAE and MAE

modes. In the typical case of the interaction of the gaps (m;1) and (m +1;1), the wave propagation takes place

on either inner or outer circumference of the plasma torus, although a more complicated geometry is also possible.

Mirnov observations in a W7-AS shot where high-frequency oscillations were observed seem to agree with the theory

predictions.

This phenomenon of the Alfvén wave trapping due to joint effect of several Fourier harmonics of the wave equation

coefficients takes place not only in the frequency range of the HAE and MAE modes but also near all crossings of

frequency gaps in the AC and seems to be typical for the three-dimensional toroidal magnetic configurations.

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