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
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
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.
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).
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
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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