Shear Alfvén waves within magnetic islands

We calculate Alfvén eigenmodes within a magnetic island (MiAE) which have been conjectured over a decade ago. Starting from a cylindrical plasma equilibrium, we calculate the complete metric of the island interior assuming an iota profile with a constant shear for Wendelstein 7-X parameters. Then, we solve the resulting Magneto-Hydrodynamic equations inside the island optionally considering Finite Larmor Radius corrections. We find various eigenmodes in the lowest gaps for n = 0. The eigenmode with the lowest frequency shows a weakly non-linear dependence on the island width which deviates considerably from an earlier estimate.


Introduction
In fusion or astrophysical plasmas magnetic islands may occur.They can originate from a saturated perturbation or be induced by an external perturbation.Having reached or being in a steady state, these islands form-simply speaking-small equilibria of itself which interact with the surrounding plasma.
In Wendelstein 7-X (W7-X), a modern stellarator [1] a chain of static islands at the plasma boundary is being used as key element of its so-called island divertor.The island chain forms when the rotational transform equals a rational value ι = − n m .It is called 'natural' if the n is a multiple of the periodicity of the device (n = 5 for W7-X) [2].There is some Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.evidence that energy losses can be attributed to mode activity in islands of W7-X which may account for 20%-30% of the total losses [3,4].The mode behavior varies from bursting to steady state oscillations with fractions of a kilohertz.On the other hand, there are direct observations of MiAE in conjunction with the observations of beta induced Alfvén eigenmodes (BAE) in J-TEXT [5].

Island metric
We start with a cylindrical plasma equilibrium.As usual, the island is introduced as a perturbation to the equilibrium: island in a W7-X FLM configuration.The continuum has been calculated with CONTI for the actual W7-X equilibrium (grey lines).The colored dots are results of a global simulation based on a cylindrical approximation of the underlying equilibrium.Among those, well converged global modes are marked with black crosses (figures 2 and 3).The blue crosses mark kinetic modes (see figure 4).
where r, ϑ and φ are the radius, the poloidal and toroidal cylindrical angle, respectively.A/a 0 gives the size of the magnetic perturbation in comparison to the main field, where a 0 is the small radius of the device.Then, new flux surfaces ψ * = const can be found from 8 (2κ 2 − 1) and α = 1 2 (Mϑ + Nφ), we can define new coordinates using κ and the central angle β which are related to the background quantities in the following way: r 2 − r 2 0 = wκ cos β and sin α = κ sin β . ( These coordinates (κ, β, φ) are well-defined (with κ = 0 being the island center and κ = 1 its separatrix) such that the elements of the metric tensor and also ⃗ B can be calculated.To allow ourselves to proceed analytically, we assume the background rotational transform to be of the following form: ι(r) = ι 0 + ῑ′ (r 2 − r 2 0 ) with ῑ′ = const.The island is located at r 0 where ι 0 = − N M .During the calculation, we drop terms of order higher than O( a0 R0 ) or O( w 2 r 4 0 ). (Note, that r 2 is the radial reference variable, such that in this work w = 2 √ 2 AR0 ῑ ′ measures the island half width in units of r 2 .)Thus, we find: The other metric elements are simply: As in earlier work [6,10,11], we transition to a straight field line system introducing a new poloidal angle β * = 2π 4K(κ) F(β, κ) with K(κ) and F(β, κ) being the complete and incomplete elliptic integral of the first kind, respectively.With the partial derivatives ) , (7) (Z(ζ, κ) being the Jacobi Zeta function) the metric can be transformed to the new coordinates (κ, β * , φ).Because β is given by the amplitude function β = am( 4K 2π β * , κ), the dependencies on β * can expressed in terms of Jacobi elliptic functions as sin β = sn( 4K 2π β * , κ) or cos β = cn( 4K 2π β * , κ).For the contravariant components of the magnetic field we obtain: R0 , while its covariant components are B κ = 0, B β * = 0 and B φ = B 0 R 0 to the relevant order (see above).The rotational transform inside the island is a function of the flux surfaces inside the island: , where K L = M/gcd(M, −N) the number of toroidal turns until the island closes in itself [12].
From the continuum calculations inside the island of a stellarator plasma it was shown that there was little change compared to a cylindrical equilibrium [12] This is also demonstrated in figure 1 where the continuum calculated using the full equilibrium information of W7-X (see [12]) is compared with the cylindrical approximation of this work.

Equations
We are invoking reduced Magneto-Hydrodynamics (MHD) for the perturbed fields were we neglect plasma compressibility: with v 2 A = B 2 µ0ϱ , denoting the Alfvén velocity where ϱ is the mass density.Note that ϕ is the electric potential, ⃗ κ the curvature vector and p 0 the equilibrium pressure.
All quantities, i.e. differential operators and fields are now referring to the island interior.Usually, in the process of island formation, there is a flattening of the profiles inside the island.So, we take flat profiles for the density as well as for the pressure inside the island.Thus, the last term involving the pressure gradient drops out in our calculations.Furthermore, we can show that the parallel current inside the island is of order (r/R) 2 .Therefore, the current term drops out as well.To estimate the radiative damping, we use the Finite Larmor Radius (FLR) correction and that of a finite parallel electric field as derived in reference [13][14][15][16].This renders equation ( 8) into: where the parallel current term and the pressure term already have been left away.The forth order differential operator contains the FLR correction ∝ ρ 2 i = M i k B T i /ZeB and a correction due to the finite parallel electric field E ∥ ∝ ρ 2 s = M i k B T e /ZeB where M i is the ion mass, Ze is the ion charge and T i and T e are the ion and electron temperature, respectively.We use  the actual expression for E ∥ invoking kinetic contributions of the bulk plasma [16] to estimate δ ≈ 0.23.This way, we get a noticeable damping of several percent of the mode frequency.From such a simple method, we cannot expect to get a precise result but a tendency where the damping of the modes gets larger or smaller and thus a measure of the likelihood of their excitation.
The equations ( 8) and ( 9) are formulated in weak form and the potential is expressed by a tensor product of cubic B-splines.Dirichlet boundary conditions have been set at κ = 0.999.The resulting generalized eigenvalue problem is solved numerically [17,18].

Results
We calculate the modes in the island interior taking the following model parameters which resemble an m = 5, n = −5 island in the W7-X FLM configuration [4] (R = 5.71 m, B = 2.43 T, a 0 = 0.523 m) at r 2 0 /a 2 0 = 0.7381 with ῑ′ 0 = 0.203 9716/a 2 0 , n e = 19 m 3 and T = 0.25 keV.The perturbation strength is A = 6.127 152 • 10 −6 m yielding an island size of approximately 2w r = 4.3 cm (full width).Although our code is general, we focus on the calculation of modes having n = 0 i.e. the same helicity as the island.This way we can address an open gap and avoid having too high mode numbers.That means that the resulting modes exist on spatial scales where MHD is still applicable (see the discussion in [12]).
The result of the global mode calculation in the cylindrical background has been compared with the continuum calculation for the full W7-X equilibrium in figure 1.The differences are hardly distinguishable.Quantitatively, they do not exceed a few percent.
The lowest gap of the Alfvén continuum (∆m = 2 and, therefore, ellipticity induced Alfvén Eigenmode gap-EAE gap) is open towards the separatrix.In this gap, even and odd modes can be found which are marked from a-f and are shown in figure 2 This notion addresses the parity of their radial components with respect to the opposite signs of the two dominating m components.The appearance of eigenmodes with a higher number of radial knots, resembles the typical anti-Sturmian behavior of stable MHD modes [19].There are also global modes in a gap named after non-circular Alfvén eigenmodes (NAE).The modes, however, show strong interaction with the continuum (see the g and h labels in figure 3).The modes that are not marked in in figure 1 belong either to the continuum, show strong continuum interaction or are not well converged.
If FLR effects and damping are included in the calculations, kinetic MiAE (k-MiAE) can be found (figure 4).The odd mode localizes itself radially and shows, unlike its MHD counterpart, only little continuum interaction and less damping than the even mode, making it likely to be observable experimentally.
In earlier works, the mode frequency was estimated using an approximated formula for the Continuum Accumulation Frequency of several MiAE in comparison with the continuum estimate from reference [6].The blue squares resemble the true CAP m=−1 at the plasma center of the global calculation.The larger the island size, the closer the modes come to the continuum.Note that wr is the island half width with respect to r measured in meters.Point (CAP) of the lowest gap [6,7,9].Differently to this estimate, we find a weakly non-linear behavior of the mode frequency on the island width which is much closer to the actual continua bounding the lowest frequency gap (figure 5).However, the frequency of the k-MiAE close to the upper gap boundary agrees better with the estimate.
Figure 6 shows that the damping of the k-MiAE does not change much with the the island width and that the odd mode always has a smaller damping rate.

Conclusions
We have calculated MiAE in cylindrical plasma which is a good approximation to a stellarator equilibrium as illustrated by the comparison of the continua.As our theory calculates the full island metric up to order w 2 /r 4 0 , we get a weakly nonlinear dependence of the mode frequency on the island width.The odd k-MiAE found shows virtually no continuum interacand appears to be less damped than its even counterpart and is very likely being excited in experiments.The estimate for the MiAE frequency from earlier work [6] is found to deviate considerably from the actual mode frequency.
It needs to be noted however, that the radiative damping is only an estimate as we have prescribed (i) a constant external damping and (ii) Dirichlet boundary conditions at the separatrix (this way suppressing the interaction with the bulk plasma).Especially the latter effect will invoke the interaction with the full bulk plasma geometry which may alter the picture.However, the solution of a fully coupled system with interfacing boundaries and an X-point geometry would be extremely demanding numerically.
Agreement No 101052200-EUROfusion).Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Commission.Neither the European Union nor the European Commission can be held responsible for them.
Jinjia Cao, was supported by a grant from the ITER Project of Ministry of Science and Technology of China (Grant No. 2022YFE03080002).

Figure 1 .
Figure 1.Continuum and global Alfvén eigenmodes in the island interior.Ideal MHD continuum and global modes inside an (5, −5)−island in a W7-X FLM configuration.The continuum has been calculated with CONTI for the actual W7-X equilibrium (grey lines).The colored dots are results of a global simulation based on a cylindrical approximation of the underlying equilibrium.Among those, well converged global modes are marked with black crosses (figures 2 and 3).The blue crosses mark kinetic modes (see figure4).

Figure 2 .
Figure 2. Global EAE-like MiAE 2,0 inside island.There are even modes (a)-(c) with a different number of radial knots.For the odd modes (d)-(f ) the gap is already closed.The modes (d)-(f ) differ by their continuum interaction at the separatrix.

Figure 4 .
Figure 4. k-MiAE 2,0 inside the island.Even mode showing strong wiggles due to FLR-modification (i) and the odd mode with virtually no continuum interaction (j).

Figure 5 .
Figure5.Frequency of several MiAE in comparison with the continuum estimate from reference[6].The blue squares resemble the true CAP m=−1 at the plasma center of the global calculation.The larger the island size, the closer the modes come to the continuum.Note that wr is the island half width with respect to r measured in meters.

Figure 6 .
Figure 6.Calculated estimate for the radiative damping of two MiAE with a prescribed δ = 0.23.