In fusion plasmas, weakly damped eigenmodes which are readily destabilized by energetic ions often reside in spectral gaps.^[1] These gap eigenmodes may degrade fast ion confinement and thereby threaten the success of the magnetically controlled fusion reaction.^[2,3] Alfvénic gap eigenmode (AGE) is one of the most dangerous gap eigenmodes being observed.^[4] Previous studies mainly focused on traditional fusion reactors which, however, are complex, expensive and unfriendly to diagnostic access. Recently, low-temperature plasma cylinder has been proposed for gap eigenmode study. Zhang et al.^[5] arranged a periodic array of magnetic mirrors on the large plasma device (LAPD) and observed a spectral gap in the shear Alfvén wave continuum. Moreover, they suggested introducing a sector of magnet with much stronger background field compared to the regular mirror field to create AGE. Inspired by this work, Chang et al.^[6,7] computed the gap eigenmodes of the radially localized helicon mode, which can be excited by energetic electrons similar to that by energetic ions, and a shear Alfvénic mode by introducing a local defect to the system’s otherwise perfect periodicity. The computations made use of a nearly infinite number of magnetic mirrors (or field periods) to obtain a clear spectral gap and gap eigenmode inside. However, in experiment this number is limited to machine length, e.g., the maximum number of magnetic mirrors on the LAPD is 4 with a length of 20 m.^[5] This brings about the practical interest to study the influence of the number of magnetic mirrors on the characteristics of AGE, especially the minimum number required to form AGE. Furthermore, the influence of depth (or modulation factor) of magnetic mirror on AGE is not yet clear so far, although its scaling with the width of spectral gap has been given.^[5–7] Therefore, this paper explores the influence of number and depth of magnetic mirror on AGE with particular reference to the LAPD experiment.

Before studying the influence of the number and depth of magnetic mirrors on the characteristics of AGE, we shall review the analytical and numerical AGE for ideal conditions, i.e., nearly an infinite number and moderate depth of magnetic mirrors.^[7]

2.1. Governing equations

To describe the Alfvénic mode propagating in low-temperature magnetized plasmas, we first consider a cold plasma cylinder immersed in a uniform magnetic field. The plasma density is non-uniform in radius but axially uniform. This cold plasma approximation ensures that longitudinal wave phase velocity is much greater than particle thermal velocity, and Landau damping is low. The linear wave equation is

where E is the electric wave field and is the speed of light, with μ₀ the permeability of free space and ε₀ the permittivity of free space. Perturbations of the form exp[i(mφ + kz − ωt)] are assumed with ω the wave frequency, t the time, and m and k the azimuthal and axial mode numbers, respectively. The cylindrical coordinate system is (r, φ, z) with r the radius, φ the azimuthal angle, and z the axial position. To derive Eq. (1), a sufficiently dense plasma is assumed to ensure that the displacement current is negligible compared with plasma current. We employ the cold-plasma dielectric tensor to link E and D (electric displacement) in the form of . For the Alfvénic mode with ω ≈ ω_ci (ω_ci is the ion cyclotron frequency), can be written as^[8]

with

Here, we have made use of the relation , with ω_pe and ω_pi the electron and ion plasma frequencies, respectively, and ω_ce the electron cyclotron frequency. Because of ω ≪ ω_pe, the parallel conductivity η is so large that the axial component of the electric field vanishes, i.e., E_z = 0. This simplifies the radial and azimuthal components of Eq. (1) as

These two equations can be combined into

For modes with sufficiently small values of k_z and ω, Eq. (4) gives

Here we choose small values of k_z to see clear effects of the radial non-uniformity in plasma density on the wave mode structure for non-zero azimuthal mode numbers.^[9] Combining Eq. (4), Eq. (6), and Eq. (7), we get the wave equation of Alfvénic mode in a cold magnetized plasma cylinder

To evidently show the effect of radial density gradient on the Alfvénic mode and avoid possible continuum damping, we utilize a step-like radial profile of plasma density

with r₀ the radius of density discontinuity, and assume that the width of this discontinuity layer is larger than the skin depth. A specific form of E_φ is chosen to be

with E₀ a constant electric field. Integrating Eq. (8) across the density discontinuity, we get the dispersion relation of the Alfvénic mode for a uniform magnetic field

with , ω_pi− and ω_pi+ the ion plasma frequencies inside and outside of the discontinuity layer respectively, and x = ω/ω_ci the normalized wave frequency. For Alfvénic modes with ω ≪ ω_ci (i.e., x ≪ 1) and uniform plasma (i.e., Ψ = 0), equation (11) turns out to be the well-known dispersion relation of ω_A = k_zv_A with ω_A and v_A the wave frequency and phase velocity of shear Alfvén waves, respectively.

Second, we consider a periodic magnetic field in the form of (B_z − B₀)/B₀ = εcos(qz), with B_z the axial component of modulated field, B₀ the equilibrium field, ε ≪ 1 the depth (or modulation factor), and q the modulation period. This is equivalent to a set of magnetic mirrors arranged periodically in a straight line. The periodic modulation introduces resonant coupling between modes with k_z = ±q/2, and it takes effect mainly through two terms in Eq. (8): (ω²/c²)ε and (ω²/c²)r∂g/∂r according to Eq. (3). Their linear Taylor expansions about x can be written as:

Here, the subscript 0 denotes values for a uniform magnetic field, and the dependences of ε and g on the frequency difference ω − ω₀ and field modulation ω_ci − ω_ci0 have been separated. We consider E_φ of the form

with F(z) and G(z) being the slow functions of z compared with cos(qz), Ψ(r) the eigenfunction for a uniform magnetic field, and κ the wave number for a periodically modulated field. Then by integrating Eq. (8) over radius and separating longitudinal Fourier harmonics, we get the spectrum of the Alfvénic mode in a periodic system with a step-like radial density profile

It is straightforward to see that waves cannot propagate along the plasma column when |ω − ω₀|/ω₀ < ε/2 thereby forming a spectral gap. Setting κ = 0, we can easily get the lower (ω₋) and upper (ω₊) edges of the spectral gap from Eq. (15):

As a result, the normalized gap width is

and is thus proportional to the depth of magnetic mirror and consistent with previous studies.^[5,6] When |ω − ω₀|/ω₀ > ε/2, waves are propagable along the plasma column, forming the so-called spectral continuum.

Third, for a periodic field with local defect, we assume that the gap eigenmode decays away from the defect location, and hence consider E_φ in the form of

with λ being the decay constant. For the boundary condition of E_φ(r, z₀) = 0, the wave frequency and decay constant of AGE can be estimated from

Equation (18) shows the analytical AGE for the step-like radial profile of plasma density. For arbitrary radial density profile, one can refer to Ref. [7] where the detailed calculation of Eqs. (4)–(18) and the analysis for another type of defect can also be found. It should be noted that only odd-parity AGE (corresponding to E_φ(r, z₀) = 0) is considered throughout the paper. The even-parity AGE (corresponding to behaves similarly with the variation of number and depth of magnetic mirror.

2.2. Ideal simulation

As a starting point for Section 3, this section reproduces the numerical AGE obtained in Ref. [7] for ideal conditions, using the electromagnetic solver (EMS).^[10]

Figure 1 shows a schematic of the computational domain. It should be noted that the glass tube which usually exists for inductive discharges has been removed. The aim is to ensure that there is only one density discontinuity in the whole chamber, namely the constructed step-like discontinuity. The enclosing chamber is assumed to be ideally conducting so that the axial and azimuthal components of electric field vanish on the wall surface. A radio frequency (RF) antenna is placed inside the plasma for mode excitation. Here, we particularly consider a shear Alfvénic mode for its simpler and more robust structure compared with other Alfvénic modes. Further, the RF antenna is twisted in the left-hand direction regarding the magnetic field to effectively drive the shear Alfvénic mode, which is left-hand circularly polarized, and the antenna is half-turn helical to mainly excite non-axisymmetric modes for better revealing the effect of radial non-uniformity on wave mode structure.^[9]

Fig. 1.

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	Fig. 1. A schematic diagram of the computational domain. The red thick line denotes a left-hand half-turn helical antenna. The machine and coordinate system axis (r = 0) is represented by a dot-dashed line. The coordinate system (r, φ, z) is chosen to be right-handed with φ. The inner radius of the antenna is labelled by a double-dot-dashed line. Dimensions are in units of mm.

We choose a step-like radial profile of plasma density, as shown in Fig. 2(a), to precisely compare with the analytical estimate in Subsection 2.1, and to eliminate possibly existing continuum damping. The on-axis plasma density is chosen to be 10²⁰ m⁻³, which is sufficiently dense as assumed for Eq. (1). A single-ionized helium plasma is particularly considered, with reference to the LAPD experiment.^[5] The on-axis electron–ion collision frequency is thereby ν_ei(0) = 4 × 10⁸ s⁻¹. For a uniform magnetic field of B₀ = 1 T, we run the EMS for various frequencies but with the same other conditions, and calculate the dominant axial mode number through a Fourier decomposition. Figure 2(b) shows the computed dispersion relation and its comparison with that from Eq. (11). The inset shows the ratio between them. We can see that they agree well when the wave frequency is far from the ion cyclotron frequency. The divergence near x = 1 is mainly caused by ion cyclotron motion which strongly effects the wave propagation but is not considered in EMS. Another cause may be the computed values of k and ω which become bigger when ω → ω_ci, contradictory with the assumption in the analytical treatment for Eq. (7).

Fig. 2.

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Fig. 2. Numerical AGE for nearly infinite number and moderate depth of magnetic mirrors: (a) radial profile of plasma density; (b) analytical and numerical dispersion relations for uniform magnetic field; (c) axial profiles of periodic magnetic field with local defect (solid line) and typically formed AGE (dashed line) for x = 0.626, together with the decay constant (dotted) from Eq. (18), and (d) surface plot of the formed AGE (on axis) as a function of z and x. The RF antenna is labeled in the red thick line.

For a periodic magnetic field with the local defect, figure 2(c) shows the field configuration and typically formed Alfvénic gap eigenmode. The periodic modulation is in a particular form of [B_z − B₀]/B₀ = 0.1cos(40z), for which q = 40 is chosen to locate the spectral gap within a frequency range of 0.5 < x < 0.9 where the decay length of Alfvénic mode is long. Moreover, according to Eq. (18), we locate the defect at cos(qz₀) = 0 to force the frequency of the Alfvénic gap eigenmode to be right at the center of the spectral gap. The possible width of Alfvénic gap eigenmode is also shortest in this case. We further reduce the collision frequency to 5 × 10⁻⁴ν_ei in order to minimize the role of collisional dissipation and thus get a sharper resonant peak. Figure 2(c) shows that the formed Alfvénic gap eigenmode is a standing wave with the wavelength nearly twice the system’s periodicity, a typical characteristic of Bragg’s reflection. While the decay length indicated by the exponential envelope agrees well with the analytical estimate from Eq. (18) on the right-hand side of antenna, a strong near field of the antenna distorts the Alfvénic gap eigenmode going through to the left-hand side and thus causes slight divergence there. Figure 2(d) gives a full view of the plasma response in the (x, z) space where a clear and sharp AGE is formed inside the spectral gap.

The number of magnetic mirrors employed above is about L/(2π/q) = 127, which is ideal to form AGE but hard to be achieved in experiment. For example, only 4 magnetic mirrors could be constructed on the LAPD despite that its length is 20 m.^[5] Therefore, it is of practical interest to understand the influence of the number of magnetic mirrors on AGE, especially the minimum number required for AGE observation. We shall decrease the length of plasma cylinder but keep all other conditions the same, which equivalently reduces the number of magnetic mirrors, to reveal this influence in detail.

We compute AGE for shortened machine lengths of 10 m, 5 m, 2 m, and 1 m, which correspondingly represent the numbers of magnetic mirrors of N = 63, N = 31, N = 12, and N = 6. Figure 3 shows the computed axial (on-axis) and radial (defect location) profiles of AGE, together with its peak magnitude in (r, z) space, as a function of the number of magnetic mirrors.

Fig. 3.

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	Fig. 3. Variations of axial profile (a), radial profile (b), and peak magnitude (c) of AGE with the number of magnetic mirrors. The RF antenna and defect location (z = 10.485 m) are labeled in the red thick line and blue arrow, respectively.

It can be seen that the structure of AGE remains nearly the same when the number of magnetic mirrors is decreased from N = 127 to N = 12, about 10 times smaller, and it peaks at the same frequency of x = 0.626. Moreover, the formed AGE is stronger for N = 31 than for others, indicating the possible existence of an optimum number for which AGE is maximum. When the number of magnetic mirrors is further decreased to N = 6, the structure of AGE changes significantly and it becomes weak to identify. There appear other modes inside the spectral gap, which are much stronger than AGE. The main reason is that the number of magnetic mirrors is so small that propagating waves suffer little reflection and thereby a weak Bragg’s effect along its path. As a result, the spectral gap is too weak to support AGE. This is shown more clearly in Fig. 4 where the surface and contour plots of AGE are given for different numbers of magnetic mirrors. We can see that, with the number of magnetic mirrors decreased, AGE is gradually suppressed by other modes inside the spectral gap which finally breaks down. The minimum number of magnetic mirrors for AGE formation here is around N = 10. It should be noted that this minimum number does not depend on the wavelength of gap eigenmode but the Bragg effect that this eigenmode undergoes. Nevertheless, a short wavelength of gap eigenmode allows a short machine length for the same number of magnetic mirrors, due to Bragg’s law.

Fig. 4.

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	Fig. 4. Surface (left) and contour (right) plots of AGE for different numbers of magnetic mirror. The RF antenna is labeled in the red thick line.

Since the Bragg effect and decay constant of AGE (Eq. (18)) are proportional to field modulation, this section explores the possibility of forming strong AGE with a limited number of magnetic mirrors (N = 6) through increasing the depth of field modulation. Before that, the influence of this depth on the AGE formation is studied for N = 12 where AGE has been clearly formed in Fig. 3 and Fig. 4. Figure 5 shows the axial profiles (on-axis), radial (defect location) profiles, and peak magnitudes in (r, z) space of AGE for different depths: ε = 0.05, ε = 0.1, ε = 0.15, ε = 0.2, and ε = 0.3. We can see that the depth of magnetic mirror does not change the structure of AGE but its frequency, which becomes smaller as the depth is increased. Recalling from Fig. 3(c) that the number of magnetic mirrors does not change the frequency of AGE, thus we infer that this frequency shift is not caused by reflections from endplates but other candidates. Two possible contributors are density variation in the axial direction and invalidity of the analytical treatment in Subsection 2.1 which assumes small field modulation. We tried to freeze the plasma to the magnetic field line by constructing the axial profile of plasma density, which was set to be uniform in previous sections (see Fig. 6 (a1)), the same as the axial profile of the defective field (shown in Fig. 6 (b1)) and found that the frequency of AGE indeed shifts to a higher value (from x = 0.55 to x = 0.58). The invalidity of analysis for big field modulation causes the divergence between the numerical frequency of AGE and the analytical center of the spectral gap (see Eq. (16)). Nevertheless, the width of spectral gap is proportional to the depth of magnetic mirror, which is consistent with the analysis in Subsetion 2.1 and previous studies.^[5,6]

Fig. 5.

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	Fig. 5. Variations of axial profile (a), radial profile (b), and peak magnitude (c) of AGE with the depth of magnetic mirror. The RF antenna and defect location (z = 2.788 m) are labeled in the red thick line and blue arrow, respectively.

Now, to enhance the Bragg’s effect for a limited number of magnetic mirrors (N = 6) and form a clear AGE, we first increase the depth of magnetic mirror from 0.1 to 0.3. Figure 6 (a2) shows the surface plot of AGE for ε = 0.3, from which we can see a clear AGE inside the spectral gap. Second, we assume that the plasma is frozen to the magnetic field line according to magnetohydrodynamic theory, so that the plasma distribution is also periodic and with local defect. As a result, the propagating waves undergo Bragg’s reflection from both the magnetic mirror and plasma mirror, leading to an enhanced Bragg’s effect. We can see from Fig. 6 (b2) that the AGE becomes even sharper. This confirms that a strong AGE can indeed be formed by increasing the depth of magnetic mirror whose number is very limited. Moreover, the decay length of AGE is much shorter than that for ε = 0.1 shown in Fig. 4, which is consistent with Eq. (18). A shorter decay length results in a sharper and stronger gap eigenmode, making it easier for experimental observation. These conclusions are particularly useful for the implementation of AGE on the LAPD.

Fig. 6.

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	Fig. 6. Axial profiles of plasma density and corresponding surface plots of AGE for ε = 0.3 and N = 6. The AGE in (a2) peaks at x = 0.55 while the AGE in (b2) peaks at 0.58. The RF antenna is labeled in the red thick line.

To guide the experimental implementation of AGE on a low-temperature plasma cylinder, along which the number of field period is usually small (e.g., about 4 on the LAPD), the influence of the number of magnetic mirrors on the characteristics of AGE is studied. We find that AGE is suppressed by other modes inside the spectral gap when the number of magnetic mirrors is below a certain value, due to the weakened Bragg effect. However, the structure and frequency of AGE remain nearly the same when this number is decreased, as long as it is still enough for the AGE formation. To explore the possibility of forming a strong AGE with a limited number of magnetic mirrors through increasing the depth of field modulation, we further study the influence of the depth of magnetic mirror on the AGE formation. It is found that the depth of magnetic mirror does not change the structure of AGE but its frequency, which becomes smaller as the depth is increased. This frequency shift is not caused by reflections from endplates, but possibly by density variation in the axial direction, which should not be uniform but varied with field lines, and the invalidity of analytical treatment which assumes small field modulation. By increasing the depth of magnetic mirror from ε = 0.1 to ε = 0.3, we succeed in forming a clear AGE for N = 6 and it is even sharper if the plasma is frozen to the magnetic field line, forming an extra plasma mirror. This success is of great importance and practical interest to the numerical and experimental observation of AGE on the LAPD. Indeed, by increasing the depth of magnetic mirror but using the same other conditions of LAPD, we have numerically observed an AGE. The proposal of experimental implementation of AGE on the LAPD employing this depth scheme is in good progress.