Frequency chirping of neoclassical tearing modes by energetic ions

The mechanism of rapid frequency chirping for neoclassical tearing modes (NTMs) is studied. Resonance between NTMs and trapped energetic ions can provide an additional torque to change the evolution of frequency. Whether the frequency rises or falls depends on the direction of island propagation. If the island propagates in the direction of ion diamagnetic drift, the frequency will be increased dramatically and rapidly. If the island propagates in the direction of electron diamagnetic drift, the frequency will be reduced to a lower value. The predicted chirping time is consistent with experimental results in DIII-D (Liu et al 2020 Nucl. Fusion 60 112009).


Introduction
It is well known that neoclassical tearing modes (NTMs) [1,2] can significantly reduce the performance of magnetically confined plasmas. When a magnetic island reaches a certain threshold, a perturbed helical bootstrap current, resulting from the pressure flattening across the island, is generated to drive NTMs. These modes can increase the local radial transport, reduce the maximum achievable β (β = 8π p/B 2 0 , where p and B 0 are the plasma pressure and magnetic field), and lead to plasma disruption if the island is large enough in a high-β plasma [3]. Thus, NTMs are of critical importance in the context of achieving steady-state and highly confined plasmas in present and future tokamak devices [4].
Energetic particles are inevitably produced in burning plasma or during external heating (such as neutral beam injection) in a tokamak. It is evident that energetic particles * Author to whom any correspondence should be addressed.
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can interact with plasma instabilities, such as internal kink, Alfvén eigenmodes [5,6], and tearing modes, including NTMs [7][8][9][10][11][12][13][14][15]. Stabilization of NTMs by energetic particles and the redistribution or loss of energetic particles due to NTMs in turn have been observed in tokamak experiments [7][8][9]. Some analytical and simulation studies [10][11][12][13][14][15] have shown that energetic particles can actually stabilize tearing modes (including NTMs). Furthermore, frequency chirping of tearing modes or NTMs has been observed on TFTR [16], ASDEX-U [17,18], EAST [19], HL-2A [20], and DIII-D [21]. On TFTR [16], during the evolution of NTMs, frequency chirping of these modes was accompanied by loss of energetic ions. Recently, frequency chirping up and back down of NTMs has also been observed on DIII-D, where the chirping time is a few milliseconds, and has been suggested that trapped energetic ions may be responsible for this [21]. Some efforts [22][23][24] have been made to theoretcially understand the underlying physics. In [22], it was pointed out that resonance between trapped energetic ions and rotating NTMs provides a torque to accelerate the rotation of the NTMs; however, the estimated chirping time is much longer than that observed in the experiments. Two recent articles [23,24] have described the bursting and chirping of tearing modes on HL-2A. In [23], it was shown that resonance between trapped energetic ions and tearing modes can drive a fishbone-like mode. Thus, the physics remains unclear. The experimentally observed frequency chirping of NTMs implies that resonance between NTMs and energetic particles is important. Since the transit frequency of passing energetic ions is much greater than the frequency of NTMs, it is very difficult for the condition for resonance between them to be satisfied.In contrast to the particle model used in [22], a drift kinetic approach is used to give a self-consistent description of resonance between NTMs and trapped energetic ions in the present work. It is thought that this resonance could provide an additional torque and thereby alter the evolution of the rotation frequency of NTMs.
In section 2, the evolution of the rotation frequency of NTMs, including energetic ion resonance effects, is described, and the frequency chirping of NTMs by energetic ions is analyzed and discussed. Finally, conclusions and a discussion are presented in section 3.

Frequency chirping of NTMs by energetic ions
Considering a plasma confined in a tokamak with large aspect ratio ( a = a/R 0 1) and low β (β ∝ 2 a ), the magnetic field is written as B = I∇ζ + ∇ζ × ∇(ψ + δψ) for NTMs [15,29], where the toroidal geometry is assumed to be axisymmetric, a is the minor radius, and R 0 is the major radius. ψ and δψ are the equilibrium and perturbed poloidal magnetic flux, respectively. δψ = δψ(0, t) cos ξ, ξ = mθ − nζ − t ω(t )dt , and m and n are, respectively, the poloidal and toroidal mode numbers. ω(t) is the rotation frequency of the island relative to the plasma. The nonlinear evolutions of the island width and the rotation frequency can be obtained by matching the solutions in the outer and inner regions based on generalized Rutherford theory [25] as follows: where Δ = (∂ ln δψ/∂r)| r + s r − s is determined from the outer region. The perturbed parallel current density δJ is determined from the island region. x = r − r s , where r s is location of the rational surface. The evolutions of the island width and the rotation frequency of the island are derived from the real and imaginary parts, respectively, of equation (1). These real and imaginary parts are given, respectively, by [26] and where Ω = 2x 2 /w 2 − cos ξ, w = 2 q s δψ/(sB 0 ) is the island width, s is the magnetic shear, and · · · = dξ/(2π) (. . .)/ √ Ω + cos ξ. Δ c and Δ s are the real and imaginary parts of Δ , respectively. They are also the contributions that are in-and out-of-phase with the island. Δ c provides the instability criterion for tearing modes. Δ s describes absorption of momentum. δJ ,R and δJ ,I are the real and imaginary parts of δJ , respectively. Here, δJ ,I comes from the resonance between NTMs and energetic ions. Therefore, if there is effective resonance between NTMs and energetic ions, the evolution of the rotation frequency will be modified dramatically.
Based on the quasineutrality equation, the parallel current satisfies Here, δJ ⊥,c represents the contributions of the background plasma (neoclassical polarization, neoclassical viscosity, etc).
the perturbation form of the Chew-Goldberger-Low [27] pressure tensor of energetic ions, with b 0 being the direction of the equilibrium magnetic field. One can make the separation δJ = δJ ,c + δJ ,h , where δJ ,c results from the contribution of the background plasma and satisfies B · ∇(δJ ,c /B) = −∇ · δJ ⊥,c . Based on equation (4), the perturbed parallel current density of energetic ions δJ ,h can be obtained as where J = (∇ψ × ∇θ · ∇ζ) −1 is the Jacobian, and κ 1 ∼ −∂ ln R/∂ψ and κ 2 ∼ −∂ ln R/∂θ are the two components of the magnetic curvature. Next, an expression for δ p h needs to be derived. The nonlinear drift kinetic equation for energetic ions is [6] ∂δG h ∂t v qR Here is assumed, and so T i T h , since n h n i , where T i,h and n i,h are the temperatures and densities of ions and energetic ions, respectively. A growth rate γ ω is assumed, which is always the case for NTMs [29]. Then, taking the orbital average (qR/v )dθ of equation (8), one obtains where ω d = −(mE/rR 0 Ω ch )M(λ) [28], Ω ch is the gyrofrequency of trapped energetic ions, λ = μ/E is the pitch angle, and μ is the magnetic moment. Here, we focus on the trapped energetic ions, and only δ Here, the velocity space integral is over the trapped region. Then, taking the magnetic flux average J dθ of equation (5), one obtains where θ b denotes the bounce point of trapped energetic ions. In this work, we focus on the evolution of the rotation frequency of the island. For simplicity, the island is assumed to be constant, since the time scale of island evolution is proportional to the resistivity diffusion time [15,25] and is much longer than the chirping time of the rotation frequency [21]. Therefore, only resonance between NTMs and trapped energetic ions is considered. To satisfy the resonance condition, on must have ωω d > 0. For a monotonic q profile, ω d < 0 is always satisfied [28], i.e. the precessional frequency of energetic ions is the same direction as the ion diamagnetic frequency. To proceed further, the slowing down distribution of energetic ions is chosen as F h0 = C 0 (r)( For trapped particles, the pitch angle (1/(1 + ) λ 0 B 0 1/(1 − )) should be satisfied. Here, a given pitch angle λ 0 is considered, for simplicity. Substituting the expressions for F h0 and δ f h and equation (9) into equation (11), one obtains the resonant part of δK, following Landau's prescription, as where H s = 3 c )],ω = ω/ω dm , ω dm is the precessional frequency with maximum energy, L nh = (∂ ln n h /∂r) −1 , and L Ec = (∂ ln E c /∂r) −1 .
It is convenient to transform the coordinates (r, θ, ξ) to the island coordinates (Ω, θ, ξ). δφ can be obtained from the ion continuity equation and the electron momentum balance equation [15,26] as δφ = (∂ψ/∂r) is determined by the effect of the island on radial transport [29]. Here, σ x denotes the sign of x and E 1/(1 + Ω) is an elliptic function. It should be pointed out here that the form of δφ in which account is taken of the contribution of energetic ions remains almost unchanged, since n h n i [15]. Then, substituting equation (12) into equation (10), one obtains This results from the resonance between NTMs and trapped energetic ions. It should be pointed out that there is no contribution of δJ h,res to the evolution of the island based on equation (2), since δJ h,res sin ξ is an odd function of ξ. Substituting equation (14) into equation (3), the rotation frequency of the island can be derived as [26] G w d dt wheret = ω A t, ω A is the Alfvén frequency, μ a is the anomalous viscosity, ω * h = k θ E m /(ω ch L nh ) is the diamagnetic frequency of the energetic ions, β ⊥h = 8π p ⊥h /B 2 0 , I Ep =  G h ∼ 0.56. In equation (15), the effects of external magnetic fields, such as the error field, are not considered, for simplicity. The first term on the right-hand side of equation (15) results from the effect of viscosity. Without energetic ions, the island rotation would reachω 0 owing to the effect of viscosity if the effects of external fields were not considered. Δ sh represents the effect of resonance between NTMs and trapped energetic ions. It affects the dynamics of frequency dramatically. As pointed out above, the resonance condition isω > 0. Then, if the rotation frequency of NTMs is the same as the ion diamagnetic frequency, ω d < 0 should be satisfied from the resonance condition, which is always the case for a monotonic q profile. Thus, Δ sh > 0 can be obtained based on equation (16). The rotation frequency of NTMs will increase. On the other hand, if the rotation frequency of NTMs is the same as the electron diamagnetic frequency, ω d > 0 to satisfy the resonance condition, which can be achieved with a reversed shear q profile or with barely trapped energetic ions. In this case, Δ sh < 0, i.e. it plays a damping role. The rotation frequency of NTMs will then decrease.
For typical tokamaks like DIII-D, the main parameters are B 2 T, R 0 = 1.7 m, a = 0.61 m, n 0 = 3.5 × 10 19 m −3 , T i ∼ T e 3.5 keV, s = 0.5, L Ec ∼ L ni = −a, L nh = 0.5L ni , the mode numbers m = 3 and n = 2, and the initial frequency ω 0 /(2π) = 3 kHz. It is assumed that E m = 65 keV, the pitch angle λB 0 = 1 for energetic ions, and the viscosity coefficient μ a = 1 m 2 s −1 . Then, based on equations (15) and (16), the frequency is plotted versust in figures 1 and 2 for s = 0.5 and −0.5, respectively. From figure 1, it can be seen that the resonance effects of energetic ions on the island propagation frequency are dramatic when the island propagates in the ion diamagnetic drift direction. When the β t ⊥h of trapped energetic ions increases above a critical value, the resonance effect dominates over the viscosity damping effect, and the frequency will chirp up. This is consistent with a DIII-D experiment [21], where frequency chirping was observed when the island propagated in the ion diamagnetic drift direction. The chirping speed of the frequency increases with the frequency. When the frequency is low, namely, much smaller than the maximal precessional frequency of trapped energetic ions, the chirping increases slowly, since the fraction of resonant energetic ions is small. When the frequency is higher, it chirps up more quickly. In the DIII-D experiment [21], it was shown that the frequency jumps up from the NTM frequency and then chirps down to the NTM frequency again within ∼1 ms. From figure 1, it can be calculated that the times of chirping up to ω/(2π) ∼ 15 kHz are about 1.1 ms, 2.3 ms, and 7.2 ms for β t ⊥,h = 1.0%, 1.2%, and 1.6%, respectively, which are of the same order as reported in reference [21]. In fact, the chirping found in TFTR [16] and ASDEX-U [18] is also on a millisecond scale. Here, the loss or redistribution of energetic ions is not considered, and so the process of chirping down cannot be shown. If the resonant energetic ions are lost, then the frequency will chirp down toω 0 quickly owing to the damping effect of viscosity. On the other hand, if the island propagates in the electron diamagnetic drift direction, ω d > 0 should be satisfied for the resonance condition with reversed magnetic shear. In this case, it can be seen from figure 2 that the resonance effect tends to decrease the frequency. When ω < ω 0 (where ω 0 is the rotation frequency of the NTM without energetic ions), the effect of viscosity tends to enhance the frequency to ω 0 . When the effects of resonance and of viscosity balance, the frequency remains constant, as can be seen from figure 2. Hence, frequency chirping of NTMs occurs only when the rotation frequency of NTMs is the same as the ion diamagnetic drift frequency. Actually, in a recent HL-2A experiment [20], it was shown that the phenomenon of frequency chirping only occurs while the rotation direction of the tearing mode changes from that of electron diamagnetic drift to that of ion diamagnetic drift. To confirm the above results, further statistical analysis of experimental results is needed. On the other hand, because the transit frequency of passing energetic ions is much larger than the frequency of NTMs, the resonance contribution from passing energetic ions is thought to be small and is not considered here. This also needs to be verified by further experiments.

Conclusion and discussion
In conclusion, the mechanism of frequency chirping of NTMs has been investigated based on drift kinetic theory. It is found that resonance between trapped energetic ions and NTMs provides an additional torque to alter the rotation frequency of the NTMs. The rise or fall of the rotation frequency depends on the direction of rotation of the NTMs. If this direction is the same as that of ion diamagnetic drift, then the resonance effect will increase the rotation frequency of NTMs rapidly. This is consistent with the experimental results reported in TFTR [16], ASDEX-U [18], DIII-D [21], and HL-2A [20]. If the direction of rotation of NTMs is the same as the electron diamagnetic drift direction, then frequency chirping does not occur. It will be reduced to a lower frequency. It can be concluded that the resonance between the island and trapped energetic ions can change the island rotation frequency dramatically and quickly, in a few milliseconds, when the island propagates in the ion diamagnetic drift direction.
In this work, the effect of the resonance contribution on the evolution of island width has not been considered. In fact, the change in frequency due to resonance will affect the neoclassical polarization, which depends on the island propagation frequency. Thus, the evolution of the island width will be affected by the resonance. This effect will become significant if the island width is small, since the neoclassical polarization then plays an important role. On the other hand, non-resonance effects will also affect the island evolution.
Rapid frequency chirping of NTMs due to energetic ions may have an impact on the dynamics of island locking. It may change the locking state of the island, and may reduce the explosive growth of the island to avoid plasma disruption [4,30,31]. It should be pointed out that resonance can also lead to redistribution and loss of energetic ions, which is not considered here. In that case, a chirping down process can occur.