The influence of electron cyclotron resonance heating on ion-driven fishbone instability

The effect of electron cyclotron resonance heating (ECRH) on fishbone instability is studied based on the generalized energy principle. Off-axis ECRH plays a stabilizing role in fishbone instability, while on-axis ECRH does not distinctly change the growth rate. The frequency of fishbone instability increases (decreases) for off-axis (on-axis) ECRH. The effect of ECRH is greatest when power is deposited near the rational surface. More concentrated power deposition has a better stabilizing effect. Furthermore, the non-resonance effect of trapped energetic electrons is the main factor behind the stabilization effect in the off-axis case, while it has a weak effect in the on-axis case. The frequency of fishbone instability is changed mainly by the Shafranov shift effect on trapped energetic ions since it can change the precessional drift frequency. The Shafranov shift effect can also affect the growth rate because the onset threshold of energetic ion beta is related to the frequency. The effects of magnetic Reynolds number and slowing-down critical energy are weak and can be neglected. This provides the possibility of using off-axis ECRH targeted to the rational surface to control fishbone instability.


Introduction
Fishbone instability is one of the most important energeticparticle-driven magnetohydrodynamic (MHD) instabilities. It was first observed in the Poloidal Divertor Experiment * Author to whom any correspondence should be addressed.
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. tokamak in 1983 with perpendicular neutral beam injection (NBI) [1], and was later observed in many other tokamaks such as DIII-D [2], JET [3], TFTR [4], JT-60U [5] and HL-2A [6]. Two theoretical models with typical frequencies ω ≈ ω d [7][8][9][10] and ω ≈ ω * i [11] were proposed in the 1980s, where ω d is the toroidal precessional drift frequency of trapped energetic ions (TEIs) and ω * i is the diamagnetic drift frequency of thermal ions. Fishbone instability is usually correlated with significant losses of energetic particles, which can reduce the efficiency of the heating system and thus limit the maximum achievable beta value (β = 8π nT/B 2 ) in tokamaks, and might even lead to serious damage to the first wall. Hence, it is important to study fishbone instability.
Among the various controlling methods in tokamaks, electron cyclotron (EC) waves are a unique and very promising tool because they are localized and can be flexibly and precisely targeted to a wide range of radial positions. Electron cyclotron resonance heating (ECRH) systems have been widely used to control instabilities such as the energetic particle mode [12], Alfvén eigenmode (AE) [13,14], neoclassical tearing mode [15,16] and so on. When it comes to ECRH, trapped energetic electrons (TEEs) need to be considered, and they are capable of affecting fishbone instabilities efficiently, since the interaction between them depends on precessional drift frequency not on mass [17]. Electron-driven fishbone instabilities have been observed in many tokamaks [18][19][20][21][22], where TEEs play the role of the driving source in these cases. Recently, the stabilization of ion-driven fishbone instability by ECRH has been observed in HL-2A [23], and related simulation work has also been carried out [24]. However, TEEs produced by ECRH, which are thought to play an important role, are not considered in that model. Actually, the stabilizing effect of energetic electrons on other instabilities, such as the toroidal Alfvén eigenmode, has been confirmed [14]. Thus, a model related to fishbone instabilities and TEEs is used and the effect of ECRH is studied systematically.
The remainder of this paper is organized as follows. In section 2, the dispersion relation is given by considering a system consisting of an internal kink mode with TEIs and TEEs. In section 3, the parameters, profiles and basic fishbone case are given. In section 4, numerical results, including the stabilizing effects on fishbone instability by ECRH, are provided. The effects of different physical parameters are investigated.

Dispersion relation
The generalized energy principle is widely used to explore the effect of trapped energetic particles (TEPs) on MHD modes in tokamaks [17,[25][26][27][28][29][30]. The dispersion relation includes waveparticle resonance and resistive layer physics, which can be written as where δ W R denotes the energy dissipation term associated with the resistive layer, δ W C is the minimized variational ideal MHD potential energy, and δ W TEI and δ W TEE represent the contribution of potential energy coming from the TEIs and TEEs, respectively. All the potential energy terms are normalized by δW = 2π R 0 |ξ r0 | 2 (r s B 0 /2R 0 ) 2 δ W, where R 0 and B 0 are the major radius and the magnetic field, ξ r0 is the perturbed fluid displacement and r s is the radial location. Here, the subscripts s and 0 denote the values defined at the rational surface and the plasma axis of the torus, respectively. δ W R [9] can be written as where s s is the magnetic shear at the rational surface, is the resistive diffusion time and η is the resistivity. Note that the Spitzer resistivity would be used. Thus the parameter S M is related to T e , where T e is the electron temperature. Here, the diamagnetic drift effect of bulk plasma is neglected, since ω * i is about 10 4 rad s −2 which is much smaller than the fishbone instability's real frequency of 10 5 rad s −2 based on the parameters in the HL-2A experiment [23].
The potential energies of TEIs and TEEs have a similar form except for the different equilibrium distribution functions. δ W TEP [7,8,10,17] for both TEIs and TEEs is written as where Ω d = ω/|ω d |, Λ = ΛB 0 , Λ = µ/E is the pitch angle, E = E/T e,0 , E denotes energy parameters, T e,0 is the value of T e at the axis, F = m p T 5/2 e,0 B 2 0 /8π −1 F, m p represents the mass of energetic particles, F is the equilibrium distribution function, K 2 = (2/ϵ r ΛB 0 ) 1/2 [2E(k 2 ) − K(k 2 )]/π and K b = (2/ϵ r ΛB 0 ) 1/2 K(k 2 )/π are the elliptic functions arising from bounce averaging, ϵ r = r/R 0 , and K(k 2 ) and E(k 2 ) refer to the complete elliptic integrals of the first and second kinds, respectively. All the frequency parameters are normalized by |ω dms0 |, where ω dms0 = ω d | E=Em,r=rs,Λ=Λ0 . E m is the maximum energy of TEIs and Λ 0 stands for the peak value of pitch angle for TEIs. The first part in the square brackets represents the adiabatic contribution while the other parts represent the kinetic contribution. ω d [32] can be written as where ω c denotes the gyrofrequency and s represents the magnetic shear. The definitions of circled letters and some details are given in the appendix. Note that the Shafranov shift effect related to α β and ∆ ′ is included in ω d . A typical slowing-down distribution function [33,34] for TEIs is used for their equilibrium distribution function, which can be written as where C I is a normalization factor, H indicates the step func- [34] is the slowingdown critical energy and β TEI ( r) = β TEI0 exp(− r/ σ i ) [30]. An anisotropic equilibrium distribution function for TEEs can be written as where C E is a normalization factor, β TEE ( r) = β TEE0 exp(−( r − r e ) 2 /(2 σ 2 r )) [14], where r e and σ r are the deposition location and width, respectively, and [14], where Λ e and σ Λ are the peak value and width of the pitch angle for TEEs, respectively. The form of energy, which mainly refers to Marchenko's work [35] for TEEs from ECRH, can be written as

Parameter setting
The typical parameters of the shot 27 527 experiment performed in the HL-2A tokamak [23] are used. The main parameters are as follows: Three equilibrium profiles are shown in figure 1. Λ 0 = 1 and σ i = 0.2 are applied in F TEI . Λ e = 1.076 and σ Λ = 0.036 are applied in F TEE , which means deeply trapped TEEs are considered. E min = 3 keV and E max = 60 keV are applied according to [35]. An ion-driven fishbone case is solved by the dispersion relation without TEEs. As shown in figure 2, an unstable internal kink mode with nearly zero frequency exists initially. Fishbone instability appears when β TEI0 exceeds a critical value β cr TEI0 (which also means the onset threshold of energetic ion beta for fishbone instability) and its growth rate increases rapidly with increasing β TEI0 . β TEI0 = 0.8% is chosen as the basic ion-driven fishbone case, in which the frequency of fishbone instability is ω = 7.4 × 10 4 rad s −2 , or around 12 kHz.

Numerical results
The main effects of ECRH are thought to be creating TEEs and increasing the electron temperature T e [36,37]. When the contribution of TEEs is considered, the perturbed potential energy of TEEs, δ W TEE , is added to the dispersion relation and the Shafranov shift effect, which is reflected in the toroidal precessional drift frequency ω d , is added. These two effects are both measured by the profile β TEE . β during ( r) = β before ( r) + β TEE ( r) is applied, where β during is the profile during ECRH and β before is that before ECRH. When the change of T e is considered, the magnetic Reynolds number S M and the slowing-down critical energy E c are changed accordingly. During ECRH, T e is increased based on the profile before ECRH by an additional part T e,ECRH . T e,ECRH = T e,ECRH0 exp(−( r − r e )/(2 σ 2 r )) has a similar shape to the β TEE ( r) profile.
According to the experimental observations [23], β TEE0 = 0.8% and T e,ECRH0 = 0.24 keV are chosen as the maximum values. Here, the parameter P ECRH is defined as a measure of the ECRH power level. With the change of P ECRH from 0 to 1, T e,ECRH0 changes from 0 keV to 0.24 keV and β TEE0 changes from 0% to 0.8% linearly.
The effects of on-axis and off-axis ECRH on fishbone instability are shown in figure 3. For the off-axis (on-axis) case, as the power level of ECRH increases, the growth rate decreases (is nearly unchanged), while the frequency increases (decreases). Thus, off-axis ECRH plays a stabilizing role in fishbone instability while on-axis ECRH does not distinctly change the growth rate. In addition, the frequency of fishbone instability is changed by ECRH.
The growth rate and real frequency of fishbone instability versus P ECRH with different power deposition locations and widths are shown in figures 4 and 5, respectively. From figure 4, when the deposition location is near the rational surface (q = 1/1) the growth rate is the smallest. Namely, the stabilization effect becomes largest when the power is deposited near the rational surface. Besides, ECRH with deposition locations inside or outside the rational surface has similar stabilizing effects when |r e − r s | is the same. From figure 5, with decreasing deposition width, the growth rate decreases and the change of frequency becomes smaller. This means more concentrated power deposition has a better stabilizing effect but a smaller effect on frequency.
In order to clarify the main mechanism of influence, the effects of the contribution of TEEs and T e will be studied separately in the following.

Effects of the contribution of TEEs
In this subsection, the effect of the contribution of TEEs is studied. The T e profile is kept unchanged, namely S M and E c are not changed.
The growth rate and frequency of fishbone instability versus β TEE0 are shown in figure 6. For the off-axis (on-axis) case, as β TEE0 increases, the growth rate decreases (is nearly unchanged), while the frequency increases (decreases). These results are similar to those when scanning P ECRH . Therefore, the contribution of TEEs is the main factor affecting fishbone instability.
An analysis based on the value of δ W is shown in figure 7. For the off-axis case in figure 7, the sign of Re[δ W TEE ] is   opposite to Re[δ W TEI ]. This means that the TEEs have a stabilizing effect on fishbone instability since the fishbone is an ion-driven mode and TEIs play a destabilizing role. For the on-axis case in figure 7, the amplitude of Re[δ W TEE ] is much smaller than Re[δ W TEI ]. Namely, the effect of onaxis TEEs is weak. This is because the real part of δ W is related to the growth rate, since δ W R = −iωs 2 s /ω H for the small-resistivity limit [7,9,30], and the growth rate represents the mode's stability. From figure 7, Im[δ W TEE ] is much smaller than Im[δ W TEI ] for both off-axis and on-axis cases, namely the resonance between TEEs and fishbone instability can be neglected. This is because the imaginary part of δ W TEP represents the measure of resonance between wave and particles, since it comes from the resonance term ω − ω d while using the Sokhotski-Plemelj formulae [26]. The small resonance is because the toroidal precessional drift frequency of TEEs is opposite to the frequency of fishbone instability. So the resonance condition is hardly satisfied. Thus, the contribution of TEEs mainly results from the non-resonance effect.
More details of Re[δ W TEE ] are shown in figure 8. For the off-axis case, the adiabatic part of the TEEs' contribution plays a stabilizing role while the kinetic part is weak. Namely, the stabilizing effect of TEEs mainly comes from the adiabatic contribution. For the on-axis case, the adiabatic part of the TEEs' contribution plays a distabilizing role while the kinetic part plays a stabilizing role. These tend to cancel each other out so the net contribution is weak.
The contribution of the TEEs contains perturbed potential energy and the Shafranov shift effect. The Shafranov shift effect is reflected in ω d , measured by the parameters α β and ∆ ′ [32]. It can change the resonance regions through the resonance term ω − ω d by changing ω d of TEPs.
The Shafranov shift effect on TEIs is studied by scanning β TEE0 without TEEs, which means δ W TEE in equation (1) is not included but the effect of β TEE0 on the toroidal precessional drift frequency of TEIs, ω d,i , is included; the results are shown in figure 9. For the off-axis (on-axis) case, as β TEE0 increases, the frequency without TEEs increases (decreases) and the trend is similar to that with TEEs. Namely, the Shafranov shift effect on TEIs is the main factor changing the frequency of fishbone instability. From figure 9, for the off-axis (on-axis) case, ω d,i becomes bigger (smaller). A simple explanation can be given here. The Shafranov shift effect modifies ω d [32]. The frequency of fishbone instability is directly dependent on the ω d,i profile owing to the resonance process with toroidal precessional drift of TEIs. Thus, the change of ω d,i due to the Shafranov shift effect leads to a change in the frequency of the fishbone instability. As shown in figure 9, for the off-axis (on-axis) case, as β TEE0 increases, the growth rate decreases (increases). This is because the critical beta of fishbone instability, β cr TEI , is in direct proportion to the average value of ω d,i [7,30]. In the off-axis (on-axis) case, the average value of ω d,i is increased (decreased) by the Shafranov shift effect. Correspondingly, β cr TEI is increased (decreased). Thus, the growth rate of fishbone instability is decreased (increased). For off-axis case this is another important stabilizing effect in addition to the adiabatic contribution of TEEs.
The Shafranov shift effect on TEEs is studied by scanning β TEE0 while keeping ω d,e unchanged, namely the Shafranov shift effect does not act on TEEs. The results are shown in figure 10. The change of frequency without a shift is similar to that with a shift for both off-axis and on-axis cases, i.e. the Shafranov shift effect on TEEs would not change the frequency of fishbone instability. This is because the mode is ion-driven and the frequency is mainly dependent on the resonance between mode and TEIs, not TEEs. From figure 10, for the off-axis case the growth rate without a shift is similar to that with a shift, while for the on-axis case the growth rate without a shift is bigger than that with a shift. This is because the Shafranov shift effect can only affect the kinetic part through resonance term ω − ω d and, as mentioned before, the kinetic part of Re[δ W TEE ] is small in the off-axis case but as big as the adiabatic part in the on-axis case. In addition, from figure 10, for the off-axis (on-axis) case |ω d,e | with a shift becomes bigger (smaller), namely the resonance condition ω − ω d = 0 for TEEs is harder (easier) to satisfy. Then the kinetic part of Re[δ W TEE ] becomes smaller (bigger) since it is in inverse proportion to the resonance term ω − ω d . For the off-axis case, the Shafranov shift effect on TEEs makes the kinetic part even smaller, so the growth rate with a shift is nearly the same as that without a shift. But for the on-axis case, the kinetic part of Re[δ W TEE ] plays a stabilizing role, as mentioned before, and the Shafranov shift effect makes it bigger. So the growth rate with a shift is smaller than that without a shift. Specially, for the on-axis case the Shafranov shift effect is so strong that it reverses the TEE drift in some regions, as shown in figure 10, i.e. in these regions ω d,e has the same direction as the mode frequency. Although the resonance condition ω − ω d = 0 still cannot be satisfied and the contribution of the TEEs is still mainly non-resonance, it can be seen from figure 7 that the resonance of TEEs in the on-axis case is a bit bigger than that in the off-axis case.

Effects of the electron temperature
In this subsection, the effect of electron temperature is studied alone without TEEs (β TEE0 = 0% is kept unchanged). S M and E c are changed respectively (in the following text S M and E c represent the values at r s ).
The growth rate and frequency of fishbone instability versus S M , when E c ≡ 13.7 keV, are shown in figure 11. When S M > 10 5 , the growth rate and frequency are almost unchanged with increasing S M . When S M < 10 5 , the growth rate decreases as S M decreases. Based on the experimental parameters, S M is around 5.7 × 10 5 to 8.7 × 10 5 , namely S M > 10 5 is always satisfied. Thus, the value of S M is large enough and the change in S M is not so large that the increasing S M hardly affects fishbone instability.
Further analysis of critical ion beta for fishbone instability, β cr TEI0 , is done and the results are shown in figure 12. This is solved by setting Im[ω] ≈ 0 and searching for Re[ω] and β TEI . The solution of β TEI0 is called β cr TEI0 . When S M < 10 5 , the critical beta β cr TEI0 decreases with increasing S M . When S M > 10 5 , the critical beta β cr TEI0 is almost unchanged as S M increases. This further confirms the above result and is consistent with previous theoretical work [9,10,30]. As pointed out above, the value of S M from experimental parameters is sufficiently large that the critical beta, β cr TEI0 , remains unchanged with increasing S M . This means that the effect of the changing S M can be neglected.
The growth rate and frequency of fishbone instability versus E c , when S M ≡ 5.7 × 10 5 , are shown in figure 13. As E c increases, fishbone instability is unaffected, namely the effect of the changing E c can be neglected.

Discussion and conclusion
In many works, a Maxwellian distribution function would be chosen as the TEEs' energy distribution function, which means that f E = exp(− E/ E E ), where E E represents the equivalent temperature of the TEEs. The real frequency and growth rate versus P ECRH with different distribution functions are shown in figure 14. The frequency and growth rate versus P ECRH are almost the same as for Marchenko's case and the E E = 10 keV Maxwellian case. Namely the Marchenko distribution function is similar to the Maxwellian distribution function with E E = 10 keV under the experimental parameters in [23]. The effects of ECRH with different E E have been discussed and the results are shown in figure 15. For E E = 40 keV, the changes of frequency are almost the same, and the stabilization effect for off-axis ECRH is weaker than for E E = 10 keV. The stabilizing effects are qualitatively consistent. In addition, on-axis ECRH tends to play a stabilizing role when E E = 40 keV. Figure 4(a) in [23] shows that the frequency of fishbone instability is around 13 kHz. While in our work the frequency of fishbone instability in the basic case is 12.5 kHz, which basically conforms to the experimental results. In [23] it was concluded that the fishbone instability is stabilized by off-axis ECRH, while in our work off-axis ECRH has a stabilizing efefct on fishbone instability, which can be seen in figure 3. The main theoretical result is essentially consistant with the experimental observations .
In conclusion, the effects of ECRH on fishbone instability have been studied. The effects of ECRH are considered to be the creation of TEEs and increasing the electron temperature, T e . The contribution of TEEs comes from their perturbed potential energy, δ W TEE and the Shafranov shift effect. The change of T e is relevant to changes in the magnetic Reynolds number, S M , and the slowing-down critical energy, E c . It is found that ECRH has a stabilizing effect on fishbone instability in the off-axis case while it does not distinctly change the growth rate in the on-axis case. In addition, the frequency of fishbone instability increases (decreases) when the ECRH power is deposited off-axis (on-axis). Moreover, the effects of ECRH are largest when power is deposited near the rational surface. ECRH with deposition locations inside or outside the rational surface has a similar stabilizing effect when |r e − r s | is the same. Also, more concentrated power deposition has a better stabilizing effect but a smaller effect on frequency.
Some details of the effects of ECRH have been studied. The effect of δ W TEE resulting from the non-resonance effect is the main factor to affect the stability of the fishbone mode. It is also found that the Shafranov shift effect, which is reflected in the change of ω d , leads to a change in the frequency of fishbone instability, and the growth rate is changed. For S M , on the one hand, the value of S M is large enough, but the change in S M is not so large that the increasing S M has hardly any impact on fishbone instability. On the other hand, the critical beta, β cr TEI0 , remains unchanged with increasing S M . Moreover, the effect of the change of E c can be neglected.  It should be pointed out that the conclusion about the effects of S M in our work is different from that in [23,24]. The conclusion in [23,24] is that the onset threshold of TEI beta for fishbone instability, β cr TEI , increases with increasing S M , so the fishbone instability is stabilized. This conclusion mainly refers to the relation β cr TEI ∝ S 1/3 M from [9], and this relation requires a big-resistivity limit according to [9]. While in our work, as noted before, the value of S M is sufficiently large, meaning that the big-resistivity limit cannot be satisfied. Hence, the explanation for the stabilizing effect on fishbone instability of S M in [23,24] fails.
In general, the direct contribution of TEEs is important for the stabilizing effect on fishbone instability. It should be pointed out that the effect of magnetic Reynold number fails to explain the stabilization of fishbone instability by ECRH, since its effect can be neglected based on the experimental parameters. The calculated results are consistent with the experimental results. Therefore, it can be predicted that off-axis ECRH with a deposition location near the rational surface can effectively stabilize fishbone instability.