Prediction of Nonlinear Evolution Character of Energetic-Particle-Driven Instabilities

A general criterion is proposed and found to successfully predict the emergence of chirping oscillations of unstable Alfv\'enic eigenmodes in tokamak plasma experiments. The model includes realistic eigenfunction structure, detailed phase-space dependences of the instability drive, stochastic scattering and the Coulomb drag. The stochastic scattering combines the effects of collisional pitch angle scattering and micro-turbulence spatial diffusion. The latter mechanism is essential to accurately identify the transition between the fixed-frequency mode behavior and rapid chirping in tokamaks and to resolve the disparity with respect to chirping observation in spherical and conventional tokamaks.

A general criterion is proposed and found to successfully predict the emergence of chirping oscillations of unstable Alfvénic eigenmodes in tokamak plasma experiments. The model includes realistic eigenfunction structure, detailed phase-space dependences of the instability drive, stochastic scattering and the Coulomb drag. The stochastic scattering combines the effects of collisional pitch angle scattering and micro-turbulence spatial diffusion. The latter mechanism is essential to accurately identify the transition between the fixed-frequency mode behavior and rapid chirping in tokamaks and to resolve the disparity with respect to chirping observation in spherical and conventional tokamaks.
In fusion grade plasmas, there is a population of energetic particles (EPs) with typical energies substantially greater than those of the thermal background. These particles provide an energy-inverted population, that through the kinetic wave-particle interaction with Alfvén waves, can induce instabilities that jeopardize plasma confinement [1,2]. The nature of these oscillations vary considerably (with the possibility of several bifurcations [3]), with two typical non-linear scenarios being: (a) the excitation of a slow evolving amplitude, nearly fixedfrequency oscillation and (b) coherent oscillations that chirp in frequency at timescales much shorter than that of the plasma equilibrium modification. These scenarios lead to dominant diffusive and convective transport of EPs, respectively. This letter addresses two outstanding and interconnected issues that, in spite of their major relevance for the transport of EPs in future-generation burning plasmas, are currently not understood. The first issue is what plasma conditions most strongly determine the likelihood of each non-linear saturation scenario in experiments. The second is why the chirping response (observed in all major tokamaks, e.g. DIII-D [4], NSTX [5,6], JET [7], MAST [8], JT-60U [9,10], ASDEX-U [11]) is much more common in spherical tokamaks than in conventional tokamaks. This classification is important in anticipating whether EP-induced instabilities in burning plasma experiments will likely lead to steady oscillations, where quasi-linear theory [12][13][14] would be expected to described EP transport or chirping, which would then require new theoretical tools to assess the consequences of the induced EP transport.
In this letter we show that a previous approach that attempted to simplify the needed input that the theory requires [15] is insightful but limited in making accurate predictions for experimental scenarios. Here we employ a generalized formulation and show that its predictions are in accordance with observations. This analysis reveals that micro-turbulence, even while producing no observable effect on beam ion transport, provides the vital mechanism in determining which non-linear regime is more likely for a mode as well as the mode transition from one regime to the other, as parameters of an experiment change in time.
We focus the analysis on the onset of a mode non-linear evolution near marginal stability. The interaction Hamiltonian between a particle (at position r, with velocity v) and a tokamak eigenmode with frequency ω can be written as qA (r, t) where A (r, t) is the perturbed vector potential along an unperturbed orbit in a gauge where the electrostatic potential vanishes, E is the unperturbed energy, P ϕ the canonical angular momentum, µ the magnetic moment (all per unit EP mass), the summation is over all integers j, ϕ a and θ a are the action angles in the toroidal and poloidal directions, q is the charge of an EP and n is a fixed quantum number for the angular response of a perturbed linear wave in an axisymmetric toroidal tokamak. V j (see Eq. 12 of Ref. [16]) accounts for the waveparticle energy exchange. Upon a suitable normalization, the amplitude C(t) has been shown to be governed by an integro-differential cubic equation that is nonlocal in time [3,15,17] [18], ∂Ω , with f being the equilibrium distribution function. We assume a low frequency mode for which µ is conserved. Then ∂/∂I ≡ −n∂/∂P ϕ + ω∂/∂E. The resonance condition is given by Ω j = ω + nω ϕ − jω θ ≈ 0, where ω θ and ω ϕ arXiv:1610.05059v1 [physics.plasm-ph] 17 Oct 2016 are the mean poloidal and toroidal transit frequencies of the equilibrium orbit. The phase-space integration is given by´dΓ... = (2π) where m EP is the mass of EPs, c is the light speed and σ accounts for counter-and co-passing particles. The effective collisional operator can be cast in the form where ν scatt and ν drag are understood to be the effective pitch-angle scattering and drag (slowing down) coefficients, defined in Eq. 6 of Ref. [15]. ν stoch is the effective stochasticity, which includes ν scatt . In equation (1), the circumflex denotes normalization with respect to γ = γ L − γ d (growth rate minus damping rate) and t is the time normalized with the same quantity. Vlasov simulation codes have shown [19,20] that the blow-up solutions of (1) are precursors to chirping behavior.
The type of nonlinear evolution of a wave destabilized by a perturbing EP drive is strongly dependent on the kernel of the integrals of Eq. (1), more specifically on the ratio between the effective stochastic relaxation felt by the EPs and the effective drag rate, as well as the linear growth rate. In Ref. [15], Eq. (1) was simplified by using characteristic values for the collisional ν scatt and ν drag and conditions for the existence and stability of solutions of the cubic equation were derived. In Fig. 1, we test for the first time this prediction against modes measured in different tokamaks. In order to determine mode properties, we employ the kinetic-MHD code NOVA [21] to compute eigenstructures and the frequency continua and gaps. Its kinetic postprocessor NOVA-K [22,23] is used to calculate perturbative contributions that can stabilize and destabilize MHD eigenmodes. In addition, NOVA-K is also employed to compute resonant surfaces in (E, P ϕ , µ) space. In order to characterize the mode being observed in the experiment, NSTX reflectometer measurements are compared to the mode structures computed by NOVA, employing a similar procedure as the one used in Refs. [6,24]. In DIII-D, similar identification is performed using Electron Cyclotron Emission (ECE) [25].
We see from Fig. 1 that about half of the chirping NSTX modes lie in a region where a stable steady mode is predicted by Ref. [15]. For the DIII-D experimental cases that produced fixed-frequency modes, the predictions of Ref. [15] are mostly in agreement although one point is borderline and another one may be unstable enough to be in a chirping regime. Hence we see that using the simplified, although elaborate, modeling akin to that used in [15], might be in satisfactory agreement with DIII-D data but is generally not satisfactory for much of the NSTX and TFTR data. This comparison indicates that the use of a single characteristic value, being representative of the entire phase space, for ν scatt (considered the only contribution to ν stoch ) and ν drag , although very insightful, appears insufficient to provide quantitative predictions  (1) while the solid line delineates the region of stability, as predicted by [15]. Modes that chirped are represented in red and the ones that were steady are in black, as experimentally observed in DIII-D (disks), NSTX (diamonds) and TFTR (square).
for practical tokamak cases. This conclusion motivated the pursuit of a general theoretical prediction to take into account important missing elements, such as spatial mode structures and local phase-space contributions on multiple resonant surfaces of the wave-particle interaction terms, all of which are needed in toroidal geometry. The appropriate weightings for the various needed quantities can be expressed in the action-angle formulation. A necessary, although not sufficient, condition for chirping solutions is that the right hand side of (1) be positive. The resonance condition, δ (ω − Ω l (P ϕ , E, µ)), allows one of the phase-space integrals to be eliminated. Upon integration over τ 1 and redefinition of the integration variable z = ν drag τ one finds the following criterion for the non-existence of steady solutions of (1): For the resonances to be linearly destabilizing to positive energy waves, Int (plotted in Fig. 2) is the only component of the criterion (2) that can be negative from the phase-space regions which contribute positively to the instability growth. N is a normalization factor consisting of the same sum that appears in Eq. (2) except for   Int. Thus we use the contributions from each resonance weighted in accord with the appropriate eigenfunction (that fits the measured field structure) and the position in phase space of the resonant interaction. We will see that this procedure produces quite a different conclusion from the less detailed method that uses a single characteristic factor, as is the case in Fig. 1. Non-steady oscillations, with the likelihood of chirping, are predicted to occur if Crt < 0 while a steady (fixed-frequency) solution exists if Crt > 0. However, we see from Fig. 2 that the magnitude of Int could be an order of magnitude larger in phase space regions where ν stoch /ν drag 1.04 compared with regions where ν stoch /ν drag 1.04. Hence, because of this disparity, it can turn out that a choice of the use of a single characteristic value for ν stoch /ν drag , would lead to a positive value for Crt while the use of the appropriately weighted average leads to a negative value for Crt. Such a change is indeed the case for all the TFTR and DIII-D modes and for most of the NSTX modes shown in Fig. 1, where ν stoch was considered simply as ν scatt . The reason for this sensitivity is that there will always be a contribution to Crt from a phase space region where ν scatt /ν drag 1 because the pitch angle scattering coefficient goes to zero as µ vanishes. Hence even when the characteristic value of ν scatt /ν drag is substantially greater than unity, one still can find that Crt < 0.
The above observation indicates that pitch-angle scattering ν scatt may not always be the dominant mechanism in determining ν stoch . Hence, we now introduce the contribution of fast-ion electrostatic micro-turbulence for the determination of ν stoch with the following procedure. The TRANSP code [26] is employed to obtain the thermal ion radial thermal conductivity, χ i (which is essentially the particle diffusivity, D i [27]) based on power balance. The heat diffusivity due to collisions is subtracted out and the remaining diffusivity is attributed to microturbulence interaction with the ions. Then the EPs diffusivity is estimated by using the scalings determined in a gyrokinetic simulation [28], which for passing particles gives D EP ≈ 5D i T i /E EP . In the experiments we ana- lyzed, the drive was mostly from the passing particles and therefore we used this relation as an estimate for D EP . The response of the resonant EPs to perturbing fields is essentially one-dimensional [17] and produces steep gradients in the EP distribution in this perturbing direction. We can then accurately account for the diffusion that is directed in all phase space directions, by projecting the actual diffusion from all these directions onto the steepest gradient path defined by the one-dimensional dynamics, using the specific relation given by Eq. (2) of Ref. [29]. Fig. 3 shows values of |Crt| 1/4 multiplyed by the sign of Crt, as a function of the ratio of phase-space averaged stochasticity and drag for modes of Fig. 1. This representation provides better visualization than simply plotting Crt, especially close to the steady/chirping boundary and is chosen because of the fourth power dependence Int ≈ 1.022 (ν stoch /ν drag ) −4 for ν stoch /ν drag 1. Fig. 3(a) shows chirping modes in NSTX and Fig. 3(b) shows steady modes in DIII-D and TFTR. The curved arrows represent how the prediction for a mode is affected by micro-turbulence-induced scattering of EPs. It has a strong effect on DIII-D and TFTR (bringing the modes to the steady region, or at least very close to it) while its effect is imperceptible for the chirping modes in NSTX. This is because, unlike in conventional tokamaks, thermal ion transport in spherical tokamaks (STs) is usually very close to neoclassical levels [30,31] even though the electron transport is anomalous. NSTX modes in Fig.  3(a) are only able to transition to the fixed-frequency region when ν stoch is artificially multiplied by a factor from 10 to 50, depending on the specific mode, which indicates the robustness of the chirping prediction.
Guided by this theory, we have then examined chirp- ing modes that rarely appear in DIII-D tokamak and we have found that chirping onset in DIII-D is observed to correlate very closely with conditions where thermal ion transport had drastically decreased, as shown in Fig.  4. This is attributed to the decrease in micro-turbulenceinduced transport, which also causes decreased EP transport. Alfvénic modes only started chirping when the thermal ion conductivity dropped to values lower than 0.3m 2 /s. An example of the evaluation of the criterion (2) is DIII-D shot 152828 (Fig. 4 (c)). Before chirping starts (at t = 920ms, when D th,i ≈ 0.55m 2 /s) the calculated criterion is Crt = +0.001. During the early phase of chirping (at t = 955ms, when D th,i ≈ 0.25m 2 /s) the value is Crt = −0.013, i.e. the mode has transitioned from the positive (steady) region to the negative region of Crt, therefore allowing chirping, in agreement with the observation.
The micro-turbulence interaction with EPs is a key factor that determines the nature of mode saturation regime (quasi-steady and chirping) and also the transition between them. It also explains the longstanding question of why chirping Alfvénic modes are ubiquitous in STs and rare in conventional tokamaks. Experimentally, the EP transport due to micro-turbulence is too low compared to Alfvénic-induced transport [32,33], yet the turbulent interaction remains capable of drastically changing the non-linear stability regime of the Alfvénic modes at their onset, when mode amplitude is low. This suggests that micro-turbulence simulations employed to predict the thermal plasma transport of future burning plasma devices must also be factored in to considerations of the drive and saturation of modes driven by EPs.
This work provides elements for choosing which of the two extreme scenarios is most likely to be relevant for predicting the character of the energetic particle transport, based on the sign of Crt. For a negative Crt the physical conditions are established to enable a nonlin-ear BGK-like mode [34] to form, where the frequency remains locked to a particle resonance frequency as particles trapped by the wave are convected in phase space which, for the Alvénic instabilities, primarily causes resonant energetic particles to flow across field lines. Alternatively, a positive Crt represents the lack of chirping and indicates that the details of the nonlinear particle transport might be described by a quasilinear diffusion theory [12][13][14]. Therefore, the application of this criterion should be important in the planning and modeling of scenarios for future fusion plasma experiments.
We acknowledge fruitful discussions with G.