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First evidence of Alfven wave activity in KSTAR plasmas

We report on first evidence of wave activity during neutral beam heating in KSTAR plasmas: 40 kHz magnetic fluctuations with a toroidal mode number of n=1. Our analysis suggests this a beta-induced Alfven eigenmode resonant with the q=1 surface. A kinetic analysis, when coupled with electron temperature measurements from electron cyclotron emission and ion/electron temperature ratios from crystallography, enables calculation of the frequency evolution, which is in agreement with observations. Complementary detailed MHD modelling of the magnetic configuration and wave modes supports the BAE mode conclusion, by locating an n=1 mode separated from the continuum in the core region. Finally, we have computed the threshold to marginal stability for a range of ion temperature profiles. These suggest the BAE can be driven unstable by energetic ions when the ion temperature radial gradient is sufficiently large. Our findings suggest that mode existence could be used as a form of inference for temperature profile consistency in the radial interval of the mode, thereby extending the tools of MHD spectroscopy.


Introduction
Instabilities such as Alfvén eigenmodes, driven by fast particles, are of programmatic concern as they can expel energetic ions from the plasma, thereby preventing heating by thermalisation. [1] In addition, such energetic particles expelled can damage the first wall, and a fusion reactor can only tolerate fast particle losses of a few per cent. [2] Another motivation for the study of Alfvén eigenmodes is their potential use as a diagnostic for the plasma, particularly through the tool of MHD spectroscopy. [3] One such class of fast particle driven instabilities that can occur at relatively low frequency are beta induced Alfvén eigenmodes (BAEs). The characteristic experimental feature of this instability are magnetic fluctuations at a frequency intermediate between the fishbone and the toroidal Alfvén eigenmode (TAE), with angular frequency ω TAE = v A /(2qR), with v A the Alfvén speed, q the safety factor and R the major radius. [4] These modes were first identified in DIII-D, [5] and have since been discovered in other tokamaks in beam, [6] ion cyclotron heated [7] and Ohmically heated discharges in the presence of a magnetic island. [8,9,10,11,12] More advanced kinetic treatments, which include corrections to kinetic theory for diamagnetic and shaping effects, as well as the inclusion of trapped particles have been used to study BAEs in ASDEX. [13] Radial profile information was measured during sawteeth in Tore Supra. [14] Recently, BAEs driven by electron populations have been observed in the tokamak HL-2A, [15] and there is some evidence that magnetic oscillations in the H-1 heliac are also driven by energetic electrons. [16] The aim of this paper is to report on first evidence of Alfvénic wave activity during neutral beam heating in KSTAR plasmas. In 2010 and 2011 campaigns KSTAR plasmas included 1.2 MW of neutral beam heating, which provided a source of heating to excite Alfvénic wave activity modes. Data from the 2010 campaign, which was fully analysed during 2011, shows 40-60kHz magnetic fluctuations. We present the first ideal MHD calculation of a core localised mode with toroidal mode number n = 1 for an experimental configuration. The mode is global with very small resonance with continuuum modes. Second, this work is the first observation of Alfvénic wave activity in KSTAR. With up to 14MW of neutral beam heating and 14MW of RF heating planned, KSTAR plasmas will become a pilot for ITER plasmas, and provide the opportunity to explore the wave-particle-plasma interaction in regimes approaching burning plasmas. Our work builds on preliminary observations of electron fishbones in KSTAR in 2009, [17] and aliased TAE activity in 2011. [18] Finally, as a spin-off, we have developed a new form of MHD spectroscopy for consistency of the temperature profile with the observation of wave activity. The remainder of the work is as follows: in Section 2 we introduce the experiments conducted in KSTAR, and in Sec. 3 present detailed modelling. Finally, Sec. 4 contains concluding remarks and discusses implications for future work.

Experiments
In 2010 a set of neutral beam injection (NBI) excitation experiments were conducted in KSTAR in an attempt to generate shear Alfvénic wave activity. Subject to operator controls, the choice of plasma conditions were optimised for this purpose: a relatively low toroidal magnetic field of 1.95 T, and maximum available NBI heating: 1.2MW of 80keV NBI. For deuterium plasmas, 80keV NBI would produce fast D neutrals with speed v ||,beam = 2.8×10 6 ms −1 . With plasma densities of up to 5×10 19 m −3 expected, the minimum Alfvén speed is v A = 4.4 × 10 6 ms −1 . While the beam speed is sub-Alfvénic, it is greater than the first sideband resonance at v A /3, and so it is possible that TAE modes may excited, as was found during early operation of MAST. [19] Alfvén cascades have also been observed on JET in reverse shear configurations with v ||,beam /v A as low as 0.2. [20] Four NBI heated discharges (#4218-#4220) were produced, with flat top plasma current I p in the range 210 < I p < 407 kA, electron cyclotron resonance heating (ECRH) power up to 200 kW, and core plasma density up to 3 × 10 19 m −3 . There was no discernible impact of ECRH heating on these plasmas. Figure 1 shows the temporal evolution of one such discharge, #4220. For this 4.5s plasma, 2s of NBI was applied from 2s during current flat-top.   Figure 2 shows magnetic oscillations of discharge #4220. The mode activity correlated with NBI heating, and a study of electron cyclotron emission data reveals that the plasma is sawtoothing throughout the heating phase. A study of signal phase versus geometric angle, computed from a toroidal Mirnov array provides weak evidence that the mode has n = 1. Unfortunately, the coils are located flush with the conducting wall, and so the signal to noise ratio is large, and there is significant n = 0 noise present. Information about the poloidal mode number is not available. Similar oscillations were observed in discharges #4218,#4219 and #4221.
A simple scoping exercise reveals that the mode activity is unlikely to correspond to a Toroidal Alfvén Eigenmode. The middle of the TAE gap lies at ω T AE = v A /(2qR). Using n i ≈ n e (0), taking R as the magnetic axis, and using q = q mn = (2m + 1)/(2n) for m = 1, 2, 3 gives 160 kHz, 100 kHz and 70 kHz, respectively. Mode activity with frequency of order 150 kHz and m = n = 1 was observed in the 2011 campaign. The observed mode reported here has a frequency of 40 kHz, which would be commensurate with a resonance of q = 5. This is the edge q of these elongated plasmas, and so the TAE frequency will be significantly greater than 40 kHz. Following Gorelenkov et al [21] we have also computed the thermal ion transit frequency ω ti = 2k B T i /m i (qR 0 ), which is the upper beta-acoustic Alfvén Eigenmode (BAAE) gap frequency. For KSTAR plasmas, the on-axis frequency ranges from 29kHz, using the central ion temperature inferred from #4229 down to 17kHz. These are below the observed wave frequency, at 40-60 kHz.
In contrast, the frequency as well as its time evolution is a closer match to the evolution of the kinetic accumulation frequency ω CAP = 1/R 0 2T i /m i (7/4 + T e /T i )), with R 0 the major radius, m i the ion mass, and T i and T e the ion and electron temperature, respectively. [22] Electron cyclotron emission (ECE) data gives the on-axis value of electron temperature T e = 1.2 keV. While not available for discharge #4220, x-ray imaging crystal spectrometer [23] data providing T i , T e is available for nearby discharge #4229 with the same level of NBI heating. This pulse also shows evidence of toroidal rotation, with core rotation up to 100 km/s, producing a core Doppler shift of up to 8kHz. Rotation of either a near stationary mode or magnetic island is thus insufficient to describe the observations.
Correcting for the offset in neutral beam heating interval for this discharge, we have computed ω CAP and over plotted the evolution in Fig. 2. The frequency match is close, as is the slight frequency drop following beam turn-on at 2 s. The drop in frequency occurs due to the initial drop in T e observed in the core ECE channel. Similar BAE frequency scaling is evident for discharges #4218 and #4219.

Detailed Modelling
We have undertaken detailed modelling of the plasma at the onset of mode activity at 2 s. By analysing data from a set of 20 electron cyclotron emission (ECE) chords we have been able to identify the inversion radius and locate the q = 1 surface. To correct the q = 1 surface of magnetics-only constrained EFIT to match ECE data we have used CHEASE [24] to remap the current profile as I * (s) → I * (s) + cI core (s) and pressure profile with p ′ (s) → λp ′ (s) so as to match the sawtooth inversion radius and EFIT β p , respectively. Here, s is the square root of normalised poloidal flux, with s = 0 the core and s = 1 the edge. The core current profile selected is a ramp, I core (s) = 1 − s. Figure  3 shows a cross section of the corrected equilibrium flux surfaces and q profile.  Figure 4 shows the Alfvén and ion sound continuum for n = 1, computed using the code CSCAS [25] with adiabatic index γ = 5/3. In Fig. 4(a) the toroidicity and ellipticity induced gaps can be identified. Using the most recent version of the ideal MHD global stability code MISHKA [26] we have computed TAE gap modes. The TAE gap mode produced by the q mn = (2m+1)/2n = 1.5 resonance at s = 0.45 has a frequency of 120 kHz, well above the measured 40 kHz oscillation. In Fig. 4(b) we have zoomed into the low frequency part of the continuum and identified different continuum branches. As shown by Gorelenkov et al , [21] the frequency of the ion sound and modified shear Alfvén continuum modes drops to zero at rational surfaces. The accumulation point of the low frequency gap introduced in the shear-Alfvén continuous spectrum because of finite beta is ω As,gap /ω A = (γβ) 0.5 , with ω A the Alfvén angular frequency at the magnetic axis. To identify a global mode separated from the MHD continuum, we have selected an s interval of 0 < s < 0.3, thereby avoiding the resonance with the m = 3 ion sound branch. The discrete mode identified in Fig. 4(b) is separated from the continuum above and below, and so forms a gap mode between singular modes. In the limit that the s interval is expanded to the full domain, the eigenfunction retains its global structure. The mode frequency is ω/ω A = −0.1371. A single channel millimeter wave interferometer system provides a measurement of the time-resolved line integrated electron density, n e . [27] Assuming a parabolic profile for the density provides a measure of on-axis density. If n i ≈ 0.8n e is also assumed, [28] the Alfvén angular frequency can be computed, yielding ω/(2π) ≈ 50 kHz at 2s. We have overplotted the frequency evolution of this estimate in Fig. 2.
The mode, which is resonant with the core of the plasma, has similar global mode structure to modes computed in DIII-D by Turnbull et al [29]. In contrast, the BAE mode modelled by Huysmans et al [30] is in the region of greater shear, and posseses more poloidal harmonics. Figure 5 shows the V 1 eigenvector, which is the ρ component of the contravariant fluid displacement velocity, for (a) a discrete mode, and (b) a nearby continuum mode. A continuum mode is one that is localised to a resonant surface with poloidal flux ψ, and a solution of the shear Alfvén dispersion relation ω(ψ) = k || (ψ)v A (ψ), with k || = k · B/B. [25] The BAE in Fig. 5(a) has a global mode structure with negligent resonant coupling with continuum modes, whereas the continuum mode in Fig. 5(b) has resonances at crossings of the continuum at s = 0.57, s = 0.66 for m = 2 and s = 0.77, s = 0.84 for m = 3. In contrast to earlier work, [29,30] the global mode of Fig. 5(a) has significantly reduced coupling to resonances with the continuum, and therefore will exhibit weaker continuum damping.  . Ion sound and Shear Alfvén continuum (shown as black points) for n = 1 modes for #4220 at 2s. Also shown is the q profile (solid black curve), normalised mass density profile ρ/ρ 0 (solid green curve), TAE frequency (solid blue curve), elliptical Alfvén eigenmode frequency (solid red curve) and accumulation point of the low frequency gap ω As,gap (solid pink curve). Figure (b) focuses on the BAE region, which also shows the BAE mode (red line).
Ideal MHD codes such as MISHKA solve for the wave structure in full toroidal geometry, but provide no information about mode drive. The particle-wave resonance condition [31] is where l is a Fourier mode number in poloidal angle θ of the particle magnetic drift velocity. The mode peaks at s = 0.16, for which q = 0.89. At this radial location the l = 0 resonance condition requires the unphysical condition v || /v ||,beam = 1.3, while for |l| ≥ 1 the resonance condition can be satisfied for v || < v ||,beam . Thus, the mode can be driven by sidebands, |l| ≥ 1. The mode resonance is broad due to the finite radial region of varying q covered by the mode width.
Meanwhile, the mode excitation threshold by energetic ions is reduced by inverse ion Landau damping connected with finite thermal ion temperature gradient. [22] In the limit of vanishing continuum damping, an estimate of the threshold to marginal stability can be made using the kinetic treatment of Zonca [22], which studied the drive due to thermal ion temperature radial gradients, and studied the relationship between kinetic ballooning modes (KBM) and BAEs. In that treatment it was shown the stability conclusions were a function of the frequency range of ω CAP relative to the core-plasma ion diamagnetic frequency ω * pi = k B T i e i B (k × b) · ∇ ln P i . Here, k B is Boltzman's constant, k = m/re θ + n/Re φ the wave vector, b = B/|B|, and P i the thermal ion pressure. If ω 2 * pi ≪ ω 2 CAP the KBM accumulation point was always stable, and the BAE may become unstable for values of η i = (∂ ln T i /∂ ln n i ) greater than a critical value η ic , given by with ω * ni = k B T i e i B (k × b) · ∇n i /n i the ion diamagnetic drift frequency. If ω * pi is increased further, the unstable BAE accumulation point is expected to smoothly connect to an unstable KBM accumulation point with exponentially small growth rate when ω 2 * pi ≫ ω CAP . The most unstable BAE/KBM accumulation point occurs when ω 2 * pi ≈ ω 2 CAP , when the BAE and KBM are strongly coupled. For the plasma conditions of KSTAR, we compute ω * pi /(2π) ≈ 4 kHz, and so ω 2 * pi ≪ ω 2 CAP , and thus plasmas with η > η ic with η ic given by Eq. (2) are unstable. In this frequency regime δE ≈ 0, consistent with ideal MHD. Indeed, we compute ω As,gap /(2π) ≈ 60 kHz at the minimum of the m = 1 continuum at s = 0.24. This frequency, and the computed BAE gap mode frequency of 50 kHz at 2s, is close to the kinetic accumulation frequency ω CAP . By ∂r , the ratio η i /η ic expands as Equation (3) is independent of the ion density profile, and so can be computed for different radial temperature profiles. X-ray crystallography of nearby discharge #4229 measures T i (0)/T e (0) = 0.75 during NBI heating, and we have used this value for #4220. Together with ECE T e data for #4220, this fixes T i on-axis. To account for the unknown core localisation of the ion temperature profile, we have expressed the temperature profile as the sum of profiles T a (s) and T b (s), with T i (s) = T a (s) + T b (s), and r ≈ sa, with a the minor radius. The profile T a (s) is an approximation to the Ohmic temperature profile. A constraint for T a (0) is provided by crystallography during the pre-NBI heating phase of #4229: that is T a (0) = 0.25T e (0). As a plausible estimate for the Ohmic ion temperature profile we have assumed n(s) ∝ T a (s), and inferred T a (s) from the equilibrium pressure profile. Finally, we have modelled the core temperature profile as with α varied to control the core radial localisation of T b (s), and term tanh(10(1 − s)) included to force T b (s) to zero at s = 1. Figure 6(a) shows a plot of different candidate modelled ion temperature profiles on the outboard radial chord, and Fig. 6(b) shows the corresponding η i /η ic values. The candidate BAE mode in Fig. 5(a), whose m = 1 Fourier magnitude is also shown in Fig.  6(b), has a peak at radial position r ≈ 0.08 m with a radial width of 0.2a = 0.1 m. For this mode, η i /η ic < 1 for a broad, Ohmic-like ion temperature profile. For sufficiently high radial temperature gradient and/or sufficiently high ion temperature, the Alfvénic ion temperature gradient driven mode [22] instability threshold will be approached, or possibly even exceeded, in the region where the mode amplitude is large, and so the mode can become unstable, due to a combination of energetic and thermal ion kinetic effects [32].

Conclusions
We have provided first evidence of beta induced Alfvén eigenmode activity in neutral beam heated KSTAR plasmas. The 40 kHz, n = 1 observed mode matches the frequency of the accumulation point of the Alfvén continuum. By using the radial localisation of sawtooth inversion radius, we have been able to identify the radial position of the q = 1 surface, and using this, constrain the equilibrium. A detailed mode analysis reveals the presence of a core localised beta induced Alfvén eigenmode. Finally, a kinetic treatment of the mode marginal stability threshold shows a range of plausible ion temperature profiles for which the mode exctiation threshold is reduced, and the mode can be driven by energetic ions. This suggests that mode existence could be used as a form of inference for temperature profile consistency in the radial interval of the mode, once beam drive  Figure 6. Variation of η i /η ic threshold with different ion temperature profiles. Panel (a) shows possible ion temperature profiles (solid) with core temperature matching crystallography data from discharge #4229, and panel (b) shows the corresponding profile of η i /η ic . In both panels the dashed line corresponds to a possible ohmic ion temperature profile, with τ taken from discharge #4229 prior to NBI heating. In panel (b) the light line is |V 1 | of the BAE in Fig. 5(a). and relevant damping contributions have been computed, thereby extending the tools of MHD spectroscopy. Our suggestion is complementary to recent work by Bertram et al , [16] who suggested mode frequency could be used to infer the temperature at the resonant surface of a BAE mode in the H-1 heliac.
Improvements in diagnosis and reliability of KSTAR plasmas will enable further exploration of new physics, in steady state plasma environments of up to 14MW of NBI heating. A priority is the introduction of kinetic and motional Stark effect constraints to EFIT, which will remove uncertainties in detailed modelling. In future work we hope to consolidate mode frequency variation with pressure and magnetic field strength. Judicious phasing of NBI during current ramp-up and ramp-down, together with new electron cyclotron current drive systems also offer the opportunity to influence the magnetic field configuration through co and counter injection. A wider sample set of modes, together with concurrent crystallography data, would enable a more thorough investigation of marginal stability thresholds and a quantitative study of mode drive and damping. Finally, as the position of the resonances are determined by the q profile and the density profile, similarity experiments with different edge q profiles and density profiles would illuminate the level of continuum damping. For instance, as the q profile flattens different m continuum mode branches are removed, and continuum damping is reduced. Alternately, as the density profile in the edge drops the location of the continuum resonance moves inward. In this case the radial separation between the resonances and gap mode decreases, the coupling of the eigenfunction to the continuum becomes stronger, and hence the continuum damping is increased. [33,34]