(cid:1) suppression of Alfvén cascade modes in the National Spherical Torus Experiment

Alfvén cascade modes have been found in low density, low (cid:1) plasmas on the National Spherical Torus Experiment (cid:3) M. Ono et al. , Nucl. Fusion 40 , 557 (cid:1) 2000 (cid:2)(cid:4) . An extension of the theory of cascade modes which includes the coupling to geodesic acoustic modes (cid:3) Breizman et al. , Phys. Plasmas 12 , 112506 (cid:1) 2005 (cid:2)(cid:4) is shown to imply their absence for typical spherical tokamak ratios of electron thermal to magnetic energy, (cid:1) . A scan in electron (cid:1) conﬁrmed a threshold for suppression of cascade modes in good agreement with theoretical predictions. © 2007 American Institute of Physics . (cid:3) DOI: 10.1063/1.2768038 (cid:4)


I. INTRODUCTION
Alfvén cascade ͑AC͒ modes are a form of Alfvén eigenmode localized to the q min region of reverse shear plasmas. They were discovered in reversed shear discharges on the Joint European Tokamak, [1][2][3][4] the Tokamak Fusion Test Reactor, 5 Alcator C-Mod, 6 and on JT-60U 7 and DIII-D, 8 where they are referred to as reversed-shear Alfvén eigenmodes ͑rsAE͒, and have been seen on many machines since. A principle characteristic of these modes is a slow ͑equilibrium timescale͒ sweep of the mode frequency, typically upward, but downward frequency sweeps have also been seen. 4 The modes provide a useful check on the measurement of the evolution of the current profile in the plasma.
The addition of a motional Stark effect ͑MSE͒ diagnostic 9 to the National Spherical Torus Experiment ͑NSTX͒ ͑Ref. 10͒ has made it possible to verify that NSTX has produced reversed shear discharges during the current ramp phase. Yet, Alfvén cascade ͑AC͒ modes were not observed in these plasmas. An explanation for their absence may be found in a recent theoretical model developed to explain the observed onset frequency of AC modes in JET as a coupling to the geodesic acoustic mode ͑GAM͒, 11 or ␤-induced Alfvén eigenmode ͑BAE͒. 12 The frequency sweeps saturate at the toroidal Alfvén eigenmode ͑TAE͒ frequency, a prediction supported by data from JET. 11 Here, we show that this theory can also explain the absence of AC modes in the relatively high ␤ ͑thermal-to-magnetic energy ratio͒ of spherical tokamak plasmas where the GAM/BAEonset frequency approaches or exceeds the TAE frequency, leaving no frequency band in which the AC modes might exist. 13

II. DESCRIPTION OF EXPERIMENT
In order to test the hypothesis that the high ␤ of typical spherical tokamak plasmas suppressed the AC modes, a very low beta, neutral beam heated regime was explored in NSTX. The operational parameters used for these experi-ments were 0.8 MA of toroidal plasma current and Ϸ4.5 kG toroidal field. The plasmas were heated with 2 MW of deuterium neutral beam injection ͑NBI͒ power at a full energy of 90 keV. The vacuum vessel was well conditioned to minimize deuterium fueling from the protective carbon tiles. The major radius of the plasma magnetic axis was Ϸ1.0 m, and the plasma minor radius was Ϸ0.65 m. NBI heating was started early during the current ramp. In Figs. 1͑b͒ and 1͑c͒, time traces of the plasma current, NBI power, and toroidal beta are shown.
Under these conditions, it was possible to operate with peaked electron density profiles and central electron density as low as Ϸ1.0− 1.2ϫ 10 13 /cm 3 . For the data shown in Fig.  1, the electron density profile was nearly constant between 0.15 s and 0.35 s, with a peak density on axis of about 1.2 ϫ 10 13 /cm 3 . The electron temperature on axis increased from about 0.6 keV to 1.4 keV over this time ͑as well as for the two higher density, higher ␤ shots discussed below͒, and the ion temperature, measured by charge exchange recombination spectroscopy ͑CHERS͒, 14 remained at about half the electron temperature. The plasma current was held constant after 250 ms and the plasma ␤ ͑␤ is the ratio of thermal to magnetic energy͒ remained below Ϸ3% through the course of the discharge. At 0.25 s and at a major radius of R = 1.2 m, the local electron ␤ was Ϸ1.2%. At these very low densities, the frequency at the center of the TAE gap at the magnetic axis is given approximately by f TAE Ϸ V Alfvén / ͑4qR͒Ϸ͑160 kHz/ q͒, where V Alfvén 2 = B 2 /4. The plasma rotation, measured by CHERS, remained low with the measured toroidal rotation frequency mostly Ͻ2 kHz except in a small region near the axis where it reached Ϸ5 kHz.
The frequency sweeps in Fig. 1͑a͒ begin above 50 kHz-80 kHz and saturate in a broad range near the center frequency of the TAE gap, as calculated using the q min deduced from AC modes and the density at the approximate location of q min ͑solid blue curve͒. Multiple modes appear sequen-tially with upward frequency sweeps lasting 20 ms to 50 ms. The toroidal mode numbers, n are indicated in the figure by color, black for n = 1, red for n = 2, green for n = 3, blue for n = 4, and cyan for n = 5. The modes propagate in the cotangential direction, that is, in the same direction as the neutral beam ions and the plasma current. The modes appear with toroidal mode numbers, n in the sequence of ͑2, 2, 3, 2, 3, 4͒, with an n = 1 mode coincident with the second n = 2 mode.
In Fig. 1, the modes appear continuous in time, due to the low time resolution used for the spectrogram. With higher time resolution, i.e., shorter sample time for the fast Fourier transform, the modes are seen as a sequence of short bursts lasting 100 s-200 s, and with the bursts separated by a similar interval as shown in Fig. 2. Occasionally the bursts also show a downward frequency chirp. The last burst visible in Fig. 2 is accompanied by a lower frequency, strongly chirping mode. Coincident with the chirping mode, the neutron rate fell by Ϸ6%. Following this burst, the AC modes are suppressed, suggesting that the strongly chirping mode affected the fast ion population driving the AC modes.
For comparison with the low ␤ discharge of Fig. 1, Fig.  3͑a͒ shows a spectrogram of a second, higher ␤ discharge. Helium gas puffing was used to raise the central density to Ϸ3 ϫ 10 13 /cm 3 by 0.25 s and the local electron ␤ to Ϸ5%. The range of the upward frequency sweeps is reduced and the mode has become continuous in time. For example, the n = 3 mode ͑green͒ appearing at Ϸ0.185 s shows two clear upward frequency sweeps, the first around 0.225 s and the second around 0.280 s. The n =2 ͑red͒, n =4 ͑blue͒, n =5 ͑cyan͒, and n =6 ͑magenta͒ modes show similar behavior. The plasma toroidal rotation frequency is increasing nearly linearly during this time, reaching Ϸ5 kHz at 250 ms at the approximate location of the modes. Doppler-shift corrections to the mode frequency evolution are necessary to interpret the data, and the mode frequency, corrected for the Doppler shift at the approximate mode location of R Ϸ 1.25 m, is shown in Fig. 3͑b͒. The modes are assumed localized near the q min location, which is at a major radius of about 1.25 m in similar plasmas The reflectometer measurements of mode amplitude are consistent with mode amplitude peaking near this radius. However, the modes are clearly global, and as such will likely be affected by the shape of the rotation profile as well.
At even higher density and higher electron ␤ e Ϸ 7% ͑at R = 1.2 m͒, the slow upward sweeps disappear altogether, and only bursting modes identified as TAE are seen. An example just at this threshold is shown in Fig. 4. The n =3 ͑green͒ and n =4 ͑blue͒ TAE appear and disappear as, presumably, the q min passes through and between low order rationals. The frequency sweeps which had previously connected the periods of TAE activity in Fig. 3 are now absent.
The internal amplitude and radial structure of the modes are measured with three fixed-frequency quadrature reflectometers. 15 In Fig. 5͑a͒ it is shown that the amplitude of the n = 3, 4 and n = 5 modes at 0.271 s peaks near the q min FIG. 1. ͑Color͒ ͑a͒ Spectrogram of a Mirnov coil signal for a low density discharge. The red and blue curves indicate the approximate GAM and TAE gap center frequencies at the assumed mode location, respectively. The q min derived from the AC modes is used for the estimate of the TAE gap center frequency. ͑b͒ Plasma current, neutral beam heating power, and ͑c͒ total toroidal beta for NSTX discharge 120103. radius. Here the n = 3 mode is near a minimum of the frequency, the n = 4 mode is sweeping down in frequency ͑plasma frame͒, and the n = 5 mode is near frequency saturation. The localization is strongest for the highest n mode ͑as would be expected for higher poloidal mode numbers͒. The reflectometer data are from the shot shown in Fig. 3 and the q profile is from a higher density shot without AC modes.
The toroidal mode numbers were measured with an array of 12 coils measuring poloidal magnetic field fluctuations. The data were acquired at a 4 MHz sampling rate. An additional array of coils, sampled at 5 MHz, measures the amplitude and relative phase of the toroidal and poloidal magnetic fluctuations, from which the mode polarization is deduced. Here, polarization refers to shear ͑perpendicular to the equilibrium field͒ vs compressional ͑parallel to the equilibrium field͒ magnetic fluctuations. Figure 6͑a͒ shows the polarization of the n = 4 mode magnetic field fluctuations, determined from this array, as an ellipse, with the vertical axis showing poloidal magnetic fluctuation and the horizontal axis showing toroidal fluctuation. The waves qualitatively have a shear polarization, that is, the magnetic fluctuations are transverse to the equilibrium field and the ellipse is nearly ͑within un-certainty͒ degenerate, forming a line perpendicular to the equilibrium field. This is consistent with the expectation that AC modes and TAE have "shear" polarization.
The pitch of the equilibrium field varies across the plasma radius as shown in Fig. 6͑b͒. The radial eigenfunction of AC modes should be dominated by the poloidal component localized near the minimum in q. Assuming that the polarization of the mode magnetic fluctuations, even as measured at the vacuum vessel wall, reflects the polarization of the dominant poloidal harmonic, the pitch of this ellipse would put the magnetic fluctuations perpendicular to the equilibrium magnetic field at a major radius of about 1.18 m, as shown in Fig. 6͑b͒. This is Ϸ5 cm inside of the radius of q min determined from MSE constrained equilibrium fits. It is FIG. 3. ͑Color͒ ͑a͒ Spectrogram of a higher density, but otherwise similar, discharge to that shown in Fig. 1. Contours are color-coded to indicate the toroidal mode number; black, red, green, and blue correspond to n =1, 2, 3, and 4, respectively. ͑b͒ The mode frequencies, corrected for the Doppler shift resulting from plasma rotation measured at the assumed mode location near 1.25 m. The red curve is the interpolated n = 2 GAM frequency evolution, the black is the local GAM frequency from Eq. ͑2͒ and scaled by 0.9 to match the data. ͑The contour range in the spectrogram is chosen to suppress noise, and unfortunately, also suppresses some periods of the n = 2 mode, which are shown in the second panel. ͒   FIG. 4. ͑Color͒ Spectrogram of Mirnov coil from discharge at the threshold beta below which Alfvén cascades are seen. Frequency sweeping is largely absent.

FIG. 5. ͑Color͒
The data from the three quadrature reflectometer channels indicating mode amplitudes peaked near q min are indicated by the red circles ͑n =5͒, blue diamonds ͑n =4͒, and green triangles ͑n =3͒. The q͑R͒ profile determined from the MSE field line pitch measurement in a similar shot is shown by a solid black line. also inside the radius of the peak in mode amplitude suggested by reflectometer measurements, although these measurements have, at present, rather limited radial resolution. It remains to be seen whether theoretical modeling can explain this apparent, albeit small, discrepancy.

III. ANALYSIS OF DATA
The modes with upward frequency sweeps in Figs. 1 and 3 have many of the characteristics of Alfvén cascade ͑AC͒ modes. The mode frequency evolution shown in these figures will next be analyzed in the context of the Alfvén cascade mode dispersion relation and shown to be consistent with the interpretation of these modes as Alfvén cascades.
Cascade modes are found in reverse-shear plasmas, localized near the radial minimum in q ͑1/q is a normalized measure of magnetic field twist͒. The simple dispersion relation for AC modes at low ␤ e is = k ʈ V Alfvén , where k ʈ = ͑m − nq min ͒ / q min R, and unlike TAE, the modes are assumed to have a single dominant poloidal mode number, m. When q min = m / n, k ʈ = 0 and as q min drops, the mode frequency, cascade starts near zero frequency and sweeps upwards. When the q min drops approximately halfway to the next lower rational value for a given n, the mode is believed to convert to a TAE and the mode frequency saturates at the TAE frequency. As the q min continues to fall, the mode may transform back into a cascade mode and the frequency will sweep downwards.
A more complete dispersion relation 11 for higher ␤ e plasmas includes coupling to the GAM or BAE and is given by In this dispersion relation, when q min = m / n, i.e., k ʈ = 0, there is a local minimum in mode frequency, As the temperature and density rise, the TAE frequency drops and the GAM frequency increases. When ␤ is high enough so that min Ϸ TAE ͓ϷV A ͑2qR͒ −1 ͔, the range of the frequency sweep is zero. The evolution of q min can be deduced from the frequency sweep of the modes in Fig. 1, assuming that the modes follow the cascade dispersion relation ͓Eq. ͑1͔͒. First the poloidal mode number corresponding to each toroidal mode number must be determined. The second n = 2 mode appearing concurrently with the n = 1 mode suggest a q min = 2 crossing at about 0.185 s, and these two modes would have ͑m , n͒ = ͑4,2͒ and ͑2,1͒. The assumption that q min is decreasing monotonically then implies the following time sequence for the cascade mode numbers: ͑m , n͒ = ͑5,2͒ , ͑4,2͒ + ͑2,1͒ , ͑5,3͒ , ͑3,2͒ , ͑4,3͒ , ͑5,4͒.
The frequency evolution of the seven distinct upward frequency sweeps in Fig. 1 is used to calculate q min ͑t͒. Equation ͑1͒ is solved for q min , using the ͑m , n͒ from above for each mode, The min term, small for this shot, is calculated from Eq. ͑2͒. In Fig. 7 the q min ͑t͒ for each of the modes seen in Fig. 1 is shown, color-coded as in the spectrograms. Of course, when the frequency saturates, the frequency evolution would no longer be described by this simple dispersion relation, and the q min ͑t͒ calculated by this method also saturates. The mode behavior in Fig. 3͑b͒ resembles the cascade mode behavior seen, e.g., in Fig. 3 of Breizman et al. 11 ͑The continuum damping is thought to be larger during the downward frequency sweeps, thus explaining the usual absence of the downward sweeping portion of the mode evolution.͒ For these data we can identify the times of the rational q min crossings as the times at which the frequency evolution hits local minima and the q min derived by this method are indicated by the black squares in Fig. 8.
Near the frequency minima, the q min ͑t͒ evolution can be calculated from the frequency evolution using Eq. ͑3͒, for the n =2, n = 3, and n = 4 mode frequency evolutions ͑where the assumed poloidal mode number decreases by one at each frequency peak͒. The poloidal mode numbers used for each toroidal mode number are indicated in the figure. For the analysis of these data, the min is found directly from Fig.  3͑b͒. The min is assumed to vary linearly over the time interval between the two minima for each frequency curve ͓e.g., red line in Fig. 3͑b͔͒. This linear approximation to the min evolution is Ϸ0.9 times the value predicted by Eq. ͑2͒ for the n = 2 mode ͓black line in Fig. 3͑b͔͒. The local ͑in time͒ evolution of q min from this method agrees well with the q min rational crossings identified from the frequency minima.
Direct measurements of the q profile evolution were not available for the examples shown in Figs. 1 and 3. However, MSE data starting at Ϸ0.27 s is available for a similar shot with slightly higher density ͑Ϸ16% higher͒, to that shown in Fig. 3. The q profile in this later shot clearly had reversed shear with the outboard q min at R Ϸ 1.2 to 1.25 m. In Fig. 8, from 0.27 s onward, the black curve shows the time evolution of q min deduced from MSE data for this slightly higher density shot. The MSE system presently measures pitch angles at 12 spatial locations approximately equally spaced from about the plasma axis to the edge. The evolution of q min inferred from multiple frequency sweeps consistent with the measured q min evolution, albeit from a different but similar shot, provides a strong argument that these modes are Alfvén cascades.
The threshold ␤ for Alfvén cascade suppression can be estimated by setting the AC minimum frequency equal to the TAE frequency, However, these dispersion relations for the TAE and GAM frequencies are approximate. We can arrive at a better estimate of the ␤ crit by taking into account the empirical correction factors for min and TAE from Fig. 3͑b͒. A scale factor of Ϸ0.9 was determined for min from the data shown in Fig.  3͑b͒, and the TAE is underestimated by a factor of Ϸ1.25, which together yields an empirical correction factor of Ϸ1.4 for the frequency ratio, or when squared Ϸ2 for ␤ crit . We can then estimate from Eq. ͑4͒ the critical ␤ as ␤ crit Ϸ͑20% -25% ͒ / q 2 for the range of ion and electron temperature ratios in these data. The ␤ e / ␤ crit may be estimated, including the ion temperature term, as ␤ e / ␤ crit Ϸ 0.06-0.2 for the data in Fig. 1, ␤ e / ␤ crit Ϸ 0.5-0.7 for the data shown in Fig. 3, and ␤ e / ␤ crit Ϸ 1 for the shot with MSE data where AC were not seen ͑Fig. 4͒.

IV. SUMMARY
In summary, the absence of Alfvén cascades in reversedshear NSTX discharges was particularly striking given the otherwise rich spectrum of fast-ion driven instabilities in NSTX. The results of the experiment reported here demonstrate that this usual absence of AC modes in spherical tokamaks is a direct consequence of their typically higher beta and is not attributable to a peculiar evolution of the q profile FIG. 7. ͑Color͒ The evolution of q min ͑t͒ as deduced from the Alfvén cascades shown in Fig. 1. Dashed line is fit to q min ͑t͒ evolution used to calculate TAE frequency.
FIG. 8. ͑Color͒ The evolution of q min ͑t͒ as deduced from the spectrogram in Fig. 3͑b͒. The black squares indicate q min deduced from frequency minima of n = 2, 3, and 4 modes, the red points are q min ͑t͒ deduced from n =2 frequency curve, green from n = 3, and blue from n = 4. The solid black curve shows the time evolution of q min ͑at R Ϸ 1.25 m͒ from equilibrium reconstructions using MSE data for the shot analyzed in Fig. 4. Also indicated are the poloidal mode numbers and time range used in the analysis. or a deficiency in Alfvén cascade theory. In fact, the clear observation of Alfvén cascades in relatively low ␤ NSTX discharges is strong confirmation of the validity of the established theory. The q min evolution deduced from the Alfvén cascades is in reasonable agreement with the q min inferred from MSE measurements in a similar, but slightly higher density shot. The modes are further shown to be localized to the q min region, as predicted by theory. These observations suggest that AC modes are likely to not be a problem during the normal operation of spherical tokamaks, nor, however, will they provide a measurement of the q min evolution.