Observation of Odd Toroidal Alfvén Eigenmodes
Physical Review Letters
Experimental evidence is presented for the existence of the theoretically predicted odd toroidicity induced Alfvén eigenmode (TAE) from the simultaneous appearance of odd and even TAEs in a normal shear discharge of the joint European torus. The modes are observed in low central magnetic shear plasmas created by injecting lower hybrid current drive. A fast ion population was created by applying ion cyclotron heating at the high-field side to excite the TAEs. The odd TAEs were identified from
... identified from their frequency, mode number, and timing relative to the even TAEs. At the time of the theoretical discovery of toroidicity induced Alfvén eigenmodes (TAE) in the low magnetic shear region of fusion plasmas, predictions were made in ideal MHD for two modes with the same toroidal mode number n in the TAE gap [1-3]. The two modes are formed by the coupling of two poloidal harmonics, m and m 1. The even mode, which resides at the bottom end of the TAE gap, is formed by the coupling of the poloidal harmonics with the same sign, whereas the odd mode, which resides at the top end of the TAE gap, has opposite signs between its two poloidal components as can be seen in Fig. 1 . Because of this difference the even TAE has a ballooning mode structure and the odd TAE has an antiballooning structure. These two core localized TAEs can exist when the following condition is fulfilled: s 2 < < s (with s r=qdq=dr the magnetic shear, r=R the inverse aspect ratio, and R the tokamak major radius) . When > s a spectrum of multiple TAEs differing in the number of radial nodes was found to exist in one gap  . The low shear TAEs can exist only when the normalized pressure gradient, ÿ2Rq 2 =B 2 p 0 (with q the magnetic safety factor, B the magnetic field strength, and p 0 dp=dr the radial derivative of the pressure) is lower than a critical value. For the even mode this critical value is given by E c 3 2s 2 and for the odd mode O c 3 ÿ 2s 2 . The odd mode exists only when the shear is sufficiently low and its critical is lower than the one for the even mode, which indicates that the odd mode ceases to exist before the even mode disappears if the central pressure gradient increases. This suggests that the even mode is more robust than the odd mode. Experimentally so far, only the even mode has been observed unambiguously in large tokamaks    , usually during sawtooth stabilization experiments with ion cyclotron range of frequency heating (ICRH). In those ICRH experiments an extended region of low shear is formed up to half of the plasma minor radius where the even TAEs reside. Until now there has been no unambiguous identification of the odd TAE in tokamak experiments despite the early prediction of its existence. Experimental observation of those predicted odd TAEs would be a fundamental confirmation of TAE theory. In this Letter we present compelling experimental evidence for the existence of the odd TAEs from the coexistence of even and odd TAEs in a joint European torus (JET) discharge. We further address the question why the odd TAEs are observed so rarely in tokamaks. In Fig. 2 a spectrogram of the magnetic fluctuations as measured with a Mirnov coil at the plasma outer midplane is shown where a number of modes with n decreasing in time from 12 to 4 show up between 190 and 210 kHz. These modes chirp down in frequency. Another set of modes with n decreasing in time from 8 to 4 and chirping up in frequency when time progresses is (c) (b) (a) JET pulse 50235 at 64.7 s n=5 odd TAE even TAE m=4 m=5 m=4 m=5 odd TAE even TAE 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 0.0 1.0 -1.0 0.0 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ψ (ω/ω 0 ) 2 ξ(ψ) ξ(ψ) FIG. 1. NOVA-K solution for an n 5 even (a) and odd (b) TAE and their locations in the Alfvén continuum gap (c) normalized to ! 0 B=R 0 p with 0 the central mass density. The modes are located at 0:1, with s 0:272 and 0:131. The dotted curves in (a) and (b) are poloidal components other than m 4 and 5.