Stabilization of Neoclassical Tearing Modes by rf Current

Neoclassical tearing mode (NTM) can degrade plasma confinement or even cause disruptions in existing tokamaks. Stabilization of the / 3 / 2 m n = NTMs by radio frequency (rf) current is investigated by the modified Rutherford equation (MRE) in this paper. In a range of parameters, the required rf current for mode stabilization is obtained, which is linearly proportional to the bootstrap current density for both modulated current drive (MCD) and non-modulated current drive (NMCD), linearly (quadratically) to the radial width of the rf current for MCD (NMCD), and quadratically to the radial deviation of the rf current from the rational surface for both MCD and NMCD.


Introduction
The neoclassical tearing mode (NTM) is expected to form a magnetic island structure which locally flattens the plasma pressure, hence losing the bootstrap current [1] [2] [3] [4].Low mode number NTMs, especially the / 3 / 2 m n = (m and n are the poloidal and toroidal mode numbers)and 2/1 modes, are observed to cause the confinement degradation and/or disruptions, limiting the achievable plasma β value well below the predictions of ideal magnetohydrodynamic (MHD) calculations for positive magnetic shear [4]- [12].
In general, NTMs can be stabilized by compensating the perturbed bootstrap governing the time evolution of the nonlinear island, has been calculated by several authors [4] [15] [16] [18].However, the predictions of the required rf current for mode stabilization are still in question and this is the purpose of this paper.

Island Evolution Equation
The MRE which describes the temporal evolution of the full width of a magnetic islandcan be written as follows [3] in which r is the rational surface and η is the plasma resistivity.0 ′ ∆ is the tearing mode stability index, defined as the logarithmic discontinuity in the radial derivative of the perturbed magnetic flux function ψ .In the vicin- ity of the rational surface the well-known "constant ψ approximation" is uti- lized.The bootstrap current perturbation term can be written as where B θ is the poloidal magnetic field, p the plasma pressure, | s q r r q L q = ≡ ′ the magnetic shear length at the rational surface, q the safety factor, and   is the characteristic island width above which the plasma pressure is flattened across the island.The prime denotes the radial derivative.
Equation ( 1) is obtained through averaging of the parallel projection of the Ohm's law over helical flux surfaces in the vicinity of the island It is convenient to introduce the normalized perturbed flux function [3] [4] [15] −Ω , and F is a general function.
In this way, the driven current term is given by ( ) where cd I is the magnitude of the rf driven current and cd w is the half width of the driven current.With a Gaussian profile for the driven current by rf waves [9] [15] the stabilization efficiency cd η is given by ( ) where is the peak amplitude of the driven current and the square box < ∆ and to 0 otherwise, defining the helical angle of the rf current deposition, where 0 h t ω = , ω is the mode rotation frequency, and h ∆ is the instantaneous wave deposition width along the helical angle.
In this paper, we neglect the modification of the equilibrium plasma current profile by the current drive and the reduced island width, which can modify the 0 ′ ∆ .

Required rf Current for Mode Stabilization
In this paper the inverse aspect ratio    The solid (dashed) curve is for MCD (NMCD).seen between the ratio / cd p I I and the radial deviation x cd there is always an approximate quadratic relationship for both MCD and NMCD.A shift of rf current deposition away from the rational surface greatly decreases the stabilizing efficiency, since the applied driven current is not around island's o-point.
The required driven current for NMCD is much more than that for MCD when 0.03 . The reason is that NMCD deposited around the island separatrix and the island X-point is destabilizing.

Conclusions
The magnetic island width changes with the pressure profile, which in turn affects the island growth.The modification of the equilibrium plasma current profile by the current drive can affect the 0 ′ ∆ .These are not considered in our cal- culations.And the numerical simulation on the required driven current for mode stabilization which consistently calculates the pressure profile and the plasma current density profile will be the future work.

5 )
World Journal of Engineering and Technology for 1 Ω > , and

in Figure 2 .
where a is the plasma minor radius and R the major radius.Results obtained from the MRE utilizing is assumed.The solid curve is the case without the driven current term.The dashed and the dot-dashed curves are for modulated current drive (MCD, 50% on-time) with of / dw dt is zero, i.e., the required driven current for mode stabilization.The solid (dashed) curve is for MCD (NMCD).It is seen the required / cd p I I depends linearly on / b p j j for both MCD and NMCD.World Journal of Engineering and Technology

Figure 1 .
Figure 1.Evaluation of / dw dt from MRE for the / 3 / 2 m n = magnetic island with / 0.173 b p j j = , / 0.41 q L a = at / 0.555 s r a = and / 0.05 cd w a = .0 / s m r ′ ∆ =− is utilized.The solid curve is the case without the driven current.The dashed and dot-dashed curves are for MCD with / 0.017 cd p I I = and for NMCD with

Figure 2
Figure 2. Required / cd p I I for mode stabilization versus / b p j j for / 0.05 cd w a = .The

Figure 4
Figure 4. Required / cd p I I for mode stabilization versus the radial deviation of the rf