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Synergistic effect of electron cyclotron current drive and poloidal shear flow on the tearing mode

The separate and synergistic effects of both the electron cyclotron current drive (ECCD) and poloidal shear flow on the tearing mode are investigated numerically by using two-dimensional compressible magnetohydrodynamics equations in slab geometry. For the misaligned ECCD, effects of radial and poloidal misalignments have been compared emphatically. It is found that the suppression effect of ECCD is weakened with the increase of malposed ratio and it is more sensitive to the radial misalignment. The stability effect of shear flow is not positively related to the flow shear; the effects of starting moment of ECCD and shear flow are similar but not identical. The synergistic stability effect of ECCD and shear flow is more effective than ECCD or shear flow acts alone on the tearing mode without considering the “flip” instability. Furthermore, the combinatorial stability effect is more obvious when ECCD has a radial misalignment as a result of the continuous poloidal shift of magnetic island.The separate and synergistic effects of both the electron cyclotron current drive (ECCD) and poloidal shear flow on the tearing mode are investigated numerically by using two-dimensional compressible magnetohydrodynamics equations in slab geometry. For the misaligned ECCD, effects of radial and poloidal misalignments have been compared emphatically. It is found that the suppression effect of ECCD is weakened with the increase of malposed ratio and it is more sensitive to the radial misalignment. The stability effect of shear flow is not positively related to the flow shear; the effects of starting moment of ECCD and shear flow are similar but not identical. The synergistic stability effect of ECCD and shear flow is more effective than ECCD or shear flow acts alone on the tearing mode without considering the “flip” instability. Furthermore, the combinatorial stability effect is more obvious when ECCD has a radial misalignment as a result of the continuous poloidal shift of magnetic island.


I. INTRODUCTION
Tearing mode instability is one of the most significant magnetohydrodynamics (MHD) instabilities associated with energy conversion and plasma transport process in various magnetic confinement device including Tokamak-type. [1][2][3][4][5][6][7] As the growing of magnetic islands, hot particles are lost more easily from the machine so that disruptions can be triggered. Therefore, the suppression or control of tearing mode instability is critical to plasma confinement. 8 In order to suppress tearing mode instability, several strategies have been conducted, such as the use of active feedback system with external coils, 9,10 localized heating, 11,12 current driven. [13][14][15][16][17][18] Among which, radio frequency (RF) wave current drive is the most widely used method in the main magnetic confinement fusion devices including EAST, 19 ASDEX-Upgrade, 20 JT-60U, 21 and DIII-D, 22 etc. The electron cyclotron current drive (ECCD) induced precisely at the X-point or O-point of the magnetic island, is certified to be one of the most appropriate approaches for suppressing the island growth. Over the last few decades, various theoretical modes including the basic nonlinear Rutherford model or its variants and the frame of MHD equations, have been widely used for investigating the influence of driven current on the tearing mode instability. [23][24][25][26][27][28][29][30][31][32][33][34] The modified MHD equations considering the non-inductive driven current has also been applied to numerically study the mitigating effect of driven current on the development of tearing mode. [28][29][30][31][32][33][34] For example, Borgogn, and Comisso et al. 32 researched the magnetic island evolution under the action of ECCD based on the reduced resistive MHD plasma model. They found that the island can be completely annihilated when the driven current is applied to a small magnetic island, but it is followed by a spatial phase shift of the island due to the "flip" instability. However, the Kelvin-Helmholtz instability is aroused when the current-drive injection in a large nonlinear island.
Plasma equilibrium flow and flow shear are widespread features in tokamaks. They can be driven by external momentum input, like neutral beam injection (NBI), wave heating and current driven processes, [35][36][37][38] or even produced intrinsically without momentum sources (see Refs. [39][40][41]. The poloidal flows have been observed in some tokamak experiments, especially in the internal transport barrier region 42,43 or near the plasma edge region (see Ref. 44). A result in DIII-D is that the actual poloidal rotation is about 10% of the total apparent velocity. 45 In recent years, a number of past studies have examined the effect of shear flows (toroidal and poloidal) on tearing modes. [46][47][48][49][50][51][52][53][54][55][56][57][58][59][60] It is widely accepted that the plasma flow and the flow shear have significant influence on the control of tearing mode stability. For example, Chandra et al. found that differential flow has a stabilizing effect on the nonlinear evolution of tearing mode and also further enhances the stabilizing influence of the pressure-curvature, whereas velocity shear has a destabilizing effect. 46 They also found that the axial flow has a destabilizing influence on the tearing mode by using the CUTIE code and the poloidal flow has a stabilizing effect in both the linear and nonlinear stages of tearing mode evolution by using the NEAR code. 47 Wei and Wang numerically investigated the roles of poloidal rotation in stabilizing the 1/1 kink-tearing mode and exciting its high-order harmonic tearing modes by using a reduced MHD model in cylindrical geometry. 48 Chen et al. 51 and Ofman et al. 52 studied the influence of sheared equilibrium flow in a simple slab geometry. They found that the resistive tearing mode is sensitive to the flow shear. The growth rate of tearing mode (γ) is not in positive correlation with the shear parameter (κ), and the growth rate γ reaches a peak value for shear parameter κ = 0.73 of shear flow with "tanh" profile which is also used in this paper. Ming et al. 54 found that the effect of the poloidal flow on the tearing mode is comparable to the effect of the toroidal flow. There are experimental evidences that flow shear has a favorable influence on both magnetic island formation and saturated island size. [55][56][57][58][59][60] For example, experiments on helical devices LHD and TJ-II observed the stabilized influence of poloidal flows on tearing modes. 59 The toroidal rotation behavior and momentum transport have been examined in NBI heated plasmas with and without electron cyclotron resonance heating (ECRH) and current drive (ECCD) in ASDEX Upgrade (AUG), 61,62 DIII-D, 63 JT-60U 64 and TCV (Ref. 65). The results in the JT-60U tokamak are that ECRH increases the toroidal momentum diffusivity and the convection velocity, and drives the co-direction (co) intrinsic rotation inside the EC deposition radius and the counter-direction (ctr) intrinsic rotation outside the EC deposition radius. 64 Experiments in the TCV tokamak show that high power central ECH and ECCD produce significant direct modification of the plasma rotation profile, as well as affecting the equilibrium current density profile. In a regime of unsteady rotation, these effects contribute to the onset of neoclassical tearing instabilities. 65 There also must be an effect of ECH and ECCD on the poloidal rotation and its effect on the tearing mode, though the poloidal rotation is less than the toroidal rotation.
In this paper, the separate and synergistic effects of ECCD and shear flow on the tearing mode are studied by using numerical simulations with a set of compressible MHD equations. We first present a model of reduced resistive MHD and the initial condition and some parameters during the simulation, and then report on the numerical results of the single and synergistic effects of ECCD and shear flow.

II. SIMULATION MODEL
Magnetized plasma can be described by the continuity Eq. (1), motion Eq. (2), together with the adiabatic state Eq. (3) and Maxwell Eqs. (4)-(6): 33 The localized current driven by RF injection can be introduced into the model by altering Ohm's law to where J is the total plasma current density and J cd is the localized driven current density. The adopted J cd with Gauss distribution 32 is shown in Eq. (8), where J d and δ cd are the amplitude and the radial full e −1 width of J cd , and the localized current driven by RF wave can be expressed as I cd = ∬ J cd (x, y)dxdy. Eq. (9) and Eq. (10) can be obtained by combination of Eqs. (2), (5), (6) and Eq. (7), The two-dimensional (2D) compressible MHD equations adopted in this article comprise of Eqs. (1), (3), (9), (10). We consider a slab model of a tokamak plasma with equilibrium magnetic field B = B T ez + ∇ψ × ez, where B T and ψ(x, y) are the guide field and magnetic flux, respectively. In consideration of 2D simulation, we assume ∂ ∂z = 0 for all variables. The parameters ν, γ and η are coefficients of viscosity, adiabatic coefficient and plasma resistivity, respectively. In the simulation, length L, time t, plasma mass density ρ, plasma pressure p, plasma velocity u, magnetic field strength B, magnetic flux ψ, are nondimensionalized by L 0 , τA = L 0 /UA, ρ 0 , p 0 , The poloidal pressure ratio expresses as β = 2μ 0 p 0 /B 2 0 . The initial magnetic field distribution and shear flow are adopted as 52 B 0y (x) = Bm tanh(κmx), where Bm = 1.0, κm = 6.7 is the magnetic shear parameter, V 0 is the shear flow amplitude in units of UA, and κv is the flow shear parameter. The initial magnetic flux is ψ 0 = − ∬ B 0y (x, y)dxdy, and the initial current density is J 0z = −∇ 2 ψ 0 (thus Ip = ∬ J 0z dxdy = − ∬ ∇ 2 ψ 0 dxdy). The numerical simulation is performed in the domain of −lx ≤ x ≤ lx (lx = 2.0) and 0 ≤ y ≤ 2ly (ly = 2.0), and calculations are carried out with a Nx × Ny grid of 801×201 points. Other parameters are η = 5.0 × 10 −5 , γ = 5/3, and ν = 10 −4 .

III. NUMERICAL RESULTS
The analysis of Ref. 24 indicates that ECCD has the greatest effectiveness when it is deposited on the O-point and the ratio of island width and its radial full e −1 width w/δ cd ∼ 1.4. Thus, in the simulation, the full e −1 width of ECCD δ cd = 0.2, and the ECCD is switched on at Ton = 100 when the island width w = 0.28, except cases in Figure 3. To be more general, we take I cd /Ip < 15% (referred Ref. 30

Effect of misaligned ECCD
The suppressing efficiency is best for a driven current precisely localized on the O-point. However, it might be experimentally difficult to carry out for such a perfect target, even if there is the theoretical method of solving misalignment issues precisely. Therefore, it is really necessary to investigate the impact of misalignment on island evolution. In Figures 1(a) and 1(b), the effect of radial (the x direction) and poloidal (the y direction) misalignment of ECCD on the maximum island width before the appearance of flip phenomenon are displayed, respectively. In the simulations, the coordinate of Opoint (xO, yO) are (0.00, 2.00), and the located coordinates of ECCD is too large. This is consistent with the results in Ref. 34. What is more, the stabilizing efficiency is more sensitive to the misalignment in radial direction. As shown in Figure 1(c), the ECCD loses its suppressing effect and destabilizes the tearing mode in the case of Δx/lx = 16.0%, but there is still some stabilization effect under the circumstance of Δy/ly = 70.0%.

Effect of flow shear
There are divers impacts on tearing mode for the shear flow with disparate parameters. The flow shear is one of the most important matters. The dependence of the maximum island widths on the ratio of flow shear to the magnetic shear is shown in Figure 2. The equilibrium magnetic field is fixed (and thus the magnetic shear) and only the flow shear is changed. It can be seen that the tearing mode is increased for small flow shear (κv/κm < 0.27) and the tearing mode becomes stable at larger flow shear (κv/κm ≥ 0.27). In addition, there is a peak value at κv/κm = 0.15. The result is consistent with the theory in slab geometry of Refs. 49-52 and the result of Ref. 53 in which the 3D resistive MHD code DEBS is used.

Effect of initial working time
The time evolution of the magnetic island width with ECCD (I cd = 12.57%Ip, loaded at Ton = 100, 150, 200 and 500 and on the O-point) but without shear flow (V 0 = 0), are shown in Figure 3(a) and that of shear flow (V 0 = 0.27, κv = 3.0, beginning at T be = 0, 120, 150, 200, 280 and 350) but without ECCD (I cd = 0) are shown in Figure 3(b). One can see that the stabilization efficiency is much more remarkable when the current deposition starts at the early phase of the growth of magnetic island, which is consistent with the results of Refs. 15 and 33. The impact of shear flow beginning to work at different times is very similar to that of ECCD, but it is not identical, as shown in Figure 3 T be = 0, 120, and 150, and the maximum island widths are smaller than that under the conditions of T be = 280 and 350. When the shear flow gets to work at the decreasing phase of the growth rate of magnetic island (T be > 280), the growth of magnetic island stops immediately. That is to say, the restraining effect of shear flow, starting to work at the linear or quasi-linear phase (t < 150), is preferable to the inhibiting effect of shear flow beginning to work at the nonlinear phase (t > 150). Moreover, the suppressing effect is almost the same when the shear flow sets to work at the linear or quasilinear phase. Because there is little difference between the value of the out-plane flow in poloidal direction, and the values of out-plane flow are all smaller than the case without shear flow, as exhibited in Figure 4(a). The width of magnetic island will quickly approach saturation at the decreasing phase of the growth rate of magnetic island. The shear flow starting to work at this period (T be > 280), can lead to a global oscillation and distortion of magnetic island (see Figure 5). Though this phenomenon brings about a bigger value of out-plane flow in poloidal direction [as shown in Figure 4(b)], the magnetic island stops growing soon.
B. The synergistic effect of ECCD and shear flow  Additionally, the absolute value of slope gets smaller and smaller when I cd ≥ 12.57%Ip because the impact of shear flow becomes relatively weak and the effect of ECCD on shear flow is displayed. Figure 7 shows the misalignment effect of ECCD on magnetic island when shear flow exists simultaneously. In Figures 7(a) and 7(b), the purple solid line marked by circle represents the time evolution of magnetic island without both ECCD and shear flow, the red dash-dot line marked by hexagram stands for the temporal evolution of island width only with ECCD (I cd = 12.57%Ip, precisely deposited on the O-point, and Ton = 100), the bold black dash line states the evolution of island only with shear flow (V 0 = 0.3, κv = 3.0, T be = 0), and the black dash line marked by square is the result of the synergistic effect of accurately localized ECCD and shear flow. In the condition of existing radial misalignment, the variation of stabilized efficiency is not positively related to the radial malposed ratio Δx/lx. When 0 < Δx/lx ≤ 15.0%, the synergistic suppressing effect of ECCD and shear flow attenuates as the increasing of Δx/lx, but it is better than that of only ECCD or shear flow. The suppressing effect becomes further weak when 15.0% < Δx/lx ≤ 22.5%, but it gets a little better while 22.5% < Δx/lx < 37.5%. Especially, when Δx/lx ≥ 37.5%, the lines of magnetic island versus time almost coincide with the curve of island with shear flow alone, that is to say, the existence of ECCD with so large radial misalignment no longer has any significance. These can be seen from Figure 7(a). Figure 7(b) gives the evolution of magnetic island width for ECCD with poloidal malposition. The stabilizing effect decreases as the increasing of Δy/ly. The impact of shear flow becomes very weak and even disappeared when Δy/ly ≥ 70.0%, and it is almost the same with the consequence of no shear flow. What needs to be particularly pointed out is that the misalignment effect of radial malposed ECCD with shear flow is very different from that without shear flow by comparing Figure 7(a) with Figures 1(a) and 1(c). The status of destabilizing effect does not appear because there is a continuously poloidal shift of magnetic island before the emergence of the "flip" effect, which can be seen from Figure 8.

IV. SUMMARY
In this work, the separate and synergistic effects of ECCD and shear flow on the tearing mode are studied without considering the "flip" instability. When there is a small misalignment, the stabilized efficiency of ECCD becomes weaker as the malposed ratio becomes bigger, but when misalignment is too large, the ECCD may strengthen the tearing mode instability. In addition, it is more sensitive to the misalignment in radial direction. The saturated island width is not linearly related to the flow shear and exists a peak value. There is a threshold for flow shear. When the flow shear is bigger than the threshold, shear flow stabilizes tearing mode, otherwise not. The optimal starting moments of ECCD and shear flow are both at the linear or quasi-linear phase, but the difference is that, the suppressing effect of shear flow starting to work during this phase is almost the same. When ECCD and shear flow synergistically impact on the tearing mode, the stability effect is better than ECCD or shear flow existing individually, particularly for the radially malposed ECCD because of the continuously poloidal shift of magnetic island. Furthermore, the ECCD (or shear flow) may be out of work, if the radial (or poloidal) malposed ratio is too large. Generally speaking, the precisely localized ECCD and shear flow (both with suitable parameters) acting together on the tearing mode can provide a good constraint effect and high efficiency.