Saturated helical mode in EAST high β hybrid plasmas

A long-lived helical saturated helical mode (LLM) with a mode frequency of 5 kHz and higher harmonics has been observed in EAST hybrid plasmas heated by NBI. The basic plasma parameters of EAST shot 63730 are presented in figure 2. The plasma discharges operate in an upper single-null (USN) configuration. The plasma current is 500 kA. The toroidal magnetic field at the magnetic axis is about 2.24 T. Line average electron density is about 3.1 * 10¹⁹ m⁻³. Ion temperature is about 2 keV. The EAST NBI system [22] has two beam lines located at Ports A and F which are toroidally separated by 112.5°. In LLM experiments, only Port A beam lines were activated. 5 MW of cocurrent direction NBI was overlapped on 2 MW of low hybrid wave injection. Plasma enters H-mode phase after NBI.

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In EAST experiments, a new method was developed to improve the accuracy of core plasma current density and q profile reconstruction on EAST [23–25]. This method is based on extracting the internal Faraday rotation measurements provided by the POINT diagnostic. Detailed comparisons of magnetic surfaces and the q profile reconstructed using external magnetic data against those using magnetic and POINT data are performed. Such a method has produced good results in reconstructing the q profile in EAST sawtooth experiments [24]. The same method is applied to reconstruct the q profiles during the LLM. As shown in figure 2(e), the q profiles during LLM exhibit a monotonic feature with a wide region of low magnetic shear in the core. In such a monotonic q profile, q₀, the q value at the magnetic axis is the same as q_min, the minimum q value. As shown in figure 4(d), q₀ is kept above unity during LLM.

The typical plasma density and temperature profile during the LLM in EAST shot 63730 is presented in figure 3. The electron density profile in the core region was measured with an HCN interferometer. The edge electron density was measured with a microwave reflectometry system. The complete electron density profile is shown in figure 3(a). The ion temperature profile was measured with CXRS as shown in figure 3(b).

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A 6 kHz oscillation with strong harmonics is observed from both core SX emission signal and boundary Mirnov signal. LLM is driven unstable as q₀ (q₀, the safety factor at the magnetic axis) gradually evolves towards unity and β_p increases significantly as shown in figure 4(d). The mode started at 3.1 s. It saturated at 3.28 s. As the mode linearly grows, the boundary magnetic perturbations are suppressed as shown in figure 4(b). The mode was terminated at 3.55 s when a minor crash occurred and q₀ was elevated well above unity. The mode frequency is close to the central bulk plasma rotation frequency as shown in figure 4(a).

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The presence of LLM would cause deleterious effects. As shown in figure 5(d), following the onset of the LLM, β_p gradually decreases, indicating a gradual loss of plasma confinement. In figures 5(b) and (c), a decrease in the neutron rate is observed despite NBI heating power being kept constant and core ion temperature being on the rise. It is an indication of fast ion losses. Figure 5(e) shows that the LLM could cause minor disruption. The minor disruption event happened at about 3.5132 s–3.5133 s.

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The toroidal rotation profile is measured by a CXRS system. As shown in figure 2(b), the external momentum introduced by NBI is added from 3.1 s and kept constant during the plasma discharge. The evolution of the toroidal rotation profile is thus caused only by internal sources. In figure 6, the toroidal rotation profiles at various time points from EAST discharge 63730 is presented. With the long-lived helical mode present, which is from 3.1 s to 3.55 s, the rotation profile inside the q  =  2 surface is flattened. The toroidal rotation profile without the presence of the LLM at 3.62 s is a monotonic one with the maximum toroidal rotation speed in the core region. Such a toroidal rotation profile evolution indicated that the LLM would cause severe toroidal rotation damping in the core. Previous research applied neoclassical toroidal viscosity (NTV) theory to calculate the breaking torque exerted by internal modes [26, 27]. In the EAST case, the flattened part of the rotation profile extends up to almost half of the minor radius, which provides excellent conditions to test NTV theory. This will be worked on in the future.

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In the laboratory frame, the MHD mode frequency is determined by plasma rotation frequency and diamagnetic frequency as ${{f}_{{\rm mhd}}}={{f}_{{\rm rotation}}}-f_{{\rm e}}^{*}$. In the EAST LLM case, the electron diamagnetic frequency $f_{{\rm e}}^{*}$ is estimated as $f_{{\rm e}}^{*}=\frac{{{T}_{{\rm e}}}}{2\pi B{{p}_{{\rm e}}}{{r}_{{\rm s}}}}\frac{{\rm d}p}{{\rm d}r}\sim \frac{{{T}_{{\rm e}}}({\rm keV})}{2\pi {{a}^{2}}({\rm m}){{B}_{\phi}}({\rm T})}\sim 0.65\,{\rm kHz}$. The electron diamagnetic frequency is small compared with the MHD mode frequency in the saturated phase, which is about 6 khz. So the MHD mode frequency is determined by plasma rotation frequency f _rotation. f _rotation is defined as ${{f}_{{\rm rotation}}}=\frac{{{V}_{{\rm toroidal}}}}{2\pi ({{R}_{0}}+r)}$. Considering the toroidal rotation speed V_toroidal is about 70 km s⁻¹ (typical value of V_toroidal in EAST NBI plasmas is up to 100 km s⁻¹) in the saturated phase, the major radius R₀ is 1.85 m, and radial position r inside the q  =  2 surface varies from 0 to 0.2 m, the mode frequency is dominantly determined by toroidal rotation velocity. A flattened toroidal rotation profile brings further implications. MHD instabilities that take place inside the q  =  2 surface, namely 1/1 and 2/1 modes, would rotate at the same angular frequency. The phase of 1/1 and 2/1 modes would be locked, i.e. toroidal mode coupling. It is safe to conclude that a flattened profile could provide conditions for toroidal mode coupling. This observation is consistent with previous works on EAST showing that 2/1 neoclassic tearing mode (NTM) is triggered by 1/1 mode via mode coupling [28, 29].

Figure 7 shows the toroidal mode numbers as found from the Fourier transform of the boundary magnetic probe measurements for EAST shot 63730. It is clear there are n  =  1 and n  =  2 harmonics present during the LLM evolution.

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A combined application of singular value decomposition (SVD) method and tomography of multi-array SXR emissions is conducted to identify the poloidal mode structure and poloidal rotation direction of LLM [30]. In figure 8, reconstructed local SXR emission intensity perturbations show a typical m  =  1 mode structure. The m  =  1 mode structure rotates along the ion-diamagnetic direction (clockwise poloidal direction). In figures 9(a) and (b), the relative amplitude of 1/1 and 2/2 perturbations are plotted in arbitrary units respectively. In figure 9(c), the amplitude of 1/1  +  2/2 perturbation is acquired by linearly adding up the amplitudes in figures 9(a) and (b). Comparing the relative amplitude of local SXR perturbation emissions, the 1/1 mode is clearly the main component.

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A novel method was developed on ASDEX Upgrade to derive robust estimates of the core displacement of 1/1 internal mode [31] from SXR tomography. Here, the same method is used to acquire the core displacement of LLM in EAST shot 63730. The displacement of the m  =  1/n  =  1 mode, ξ_1/1, is evaluated as the average radius of the area (figure 10(a)) enclosed by the trajectories of SXR maximum in one oscillation period (figure 10(b)). This method can overcome two major shortfalls of the naive method, which uses the distance between the magnetic axis and the SXR maximum: (1) the uncertainty in the magnetic axis attained from an equilibrium reconstruction, and (2) the errors resulting from severe noncircular plasma shape. The trajectories of the SXR maximum (referred to as a 'hot core') during one LLM period show that the 'hot cores' circulate in the ion diamagnetic direction in the R–Z plane and that the displacement ξ_1/1 is about 2.6 cm in the saturated phase. Figure 10(c) shows the radial displacement ξ with respect to time. The mode growth rate γ is estimated as 1/ξ * dξ/dt in figure 10(d).

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This novel method of extracting plasma core displacement is based on the assumption that the trajectory of the SXR maximum is a reliable proxy of the trajectory of the displaced core. This assumption is valid upon a few conditions [31]. The mode frequency f  should change slowly. The mode growth rate γ should change slowly. The growth rate γ should be small compared to mode frequency f. That is to say, the method is most accurate when the mode is in saturated phase. If the mode is fast growing or decaying, though inaccurate, as long as the trajectories of SXR maximum could still enclose an area like the one shown in figure 10(a), a value for mode displacement could be estimated. As figure 10(d) shows, the absolute value of growth rate γ is rather large compared to mode frequency from 3.25 to 3.26 s, the corresponding displacement is thus inaccurate. The same would apply to radial displacement ξ in figure 10(c) at 3.36–3.37 s. Note that this method is based on SXR tomography. The limited resolution of SXR tomography would cause errors at 3.1–3.25 s and 3.37–3.55, when 1/1 displacement is small the trajectories of SXR maximum could not enclose an area like the one shown in figure 10(a). These values for mode displacement ξ are set to be zero, though in reality, they are not (see figure 4(c), the central chord SXR perturbation).

Given all these, the reliable estimation of the amplitude of the radial displacement ξ should be found at 3.26–3.36 s. The minimum reliable amplitude of the radial displacement ξ should be found at about 3.26 s. It is safer to choose a time point well after 3.62 s, say 3.27 s. The corresponding displacement is 1.5 cm as shown in figure 10(c).

A new combined method for investigation of MHD activity is developed by Igochine et al [32]. This new method develops a MHD interpretation code (MHD-IC) [33] and makes it possible to compare the mode structure predicted by theoretical models directly with experimental data to accurately determine the mode eigenfunction. The same approach is applied here to interpret experimental SXR data at the saturated phase of the LLM. The MHD-IC code would require an initial guess for the form of radial displacement ξ. This initial guess took a form of parameterized theoretical models. Then the MHD-IC will fit the theoretical mode structure to experimental data. In this reconstruction, the initial guess is provided by a parameterized internal kink mode model. Theoretically, a safety factor profile as shown in figure 2(e) would be more likely to produce a quasi-interchange mode other than an internal kink mode. However, due to limited special resolution, the experimental SXR data used in the reconstruction would not be able to provide fine details to distinguish the quasi-interchange mode structure from internal kink mode structure. Seen also in previous work, an ideal internal kink model was applied to calculate the mode eigenstructure in MAST LLM plasmas [8].

The reconstructed results are presented in figure 11. The radial displacement of 1/1, 2/2 and 3/3 components in figure 11(c) are obtained by applying the best fitted eigenfuntions in figures 11(a) and (b). The maximum ξ_1/1 is about 2.6 cm, which is consistent with the results in figures 10(a) and (c).

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3D resistive nonlinear simulations using the M3D code [34] have been applied to reproduce the LLM. Resistive MHD equations including viscosity are detailed in section 3 of [35]. Plasma rotation is not taken into account in the model. In this simulation, the experimental plasma density profile (figure 3(a)), temperature profile (figure 3(b)) and safety factor q profile (figure 2(e)) of EAST shot 63730 are employed to provide a realistic tokamak configuration. Note that there is no Thompson scattering diagnostic support in EAST shot 63730 for electron temperature profile measurements. The electron temperature profile is assumed to be identical to the ion temperature profile shown in figure 3(b). Figure 12 shows the mesh in one arbitrary poloidal cross section. The simulation is performed in fixed boundary conditions constrained by the last closed flux surface of EAST shot 63730 at 3.3 s.

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M3D is an initial value code. A perturbation is added into the background equilibrium plasma. If the equilibrium is unstable, the perturbation will evolve in space and time until a spatial structure is produced corresponding to the unstable mode eigenfunction. To initiate the simulation, the toroidal mode numbers are set to be 1–8 and the poloidal mode numbers are set to be 1–4. LLM is reproduced when the value of viscosity µ is set to be 10⁻⁵ and the resistivity η is set to be 10⁻⁷. Simulation results are summarized in figure 13. Figure 13(a) shows the total kinetic energy versus time. Time is denoted as ${{\tau}_{{\rm A}}}=\frac{\sqrt{3}}{{{s}_{1}}}\tau _{{\rm A}}^{*}$, where $\tau _{{\rm A}}^{*}$ is the Alfvén time, s₁ is the magnetic shear at the q  =  1 surface. Figure 13(b) shows the average amplitude of magnetic perturbations (denoted as δφ) as a function of toroidal mode number at initial phase (τ_A  =  0.246), linear phase (τ_A  =  245.69) and saturation phase (τ_A  =  368.09).

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The simulated stream function U [35] and pressure perturbations confirmed a 1/1 helical mode structure (figures 14(a) and (b)). However, the magnetic perturbation of the mode has very strong m  =  2 components that are comparable to the n  =  1 component (figure 14(c)). This result is consistent with the experimental observation that a dominating m  =  2 component is shown in SVD analysis of boundary Mirnov coil signals. Note that 1/1 kink mode always has 2/2 components in a torus configuration (e.g. toroidal effects), which agrees with previous nonlinear simulations [34].

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In figure 15(a), the spectrogram of central chord SXR signal shows that 1/1, 2/2 and 3/3 mode are driven unstable successively in EAST shot 63730. It is possible to estimate the relative amplitude of the different toroidal mode number modes during the long-lived helical mode evolution from the SXR fluctuation data. By taking a fast Fourier transformation of SXR data from a small time window, the amplitude of the fluctuation can be calculated as a function of frequency. The frequency spectrum peaked at n  =  1 and n  =  2 frequency, respectively. So the relative amplitude of n  =  2 and n  =  1 modes can be estimated at this small time window. By moving the small time window along the time axis, the amplitude ratio of the n  =  2 mode to the n  =  1 mode against time is acquired (figure 15(b)). It is clear that as q_min (figure 4(d)) gradually approaches unity, the n  =  2 amplitude increases substantially with respect to n  =  1 amplitude. Conversely, as q_min is driven further above unity, n  =  2 amplitude decreases with respect to n  =  1 amplitude.

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The simulated magnetic perturbations agree very well with similar evolution. In figure 13(b), as LLM grows, the n  =  2 amplitude increases immensely with respect to the n  =  1 amplitude. When LLM saturates at τ_A  =  368.09, the n  =  2 amplitude becomes the dominant component. The corresponding 2/2 harmonic component of the magnetic perturbation at saturated phase is presented in figure 15(c). The simulation also shows that when the 2/2 component become dominant in the saturated phase, the initial weak shear safety factor profile in the core becomes completely flat (figure 15(d)) This change of safety factor profile in the core corroborates a self-regulating magnetic flux pumping mechanism that was considered responsible for maintaining stationary hybrid discharges.