Effects of diamagnetic drift on nonlinear interaction between multi-helicity neoclassical tearing modes

A numerical study of the diamagnetic drift effect on the nonlinear interaction between multi-helicity neoclassical tearing modes(NTMs) is carried out using a set of four-field equations including two-fluid effects.The results show that,in contrast to the single-fluid case,5/3 NTM cannot be completely suppressed by 3/2 NTM with diamagnetic drift flow.Both modes exhibit oscillation and coexist in the saturated phase.To better understand the effect of the diamagnetic drift flow on multiple-helicity NTMs,the influence of typical relevant parameters is investigated.It is found that the average saturated magnetic island width increases with increasing bootstrap current fraction f_(b) but decreases with the ion skin depth δ.In addition,as the ratio of parallel to perpendicular transport coefficients χ_(‖)/χ_(⊥) increases,the average saturated magnetic island widths of the 3/2 and 5/3 NTMs increase.The underlying mechanisms behind these observations are discussed in detail.

Effects of counter-current driven by electron cyclotron waves on neoclassical tearing mode suppression

Through theoretical analysis,we construct a physical model that includes the influence of counter-external driven current opposite to the plasma current direction in the neoclassical tearing mode(NTM).The equation is used with this model to obtain the modified Rutherford equation with co-current and counter-current contributions.Consistent with the reported experimental results,numerical simulations have shown that the localized counter external current can only partially suppress NTM when it is far from the resonant magnetic surface.Under some circumstances,the Ohkawa mechanism dominated current drive(OKCD)by electron cyclotron waves can concurrently create both co-current and counter-current.In this instance,the minimal electron cyclotron wave power that suppresses a particular NTM was calculated by the Rutherford equation.The result is marginally less than when taking co-current alone into consideration.As a result,to suppress NTM using OKCD,one only needs to align the co-current with a greater OKCD peak well with the resonant magnetic surface.The effect of its lower counter-current does not need to be considered because the location of the counter-current deviates greatly from the resonant magnetic surface.

Effects of plasma radiation on the nonlinear evolution of neo-classical tearing modes in tokamak plasmas with reversed magnetic shear

Effects of plasma radiation on the nonlinear evolution of neo-classical double tearing modes(NDTMs)in tokamak plasmas with reversed magnetic shear are numerically investigated based on a set of reduced magnetohydrodynamics(MHD)equations.Cases with different separations △_(rs)=|r_(s2)-r_(s1)|between the two same rational surfaces are considered.In the small △_(rs)cases,the plasma radiation destabilizes the NDTMs and makes the kinetic energy still grow gradually in the late nonlinear phase.Moreover,the NDTM harmonics with high mode numbers reach a high level in the presence of plasma radiation,forming a broad spectrum of MHD perturbations that induces a radially broadened region of MHD turbulence.As a result,the profiles of safety factors also enter a nonlinear oscillation phase.In the intermediate △_(rs)case,the plasma radiation can advance the explosive burst of kinetic energy that results from the fast driven reconnection between the two rational surfaces,because it can further promote the destabilizing effects of bootstrap current perturbation on the magnetic island near the outer rational surfaces.In the large △_(rs)case,through destabilizing the outer islands significantly,the plasma radiation can even induce the explosive burst in the reversed magnetic shear configuration where the burst cannot be induced in the absence of plasma radiation.

Validation of the current and pressure coupling schemes with nonlinear simulations of TAE and analysis on the linear stability of tearing mode in the presence of energetic particles

Both current and pressure coupling schemes have been adopted in the hybrid kinetic–magnetohydrodynamic code CLT-K recently.Numerical equivalences between these two coupling schemes are strictly verified under different approximations.First,when considering only the perturbed distribution function of energetic particles(EPs),the equivalence can be proved analytically.Second,when both the variations of the magnetic field and the EP distribution function are included,the current and pressure coupling schemes numerically produce the same result in the nonlinear simulations.On this basis,the influences of co-/counter-passing and trapped EPs on the linear stabilities of the m/n=2/1 tearing mode(TM)have been investigated(where m and n represent the poloidal and toroidal mode numbers,respectively).The results of scanningβh of EPs show that the co-passing and trapped EPs are found to stabilize the TM,while the counter-passing EPs tend to destabilize the TM.The behind(de)stabilization mechanisms of the TM by EPs are carefully analyzed.Furthermore,after exceeding critical EP betas,the same branch of the high-frequency mode is excited by co-/counterpassing and trapped EPs,which is identified as the m/n=2/1 energetic particle mode.

Sawtooth-like oscillations and steady states caused by the m/n=2/1 double tearing mode

The sawtooth-like oscillations resulting from the m/n=2/1 double tearing mode(DTM)are numerically investigated through the three-dimensional,toroidal,nonlinear resistive-MHD code(CLT).We find that the nonlinear evolution of the m/n=2/1 DTM can lead to sawtooth-like oscillations,which are similar to those driven by the kink mode.The perpendicular thermal conductivity and the external heating rate can significantly alter the behaviors of the DTM driven sawtooth-like oscillations.With a high perpendicular thermal conductivity,the system quickly evolves into a steady state with m/n=2/1 magnetic islands and helical flow.However,with a low perpendicular thermal conductivity,the system tends to exhibit sawtooth-like oscillations.With a sufficiently high or low heating rate,the system exhibits sawtooth-like oscillations,while with an intermediate heating rate,the system quickly evolves into a steady state.At the steady state,there exist the non-axisymmetric magnetic field and strong radial flow,and both are with helicity of m/n=2/1.Like the steady state with m/n=1/1 radial flow,which is beneficial for preventing the helium ash accumulation in the core,the steady state with m/n=2/1 radial flow might also be a good candidate for the advanced steady state operations in future fusion reactors.We also find that the behaviors of the sawtooth-like oscillations are almost independent of tokamak geometry,which implies that the steady state with saturated m/n=2/1 islands might exist in different tokamaks.

Basic features of the multiscale interaction between tearing modes and slab ion-temperature-gradient modes

Nonlinear interaction between tearing modes(TM) and slab ion-temperature-gradient(ITG) modes is numerically investigated by using a Landau fluid model. It is observed that the energy spectra with respect to wavenumbers become broader during the transition phase from the ITG-dominated stage to TM-dominated stage. Accompanied with the fast growth of the magnetic island, the frequency of TM/ITG with long/short wavelength fluctuations in the electron/ion diamagnetic direction decreases/increases respectively. The decrease of TM frequency is identified to result from the effect of the profile flattening in the vicinity of the magnetic island, while the increase of the frequencies of ITG fluctuations is due to the eigenmode transition of ITG induced by the large scale zonal flow and zonal current related to TM. Roles of zonal current induced by the ITG fluctuations in the instability of TM are also analyzed. Finally, the electromagnetic transport features in the vicinity of the magnetic island are discussed.

Multi-scale interaction between tearing modes and micro-turbulence in the HL-2A plasmas

The influence of m/n=2/1(m and n are poloidal and toroidal mode numbers)tearing modes on plasma perpendicular flows and micro-fluctuations has been investigated in HL-2A neutral beam injection heated L-mode plasmas.It is found that the local perpendicular rotation velocity and turbulence energy are modulated by the alternation between the island X-point and O-point of the naturally rotating tearing modes.Cross-correlation analysis indicates that the modulation of density fluctuations by the tearing mode is not only limited to the island region,but also occurs in the edge region near the last closed flux surface.The turbulence exhibits distinct spectral characteristics inside and outside the island region.In addition,it is observed that the particle flux near the strike point is also significantly impacted by the tearing modes.The experimental evidence reveals that there are strong core-edge interactions between the core tearing modes and the edge transport.

Application of Galerkin spectral method for tearing mode instability

Magnetic reconnection and tearing mode instability play a critical role in many physical processes.The application of Galerkin spectral method for tearing mode instability in two-dimensional geometry is investigated in this paper.A resistive magnetohydrodynamic code is developed,by the Galerkin spectral method both in the periodic and aperiodic directions.Spectral schemes are provided for global modes and local modes.Mode structures,resistivity scaling,convergence and stability of tearing modes are discussed.The effectiveness of the code is demonstrated,and the computational results are compared with the results using Galerkin spectral method only in the periodic direction.The numerical results show that the code using Galerkin spectral method individually allows larger time step in global and local modes simulations,and has better convergence in global modes simulations.

Neoclassical tearing mode stabilization by electron cyclotron current drive for HL-2M tokamak

Investigation of neoclassical tearing mode and its suppression by electron cyclotron current drive(ECCD)has been carried out in HL-2 M tokamak.The current driving capability of the electron cyclotron wave is evaluated.It is found that the deposition location can be effectively controlled by changing the poloidal angle.The validation of electron cyclotron wave heating and current driving has been demonstrated for the upper launcher port.We show that 3.0 MW and2.5 MW modulated ECCD can completely stabilize(2,1)and(3,2)NTMs,respectively.The non-modulated ECCD,radial misalignment as well as current profile broadening have deleterious effect on the NTM stabilization.The time required for suppression of(3,2)mode is shorter than that required for the suppression of(2,1)mode.Moreover,the time needed for complete stabilization at different initial island width has been quantitatively presented and analyzed.

Linear tearing modes in an electron-positron plasma

The general dispersion of tearing modes due to the effects of electron inertia and resistivity in pair plasmas is derived analytically,and is discussed in two cases：Δ＇〉〉 1 andΔ＇〈〈 1,where Δ’ is the instability criterion of the tearing mode.It is found that the conditions under which either resistivity or electron inertia dominates depend strongly on the limit of Δ＇considered.

Nonlinear evolution and secondary island formation of the double tearing mode in a hybrid simulation

Double tearing modes(DTMs),induced by double current sheet configurations or two neighboring rational surfaces with the same safety factor in tokamaks,are widely observed in solar,space,and fusion plasmas.In this paper,the evolution of DTMs without a guide field is investigated numerically using a hybrid model(electron fluid+ion PIC).The overall evolution processes of DTMs are qualitatively consistent with previous works using other models.The particle dynamics during the evolution of DTMs is analyzed in detail.Behaviors of ions and electrons present different characteristics around the reconnection region which gives rise to Hall effects producing the out-of-plane quadrupole magnetic field.In the explosive reconnection process with interactions between two DTMs islands,the asymmetric drive and the thin current layer feature lead to the emergence of secondary magnetic islands which develop with the late evolution of the DTMs.

Experimental study of double tearing mode on EAST tokamak

The double tearing mode(DTM)in a high βN(βN>1.5)discharge with internal transport barrier on EAST was investigated.A 15 kHz tearing mode(TM)(m≥3,n=3)appears at outer q=2 surface at first,which is stable in the highβN phase.Then a 2 kHz TM(m=2,n=1)occurs at inner q=2 surface.Soon after,high βN collapsed with the crash of ELM,and DTM formed during the collapse of βN.The positions of the two islands of the DTM are consistent with the q=2 surface.The temperature fluctuations are the strongest between the two magnetic islands.A statistical analysis of high βN discharges operating with the reversed magnetic shear configuration in the EAST 2015-2018 campaign revealed the existence of the DTM in many discharges.During the DTM phase,all βN are reduced by 10%-30% within 0.1 s.However,there are two different βN behaviors thereafter-with and without βN recovery.Studying the physical mechanism of βN recovery during the DTM phase will benefit steady-state operation with reversed shear configurations in the future.

Effects of m=0 Harmonics on Quasi-Linear Stage of m=1 Double Tearing Mode

Effects of tearing modes are inves shows that the effect of the m = 0 harmonics on the early quasi-linear stage of m = 1 double tigated. The numerical calculation with the harmonics m = 0 included the m = 0 harmonics on the mode is negligible in the linear stage. As the mode begins to grow nonlinearly, both the current and flow profiles are pinched due to the m = 0 harmonics. To make a comparison we also carry out the calculation with m = 0 harmonics turned off. The profiles of the total current, poloidal magnetic field, and poloidal shear velocity in the cases with or without the m = 0 harmonics are compared and discussed. In addition, the formation of a poloidal velocity shear is found and its mechanism is investigated.

Effects of plasma radiation on the nonlinear evolution of neo-classical tearing modes in tokamak plasmas

The effects of plasma radiation on the nonlinear evolution of neo-classical tearing modes are investigated based on a set of reduced magnetohydrodynamic equations.It is found that the radiation can reduce the pressure near the rational surface.During the nonlinear evolution,the magnitude of perturbed bootstrap current is drastically enhanced in the presence of the radiation.Besides,the radiation can increase the growth rate of the magnetic islands by diminishing the pressure,such that the magnetic islands do not saturate compared with that without radiation.On the other hand,with the increase of the ratio of parallel to perpendicular transport coefficientχ‖/χ⊥,the reduction of pressure can further increase the growth rate of magnetic islands in the presence of plasma radiation.Finally,the mechanisms of the destabilizing effects driven by the radiation are discussed in detail as well.

A brief review:effects of resonant magnetic perturbation on classical and neoclassical tearing modes in tokamaks

This paper reviews the effects of resonant magnetic perturbation(RMP)on classical tearing modes(TMs)and neoclassical tearing modes(NTMs)from the theory,experimental discovery and numerical results with a focus on four major aspects:(i)mode mitigation,where the TM/NTM is totally suppressed or partly mitigated by the use of RMP;(ii)mode penetration,which means a linearly stable TM/NTM triggered by the externally applied RMP;(iii)mode locking,namely an existing rotating magnetic island braked and finally stopped by the RMP;(iv)mode unlocking,as the name suggests,it is the reverse of the mode locking process.The key mechanism and physical picture of above phenomena are revealed and summarized.

Influence of toroidal rotation on the tearing mode in tokamak plasmas

The stabilizing mechanism of toroidal rotation on the tearing mode is studied using the 3 D toroidal resistive magnetohydrodynamic code M3 D.It is found that the dominating mechanism,either the centrifugal effect or the Coriolis effect, depends on the specific pressure β and rotation frequency Ω.On the premise that Ω is sufficiently large, when β is greater than a critical value,the effect of the centrifugal force is dominant, and the stabilizing effect mainly comes from the modification of equilibrium induced by the centrifugal force;when β is less than a critical value,the stabilizing effect from the Coriolis force overcomes that from the centrifugal force.However,if Ω is small, then the effect of equilibrium modification due to the centrifugal force is not significant even if β is large.Finally, the results showed that toroidal rotation shear enhances the stabilizing effect.

On theoretical research for nonlinear tearing mode

The analytical approaches for nonlinear tearing mode have been reviewed.It is shown that Rutherford＇s model has triggered numerous studies on the nonlinear tearing mode.Its physical picture is clear meanwhile its mathematical method is ingenious but still puzzling to understand.It is trying to find how the ＇nonlinear behavior＇ resulted from the linear equation by a nonlinear transform.It is indicated that Li＇s model for the tearing mode includes the linear growth,Rutherford＇s behavior and the new behavior.It was found that the quasilinear modification of magnetic field provided a new damping mechanism for nonlinear growth.The new behavior w~ t1/2 becomes dominant if the mode is weakly unstable.It is shown that many analytical methods have been developed to calculate the criterion parameter D＇ of the tearing mode.Li＇s instability criterion can cover the previous results in the limit cases.

Suppression of Neoclassical Tearing Modes Towards Stationary High-Beta Plasmas in JT-60U

Results from stabilization of neoclassical tearing modes （NTMs） in JT-60U are described. NTM stabilization and confinement improvement have been demonstrated by employing a real-time NTM stabilization system, where the identification of the location of an NTM and the optimization of the injection angle of the electron cyclotron wave are performed in real time. Also, a high-beta plasma with the normalized beta of 3 has been sustained by suppressing NTM by applying the electron cyclotron current drive （ECCD） before the onset （＇preemptive ECCD＇）. In addition, a simulation code for analysis of the NTM evolution has been developed by combining the modified Rutherford equation with the transport code TOPICS. It is found that the simulation well reproduces the NTM behavior in JT-60U. The simulation also shows that the ECCD width is also important for NTM stabilization, and that the EC wave power for complete stabilization can be reduced by narrowing the ECCD profile.

Effect of Hyper-Resistivity on Nonlinear Tearing Modes

We analytically investigate nonlinear tearing modes with the anomalous electron viscosity or,as it is normally called,hyper-resistivity.In contrast to the flux average method used by previous work,we employ the standard singular perturbation technique and a quasilinear method to obtain the time evolution equation of tearing modes.The result that the magnetic flux grows with time in a scaling as t^（2/3）demonstrates that nonlinear tearing modes with the hyper-resistivity effect alone have a weaker dependence on time than that of the corresponding resistive case.

Linear analysis of plasma pressure-driven mode in reversed shear cylindrical tokamak plasmas

The linear behavior of the dominant unstable mode(m=2,n=1)and its high order harmonics(m=2n,n≥2)are numerically investigated in a reversed magnetic shear cylindrical plasma with two q=2 rational surfaces on the basis of the non-reduced magnetohydrodynamics(MHD)equations.The results show that with low beta(beta is defined as the ratio of plasma pressure to magnetic field pressure),the dominant mode is a classical double tearing mode(DTM).However,when the beta is sufficiently large,the mode is driven mainly by plasma pressure.In such a case,both the linear growth rate and mode structures are strongly affected by pressure,while almost independent of the resistivity.This means that the dominant mode undergoes a transition from DTM to pressure-driven mode with the increase of pressure,which is consistent with the experimental result in ASDEX Upgrade.The simulations also show that the distance between two rational surfaces has an important influence on the pressure needed in mode transition.The larger the distance between two rational surfaces,the larger the pressure for driving the mode transition is.Motivated by the phenomena that the high-m modes may dominate over low-m modes at small inter-resonance distance,the high-m modes with different pressures and q profiles are studied too.