Enhancement of ECCD by the current condensation effect for stabilizing large magnetic islands caused by neoclassical tearing modes in tokamak plasmas

The radio frequency current condensation effect reported in Reiman and Fisch (2018 Phys. Rev. Lett. 121 225001) is modeled in the nonlinear resistive magnetohydrodynamic code. A series of numerical investigations have been performed to investigate the enhancement of electron cyclotron current drive (ECCD) by the current condensation effect during the control of neoclassical tearing mode (NTM) in tokamak plasmas. In the numerical model, both the parallel transport and the perpendicular transport of electron temperature are considered. The EC driven current and driven perturbed electron temperature can nonlinearly evolve within the given magnetic configuration and eventually reach saturation states. The input power threshold of ECCD and the fold bifurcation phenomenon are numerically verified via nonlinear simulations. The numerical results show good agreements with the analytical results. Moreover, spatial distributions of EC current for the two solutions at different condensed level are displayed. The control effectiveness of ECCD for large NTM islands has been evaluated while considering the current condensation effect. While taking into account current condensation effect, for a sufficiently large input power, a larger island can be more effectively stabilized than a smaller one, which suggests a reassessment of the previous idea that the ECCD should always be turned on as early as possible. The potential physics mechanism behind the ECCD control have all been discussed in detail. Furthermore, the condensation effect is found to have favorable effects on the radial misalignment of ECCD. In the consideration of the situation for extremely localized control needs, a highly peaked heating profile is adopted to verify that the fold bifurcation phenomenon still exists and the current condensation effect can still take effect in this extreme condition.


Introduction
In order to achieve self-sustained burning in ITER and future large-scale devices, good confinement will be essential.These devices will operate at low collisionality and at sufficiently high pressure to drive substantial bootstrap current.The bootstrap current will need to be particularly large in devices that operate in steady-state [1].The large bootstrap current poses a threat in the form of neoclassical tearing modes (NTMs).The resulting breaking of the equilibrium magnetic topology [2], and the formation of deleterious magnetic islands within the core region of plasmas is highly undesirable [3][4][5][6].The NTM, a very dangerous magnetohydrodynamic (MHD) instability, is usually driven due to the missing local bootstrap current after the pressure is flattened around the O-points of islands [7][8][9].Large NTM islands can completely destroy the well-nested equilibrium flux surfaces and even lead to major disruptions [10][11][12].Major disruptions of plasmas, marked as a considerable threat, can badly damage the future large-scale devices and cause economic loss.A statistical study in JET show that NTM is the most detrimental physics reason that is responsible for plasma disruptions [13].
With the aim of the mitigation and/or active control of NTM for disruption avoidance, different sorts of techniques have been developed [14].For instance, multi-mode threedimensional MHD spectroscopy is developed and applied in DIII-D, EAST and KSTAR to real time evaluate plasma stability for the purpose of predicting plasma disruptions [15,16].Externally applied resonant magnetic perturbation (RMP) [17][18][19][20] and electron cyclotron current drive (ECCD) [21] are developed for the suppression and active control of NTMs.Fisch first suggested that continuous toroidal current may be driven by injecting radio frequency (RF) waves into plasmas [22].The RF driven current is proposed to be used for the stabilization of magnetic islands for the first time by Reiman [23].As one of the RF techniques, the ECCD is usually regarded as an effective method for fully stabilizing NTMs via directly compensating missing bootstrap current.Due to its good localized properties, ECCD can be very flexible for a variety of control strategies.It has been achieved in many tokamaks all over the world that large magnetic islands caused by NTMs are completely stabilized by means of ECCD, such as in ASDEX Upgrade [24], in DIII-D [25,26] and in JT-60U [27].To investigate the potential mechanisms behind the ECCD control of NTMs, extensive numerical studies have been implemented via nonlinear resistive MHD simulations [28][29][30].It is found that the main stabilizing effect of ECCD is twofold including the equilibrium component (m/n = 0/0, with m and n being poloidal and toroidal mode numbers, respectively.)and perturbed components [31][32][33].Given all the advantages, the upper electron cyclotron (EC) launchers in ITER have been designed primarily for the control of NTMs and are expected to be used routinely for this purpose in high performance plasmas [34][35][36][37].
Although ECCD is proven to be effective to stabilize NTMs both in experiments and in simulations, off-normal events that can cause extremely large islands is still challenging to control in future large-scale devices such as ITER.The plasma does not disrupt in normal operation.However, the normal plasma can be affected by operational factors or/and instabilities such as excursions in density, and then will become a pre-disruptive plasma.The abrupt events that will lead to predisruptive plasma are so-called off-normal events.Fortunately, Reiman and Fisch recently found a RF current condensation effect that can nonlinearly enhance the stabilizing efficiency of ECCD so that the extremely large islands are expected to be effectively controlled [38].The sensitivity of the current drive and power deposition to small changes in the electron temperature can lead to an effect that largely concentrates the current density around the O-point of the magnetic island.Thus, the efficiency of the stabilization can be greatly improved.This is called a 'current condensation' effect.At present, the local electron acceleration is assumed to be unaffected within islands.Nonetheless, the local power deposition is sensitive to the temperature perturbation.The existence of magnetic islands can evidently modify the behavior of temperature perturbation.As a matter of fact, the previous work discussed the sensitivity of the deposition to perturbation of the temperature profile [23], but the nonlinear enhancement of the perturbed temperature was not recognized, which is a key part of the condensation effect.It is estimated in theory that the effect on the local deposition is considerable when the temperature perturbation exceeds a certain threshold.In TEXTOR tokamak, 20% fractional temperature perturbations inside islands have been observed in EC resonance heating experiments [39], in which the critical threshold is expected to be encountered.The sensitivity of the current drive to the temperature perturbation can lead to a current condensation effect that largely enhance the stabilizing efficiency of ECCD.Therefore, a larger island can be completely suppressed by a given input power of ECCD.
In view of the above elucidation, it is necessary to consider the aforementioned current condensation effect in nonlinear simulations and investigate the enhancement of the current condensation effect on the ECCD stabilization of large NTM islands in tokamak plasmas.A series of papers addressing various physics issues associated with RF condensation have been published since the original paper [40][41][42][43][44][45][46].The work described here is the first to use a nonlinear resistive MHD code in the modeling of the current condensation effect, the MHD@Dalian (MD) Code [47][48][49][50][51].In comparison with previous work, this numerical study has fewer limitations.Thus, in this work, the parallel transport of electron temperature is included, which is assumed as an instantaneous process in previous studies.The EC driven current and driven perturbed electron temperature are nonlinearly evolving according to specific magnetic configurations until the saturation stage, which are dealt with as a problem of stationary solutions in previous studies.Moreover, the direct ECCD control effectiveness of large NTM islands has been evaluated between the cases with and without current condensation effect.The analytical results reported in [38] are numerically verified via nonlinear simulations using MD Code.Numerical results of ECCD control of NTMs show that, when current condensation effect is considered, for a sufficiently large input power, a larger island can be more effectively stabilized than a smaller one, which suggests a reassessment of the concept that the ECCD should always be turned on as early as possible [50].The potential physics mechanism of ECCD control have all been discussed in detail.In addition, the condensation effect with an extremely peaked heating profile is also investigated in this work, which has received only limited study.

Model for the NTM
The nonlinear evolution of the NTMs can be described via a group of reduced MHD equations [52][53][54] in cylindrical geometry (r, θ, z).The bootstrap current and EC driven current are subtracted from the total plasma current and coupled into the model via modified Ohm's law.The normalized modeling equations are displayed as follows where ψ , ϕ and p represent the magnetic flux, streaming function, and plasma pressure, respectively.j = −∇ 2 ⊥ ψ and u = ∇ 2 ⊥ ϕ denote the plasma current density and vorticity along the axial direction, respectively.The bootstrap current density is charicterized as where f (r, β) = ´a 0 j b0 rdr/ ´a 0 j z0 rdr is a function of radius r and plasma β measuring the fraction of bootstrap current, with ε = a/R 0 being the inverse aspect-ratio and B θ being the poloidal magnetic field.The EC driven current density j d nonlinearly evolves along the direction parallel to magnetic field lines.The evolving behaviors of EC driven current are described through a group of equations that will be introduced in the following subsection.The radial coordinate r is normalized by the plasma minor radius a.Time t and velocity V are measured in units of Alfvén time τ A = √ µ 0 ρa/B 0 and Alfvén velocity V A = B 0 / √ µ 0 ρ, respectively.S A = τ η /τ A and R e = τ υ /τ A are the magnetic Reynolds number and Reynolds number, respectively, where τ η = a 2 µ 0 /η and τ υ = a 2 /υ are resistive diffusion time and viscosity diffusion time, respectively.χ // and χ ⊥ denote the parallel and perpendicular thermal transport coefficients, respectively, which are normalized to a 2 /τ A .The source terms E z0 = S −1 A (j 0 − j b0 ) and S 0 = −χ ⊥ ∇ 2 ⊥ p 0 in equations ( 1) and ( 3) are adopted to balance the diffusions of initial profile of Ohm current and pressure, respectively.j 0 , j b0 and p 0 represent the total equilibrium plasma current, equilibrium bootstrap current and equilibrium pressure, respectively.The Poission bracket is defined as [f, g] = ẑ • ∇f × ∇g.
Each variable f (r, θ, z, t) in equations ( 1)-( 3) can be written in the form f = f 0 + f (r, θ, z, t) with f 0 and f being the timeindependent initial profile and the time-dependent perturbation, respectively.By applying the periodic boundary conditions in the poloidal and axial directions, the perturbed fields can be Fourier-transformed into with R 0 being the major radius of the tokamak.Given the initial profiles, equations ( 1)-( 3) can be solved by the initial value code: MD code.The MD code has been repeatedly benchmarked with the codes employed in [52,53].

Model for EC resonance heating
To model the current condensation effect inside an island, it is necessary to describe the evolving behavior of perturbed electron temperature resulting from electron cyclotron resonance heating (ECRH).It is assumed that ECRH operation generates a heating source T h within the island, and the resulting perturbations in electron temperature can transport, according to given magnetic topology, along magnetic field lines and traverse magnetic field lines.The perturbed electron temperature can be represented as follow where T e represent the perturbed electron temperature caused by ECRH.χ f// and χ f ⊥ denote the parallel and perpendicular thermal transport coefficients of electron temperature, respectively, which are normalized to.1/ν f is slowing-down time of the fast electrons.T h is the heating source term adopted here as Gaussian distributions in both radial direction and helical angle direction as follow where T h0 is the magnitude of the source, in proportion to the input power of EC waves.Considering the heating efficiency is related to island width w i , T h0 is simply set in linear relation with island width w i [38].w 0 = v 0 /v T0 , with v 0 being the electron speed at the location in velocity space of greatest power deposition and v T0 being the equilibrium electron thermal velocity, measures the strength of condensation effect.This quantity can have a range of values, depending on the location and angle at which the EC waves are launched.For efficient ECCD, w 2 0 can approach 10 [55].T e0 is the electron temperature around the O-points of islands without ECRH.r h0 and ξ h0 are the deposition region center of radial direction and helical angle direction, respectively.∆r h and ∆ξ h represent the deposition region width of radial direction and helical angle direction, respectively.

Model for ECCD
According to Fisch-Boozer effect [56], while injecting EC waves into plasmas, a perturbation in velocity space can be induced near the vicinity of the resonant parallel velocity.The perturbation can be characterized by a bulge at high perpendicular velocity of electrons side and a hole at low side.Due to the particle collision, the perturbation in velocity space will gradually fade away.Assuming that the collision pertains Krooktype, the collision frequency attenuates with increasing velocity in a relation of v −3 .Therefore, the velocity space hole at low velocity side is filled in more quickly than the bulge at high velocity side decays.Gross effect eventually generates a net current.A closure relation is obtained originated from above process, modeling the evolving behaviors of EC driven current j d [57].Here, the EC driven current is coupled into the reduced MHD model via the modified Ohm's law.The nonlinear evolution of the EC driven current can be described as the following equations [50,57] where j d1 and j d2 represent the perturbed current density driven by EC wave through the velocity space hole filling in and the bulge decaying, respectively.j d is the net EC driven current density.ν 1 and ν 2 denote the collision frequencies during the velocity space hole filling in and the bulge decaying, respectively.V pres represents the velocity of parallel resonant electrons.Collision frequencies ν 1 (ν 2 ) and velocity V pres are measured in units of Alfvén time τ A = √ µ 0 ρa/B 0 and Alfvén velocity V A = B 0 / √ µ 0 ρ, respectively.The source term S j in the equations ( 7) and ( 8) with opposite signs is the balance between two current density perturbations driven in opposite directions.And the form of S j is adopted with Gaussian distributions in both radial direction and helical angle direction as follow , (10) where S j0 is the magnitude of the source.w rf is corresponding to the resonant velocity producing the maximum current.For high current drive efficiency, w 0 ≈ w rf , with w 0 mainly in the parallel direction [38].For simplicity, in this work we would assume w 0 = w rf .r d0 and ξ d0 are the deposition region center of radial direction and helical angle direction, respectively.∆r d and ∆ξ d represent the deposition region width of radial direction and helical angle direction, respectively.

Numerical verification of current condensation effect
To begin with, a sufficiently large magnetic island is required to form an initial magnetic topology.Considering that m/n = 2/1 is the most dangerous helicity, initial safety factor and pressure profiles are designed to obtain an unstable m/n = 2/1 NTM with bootstrap current fraction f b = 0.3.The parameters of a middle-sized tokamak such as HL-2A [6] have been used in this work.Major radius and minor radius are R 0 = 1.65 m and a = 0.41 m, respectively.The equilibrium magnetic field at the magnetic axis is assumed to be B 0 = 1.2T.Plasma current is about 400 kA.The equilibrium electron density and electron temperature are n e0 = 6 × 10 19 m −3 and T e0 = 5 keV.The total power of EC wave is 5 MW.The threshold for bifurcation is approximately given by w 2 0 Te Te0 ≈ 1.For top launch ECCD, w 2 0 can be as large as 10.This gives a threshold of about 10% for the temperature perturbation in the island.To determine an approximate threshold power, we solve a 1D diffusion equation in an island region: χ ⊥ n e0 d 2 Te dx 2 = −P i , where P i is the absorbed power density in the island.This gives approximately T e = w 2 i Pi 8χ ⊥ ne0 .The total power deposited in the island is approximately P ≈ 4π 2 aR 0 w i P i , giving T e ≈ wiP 32π 2 aR0χ ⊥ ne0 .Here, we take the transport coefficient χ ⊥ = 0.2 m 2 s −1 , T e = 500 eV and w i = 0.2a.We get the threshold is about 2.5 MW.
Normalized typical parameters for the simulation are set as ε = a/R 0 = 0.25, S −1 A = 10 −6 , R −1 e = 10 −7 , χ // = 10 and χ ⊥ = 10 −7 .After the NTM islands are saturated, the righthand terms of equation ( 1) are set to zero so that the structure of islands is fixed from this moment.On the basis of this equilibrium configuration, the ECCD and ECRH are turned on at this point.To get a simple baseline for easy comparison in the following radial misalignment researches, the ECCD and ECRH are aiming at the initial q = 2 rational surface first.Typical parameters are set as  After turning on ECCD and ECRH, the EC driven current and perturbed electron temperature will grow and evolve inside m/n = 2/1 islands.Figure 1(a) shows the temporal evolution of total deposited EC current under different conditions.j 1 and j 2 represent different magnitudes of EC current corresponding to that in figure 1(b).It is found that, regardless of current condensation effect (w 2 0 = w 2 rf = 0), the EC current will soon saturate in a short period of time.Taking into account current condensation effect, the saturated EC current become larger than that without condensation for the same magnitude of ECCD and ECRH sources.Increasing EC sources' magnitudes (generally, EC wave input power P 0 ∝ S j0 ∝ T h0 ), the input power threshold reported in [38] is encountered.Beyond the threshold, instead of saturation, the total deposited EC current will grow to infinity.(Incorporating additional physics, such as depletion of the wave energy, will cause saturation at a finite level.There is then a discontinuous jump in the saturated value at the threshold.)By scanning EC sources' magnitudes, relative magnitude of contribution of the condensation effect to the saturated EC current R j = ´a 0 rdr ¸dξ j d( w 2 0 =w 2 rf =10) ´a 0 rdr ¸dξ j d( w 2 0 =w 2 rf =0) − 1 can be obtained as shown in figure 1(b).It is noted that the relative magnitude R j here is similar to that reported in [38].Due to the fact that the EC current is nonlinearly evolving along the magnetic field lines in this work, the steady deposition of EC current will self-consistently present an island shape.Thus, directly integrating the EC current in the total space can be used for simply evaluating the stabilization efficiency of ECCD.It is found the numerical results show good agreements with the analytical results reported in [38].
Figure 2 versus ECCD magnitude, showing the fold bifurcation reported in [38].It is noted that, inside magnetic islands, the electron temperature profile is flattened so that the electron temperature near the island separatrix T s is the same as that at O-points of islands.Here, the lower solution can be easily found by naturally scanning EC sources' magnitudes.In order to find the higher solution, a relatively large electron temperature perturbation should be given at the beginning of the simulation.By repeatedly attempting for different initial electron temperature perturbations, the higher solution can be found under an appropriate perturbation.It is noted that the higher solution is physically unstable.For an initial perturbation that lies just below the solution, the temporal evolution of U 0 should gradually decrease until the lower root is reached.
While for an initial perturbation that lies just above the solution, the temporal evolution of U 0 should increase to infinity.(Again, incorporating additional physics, such as depletion of the wave energy, will cause saturation at a finite level.)This phenomenon is displayed in figure 2  sources' magnitudes, the deposition of EC current is gradually condensed into O-points of islands for the lower solution.While for the higher solution, the current condensation is strengthened as EC sources' magnitudes decrease.For the higher solution at a relatively small ECCD, there exist hollows around O-points of islands.This is because the current condensation is so strong that the region of high EC current deposition rate is too localized.And here the ECCD and ECRH are aiming at initial q = 2 rational surface instead of O-points as aforementioned.Thus, the high deposition region of EC current cannot cover O-points.Figure 4 shows a poloidal section of m/n = 2/1 islands.When the island width is large, the radial position of islands' O-points (the 'X' marks) would inwardly deviate from the initial rational surface (the dot-dashed line).Similar phenomenon has been observed in the previous researches of NTM control for radial misalignment investigations [50].It should be noted, however, that the higher solution is not likely to be seen in practice because it is physically unstable.To further compare the numerical results with the analytical results, the dependence of heating intensity threshold on critical parameters are investigated.Figure 5(a) shows that the threshold decreases as the ratio of parallel and perpendicular transport coefficients increases.(Here, to better compare with the analytical results, the χ // is fixed as 10 and χ ⊥ is adjusted for the increase of the ratio).According to the analytical results reported in [38], the threshold should decrease with decreasing perpendicular transport coefficient, while the parallel transport coefficient is sufficiently large.Thus, the analytical results are in good agreement with the numerical results.shows the ECCD magnitude threshold versus helical angle width.It is found that the ECCD magnitude (total EC driven current) threshold S c (j dc ) decreases (increases) as helical angle width increases.Narrower helical angle width corresponds to lower j dc , which means the enhancement effect can be relatively easier to achieve.Thus, it is expected to lower the required input power for completely stabilizing large islands via narrowing down the helical angle width when the condensation effect is important.This has implications for the stabilization of locked vs rotating islands, and for the effect of modulating the ECCD.For a rotating island with continuous RF current drive, the helical angle deposition width is broad.The width can be narrowed by modulating the RF, so that it is on when the ray trajectory passes near the O-point of the island, but off when it passes near the X-point.The helical angle deposition width can be particularly narrow for a locked island, if the island is locked at the appropriate phase.This can be arranged by proper adjustment of the current in the tokamak error field correction coils.Almost all large contemporary tokamaks have non-axisymmetric error field coils to compensate for finite tolerances in construction, and ITER will also have such coils.

Enhancement of ECCD by condensation effect for NTM control
In this subsection, the enhancement of ECCD by condensation effect is investigated during the control of NTM.Previous results showed that, as the island width increases, the required input power of ECCD for completely stabilizing NTM islands also increases [29,50].Thus, the ECCD is suggested to be turned on as early as possible [29,50].Figure 6(a) shows the temporal evolutions of island width with ECCD turned on at different moments.The NTM islands can be completely stabilized as long as the ECCD is large enough.For a magnetic island with island width W = 0.05, slightly increasing the ECCD magnitude from S j0 = 7256 kA m 2 ms to can effectively stabilize it.For a magnetic island with island width W = 0.10, slightly increasing the ECCD magnitude from S j0 = 9410 kA m 2 ms to S j0 = 9977 kA m 2 ms can effectively stabilize it.The required magnitude of ECCD for completely stabilizing NTMs versus island width at the turn-on moment of ECCD is displayed in figure 6(b) indicated by the solid blue curve.The required ECCD magnitudes go through a nearly linear increase with island width and then reach the maximum.Taking into account the condensation effect, the difference between the minimum and maximum ECCD magnitudes becomes lower with increasing temperature differential inside islands.And the turning point of the ECCD magnitudes is brought forward, indicating that the enhancement effect by condensation is large when island width is large.
With the aim of better understanding the difference between the cases with and without condensation effect, the temporal evolutions of island width under control of ECCD at the same island width are compared in figure 7. The ECCD is turned on at the moment island width reaching w = 0.175 and w = 0.200, respectively.Figures 7(a) and (b) represent the case without and with condensation effect, respectively.The ECCD magnitudes are set as S j0 = 17 006 kA m 2 ms for the case without condensation effect and S j0 = 9070 kA m 2 ms for the case with condensation effect.Corresponding growth rates of island width dw/dt are displayed in figures 7(c) and (d).It is found that, for the case without condensation effect, the growth rate of island width almost has no relation with the turn-on time of ECCD.The growth rate of island width is almost the same under the same island width.However, for the condensation effect case, the stabilizing effect of ECCD is largely enhanced when turning on ECCD at a large island width.Figure 7(d) shows that there is an evident increase of the shrinking rate of island width at large island width.And the enhancement effect gradually disappears with the decreasing of the island.It appears that when the island width is small, the shrinking rate of the island width is less for the case with condensation than that for the case without condensation.This is due to the fact that the condensation effect is no longer present at small island width, and the adopted ECCD magnitude is larger in the case without condensation effect.To compare the deposition intensity of EC current at the same island width, the contour plots of helical magnetic flux and EC current at t 1 − t 4 in figure 7 are shown in figure 8. Contours of helical magnetic flux in the first-row show that the island widths are all the same.The EC current at t 1 and t 2 are almost the same.However, the EC current at t 4 , with ECCD turned on at large island width, is obviously stronger than that at the t 3 .Moreover, the deposition regions of EC current are more localized at t 3 and t 4 comparing to that at t 1 and t 2 due to the condensation effect.
In consideration of the condensation enhancement of ECCD at large island width, the previous idea of turning on ECCD as early as possible may not always be the best choice for stabilizing NTM islands.Here, we focus on the total stabilizing time (time interval between the moment the ECCD is turned on and the moment the NTM island is completely stabilized) to show that turning on ECCD at a relatively large island may provide a better stabilizing effect on NTM islands.Figure 9 shows the temporal evolutions of island width under control of ECCD with condensation effect and with different ECCD magnitudes.It is found that, with increasing ECCD magnitude, the total stabilizing time of turning on ECCD at a relatively large island can be shorter than that at a small island (∆t 4 < ∆t 3 ).Figure 1(b) shows that the enhancement effect of ECCD by the RF current condensation is nonlinearly related to the input power.A large input power will lead to extremely strong condensation effect while approaching to the threshold.When the island starts to shrink, the enhanced driven current at large island width will not disappear immediately, presenting a hysteresis-like behavior.Thus, the stabilizing effect can be more effective when the input power is larger.This result provides another idea in the control of NTMs via ECCD that, taking into account condensation effect, appropriately postponing the turn-on time of ECCD sometimes can improve control efficiency.Also, it has previously been argued that there are circumstances where it is preferable to wait for an island to grow large enough to lock before attempting ECCD stabilization [58].The possibility of increased efficiency of ECCD stabilization for locked islands has been discussed in section 3.1 in connection with the results shown in figure 5(b).However, larger NTM islands may impair the confinement and even lead to major disruptions.There is a tradeoff that needs to be evaluated.
In experiments, the aiming point of ECCD is not always that accurate, the radial misalignment of ECCD is inevitable on some level due to the realistic limitations.Here, the effect of radial misalignment is investigated via comparing the cases with and without condensation effect.Figure 10(a) shows temporal evolutions of island width under control of ECCD with different radial misalignments.∆r = r 0 − r s represents the radial distance between the initial q = 2 rational surface and the ECCD aiming point, with r 0 being the radial position of the ECCD aiming point and r s being the radial position of the initial q = 2 rational surface.∆r = 0 represents that the ECCD is aiming at the initial q = 2 rational surface.∆r < 0 and ∆r > 0 represent the aiming point is inwardly and outwardly deviated from the initial q = 2 rational surface, respectively.It is found that, under radial misalignment, the nonlinear evolution behavior of NTMs can be changed and the stabilizing effect of ECCD can be impaired.Figure 10(b) shows saturated island width versus different radial misalignments.The best control efficiency occurs at the radial position that inwardly deviates from the initial q = 2 rational surface.This is because the radial position of the magnetic islands' O-points has a slight deviation from the initial q = 2 rational surface, and the radial position of the best control efficiency should be in the vicinity of the O-points [50].In comparison with the situations without condensation effect (the blue curve with triangular marker), the stabilizing effect is improved in all the radial misalignment cases after considering condensation effect (the red curve with diamond marker).Especially in the vicinity of O-points, the stabilizing effect is largely improved.Corresponding depositions of EC current are shown in figure 11.The deposited intensities of EC current are apparently stronger in the condensation effect cases than that without condensation effect.
Here, another situation is considered.Adjusting ECCD magnitude to make sure the saturated island widths stay the same for different temperature differential when ∆r = 0. On the basis of same control effect, the radial misalignment effect is investigated.Figure 12(a) shows the temporal evolutions of island width for baseline cases.It is found that the shrinking  rate are different for different cases but the saturated island widths are kept the same.Increasing temperature differential accelerates the shrinking rate of island width at the early stage.Figure 12(b) shows the saturated island width with different radial misalignments.With increasing temperature differential, the stabilizing efficiency is somewhat impaired.This is due to the fact that the nonlinear condensation effect makes the deposited EC current more localized around the O-point of the islands.At, ∆r = 0 the increased efficiency at larger U 0 is compensated by reducing the ECCD magnitude.When ∆r is nonzero, the more localized deposition profile at larger U 0 makes that the advantage of the condensation effect is now not large enough to compensate for the reduced magnitude of the ECCD.

Extension for peaked heating profile
In real experiments, the heating profile may be peaked when the RF system is adjusted to deal with extremely localized control needs.A quasianalytic model that has been used to study RF condensation [38] is specialized to relatively broad deposition profiles.More peaked deposition profiles mean the heating power is more concentrated, which lowers some of the  T. Liu et al normalized thresholds calculated in the analytical work.Here, an extremely peaked heating profile is chosen for the numerical study to verify that the condensation effect still works in this extreme condition.Figure 13 displays the numerical results about the relative magnitude of contribution of the condensation effect to the saturated EC current versus ECCD magnitude and the fold bifurcation phenomenon.Although the large solution is hard to be found when the ECCD magnitude is low due to the numerical limitation, the variation trend of the temperature differential indicates that the fold bifurcation still exists.Here, the total EC current threshold j dc with peaked profile is calculated.It is found that the ratio of j dc with ∆r h = 0.01 and j dc with ∆r h = 0.10 is about 0.44, meaning the threshold decreases 56% after using peaked heating profile in this case.

Summary and discussion
In this work, the RF current condensation effect recently suggested by Reiman et al via analytical study [38], is modelled for the first time with a nonlinear resistive MHD code.A series of numerical investigations have been implemented to study the enhancement of ECCD by current condensation effect during the NTM control.In their analytical study, the parallel transport of electron temperature is assumed as an instantaneous process and only perpendicular transport is considered.The condensation effect is dealt with as a problem of stationary analytical solution.In this resistive MHD model, both the parallel transport and the perpendicular transport of electron temperature are included.With given information of source terms, the EC driven current and driven perturbed electron temperature can nonlinearly evolve along the given magnetic field lines and eventually reach saturation states.At first, a large magnetic island with a fixed width is obtained to serve as the magnetic configuration for the verification of the analytical results.Considering the current condensation effect, the saturated EC driven current is larger than that without condensation for the same magnitude of ECCD and ECRH sources.Increasing EC sources' magnitudes, the input power threshold reported in [38] is encountered.Beyond the threshold, instead of saturation, the total deposited EC current will grow to infinity.(Incorporating additional physics, such as depletion of the wave energy, will cause saturation at a finite level.There is then a discontinuous jump in the saturated value at the threshold.)Moreover, the fold bifurcation phenomenon is also verified via numerical simulations.Below the input power threshold, two solutions of electron temperature perturbation can be found.The numerical results show good agreements with the analytical results.Moreover, spatial distributions of EC current for the two solutions at different condensed level are displayed here.It is found that, with increasing EC sources' magnitudes, the deposition of EC current is gradually condensed around O-points of islands for the lower solution.While for the higher solution, it is on the other way around.The higher solution is unstable, and is therefore not likely to be seen in practice.
Beyond numerical verifications, the direct ECCD control effectiveness of large NTM islands has been evaluated via comparing the cases with and without current condensation effect.For the case without condensation effect, the shrinking rate of island width almost has no relation with the turn-on time of ECCD.However, for the condensation effect case, the stabilizing effect of ECCD is evidently improved at a large island width.While the ECCD magnitude is sufficiently large, the time for completely stabilizing NTM islands can be shorter for turning on ECCD at a relatively large island than that at a small island.The numerical results suggested that, taking into account condensation effect, appropriately postponing the turn-on time of ECCD may help to improve the efficiency and effectiveness in the control of NTMs via ECCD, which suggests a reassessment of the idea that the ECCD should be turned on as early as possible [50].Also, it has previously been argued that there are circumstances where it is preferable to wait for an island to grow large enough to lock before attempting ECCD stabilization [58].The possibility of increased efficiency of ECCD stabilization for locked islands is discussed in section 3.1.However, larger NTM islands may impair the confinement and even lead to major disruptions.Therefore, there is a tradeoff between high control efficiency and the increasing risk of causing major disruptions.It is noteworthy that offnormal events, such as large magnetic islands abruptly occur out of nowhere, can be encountered during the tokamak discharges.Such events are usually unexpected and may not be caught at small island width.Regardless of the control efficiency, the current condensation effect is expected to play significant roles under such situations.
In experiments, the radial misalignment of ECCD aiming point is inevitable due to the realistic limitations.Thus, the effect of radial misalignment is also investigated here.In comparison with the cases without condensation effect, the stabilizing effect is improved in all the radial misalignment cases while considering condensation effect, especially for the case aiming ECCD in the vicinity of O-points.Under some circumstances, the heating profile may be peaked for extremely localized control needs.Here, an extremely peaked heating profile is adopted to verify whether the condensation effect still works in this extreme condition.It is found in the temperature differential that the fold bifurcation still exists.
The investigations carried out in this work manifested that the current condensation effect can be more effective for the ECCD control of large NTM islands.Considering the larger the island is, the effectiveness and efficiency of ECCD is more enhanced, the condensation effect is expected to play important roles in dealing with off-normal events in future largescale devices.For example, the explosive burst triggered by ill-advised ECCD reported in [59] may be mitigated on some level while considering the condensation effect.Moreover, the synergistic application of RMP and ECCD [60] may bring other more effective control strategies, which requires more complicated design and will be investigated in future work.resources.This work is supported by National Natural Science Foundation of China (Grant Nos.11925501 and 12105034).

Figure 1 .
Figure 1.(a) Temporal evolution of total deposited EC current under different conditions.(b) Relative magnitude of contribution of the condensation effect to the saturated EC current versus ECCD magnitude.(c) and (d) display the same physical quantities as (a) and (b) in dimensional units, respectively.

2 s
(a) shows the temperature differential U 0 = w To Ts (b).Figure2(c)shows the radial profiles of perturbed electron temperature inside the magnetic island.Furthermore, corresponding spatial distributions of EC current at different ECCD magnitude of figure2(a) are shown in figure3.It is found that, with increasing EC

Figure 2 .
Figure 2. (a) Temperature differential U 0 versus ECCD magnitude.(b) Temporal evolution of temperature differential U 0 after given initial large electron temperature perturbation at j 3 in figure (a).(c) The radial profiles of perturbed electron temperature inside islands at j 3 in figure (a).(d)-(f ) display the same physical quantities as (a)-(c) in dimensional units, respectively.

Figure 4 .
Figure 4.The m/n = 2/1 magnetic island in a poloidal cross section plotted in a rectangular plan (r, θ) → (x, y).The dot-dashed line indicates the q = 2 rational surface.The 'X' marks indicate the position of the O-points of the islands.

Figure 5 (
Figure 5(b)  shows the ECCD magnitude threshold versus helical angle width.It is found that the ECCD magnitude (total EC driven current) threshold S c (j dc ) decreases (increases) as helical angle width increases.Narrower helical angle width corresponds to lower j dc , which means the enhancement effect can be relatively easier to achieve.Thus, it is expected to lower the required input power for completely stabilizing large islands via narrowing down the helical angle width when the condensation effect is important.This has implications for the stabilization of locked vs rotating islands, and for the effect of modulating the ECCD.For a rotating island with continuous RF current drive, the helical angle deposition width is broad.The width can be narrowed by modulating the RF, so that it is on when the ray trajectory passes near the O-point of the

Figure 5 .
Figure 5.The ECCD magnitude threshold (blue) and the total EC driven current threshold (red) versus (a) transport coefficients and (b) helical angle width.Here, the parallel transport coefficient is set as χ // = 10, and the perpendicular transport coefficient χ ⊥ is varied accordingly.(c) and (d) display the same physical quantities as (a) and (b) in dimensional units, respectively.

Figure 6 .
Figure 6.(a) Temporal evolutions of island width under different turn-on time of ECCD and different ECCD magnitudes.For the cases with ECCD turned on at W = 0.05, the small (blue solid curve) and large (red dot-dashed curve) ECCD magnitudes are S j0 = 7256 kA m 2 ms and S j0 = 7369 kA m 2 ms , respectively.For the cases with ECCD turned on at W = 0.10, the small (yellow solid curve) and large (green dot-dashed curve) ECCD magnitudes are S j0 = 9410 kA m 2 ms and S j0 = 9977 kA m 2 ms , respectively.(b) The ECCD magnitude for completely stabilizing magnetic islands versus island width under different U 0 .(c) and (d) display the same physical quantities as (a) and (b) in dimensional units, respectively.

Figure 7 .
Figure 7. Temporal evolutions of island width under control of ECCD (a) without condensation effect and (b) with condensation effect.Decreasing rate of island width under control of ECCD versus island width (c) without condensation effect and (d) with condensation effect.The ECCD magnitudes are set as 17 006 kA m 2 ms for (a) and (c), and S j0 = 9070 kA m 2 ms for (b) and (d).

Figure 9 .
Figure 9. Temporal evolutions of island width under control of ECCD at different island widths with condensation effect of (a) S j0 = 7936 kA m 2 ms and (b) S j0 = 10 204 kA m 2 ms .

Figure 10 .
Figure 10.(a) Temporal evolutions of island width under control of ECCD with different radial misalignments.(b) Comparison of saturated island width with different radial misalignments between with and without condensation effect.

Figure 12 .
Figure 12.(a) Temporal evolutions of island width under control of ECCD with the same saturated island width.(b) Comparison of saturated island width with different radial misalignments between with and without condensation effect.

Figure 13 .
Figure 13.(a) Comparison of heating profile between broad profile and peaked profile.(b) Relative magnitude of contribution of the condensation effect to the saturated EC current versus ECCD magnitude under peaked heating profile.(c) Temperature differential U 0 versus ECCD magnitude under peaked heating profile.(d)-(f ) display the same physical quantities as (a)-(c) in dimensional units, respectively.