Effect of resonant magnetic perturbation on edge–core turbulence spreading in a tokamak plasma

Turbulence spreading from the edge to the core region of a tokamak with a resonant magnetic perturbation (RMP) is investigated using an electromagnetic Landau-fluid model in a toroidal geometry. As a RMP field with an appropriate amplitude is applied, long-wavelength fluctuations around the resonance surface can be excited due to forced magnetic reconnection. Strong shear flow at the magnetic island separatrix is observed, which can break the radially elongated vortex structures of the turbulent fluctuation. Inward turbulence spreading can be blocked by this shear flow, and the saturation level of turbulence intensity in the core region declines.


Introduction
Turbulent transport [1,2] in toroidal plasmas, mainly induced by drift wave type instabilities, is one of the key topics in magnetic fusion research.As a typical transport nonlocality [3,4], the radial propagation of fluctuations, or turbulence spreading [5], has been extensively studied in the past several decades.Turbulence spreading can transfer free turbulent energy from strongly driven to weakly driven regions and redistribute the turbulence intensity profile.For current devices of moderate size, the variation of transport scaling from gyro-Bohm scaling has been observed in experiments [6] and numerical simulations [7,8].Turbulence spreading is believed to contribute to that variation through mesoscale dynamics [9][10][11][12], such as linear toroidal coupling and nonlinear coupling between zonal flow and turbulence, which are missing in local or quasilocal models [13].A theory with mesoscale dynamics should therefore be established to include turbulence intensity or transport flux, profile modification, shear flow and other nonlocal effects.
An important application of turbulence spreading theory is the dynamics of edge-core interaction and coupling [5,14,15], which are traditionally treated as independent.For Lmode plasmas, the turbulence intensity usually increases radially outward since the smaller scale length of density and temperature profiles at the edge and micro-instabilities are easier to excite.The strong resistive turbulence at the edge or the invasion of turbulence from the scrape-off layer can also be treated as influxes in turbulence spreading.Therefore, inward turbulence spreading is common in this situation, although the definition of the edge-core boundary in this context may be ambiguous.For H-mode plasmas, the quenching of turbulence in the core region that originates at the edge during L-H transition can be found [16].During the H-L back transition, fluctuations of turbulence in the core region can reach a high level accompanied by disappearance of the edge transport barrier and the collapse of edge flow shear [17].Moreover, the properties of turbulence spreading are thought to have a significant influence on the pedestal height and width [14].Similarly, a transport barrier can also be produced in the core region, and is referred to as an internal transport barrier (ITB).Magnetic shear and flow shear are reported to be important for ITB formation, and the properties of turbulence spreading in the weak magnetic shear region [18] and through the strong flow shear region [19,20] have been reported.Furthermore, turbulence propagation throughout the magnetic island [21][22][23] and stochastic magnetic layer [24] have also received attention.
Due to the multi-scale nature of plasma, the electric and magnetic fields are also affected by other macro-scale structures, such as magnetohydrodynamic (MHD) activity.A dangerous type of MHD instability, edge localized modes (ELMs) [25], can cause partial collapse of the pedestal profiles in H-mode, thus inducing significant heat and particle fluxes.Resonant magnetic perturbations (RMPs) have been shown to mitigate or suppress ELMs [26,27], and it is reported that radial electric field shear could be reduced [28][29][30] and a quasi-coherent mode could be driven [31] at the pedestal top so that turbulence spreading is enhanced.Theories about the effect of a RMP field on the turbulence-zonal flow system have also been developed [32,33].Furthermore, RMP is also utilized to modulate MHD instabilities, such as tearing mode (TM) [34,35] and neoclassical tearing mode [36][37][38], and prevent disruption.The modification of both the radial electric field [39] and the magnetic island induced by the locked TM could affect the properties of turbulence, and these have been studied in various experiments and electrostatic turbulence simulations [40,41].
In this work, the effect of RMP on the inward spreading of turbulence during pedestal collapse is investigated based on the global electromagnetic Landau-fluid simulation [42], which has been used to simulate the effect of plasma beta on turbulence spreading, as shown in [43].Unlike electrostatic simulations with an embedded magnetic island or radial electric field, the electromagnetic perturbations are calculated consistently.The remainder of this paper is organized as follows.In section 2, the global electromagnetic Landau-fluid model is briefly introduced.Characterization of the inward propagation of the electromagnetic turbulence front is analyzed in section 3. The effect of RMP on inward turbulence spreading is analyzed in section 4. Finally, a short summary and discussion are given in section 5.

Model
The Landau-fluid model is employed in this work; it consists of evolution equations for the perturbed density ñe , vorticity ⊥ φ, parallel ion velocity ṽ∥i , parallel magnetic vector potential Ã∥ and ion temperature Ti where Here, the variables with subscript eq are time-independent equilibrium variables.Pressures p i = p i,eq a/ρ i + n eq Ti + T i,eq ñi and p e = p e,eq a/ρ i + T e,eq ñe are the ion and electron pressures, respectively.ϵ = a/R 0 is the inverse aspect ratio and a and R 0 the minor and major radii, respectively.τ = T e,eq /T i,eq is the ratio of electron and ion equilibrium temperatures.θ, ζ) is employed in the numerical simulation, where r is the radius of the magnetic surface and θ and ζ are poloidal and toroidal angles, respectively.The Poisson bracket [ f, g] = (∂ r f∂ θ g − ∂ r g∂ θ f)/r.The variables in the equations are normalized as follows: where n c , T c , B c are values at the axis, v ti = √ T c /m i and ρ i = m i v ti /eB c .Parameter β is defined as half of the ratio of the thermal ion pressure to the magnetic pressure in the center, 1)-( 5) are numerically solved as an initial value problem.The finite difference method is used in the r direction and the Fourier expansion in poloidal and toroidal directions.The inner boundary condition is given by ∂ r f0,0 | ρ=0 = 0 for the m = 0, n = 0 mode and fm,n = 0 for m ̸ = 0 or n ̸ = 0.Here ρ = r/a.The zero Dirichlet condition is imposed for all perturbations on the outer boundary.The equilibrium profiles of safety factor, density and ion temperature profiles that were used are shown in figure 1, where n eq and T eq are analytical expressions combining the quadratic or exponential function at the core region and the hyperbolic tangent function at the edge, and q is a simple quadratic function [43].Steep gradients exist in the edge density temperature profiles.To simulate the turbulence in the pedestal region, the main radial domain covers the whole radius up to the last closed flux surface, ρ ∈ [0, 1.0], which is bounded by buffer zones ρ ∈ (1.0, 1.15] with an artificially increased hyper-diffusivity, which represents the region with a pressure sink [44].With a steep gradient in the edge, spreading of turbulence into the quiescent core region can be observed, and the effect of the RMP field on the process of spreading is the main focus of this study.The RMP field is approximated by A ∥,RMP = A M,N (r) = ψ a,M/N A ∥,eq (r = a)ρ M , added to the equilibrium magnetic field, where ψ a is the amplitude of the field.M and N are, respectively, the poloidal and toroidal mode numbers of the RMP field.Other equilibrium parameters are set as ϵ = 0.25, ρ i /a = 0.0125, τ = 1 and β = 0.1%.

Characteristics of turbulence spreading
In the absence of a RMP field, the dominant instabilities are drift wave type instabilities driven by the steep density or temperature profiles, increasing first at the edge region and saturating before the subdominant instabilities increase in the inner region.A detailed analysis of the instabilities is provided in figures 2 and 3 in [43], which shows the dominant electron drift wave type instabilities at the edge as well as the subdominant ion temperature gradient instabilities and kinetic ballooning mode at the core.Figures 2(a) and (b) show the temporal evolution of the kinematic energies and heat flux of the perturbations, respectively.Here, the kinematic energy E K = ´|∇ ⊥ ϕ| 2 ρdρ/2 and heat flux Q i = ⟨ Ti ṽEr ⟩, with ⟨. ..⟩ denoting the flux surface average.Before t = 70, the turbulence fluctuations are driven linearly and zonal flow is driven through modulation instability and/or the parametric decay process, which has little influence on turbulence fluctuations.Due to the steep gradient, fluctuations at the edge grow first and zonal flows are excited here, as shown in figure 2(c); the saturation of turbulence is induced at around t ∼ 70.After saturation, the fluctuations start to spread into the core region as inward flux, as shown in figure 2(b).
Ballistic propagation of the flux front defined at a constant level is indicated by the white curves in figure 2(b); it has a nearly constant velocity of u r ∼ 0.72(ρ i /a)v ti .The linear coupling of poloidal harmonics and nonlinear coupling between zonal flow and turbulence can contribute to the ballistic propagation of the flux front, as per the theory in [5,10].In figure 2(b), two white curves correspond to the locations of 1% and 14% of the maximum heat flux.The inward spreading speeds of the turbulence are almost the same for the two cases tracing 1% and 14% of the maximum heat flux profiles.A small difference is that the spreading velocity gradually increases as the fluctuations propagate in the core region for the case tracing 14% of the maximum heat flux.This phenomenon may be caused by the features of the instabilities in the core region, since for the core fluctuations the streamer structures are more prevalent, which could enhance the perturbation spreading compared with the edge instabilities.
Radially elongated vortex structures can be observed in the spreading process, as shown in figure 2(d1).These radially elongated vortices could be driven by the eigenstructures of the instabilities at the core, and convection through this contributes to the inward spreading.Since no heat sources are considered in the present study, profile relaxation occurs after turbulence saturation at the edge region, and the fluctuations are mainly concentrated in the core region after inward turbulence spreading, as shown in figure 2(d2).Thus, the saturation level of heat flux is higher in the core when the influx arrives at the core region in figure 2(b).The zonal flow structures also have the same inward spreading speed as the turbulence fluctuations, as indicated by the black curve in figure 2(c).An E × B staircase [45] can be observed.In the core region, zonal flow is weaker, which also contributes to the high saturation level of turbulence.

Blocking of spreading
When the appropriate amplitude of RMP, ψ a , is set, penetration of the RMP field has a great impact on turbulence spreading between the edge and core regions.Here ψ a = 5 × 10 −3 is chosen, which corresponds to a radial magnetic field δB r /B T ∼ 6.7 × 10 −4 on the last closed flux surface.As shown in figures 3(a) and (b), the turbulence stops spreading when a 2/1 or 3/1 RMP field is employed.In the edge region, the fluctuations propagate at the same rate as the case in the absence of RMP, as indicated by the flux front locations with white curves in figure 3.However, when the turbulence arrives at the corresponding resonance surface q = M/N with a RMP with the same helicity, indicated by the black dashed line in figure 3, the properties of the spreading become different.Firstly, as indicated by the front locations during t ∈ (110, 140), the speed of turbulence spreading obviously declines.Secondly, the amplitude of heat flux at the inner region of the resonance surface also decreases compared with the case without RMP, indicating that the turbulence saturation level in the core region decreases due to the effect of the RMP field.The variation of the electric and magnetic fields around the resonance surfaces is the cause of the change in the characteristics of turbulence propagation.
The long-wavelength fluctuations increase drastically as RMPs are turned on, as shown by the n = 1 component in figure 3(c) and electrostatic potential structures in figure 4(a).Magnetic islands could be formed during this procedure and modification of the local radial electric field could also have a possible impact on the turbulence.An interesting observation is that strong shear flow is induced around the resonance surface [39,46], as shown in figure 3(d) for the 3/1 RMP case.Compared with the case without RMP, shear flow around the q = 3 surface is excited after the RMPs are turned on; this increases gradually and becomes stronger as the turbulence front arrives.The helical flux function Ψ H = − ´r/qdr + r 2 /(2q s ) + βA ∥ /ϵ with 3/1 perturbations is given in figure 4(b).An obvious magnetic island can be seen with a width of several ρ i .Actually, with this magnetic island the shear flow is mainly excited at the location of the inner separatrix of the island, similar to [39].This strong shear flow reduces the scale length of the turbulent eddies, as shown in figure 4(c).Around the q = 3 surface, the radially elongated eigenmode structures in the absence of RMP are broken up, and decline of the radial length of turbulence fluctuations could decrease the transport.Thus, blocking of turbulence spreading from the edge to the core region can be observed.As a result, the fluctuations are mainly concentrated in the outer region of the q = 3 surface and the saturation level of fluctuations decreases a lot in the inner region compared with the case without RMP field, as shown in figure 4(d).Correspondingly, the amplitude of zonal flow generated from the turbulence should be also small on the inner side of the q = 3 surface.
To validate the driving mechanism of zonal flow with a RMP field, driving stress analysis is carried out, as shown in figure 5.The zonal flow energy equation is obtained from flux surface average of the vorticity equation where the first term on the right-hand side is Reynolds stress and the second term is Maxwell stress.The terms related to geodesic curvature and viscosity are neglected here.In the absence of a RMP field, as a result of the steep gradient in the edge zonal flow is primarily driven by the Reynolds stress at the edge region at t ∈ [100, 130].Dominant fluctuations have a mode number n > 1 and broad spectra contribute to the excitation of zonal flow.Maxwell stress contributes to the excitation of zonal flow as a sink term.With a RMP field, after the long-wavelength fluctuations increase, zonal flow is strongly excited at the inner separatrix of the magnetic island.Correspondingly, the Reynolds stress and Maxwell stress both peak there.It is also found that the contribution of the n = 1

Conclusion
In summary, a global electromagnetic Landau-fluid simulation is carried out to study the effect of RMP on turbulence spreading from the edge to the core in tokamak plasmas.In the absence of a RMP field, ballistic propagation of turbulence is observed, which has nearly constant velocity.After profile relaxation in the edge region, turbulence fluctuations are mainly concentrated in the core region in the saturation phase.A RMP field can affect the process of turbulence spreading through the excitation of long-wavelength fluctuations due to forced magnetic reconnection.When a RMP field with an appropriate amplitude is employed in the simulation, an obvious magnetic island along with strong shear flow at the island separatrix can be observed.We have identified that the strong shear flow is induced by the n = 1 mode through Maxwell stress.Moreover, turbulence spreading could be blocked by this strong zonal shear flow.The radially elongated vortex structures are broken by the shear flow, and the corresponding transport levels are reduced.With this blocking effect, the saturation level in the core region also declines substantially.These qualitative features seem to be robust consequences of our simulations when varying the poloidal and toroidal mode numbers of RMP.Thus, the results have a potential impact on the study of multi-scale interaction between micro turbulence, zonal flow and long-wavelength MHD mode.

Figure 1 .
Figure 1.The equilibrium profiles of safety factor q, density neq and temperature T i,eq .

Figure 3 .
Figure 3.Time evolution of ion heat flux profiles with a 2/1 (a) and 3/1 RMP field (b).Time evolution of kinematic energies (c) and zonal flow (d) with a 3/1 RMP field.

Figure 5 .
Figure 5. Distribution of Reynolds stress and Maxwell stress averaged during t ∈ [100, 150] with no RMP field (a) and with a 3/1 RMP field (b).