The magnetic coherent mode with shifted Alfvén gap frequency destabilized by the thermal trapped electron resonance in the pedestal of high-confinement tokamak plasmas

The magnetic coherent modes (MCM) with toroidal mode number n about 1 (Chen R. et al 2018 Nucl. Fusion 58 112004) frequently appear in the edge pedestal of high-confinement tokamak plasmas on EAST in the absence of energetic particles. Although these modes are experimentally compatible with the steady-state operation of the pedestal, the driving mechanism without energetic particles of MCM is a long-standing mystery. To reveal the excitation mechanism, a fluid-drift kinetic hybrid local linear model has been developed. It is found that MCM is a new Alfvén eigenmode with a gap frequency much lower than the ideal Toroidal Alfvén Eigenmodes (TAEs) with two significant properties: (1) due to the unique steep pressure gradient in the pedestal region, the diamagnetic frequency becomes comparable to the ideal TAE frequency, which makes the Alfvén continuum in this region move significantly in the ion diamagnetic direction and form a gap of lower frequency; (2) due to the bounce frequencies of thermal electrons becoming also comparable to the ideal TAE frequency in the pedestal region, the free energy of the pressure gradient can be fed into the MCM through the thermal electron bounce resonance excitation, which is essentially the coupling between the shifted TAEs and low-n trapped electron modes. The low-n MCM is proved to be a shifted

Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Introduction
This paper is to report the modeling research on the excitation mechanism of the low-n magnetic coherent mode (MCM) in tokamak plasmas.Here, n is the toroidal mode number.MCM as an experimental phenomenon appears locally in the H-mode pedestal region, which has been analyzed in detail based on the EAST experimental data [1].The H-mode pedestal is characterized by a transport barrier with steep pressure gradient at the plasma edge region [2,3].MCM does not break the H-mode pedestal structure by experimental observations, which is not only compatible with high confinement performance with a wide operating range of pedestal collisionality but also exhibits many unique properties, such as [1]: (a) The appearance of the MCM is independent on heating schemes (i.e.pure radio-frequency (RF) waves heating including the combination of LHW [4], ICRH [5],and ECRH [6]; or pure neutral beam injection heating [7]), which implies that energetic particles should not be the driving factor of this mode.(b) The existence of the MCM does not depend on the wall conditions (i.e.carbon wall, metal wall, the discharge with silicification or lithiation), which implies that impurity components do not play an important role in the mode excitation.(c) The magnetic fluctuations of the MCM with similar amplitudes can be detected simultaneously on the high-and the low-field-side midplanes of the tokamak plasma, which implies that the mode is not a ballooning-like electromagnetic mode localized only in the bad curvature region of the low-field side [8,9].(d) The characteristic frequency of MCM is qualitatively in line with the typical Alfvénic parameter dependencies on both edge electron density and edge safety factor (i.e.f MCM ∝ 1/q m √ n e ), which implies that the mode has the nature of a low-frequency shear Alfvén eigenmode [10].This is also the starting point of this paper to explore the mode, but it must be taken into account of the aforementioned property (a), that is, non-energetic particles driven.
MCM with n about 1 on EAST [1] is one of the coherent modes induced by benign instabilities.However, the driving mechanism of MCM is different from those reported coherent modes [11][12][13][14][15], which are not as problematic instability as ELM [16] and will not cause the collapse of the pedestal.Different from the coherent magnetic fluctuations in the high recycling steady regime on JFT-2M tokamak [11,17], MCM does not require high neutral particle recycling or neutral pressure at the plasma edge.Different from the edge harmonic oscillation (EHO) on DIII-D tokamak [12], where EHO is currently considered as a low-n kink mode driven by the edge current gradient [18,19], MCM exhibits the characteristics of the Alfvén-like eigenmode and propagates in the electron diamagnetic direction.Different from the quasi-coherent mode (QCM) on Alcator C-Mod tokamak [13] or high frequency coherent (HFC) modes on DIII-D [14] both with n greater than 10, where the QCM and the HFC are considered to have the properties of resistive ballooning mode [13,20] and kinetic ballooning modes, respectively, MCM is a low-n mode with magnetic fluctuation and is not a ballooning-like mode.Furthermore, on EAST tokamak, MCM is an electromagnetic mode dominated by magnetic fluctuations [1,21], which can coexist with an electrostatic mode ECM [15] in a relatively high range of the normalized electron collisionality at the pedestal region, but more importantly, MCM could survive the low-collisionality condition that future fusion devices could achieve [22,23].Then, the modeling research here primarily aims to understand the plasma parameter space in which MCM could be excited, which could imply that this mode may have a potential application of a stable operating scenario that does not cause the pedestal collapse.
In fact, without exact mechanism analysis, the low-n electromagnetic modes exhibiting some similarities to the MCM have been also observed in the pedestals on other devices such as JET [24] and JFT-2M [17].In addition, the previous numerical analysis [25] considered the low-frequency electromagnetic modes with n = 1 as energetic-particle continuum modes [10] excited by energetic electrons provided by the radio frequency waves due to the real frequencies falling in the shear Alfvén wave continuum region.Although the excitation mechanism in [25] does not match the experimental analysis that MCM is independent on the energetic particles under various plasma heating conditions [1], its numerical calculation clarifies that the low-n mode is localized in the edge pedestal region, which will be an important reference evidence for simplifying the mode structure in following model research in this paper.Experiments on EAST indicate that the parameter dependence of MCM consistent with the frequency scaling of Toroidicity-induced shear Alfvén Eigenmode (TAE) [26], but the real frequency value about 20-60 kHz is always lower than the local TAE gap frequency and higher than the Beta-induced Alfvén Eigenmodes (BAEs) gap frequency [27][28][29].So, what is the mystery of this mode and its plasma parameter space and how to understand its excitation without energetic particles?Unraveling these mysteries could improve the understanding of pedestal physics and provide a reference for the potential applications of MCM in future devices.
This paper is structured as follows.In section 2, the whole physical model for the mode of maximum linear growth rate (i.e. a Fluid-Drift Kinetic Hybrid local Linear (FDKH-L) model) is presented, where the drift kinetic resonance of the thermal electrons is coupled with the magnetic fluid model through the perturbed kinetic pressure to analyze the pedestal instability.The FDKH-L model does not involve energetic particles but includes the coupling between the fluid and drift kinetic modules: (1) the fluid module determines the real frequency of the mode, where the pressure gradient provides free energy; (2) The drift kinetic module determines the growth rate of the mode, where thermal trapped electron bounce resonance provides energy channels for mode excitation.In section 3, the numerical investigations for the dependence of the MCM on the plasma parameters are demonstrated.The FDKH-L model demonstrates that MCM in nature is a new toroidicity-induced shear Alfvén eigenmode that exists under the special conditions of pedestal, where the mode has a gap frequency significantly lower than TAE due to the strong diamagnetic drift in the pedestal.It is noted that the steep pressure gradient in the pedestal makes the diamagnetic frequency close to the frequency of the TAE gap and the averaged effective trappedelectron bounce frequency, where TAEs can couple to low-n trapped electron modes (TEMs) through non-adiabatic electron density perturbations.Finally, conclusions and remaining issues are summarized in section 4. The mode is predominantly destabilized by trapped electron bounce resonance which releases the free energy stored in the steep pressure gradient of the pedestal.Therefore, this mode is named as Trapped Electron Toroidal Alfvén Eigenmode (TETAE).Since the physical conditions that this mode could be destabilized with will persist in low collision rate plasmas, the mode could be anticipated to exist in future high-performance devices.

The model structure
The steep pressure gradient region of H-mode pedestal is the modeling domain, which has been shown by the shadowed area in figure 1.For the common discharges of the EAST Hmode plasmas [30] with minor radius a ∼ 0.45 m and major radius R 0 ∼ 1.9m, toroidal magnetic field B 0 ∼ 2.5T at the magnetic axis, the typical radial equilibrium profiles of plasma electron density and temperature (by the modified tanh-fitting [31]) as well as the corresponding pressure profile are shown in figure 1(a), where r is the minor radius of a magnetic surface.Based on these profiles, the electron diamagnetic frequency ω * e due to the pressure gradient and the effective thermal trapped-electron bounce frequency ω be,eff become close to the local TAE frequency ω A in the steep gradient region, as shown in figure 1(b), which offers a unique opportunity for the coupling of the drive of electron pressure gradient to the TAE and inspires to investigate the excitation mechanism of the model from the perspective of the kinetic properties of thermal particles.The typical safety factor and radial mode structure are shown in figure 1(c), where in the pedestal region with the toroidal mode number n = 1 and the poloidal mode number m, the characteristic distance between two adjacent rational magnetic surfaces (i.e.q ∈ m n , m+1 n ) is close to the width of the pedestal.According to the theory of magnetohydrodynamics (MHD) [26,32,33], it is assumed that the radial mode structure is localized between two adjacent rational magnetic surfaces in the pedestal region, which is consistent with the experimental and previous numerical analysis [1,25].As shown in figure 1(c), r s is the minor radius (or the rational surface) of the radial mode structure peaking, correspondingly, which is also the location where the local linear analysis performed in the model.
By the preset radial mode structure, the corresponding dispersion relation for the linear mode growth rate can be obtained by the determinant of the coefficient matrix after the linearization of the model equations [34], where the detailed process will be introduced in the following sections.The reduced model flowchart for the coupling on FDKH-L is given by figure 1(d), where the Alfvén continuum is determined by the fluid module alone, and the excitation of the eigenmode of maximum growth rate requires the coupling of the kinetic and the fluid modules to be solved iteratively.Due to the fact that electron perturbed kinetic pressure pe,K calculated in the drift-kinetic module is a function of the mode frequency ω calculated in the fluid module (ω = ω Re + iγ is a complex number with its real frequency ω Re and growth rate γ), it is necessary to recalculate the perturbed kinetic pressure based on the former calculated mode frequency.The updated perturbed kinetic pressure will provide a new mode frequency.So the dispersion relationship matrix of FDKH-L is iteratively calculated to obtain the convergent frequency.As labeled in figure (d), ω U and ω D are the upper and lower frequency of the Alfvén continuum gap.If the real frequency of the excited eigenmode is in the continuum gap (i.e.ω D < ω Re < ω U ), the eigenmode is considered to be a gap mode.The details of the whole model will be discussed in detail in the next subsections.

Linearization of a reduced electromagnetic fluid model
Since MCM is an electromagnetic mode dominated by magnetic fluctuations in the H-mode pedestal region [1,21], the better strategy for the fluid module of FDKH-L is first to refer to the well-known electromagnetic fluid model [19,[35][36][37][38].Then, a reduced electromagnetic fluid model focusing on the electron perturbation channel is developed, which is mainly governed by the following equations: the vorticity equation, the generalized Ohm's law in the direction parallel to the equilibrium magnetic field line and the electron density continuity equation.For more detailed simplification and linearization of these initial equations, please refer to appendix.The physical quantities have been divided into equilibrium and perturbation parts, such as the electron density n e = n e0 + ñe the electron temperature T e = T e0 + Te , the electrostatic potential φ = φ 0 + φ , and the parallel current with respect to the equilibrium magnetic field j ∥ = j ∥0 + j∥ , etc. Noting that this study is on the linear analysis, thus all nonlinear terms have been neglected.
The linearized vorticity equation due to the quasi-neutrality condition is written as, where m i is the ion mass, n i0 is the main ion density and ω * i is the ion diamagnetic drift frequency.d t is the convective time derivative associated with the equilibrium E × B drift velocity, which is discussed later and can also be found in appendix.The introduction of the ion diamagnetic drift effect in the vorticity term causes a shift of the mode real frequency towards the ion diamagnetic drift direction and a stabilization effect on the high-n modes [35,39,40].B is the unperturbed equilibrium magnetic field and its corresponding unit vector is b ≡ B/ B. Br is the perturbed magnetic field in the radial direction.B −2 B × κ • ∇ is the curvature drift operator [19,41], where κ ≡ b • ∇b is the field line curvature.The perturbed parallel current j∥ = −µ −1 0 ∇ 2 ⊥ Ã∥ = −en e0 ṽ∥e with Ã∥ being the parallel to B component of the perturbed magnetic vector potential is mainly induced by the electron parallel motion with a velocity ṽ∥e , where the ion parallel motion is ignored with ṽ∥i = 0, as the timescale of ion acoustic response is much longer than the time scale of MCM.
As the timescale of electron thermal conductance in the parallel direction is much shorter than the time scale of MCM, the electron is assumed to be isothermal along the field line, As electrons are isothermal along the field line, Te does not appear in the linearized parallel generalized Ohm's law, where η ∥ = η sp F 33 (f t33 , Z eff ) is the parallel resistivity with η sp being the Spitzer resistivity and the coefficient F 33 (f t33 , Z eff ) being a function of the effective charge number Z eff and f t33 which is a coefficient in the Sauter model [42,43] reflecting effective trapped electron fraction.The last term in equation ( 3) is the electron Landau damping [35,[44][45][46] with m e being the electron mass and v Te ≡ T e0 / m e being the electron thermal velocity.The kinetic effects of electrons, such as the trapped electron bounce resonance, are introduced through the perturbed electron pressure pe , which consists of adiabatic (fluid) and non-adiabatic (kinetic) responses two parts, pe = ñe,F T e0 + pe,K , ( where the perturbed kinetic pressure pe,K due to the trapped thermal electrons will be discussed in the next section.It should be emphasized that the drift kinetic effect of the trapped thermal electrons is coupled into the fluid equation through the perturbed kinetic pressure pe,K , which enables the model to drive modes with Alfvénic properties in close compliance with the observations of MCM in the experiment.Without pe,K , the fluid model alone provides only modes dominated by the drift wave [34].The perturbed kinetic pressure pe,K , as a key physics for understanding MCM, will be further discussed in the following section of the modeling results.The adiabatic response ñe,F follows the linearized fluid electron density continuity equation, where ṽEr is the perturbed E × B convection velocity in the radial direction.The electron drift waves are contained in the model through the coupling of the fluid electron density continuity equation with equations ( 1) and ( 3).Briefly, the reduced electromagnetic fluid model consists of equations ( 1), ( 3) and ( 5) with unknown perturbed physical quantities φ , Ã∥ , pe .In order to highlight the physical process and simplify the calculation of the model, the circular magnetic configuration and a local helical coordinate system are effective tools to linearize the model [19,41].
For the large aspect ratio approximation of a tokamak in a toroidal coordinate system (r, θ, ϕ ), the equilibrium magnetic field has only toroidal component B ϕ = B0 1+ε cos θ and poloidal component B θ = ε q B ϕ , where ε = r/ R 0 << 1 is the inverse aspect ratio, q is the equilibrium safety factor, R = R 0 (1 + ε cos θ) is the major radius.ϕ and θ are the toroidal and poloidal angle in Euclidean geometry, respectively.For the lowest order of ε, θ ∼ αB 0 is satisfied in this study with α = (1 − ε cos θ) by preserving the expansion factor to the first order of O (ε).For the specific linearization of time and space differential operators, a local helical coordinate system [19,41] x = r − r s , y = l θ + εq −1 l ϕ , z = εq −1 l θ − l ϕ corresponding to the magnetic field configuration is adopted for the plasma pedestal region, where x is the radial coordinate, l θ = rθ is the poloidal arc length, l ϕ = R 0 ϕ is the toroidal arc length.So locally, the helical coordinate satisfies the right hand relation, e x × e y = e z with the unit vector e, where correspondingly the magnetic field is in the z direction, and the electron diamagnetic direction is in the positive direction of y.For the radial mode structure peaks at r s between two adjacent rational magnetic surfaces [26,32,33] in the pedestal region, the Eigen function of the mode in this coordinate system could be expressed as the following form, where the radial profile of φ with amplitude C is assumed to be a Gaussian distribution function centering at a rational surface with a radial decay length of λ m .And is the wavenumber in the z direction, k θ = m/r is the poloidal wavenumber and k ϕ = n/R 0 is the toroidal wavenumber.Apparently, differential operators ∂ y = ik y or ∂ z = ik ∥ are the gradients on the magnetic flux surface perpendicular or parallel to the direction of the magnetic field, respectively.
Under the drift approximation, the leading order convective time derivative is given by velocity in the y direction.Here, positive v Ey0 is in the electron diamagnetic direction.The partial time derivative ∂ t = −iω is calculated in the laboratoryrest frame of reference, thus the time derivative of is for the mode frequency in the plasma-moving frame associated with the E × B shift frequency ω E .It should be noted that for the low-n mode, the corresponding Doppler frequency shift could be ignored due to the weak rotation at the plasma edge region [47].According to the frequency observation of MCM in the experiment [1], the relationship ω ≫ ω E holds in the Hmode pedestal of EAST, so ω ≃ ω is assumed in the model analysis.
In the helical coordinate system, the gradient along the magnetic field line is given by where B⊥ is the perpendicular magnetic perturbation.Its radial and y-direction component is Br = ∂ y Ã|| = ik y Ã|| and By = −∂ r Ã|| , respectively.In the helical coordinate system with circular cross-section, the curvature drift operator could be expressed as with the normal curvature factor C n ≈ cos(θ) − ε corrected to O(ε), and the geodesic curvature factor C g ≈ sin(θ) corrected to O(ε 0 ) [41], respectively.Here, the average curvature factor over poloidal angle is set as is adopted in the model.For the large aspect ratio approximation, the average over the integral angle could be simplified as ⟨F⟩ θ = ¸Fdθ/ ¸dθ for any F.For this local model in the pedestal region, the mode perturbation peaks on a rational surface at r s , and its characteristic width λ m is similar to the pedestal scale length L p .Then, the vertical component of the local Laplace operator can be approximated as This study focuses on the continuum gap mode with (m, m + 1) coupling in the pedestal region, where the safety factor is q s = (m + 0.5)/ n at r s .
Eventually, the fluid module above could be reduced to a local linear analysis where the radial mode structure peaks at r s with the safety factor q s .After linearization and normalization, the coupled equations are shown as follows, where all physical quantities are normalized by the equilibrium values on the rational surface at r s , such as Φ ≡ e φ / T e0 , Ã|| ≡ e Ã|| T e0 , Pe ≡ pe / n e0 T e0 , Ñe ≡ ñe / n e0 , Ñe,F ≡ ñe,F / n e0 .And Pe = Ñe = Ñe,F + Pe,K is always satisfied for the isothermal condition along the field line in the model.
is the Alfvén velocity with µ 0 being the vacuum permeability, k ∥,s = n/ (2m + 1) R 0 is the local parallel wavenumber of TAE.k y,s = (m + 0.5)/ r s is the wavenumber perpendicular to the field line on the magnetic flux surface.ω ci ≡ Z i eB 0 / m i is the ion gyro-frequency with charge number Z i = 1 for hydrogen isotopes, and ρ s = c s / ω ci is the ion Larmor radius with ion sound speed c s ≡ T e0 / m i .The density ratio between ions and electrons is defined as is the diamagnetic frequency due to the electron density gradient with its density-gradient scale length L ne = −n e0 /∂ r n e0 .By the ratio of scale length between density and temperature η e ≡ L ne / L Te , ω * Te = η e ω * ne and ω * e = (1 + η e ) ω * ne are estimated as the diamagnetic drift frequencies due to the temperature and pressure gradients, respectively.Meanwhile, the ion diamagnetic frequency ω * i ≃ −ω * e is reasonable for the similar scale length L p = −p e0 / ∂ r p e0 at the same the pedestal region.By Ampere's equation µ 0 j ∥0 ≃ (1/ r) ∂ (rB θ )/ ∂r for the large aspect ratio approximation, the approximation expression µ 0 ∂ r j ∥0 ≃ − 2B0 rLs is valid for keeping the first derivative of the safety factor q profile, where L s = q 2 εdrq is the magnetic shear length [19].It should be noted that f i and Z eff are not independent of each other.For simplicity, considering the equivalent impurity with charge number of Z I (e.g.carbon as the main impurity [49][50][51]), the approximate relationship of |Ze|(ZI−Z eff ) could be obtained, where Z e = −1 is the electron charge number.
This section mainly introduces the linearization analysis of the fluid part of the model, but the fluid model alone does not give the mode growth of MCM.Therefore, the following part will introduce the drift kinetic model for calculating the perturbed kinetic pressure Pe,K .

The perturbed kinetic pressure of trapped thermal electrons
Since the kinetic module of the model focuses on the kinetic resonance behavior of thermal electrons, the corresponding calculations of the perturbed kinetic pressure are based on the Maxwellian distribution f 0 for the equilibrium particle distribution function.For the ordering system for low-frequency largescale MHD (i.e. the radial mode structure is relatively large in comparison with the ion gyro-radius ρ s k ⊥ ≪ 1, and the mode frequency is much smaller than the ion cyclotron frequency ω/ ω ci ≪ 1) [52,53], the drift-kinetic theory is valid for this study.Then, based on the linear analysis of driftkinetic theory [53], the total distribution function of thermal electrons could be expressed as f = f 0 + fadi + h including the equilibrium term f 0 , the adiabatic perturbation term fadi and the non-adiabatic perturbation term h.It should be pointed out that the perturbed physical quantities in the previous fluid module could be exactly obtained by taking moments of the adiabatic perturbation term fadi [34], which will not be discussed in this drift kinetic module.
This drift kinetic module plays a crucial role in the model to calculate the perturbed kinetic pressure pe,K , which involves the integration of h over the velocity space.Specifically, for a certain equilibrium particle distribution function f 0 , the non-adiabatic perturbation distribution function h could be expressed as a form of h = ∂f0 ∂E R l L [53,54].E is the total energy detailed later, and L is the perturbation of Lagrangian.R l represents the mode-particle resonance condition for the number of bounces l [53,55] (e.g.l = 1 for resonance with a single bounce).Furthermore, via a systematic ordering analysis [52], the revised perturbation of Lagrangian L to the leading order for the low-frequency dynamics is adopted in this model as follows, which is consistent with the gyrokinetic form [10], Eventually, the non-adiabatic perturbed electron pressure could be derived by solving analytically the drift kinetic equation in a perturbative procedure [53,55], where the integration is taken in the velocity space Γ.And v ∥ and v ⊥ are the unperturbed parallel and perpendicular velocities of an electron.Furthermore, for the new perturbed Lagrangian, if the terms related to the perturbed magnetic field is defined as LA = Z e e Ã∥ v ∥ and the terms related to the perturbed electrostatic potential is defined as Lφ = −Z e e φ , then the following expression can be obtained naturally as L = LA + Lφ and pe,K = pe,A + pe,φ correspondingly.
In particular, the equilibrium distribution function f 0 is assumed to be Maxwellian for thermal electrons, given by is the kinetic particle energy.Thus ∂f 0 /∂E = −f 0 /T e0 is satisfied, where E = E k + Z e eφ 0 is the total energy with the equilibrium electrostatic potential φ 0 .By the Maxwellian distribution function for trapped electron bounce frequency resonance, Then R l appears to be associated with the diamagnetic drift frequencies of the fluid model in the following form [54,55], where Êk ≡ E k /T e0 ∈ (0, ∞) is the normalized kinetic energy, ω is the mode frequency in the plasma frame of reference consistent with the fluid model mentioned above.Note that generally ω = ω Re + iγ is a complex number with the real frequency ω Re and the linear growth rate γ > 0. ω be denotes the bounce frequency of trapped electrons.ν eff = ν ei / ε is the effective collision frequency [55] to the electron-ion collisions ν ei .Note that here the electron precession frequency and the transit frequency of passing electron have been ignored as the precession frequency and the transit frequency are much smaller and larger than the mode frequency, respectively.In order to obtain the perturbed kinetic physical quantities, the same toroidal coordinate system (r, θ, ϕ ) is adopted in this drift kinetic module.For the large aspect ratio approximation to the lowest order of ε, the bounce frequency of trapped particles is analytically given by [55], is the normalized complete elliptic integral of the first kind.By the velocity space averaging, the effective thermal trapped-electron bounce frequency could be estimated as ω be,eff = v Te √ ε/ R 0 q.Also to the lowest order of ε, the equivalent form of parallel velocity is obtained as ⊥ 2B is the magnetic moment with B = αB 0 .Then the integration over the velocity space could be carried out conventionally over the particle kinetic energy and the pitch angle [55], By normalization, the two terms of the nonadiabatic perturbed electron pressure corresponding to the perturbed Lagrangian can be expressed as the following integral form of Pe,A = − Ze k (exp(− Êk ))R l d Êk , respectively.Two points need to be emphasized: (1) since the bounce frequency of trapped electrons ω be is independent of the sign of the parallel velocity, the integral term Pe,A = 0 is cancelled in the sign summation of σ; (2) considering that the trapped electron fraction f t [56] is affected by plasma collision rate and effective charge number, denoted as the effective trapped electron fraction f t33 [42,43], the ratio between them is c f = f t33 /f t .Then, taking the collision effect into account, the kinetic perturbation of pressure due to the trapped electrons could be empirically reduced by a ratio of the trapped fraction and expressed as Pe,K = c f Pe,φ .Finally, taking the average over the poloidal angle [55] to obtain a comprehensive integration factor of 1−αλ I φ dλ⟩ θ , the perturbed kinetic pressure related to the potential perturbation term is expressed as follows, where the integration of R e,φ requires the input of the mode frequency ω and other plasma parameters, and involves the integration of the generalized plasma dispersion function [57], which can ultimately be achieved through numerical calculation.
The perturbed kinetic pressure of the trapped electrons is finally expressed as a function of the perturbed electrostatic potential in the real space after integrating over the velocity space, and the potential perturbation is determined by the previous fluid equations.Therefore, the FDKH-L model for MCM has been closed and can be solved iteratively.The next section will briefly discuss the final form of the hybrid model and its related model parameters.

The matrix of the fluid-kinetic hybrid model for MCM
After linearization and normalization, the local coupled equations ( 7) and ( 12) become a matrix eigenvalue problem of MX T = 0, where X = Φ , Ã|| , Ñe,F is for the perturbed physical quantities.By the definitions of Ω = ω/ ω A , The coefficient matrix M contains the destabilization mechanism due to the kinetic effect of the trapped thermal electron bounce resonance.The kinetic effect causes the electron pressure perturbation pe,K ∝ R e,φ which drives the instability through the parallel electron pressure gradient force in the parallel generalized Ohm's law and the curvature drift coupling in vorticity equation.The mode of maximum growth rate can be obtained numerically by the determinant of the coefficient matrix with |M| = 0, which requires specific geometric parameters and plasma parameters in the pedestal region.For the convenience of viewing, the main model parameters are divided into different types and listed in the table 1, where n e0 , T e0 , L ne and L Te are indicate their local equilibrium values at the peaking position of the mode structure in the pedestal region, which will determine the parameter space of the excited mode.The following section of model calculation will discuss whether the excited mode conforms to the experimental observations of MCM, which will determine if the trapped thermal electron resonance could serve as a possible mechanism for exciting MCM.
In fact, the FDKH-L model for the linear mode growth rate of MCM does not need to consider the toroidicity-induced mode coupling.However, in order to identify the relative position of the mode real frequency in the Alfvén continuum, it is necessary to evaluate the Alfvén continuum structure that the fluid module can exhibit.For simplicity, focusing on the ideal MHD part in equation ( 7) with Ã|| = k ∥,m ω Φ , which does not consider non-ideal terms such as driving and damping effects etc, the following simplified equation is obtained, where the ion diamagnetic drift frequency ω * i will cause a shift of the mode real frequency towards the ion diamagnetic drift direction [39], which is the key effect on Alfvén continuum in this model.According to the well-known algorithm for the Alfvén continuum [33,58,59], by expanding the toroidicity effects to the lowest order of ε and keeping only the two dominant poloidal modes of m and m + 1 coupling for the TAE mode, the coefficient matrix M m,m+1 of equation ( 14) for the perturbed physical quantities Φ m , Φ m+1 is shown as follows, Taking the typical pedestal parameters of EAST as an example (i.e. the profiles in figure 1), the gap structures of the Alfvén continuum due to the toroidicity effects are shown in figure 2(a) by numerical calculation of equation (15).With the same EAST edge safety factor profile, when the ion diamagnetic drift is not considered for the case of |ω * i / ω A | = 0, the model calculation is consistent with the reference results of TAE gap in ideal MHD limit [58].It should be emphasized that λ m like k ∥ is not a free parameter, and theoretically its value is determined by the distance between two adjacent rational surfaces.Typically, λ m about 2 cm is obtained for this EAST case.If the value of λ m is arbitrarily changed, the Alfvén continuum structure will deviate from the ideal MHD theory.Therefore, this local model requires the value of λ m to be consistent with the safety factor profile in the pedestal region to provide an accurate benchmark with ideal MHD limit.However, Only changing ω * i while keeping other parameters fixed, i.e. for the case of |ω * i / ω A | ∼ 1, the gap frequency in the electron diamagnetic direction dramatically shifts towards the ion diamagnetic direction, which reveals that the final gap frequency after shift is significantly smaller than the ideal TAE gap frequency.This new shifted frequency gap matches the real frequency of MCM under similar plasma parameters for the EAST discharge #74204 [1,30], which implies that MCM could be excited as a new gap mode in the pedestal region.This mechanism that the modification of Alfvén continua by the ion drift frequency was also observed in W7-AS [60] could be universal, which is the key point of the fluid module in the model.
At the local position of the safety factor q s = 7.5 as shown in figure 2(a), one of the eigenfrequencies of equation ( 14) without the poloidal modes coupling in the electron diamagnetic direction can be analytically expressed as ω = Briefly, the mode with the maximum growth rate as well as the real frequency will be determined by the iterative calculation for equation ( 13), while the upper and lower limits of the shifted gap frequency will be estimated by equation (15).By the comparison between ω Re and ω D , ω U , the FDKH-L model is able to determine whether the excited mode belongs to the gap mode.In the following sections, according to the experimental properties of MCM, the numerical results of the model based on the local pedestal parameters will be discussed.

The influence of free parameters on the modeling results
Specifically, the FDKH-L model aims to perform a local linear analysis at the location r s of the radial mode structure peaking for the dominant poloidal modes of m and m + 1 coupling.For the typical pedestal profiles of EAST as shown in figure 1(a), where the pedestal steep gradient region is largely in between q m = 7 and q m+1 = 8 magnetic surfaces, the local equilibrium pedestal parameters shown in the table 1 as the standard reference values at r s are n e0 ≃ 2 × 10 19 m −3 , T e0 ≃ 350 eV and L ne = L Te ≃ 1.5 cm, respectively.However, besides these local equilibrium pedestal parameters, in order to obtain the mode growth rate, there are some free parameters (i.e.q s , n, C κ , L s and Z eff ) in the model for equation ( 13) that need to be estimated.
For medium-sized devices like EAST, the effective charge number is set to Z eff = 2.2 at the pedestal region according to the [50,51], which is reasonable compared to the EAST experiment [30].Meanwhile, the local magnetic shear length is about L s ≃ 2.7 m for the typical safety factor profile as shown in figure 1(c), which is directly related to the equilibrium current gradient representing the kink drive [19].For the mode that can propagate on the high-and the low-fieldside of the tokamak plasma, it experiences only the average curvature which is usually favorable in a tokamak, therefore the curvature drive term in this model is proportional to an average curvature factor of C κ ≃ −ε.
In order to more clearly understand the influence of these two parameters, L s and C κ , on the growth rate of the excited mode, firstly, for the typical case of MCM with n = 1, q s = 7.5, figure 3 shows the influence of the change of C κ in the order of ε on the real frequency and growth rate of the mode.If without the kink drive term for the case of L s → ∞, the growth rate of the excited mode increases and gradually saturates with the increase of C κ , while the real frequency in the electron diamagnetic direction shifts slightly towards the ion diamagnetic direction.For the case where L s is close to the experimental value, that is L s → Exp., the overall trend of the model results is similar to that of L s → ∞, but the growth rate reaches saturation at a small positive curvature and decreases with the further increase of C κ , which should be related to the competition between current gradient drive and magnetic shear stabilization.
However, it should be emphasized that for the case of C κ / ε ∼ −1, the real frequency calculated by the model is within the gap and consistent with the experimental value by introducing the L s corresponding to the experimental q profile.The corresponding ratio of growth rate to ω A is consistent with the theoretical scaling of γ/ ω A ∼ 1 × 10 2 [10].Therefore, determining L s according to the q profile and setting C κ ≃ −ε as the average favorable curvature will serve as the basic setting for the pedestal parameter space exploration in the next section.
In addition, it is necessary to briefly point out that the real frequency in figure 3(a) is not much different from the value in the case of C κ / ε = 0 which is close to the MCM frequency range in the experiments.Since the case of C κ / ε = 0 corresponds to the model completely ignoring the contribution of the curvature term, it can be inferred that the main driving mechanism for the mode to be excited is the coupling of the kinetic effect and the generalized Ohm's law.Meanwhile, for the parameters in figure 3, the model calculation shows that the Landau damping term in equation ( 13) does not affect the real frequency of the mode, but slightly reduces the mode growth rate.
Although experimentally, the toroidal mode number of MCM is about n ∼ 1.However, from the perspective of model analysis, the variation of the properties of the excited mode with the toroidal mode number should be investigated under the similar condition of pedestal parameters.For the same typical pedestal profiles as shown in figure 1 with C κ ≃ −ε, figure 4 demonstrates the influence of the change of toroidal mode number, n, on the real frequency of the most unstable mode and its growth rate for different edge safety factors, q m in the pedestal region.The modeling results in figure 4 show the transitions of two different types of modes: (1) The Alfvénic low-n gap mode (i.e.ω Re ∝ ω A ) which is consistent with the properties of MCM in the experiment.
For larger q m , the most unstable modes are gap modes with n = 1.However, when q m is relatively small, the most unstable mode of n = 2 could satisfy the property of the gap mode.Due to the gap mode, the real frequency (f = ω Re / 2π ) could be consistent with the Alfvén frequency, so there is an inverse relationship with q m , as shown in the subfigure on figure 4(a).( 2) The drift-wave-like mode (i.e.ω Re / ω * ne ∼ 1) where the real frequency of the most unstable mode is the locked to local electron density diamagnetic frequency ω * ne with the increase of n.Due to the diamagnetic magnetic stability in the model, the linear growth rate of the drift-wave-like mode calculated by the model will gradually decrease with the increase of n, as shown in figure 4(b).However, it should be noted that the drift-wave-like mode with higher n does not match the properties of MCM, which will not be discussed further in this study.
As mentioned earlier, that the perturbed kinetic pressure pe,K is a key physics for understanding MCM.In order to gain a clearer understanding of the role of this perturbed kinetic pressure pe,K , the influence of pe,K on the excited mode frequency is demonstrated in figure 5 by forcibly switching pe,K in the model.Regardless of varying with local equilibrium density n e0 or local equilibrium temperature T e0 , without pe,K , the model can only obtain drift-wave-like modes, where their real frequency is locked to the electron density diamagnetic frequency, ω Re / ω * ne ∼ 1.When pe,K is coupled in the mode, the modes with Alfvénic property are obtained, where their real frequency can be within the shifted Alfvén continuum gap and consistent with the experimental observations of MCM.
In the following sections, the reasonable free parameters will be specified in the FDKH-L model to examine the plasma parameter space where MCM could be excited as a gap mode in the pedestal region.

Parameter space of MCM on EAST predicted by the model
With the model, the parameter space of MCM on EAST could be explored by scanning four parameters and keeping the others fixed: local electron density n e0 , local electron temperature T e0 , electron density gradient length L ne and electron temperature gradient length L Te .The solution space can be divided into three regions: S1, S2 and S3.There is a parameter window with moderate electron density, temperature, density gradient and temperature gradient, i.e. the S2 region, a solution can be found and the real frequency is in the new shifted gap, where the local normalized electron collisionality [23,43] ν * e is limited in a range of 3-6 with Z eff = 2.2 and the local electron gradient length ratio η e ≡ L ne / L Te is limited in a range of 0.7-1.5, as indicated by the color bars in figure 6.The S2 region is consistent with the EAST experimental parameter space quite well where the MCM is frequently observable.In the S1 region with low density, low temperature, low density gradient or low temperature gradient, where the electron diamagnetic frequency is roughly below a threshold ω * e / ω A < 0.8, there is no numerical solution.This explains why in the EAST experiments the MCM is never seen in the L-mode plasmas and can only be seen in the H-mode pedestal where a steep pressure gradient presents as a necessary condition.The linear growth rate generally increases with the electron diamagnetic frequency, γ/ ω A ∝ ω * e / ω A .This is understandable as the electron pressure gradient is the main free-energy source and driving force of the mode.The linear growth rate saturates at a local electron density n e0 ∼ 2 × 10 19 m −3 , due to the balance between the pressure gradient driven and the collisional damping which increases with density.The collisional damping mainly comes from the effective collision frequency ν eff in the denominator of expression (10).In the S3 region with high density, high temperature, high density gradient or high temperature gradient, the MCM real frequency will be above the upper frequency of the new shifted gap, the mode may not appear as it will be in the continuous spectrum and subject to strong damping.In addition, the local effective thermal trapped-electron bounce frequency ω be,eff / ω A is just slightly above the mode real frequency in the whole parameter space, thus the resonance condition for trapped electrons in the pedestal is easily satisfied.
Specifically, in accordance with the statistics of the EAST pedestal width [61], if the local gradient scale lengths are fixed, figure 7 demonstrates the density-temperature parameter space of MCM as the gap solution.Obviously, for the certain gradient scale lengths, the local density and temperature which are related to the electron diamagnetic frequency need to exceed a certain threshold to excite MCM.However, as the diamagnetic frequency increases, the width of the new shifted gap will gradually shrink, so the real frequency of the excited mode will be greater than the upper frequency limit of the gap, where under this condition, the mode should not be observed in experiments due to continuous spectral damping.
Based on the understanding of MCM on EAST, the related properties of this model under the condition of ITER-like parameters will be briefly discussed next.

Parameter space of MCM on ITER predicted by the model
For the baseline scenario for ITER [2,49] with minor radius a ∼ 2m and major radius R 0 ∼ 6.2m, toroidal magnetic field B 0 ∼ 5.3T at the magnetic axis, the edge safety factors q m could reach 5 or 6 [2,62] at the radial position of r/ a ∼ 0.98.According to the related prediction [62,63] for the ITER pedestal structure, the local electron density and temperature [63] are about n e0 ∼ 6 × 10 19 m −3 and T e0 ∼ 1.2 keV, respectively.Considering the relatively low pedestal fueling due to the opacity of the ITER pedestal to neutral particles [64], The density-gradient scale length will be longer than the temperature gradient scale length.Then, the local density-gradient scale length about L ne ∼ 3 cm and the scale length ratio about η e ∼ 2 will be adopted as a reference case in the model for ITER-like parameters, which are consistent with the profile structure mentioned in the [62,63].Furthermore, the effective charge number predicted for ITER is about Z eff ∼ 2 [65,66] in the pedestal region, which is consistent with a radiation mantle in the pedestal with impurity seeding [67].
First, by the scanning of the toroidal mode number, n, similarly, the most unstable mode will be dominated by the driftwave-like mode with the increase of n, featured by that the real frequency will be locked to local electron density diamagnetic frequency ω * ne with relatively high saturated growth rate.However, the most unstable modes for MCM are gap modes for q m = 5with n = 3 or for q m = 6with n = 2 as shown in figure 8(a), which are different from the mode with dominant toroidal mode number of n = 1under EAST parameters.This is because when the edge safety factors, q m is small, the corresponding diamagnetic frequency ω * e will be too small to provide free energy to drive the MCM.
With the fixed local gradient scale lengths, figure 9 demonstrates the density-temperature parameter space of MCM as the gap solution relevant to the ITER-like pedestal for q m = 6 with n = 2. Similar to the results of EAST parameters, the numerical solutions of the model also show three types of solution regions (i.e.region of S1, Gap solution and S3).For the certain gradient scale lengths, unstable gap solutions are obtained at relatively high-temperature and high-density range to satisfy the threshold ω * e / ω A > 0.8 so that the instability drive, i.e. electron pressure gradient in the pedestal, should be strong enough.It should be pointed out that the MCM as the gap mode can also exist under the relatively low-collisionality condition of ITER pedestal, where the local normalized electron collisionality could be near 1 as shown in figure 9(c).Note that in figure 9(a) the mode frequency does not increase with T e0 , which imply MCM clearly does not follow the temperature dependence of BAE [27,68] in ITER-like parameters.Due to higher T e0 in the ITER-like pedestal than EAST, the effective trapped-electron bounce frequency ω be,eff is higher than the TAE Alfvén frequency ω A .Therefore, the trappedelectron bounce resonance condition ω ∼ ω be is relatively weak in the parameter space relevant to the ITER pedestal, which leads to a lower overall growth rate of MCM as a gap mode.In general, the model predicts that the driving mechanism responsible for MCM exists in the parameter conditions of ITER pedestal, which implies that the mode may have potential applications in future devices.

Conclusions and discussions
In tokamak or stellarator toroidal plasmas, Alfvén waves are usually excited by energetic particles.The mechanism of Alfvén mode excitation in the absence of energetic particles is a long-standing mystery.Here, FDKH-L model demonstrated for the first time that a TAE can be destabilized by resonance with the thermal trapped electrons, thus providing a new solution to the problem of Alfvén mode excitation in the absence of energetic particles.The main features of this model are as follows: (1) Fluid module: the inclusion of the ion diamagnetic drift frequency ω * i in the vorticity equation makes the frequency of the new gap, so-called TETAE gap, significantly lower than the conventional TAE gap, because the diamagnetic drift in the pedestal is usually very strong and significantly higher than that in the plasma core region.(2) Drift-kinetic module: in the H-mode pedestal steep pressure gradient region, the effective electron bounce frequency ω be,eff and the electron diamagnetic frequency ω * e are close to the TAE Alfvén frequency ω A , which offers a unique opportunity for the coupling of the electron bounce resonance drive and electron pressure gradient to the TAE.
Thus, the FDKH-L model proved that the MCM is a toroidal Alfvén eigenmode with a shifted gap frequency, named TETAE, that appear in the pedestal region.The mode real frequency and parameter space predicted by the model are consistent with the EAST experimental observations quite well, where there are certain upper and lower thresholds for density temperature and its gradient length.Furthermore, the model predicts that the MCM is very likely to appear in the pedestal region of ITER for future devices with higher plasma performance and lower collision rate, but the toroidal mode number may be slightly greater than 1 for the lower edge safety factors.It should be clarified that although the non-adiabatic resonance effect of thermal particles in the FDKH-L model could be universal, there are still uncertainties about whether MCM could be detected in a real device with parameters similar to ITER.However, the parameter space obtained by the linear analysis of the model could at least provide a reference for future experimental observations.
For discussions, The MCM in nature of TETAE is a relatively mild type of instability with moderate linear growth rate, which may produce moderate radial transport but do not cause the collapse of the pedestal.The natural ELM-absent H-mode with MCM in pedestal region that frequently appears in EAST [1,21,30] could also exist in ITER as an application of steadystate operation scenario.Therefore, whether MCM could directly affect the pedestal structure and maintain the ELM-free operation is worth further research, but this is a non-linear issue beyond the scope of the current paper.Because of the importance of ELM suppression for reducing erosion on the inner wall of fusion reactors and achieving high-confinement steady-state operation, it is of great importance to reveal the physical mechanism of MCM by linear model first.Finally, it is worth adding that the attempt of pure first-principle largescale particle simulation has not yet yielded results consistent with the experiment.How to reproduce the properties of low-n MCM from first-principles is undoubtedly a meaningful challenge.

Appendix. The initial equations for the fluid module
The fluid module of FDKH-L model is a reduced electromagnetic fluid model focusing on the electron perturbation channel.Referring to the well-known electromagnetic fluid model [19,[35][36][37][38], the initial equations are composed of three main coupled equations: where equation (A1) is the quasi-neutrality condition related to the vorticity equation.Equation (A2) is the generalized Ohm's law in the direction parallel to the equilibrium magnetic field line.Equation (A3) is the electron density continuity equation.
Next also for the large aspect ratio approximation of a tokamak with a local helical coordinate system [19,41], the linearization procedures for each equation will be briefly introduced to obtain the linearized equations in section 2.1.As mentioned before, the physical quantities have been divided into equilibrium and perturbation parts (i.e.A = A 0 + Ã), and all nonlinear terms have been neglected.The physical meaning of the relevant symbols are consistent with section 2.1.
For the linearization of equation (A1), by the well-known gyro-viscous cancellation [34], the divergence of the perturbed polarization and gyro-viscosity current is given by, ∇ • jpi + ∇ • jπ i ≈ ∇ • m i n i0 B −2 (d t + v * i • ∇) Ẽ⊥ .Simply Ẽ⊥ = −∇ ⊥ φ is the perpendicular electric field perturbation.Since the pressure gradient cannot be ignored in the pedestal region, the contribution of the ion diamagnetic drift velocity v * i is retained in the convective time derivative operator [34].For the large aspect ratio approximation of a tokamak, v * i ≡ b × ∇p i0 / Z i en i0 B is in the y direction referring to the local helical coordinate system.Then v * i • ∇ = ik y v * i = iω * i is satisfied in the model.Ultimately, the term of Br • ∇ r T e0 = 0 [34].
Then the perturbed term of ) is derived.For this study on low-frequency largescale MHD modes, the term associated with the electron inertia has been dropped [34].However, in order to make the model also applicable to the plasma with low collision rate, the perturbed electron Landau damping effects with form of Le,damp = − me e π 2 v Te k ∥ ṽ∥e has been introduced accordingly [35,46].For the linearization of equation (A3), the perturbed form of ∇ • (n e v E ) with the E × B drift velocity v E is linearized as v E0 • ∇ñ e + ṽEr • ∇n e0 , where ṽEr is the perturbed radial E × B convection velocity and v E0 is the equilibrium E × B drift velocity in the y direction referring to the local helical coordinate system.Under the drift approximation, the leading order convective time derivative is given by d t ≡ ∂ t + v E0 • ∇.Considering the expression of parallel current j || = −en e v ||e , then the term of ∇ • n e v ||e could be rewritten as the form of −e −1 ∇ • j || , which is linearized as a form of −e −1 B • ∇ j∥ B − e −1  Br • ∇ r j ∥0 B .By integrating the above linear analysis, the exact linearized equations in section 2.1 can be obtained as follows, where ñe,F specifically represents the perturbed density that do not consider non-adiabatic kinetic effects: Finally, it should be clearly pointed out that the perturbed electron pressure pe = ñe,F T e0 + pe,K consists of fluid and kinetic responses two parts, where the perturbed kinetic pressure pe,K due to the trapped thermal electrons will be discussed in the section 2.2.Without pe,K , the fluid model alone provides only drift-wave-like modes [34].Again the perturbed kinetic pressure pe,K , contains a key physics for understanding MCM in this modeling study.

Figure 1 .
Figure 1.(a) The typical radial (ρ for the normalized minor radius) equilibrium profiles of plasma electron density n e0 and temperature T e0 (by the modified tanh-fitting [31]) as well as the corresponding pressure profile p e0 for a common EAST H-mode pedestal, (b) the radial distribution of the ratio of the electron diamagnetic frequency ω * e and the effective thermal trapped-electron bounce frequency ω be,eff to the local TAE frequency ω A , (c) schematic diagram of the radial mode structure and the safety factor q profile, where the amplitude of the mode structure is arbitrary, but the safety factor is consistent with the experiment.(d) The reduced model flowchart of FDKH-L for the fluid-kinetic coupling.The modeling domain (II) is shown by the shadowed area with ω * e/ ω A ∼ 1 in contrast to the inner region (I) with ω * e/ ω A ≪ 1.

Figure 2 .
Figure 2. Taking the typical pedestal profiles as shown in figure 1 as an example: (a) A demonstration of the TAE gap frequency shifted by the ion diamagnetic drift.The frequency ratios are plotted as a function of safety factor q. The magenta bar indicates experimentally measured MCM frequency range.(b) At the local position of the safety factor qs, different frequency shifts vary with the local ratio of |ω * i / ω A |, where ∆ωgap = |ω D − ω U | is the gap frequency width, ∆ω shift,U = |ω U − ω U0 | and ∆ω shift,D = |ω D − ω D0 | are the shifts of upper and lower frequency of the gap, while ∆ω shift,ana is the analytical frequency shift without the poloidal modes coupling.

1 2 ω
* i + 1 4 ω 2 * i + ω 2 A ,which has a frequency shift ∆ω shift,ana = |ω − ω A | caused by ω * i shown in figure 2(b).At the same location of q s as labeled in figure 2(a), ω U and ω D are the upper and lower frequencies of the gap varying with ω * i , while ω U0 and ω D0 are the upper and lower frequencies of the gap without ω * i , respectively.Then, compared with ∆ω shift,ana , varying with the local ratio of |ω * i / ω A |, the shift of upper frequency of the gap defined as ∆ω shift,U = |ω U − ω U0 | is about twice as much as the shift of lower frequency of the gap defined as ∆ω shift,D = |ω D − ω D0 |, which means that the gap frequency width ∆ω gap = |ω D − ω U | will decrease with the increase of |ω * i / ω A |, as shown in figure 2(b).

Figure 3 .
Figure 3. Taking the typical pedestal profiles as shown in figure 1: A demonstration of (a) the real frequencies in the electron diamagnetic direction and (b) the mode growth rates vary with the average curvature factor of Cκ, respectively.The magenta bar indicates experimentally measured MCM frequency range.Ls → Exp.refers to the value of Ls being close to the experimental value, Ls → ∞ refers to ignoring the equilibrium current gradient term.ω U and ω D are the upper and lower frequency of the new shifted gap, respectively.

Figure 4 .
Figure 4. Taking the typical pedestal profiles as shown in figure 1: A demonstration of (a) the real frequencies of the most unstable modes normalized by the local electron density diamagnetic frequency ω * ne and (b) their mode growth rates normalized by the local TAE frequency ω A vary with the toroidal mode number, n, respectively.Where the real frequency ( f = ω Re / 2π ) of the gap mode is inversely proportional to qm.

Figure 5 .
Figure 5.A demonstration of the influence of the perturbed kinetic pressure pe,K on the excited mode frequency: the real frequencies and their mode growth rates of the most unstable modes normalized by the local electron density diamagnetic frequency ω * ne vary with (a) local equilibrium density or (b) local equilibrium temperature, respectively.

X.Q. Wu et al Figure 6 .
Figure 6.The parameter scanning of MCM in EAST pedestal predicted by the model for the standard reference values (n e0 ≃ 2 × 10 19 m −3 , T e0 ≃ 350 eV, Lne ≃ 1.5 cm, ηe = 1).The MCM real frequency (filled dot), growth rate (filled triangle), electron diamagnetic frequency (hollow square) and the effective thermal trapped-electron bounce frequency (hollow star) normalized by the Alfvén frequency are plotted as functions of (a) electron density, (b) electron temperature, (c) electron density gradient length and (d) electron temperature gradient length.The dashed curve and the dash-dotted curve indicate are the upper and lower frequency of the new shifted gap.There is no numerical solution in the S1 region.The S2 region is where a solution can be found and the real frequency is in new shifted gap.The S3 region is where the real frequency is above the gap upper frequency limit although a solution can be found.The color in (a) and (b) represents the local normalized electron collisionality and the color in (c) and (d) represents the local electron gradient length ratio ηe ≡ Lne/ L Te .

Figure 7 .
Figure 7.The demonstration of (n e0 , T e0 ) parameter space of MCM in EAST pedestal predicted by the model.(a) Real frequency and (b) linear growth rate normalized by the local TAE Alfvén frequency.(c) Local normalized electron collisionality.(d) Electron diamagnetic frequency and (e) local effective thermal trapped-electron bounce frequency normalized by the local TAE Alfvén frequency.

Figure 8 .
Figure 8. Taking the typical ITER-like parameters: A demonstration of (a) the real frequencies of the most unstable modes normalized by the local electron density diamagnetic frequency ω * ne and (b) their mode growth rates normalized by the local TAE frequency ω A vary with the toroidal mode number, n, respectively.

Figure 9 .
Figure 9.The demonstration of (n e0 , T e0 ) parameter space of MCM in ITER-like pedestal predicted by the model.(a) Real frequency and (b) Linear growth rate normalized by the local TAE Alfvén frequency.(c) Local normalized electron collisionality.(d) Electron diamagnetic frequency and (e) local effective thermal trapped-electron bounce frequency normalized by the local TAE Alfvén frequency.

Table 1 .
The main parameters in the model.