Plasma elongation effects on energetic particle-induced geodesic acoustic modes in tokamaks

Plasma elongation effects on energetic particle-induced geodesic acoustic modes (EGAMs) are theoretically investigated by using gyro-kinetic equations and the Miller local equilibrium model. Including an arbitrary elongation κ and a finite radial derivative s κ = r ∂ r κ/κ , a general EGAM dispersion relation is obtained for an arbitrary energetic particle (EP) distribution. In particular, we obtain analytical EGAM dispersion relations for both the double-shifted Maxwellian distribution and the standard slowing-down distribution of EPs. In both cases, the frequency of the unstable EGAM branch decreases slowly with increasing elongation, while its growth rate decreases rapidly with κ when the ratio of the EP to the bulk ion density, n h / n i ⩾ 0 . 1. These trends agree well with previous GENE and ORB5 simulations (Di Siena et al 2018 Nucl. Fusion 58 106014), but differ significantly from the elongation effects on geodesic acoustic modes (GAMs) (Gao et al 2009 Nucl. Fusion 49 045014). The portion of the EGAM dispersion relation accounting for the first-order finite-orbit-width shows greater sensitivity to frequency compared to that of GAM, which explains the smaller variations in the frequency of EGAM as κ changes. When the EP number is small (typically, n h / n i ≈ 5 % ) in the double-shifted Maxwellian case, the growth rate of EGAMs first increases with the increasing elongation and then decreases, while it monotonically increases with κ in the slowing-down case. Furthermore, the effects of s κ on EGAMs are similar to the elongation κ effects but weaker.


Introduction
The geodesic acoustic modes (GAMs) are well-known oscillatory flow phenomena in tokamak plasmas and are generally accepted as playing a critical regulatory role in moderating plasma turbulence and turbulent transport [1].The typical frequency of GAMs was first derived by Winsor et al in 1968 [2], with assumptions of an infinite aspect ratio and circular crosssection.However, shaping effects can significantly influence GAMs and consequently have been widely investigated in experiments [3,4], simulations [5,6] and theoretical analyses [7][8][9][10], to study the impacts of elongation κ, inverse aspect ratio ε, triangularity δ, Shafranov shift ∆ and their radial derivatives.Through the gyro-kinetic equations and widely used Miller local equilibrium [11], Gao investigated the elongation effects on GAMs [7,8] in the high safety factor limit.Starting from the ideal magnetohydrodynamic equations, Wahlberg and Graves also explored the elongation effects [9] as well as the triangularity effects [10] in a global shaping model.Recently, Chen and Ren [12] obtained the GAM dispersion relation by fully considering shaping effects in gyro-kinetic model, especially triangularity effects, which showed basic consistency with Tokamak à Configuration Variable (TCV) experimental observations [10].
The physics of energetic particles (EPs) is a core issue in magnetic confinement fusion, especially in future burning plasmas with large populations of energetic alpha particles from fusion reactions.GAMs can be driven by EPs known as the energetic particle-induced geodesic acoustic modes (EGAMs), which were first theoretically predicted by Fu in 2008 [13] and discovered experimentally almost simultaneously [14].EGAMs have been observed in nearly all main tokamaks [15][16][17][18][19][20][21] and are generally accepted as providing a new pathway for energy and momentum transfer between EPs and the bulk plasma [22][23][24].In particular, direct evidence of the EGAMs impact on turbulent transport was reported in 2013 by Zarzoso et al using a flux-driven 5D gyro-kinetic simulation [25].For analytical work investigating the effects of EPs on the EGAMs, various distributions of EPs are employed, including the standard slowing-down distribution [13,26], the bump-on-tail distribution [27][28][29], and the double-shifted Maxwellian distribution [30,31].However, most of these analytical studies of EGAMs assumed infinite aspect ratio and circular cross-section, more or less for simplicity, and do not capture the geometry that is more typical in experiments.Although the shaping effects have been proven to strongly affect GAMs, these effects on EGAMs have not been widely discussed.In 2018, Di Siena et al [32] reported that plasma elongation would significantly weaken the EGAM growth rate and barely decrease the EGAM frequency by using the gyro-kinetic codes GENE and ORB5, which directly motivated the present work.In this paper, we focus on the elongation effects on EGAMs and consider its finite radial derivative s κ ≡ r∂ r κ/κ, based on the fact that κ and s κ are the two shaping parameters most strongly affecting GAMs [1,8,12].
The rest of this paper is organized as follows.In section 2, we present Miller local equilibrium [11], a closed set of gyro-kinetic equations to describe EGAMs, and derive a general EGAM dispersion relation including an arbitrary κ and a finite s κ with an arbitrary EP distribution.In section 3, motivated by the desire to compare with previous simulations [32], EPs are first modeled using the double-shifted Maxwellian distribution as done in [32], to investigate the elongation effects.Then, in section 4, we use the more physically motivated standard slowing-down distribution of EPs to perform a similar study on the impact of elongation on EGAMs.Finally, conclusions are given in section 5.

General dispersion relation
We consider the widely used Miller local equilibrium [11] with the flux surface (R, Z) written as in which R 0 is the major radius, r is the flux label chosen as the flux surface half diameter on the equatorial midplane, θ is the generalized poloidal angle, κ and δ stand for the elongation and triangularity, respectively.In Miller equilibrium, the tokamak magnetic field ⃗ B = I(ψ)∇ξ + ∇ξ × ∇ψ can be rewritten as [7] where ξ is the toroidal angle, ψ is the magnetic flux, q is the safety factor, J = (∇r × ∇θ • ∇ξ) −1 is the Jacobian and dl/dθ = (dR/dθ) 2 + (dZ/dθ) 2 is the differential of the poloidal arc length with respect to the poloidal angle.In equation ( 2), the toroidal magnetic field B t is reached by assuming I(ψ) = B 0 (r)R 0 (r), and the poloidal magnetic field B p is obtained based on q = 1 2π ¸⃗ B•∇ξ ⃗ B•∇θ dθ.The perturbed distribution of species j (j = i, h represent bulk ions and EPs respectively) is determined by the gyrokinetic equation [33], and here we will consider only electrostatic perturbations where the perturbed guiding center distribution function for species j is given by in which the nonadiabatic part h j is governed by the electrostatic gyro-kinetic equation Here, F j 0 is the equilibrium distribution of species j, q j is the charge of species j, E is the kinetic energy, ϕ is the perturbed electrostatic potential, J 0 is the zeroth-order Bessel function, k r is the wavenumber in the flux coordinate system, ρ j = v j ⊥ /ω j c is the Larmor radius with gyro frequency is the parallel (perpendicular) velocity with respect to the magnetic field, and is the drift velocity of species j.In the right-hand side of equation ( 4), the contribution of a typical linear drive − ⃗ b/B × ∇(J 0 ϕ) • ∇F j 0 is disregarded as we are dealing with GAMs/EGAMs and k ⊥ is dominated by k r .In the following of this paper, EPs are selected as the same specie as bulk ions with unity charge, that is, q h = q i = e, m h = m i .Finally, the quasi-neutrality condition ´Fi gives the governing equation of EGAMs.All perturbations, A, are assumed to have the form (appropriate to GAMs/EGAMs) Σ m A m e imθ e i(krr−ωt) where m is poloidal mode number and k ⊥ is dominated by the radial wavenumber k r .
Equations ( 1)-( 4) and the quasi-neutrality condition are a closed set of equations to describe EGAM for arbitrary values of shaping parameters in the Miller local equilibrium model.Compared with the analytical work of GAMs with a circular cross-section, two terms in the above equations deserve more attention.One is J 0 (k r ρ j |∇r|) in equations ( 3) and ( 4), and the other is ∂ µ F j 0 in equation ( 3).The Larmor radius ρ j is a physical length, while k r is a radial wavenumber in the flux coordinate r. ∂ µ F j 0 is negligible for GAMs in isotropic tokamaks, where electrons and ions are adopted as standard Maxwellian distributions and thus ∂ µ F e,i 0 = 0.However, for anisotropic plasmas [34], especially for EPs, terms of ∂ µ F j 0 matter and cannot be disregarded directly.
According to the shaping parameters of most realistic tokamak plasmas, the assumption s κ ∼ δ ∼ s δ ∼ ∂ r R 0 ∼ ε ∼ 1/q ≪ 1 is usually adopted to obtain more specific results [12,35], where ε ≡ r/R 0 and s δ ≡ r∂ r δ/ √ 1 − δ 2 .However, the derivation of GAMs clearly shows that δ, s δ , ∂ r R 0 , ε effects on GAMs dispersion relation are all on the order of O(ε 2 ) while s κ effects are of O(ε) [12].Consequently, in the following part of this paper, we only focus on the elongation effects (including s κ ≪ 1), and set δ = s δ = ∂ r R 0 = ε ≈ 0. Our shaping assumptions are equivalent to the 'elliptic nest surface with the infinite aspect ratio' in [8].It is noted that the effects of trapped particles are neglected here due to the infinite aspect ratio assumption.Based on the above analysis, equation ( 4) is simplified to whose general solution is where the Jacobian J, |∇r|, the transit frequency and the drift frequency are given by The derivation of equation ( 7) is presented in appendix A in detail.
To solve equation ( 6) for bulk ions and EPs, we adopt assumptions as follows, k r ρ i ≪ 1 and k r ρ h ≪ 1.In the absence of EPs, one has ϕ l /ϕ 0 ∼ O (k r ρ i ) |l| [36], and only ϕ 0 and ϕ ±1 should be taken into account when we disregard the effects of high order finite-Larmor-radius (FLR) and finiteorbit-width (FOW) [37,38].In the presence of EPs, we still only consider ϕ 0 and ϕ ±1 , and our following results prove Correspondingly, only the first-order FOW and the first-order FLR effects are included in our work, i.e. solving equation ( 6) by keeping terms of order O (k r ρ i ) 2 and O (k r ρ h ) 2 .Equivalently speaking, we truncate modes at m = ±1, which means ϕ m with |m| ⩾ 2 is neglected.Although the background plasma has m = 2, 3 component naturally (see, |∇r|, ω j t , ω j d in equation ( 7)), detailed calculations prove that terms related to ϕ ±2,±3 only introduce modification of O(s 2 κ ) on EGAM dispersion relation, as shown in appendix B, while the leading order is O(s κ ).Therefore, it is reasonable to neglect ϕ m with |m| ⩾ 2, which was also adopted in the theoretical investigation of elongation effects on conventional GAM [8].There also exist theoretical papers without the mode truncation, and for example, Sasaki et al considered all ϕ m to numerically investigate a new unstable EGAM branch [39], which is excited by magnetic drift resonance in the limit of ω h d ≫ ω h t .However, in this work, we adopt ω h d /ω h t ∼ qk r ρ h ≪ 1, and the EGAM instability is actually due to the EP transit resonance [13,26,28,40].In addition, it is noticed that for terms of exp(±i ´ω j d /ω j t dθ) in equation ( 6), calculations can be simplified by using Taylor expansion directly instead of the Bessel function, as done in [12].Thus, equation (3) can be written as in which we denote In the derivation of equation ( 8), bulk ions are selected as standard Maxwellian distribution, while the EP distribution is just assumed to be The perturbed electron distribution can also be obtained from equation ( 8) by replacing j with e, and assuming the electron equilibrium distribution function is a standard Maxwellian.However, considering m e ≪ m i and ω ∼ qω i t , we have k r ρ e → 0 and ω/ω e t ≪ 1.As a result, we obtain ⟨F e 1 ⟩ s = 0 by utilizing equations ( 3), ( 5) and (7), where ⟨F e 1 ⟩ s represents the flux surface average of F e 1 and is defined as Then, the perturbed electron distribution is approximately given by the Boltzmann relation [41] where T e is the electron temperature.It should be noted in the process of obtaining equation ( 11), terms of order on O( m e /m i ) are neglected.These terms are negligible for electrostatic EGAMs/GAMs but are significant for electromagnetic modes, because the Ampere's law is utilized in electromagnetic modes.Based on the above perturbed distribution, the EGAM dispersion relation is determined by the flux surface average of the quasi-neutrality equation: where we denote The theory in equation ( 12) assumes s κ ≪ 1 and includes leading order terms O(s κ ) that appear in G 1 , G 2 : terms O(s 2 κ ) are neglected.To derive the EGAM dispersion relation, we employed two formulas, D j n and L j n , as defined in equation (12).D j n ∼ (ω j d /ω j t ) n ∼ (qk r ρ j ) n accounts for radial particle excursion driven by magnetic drift ⃗ v j d , representing FOW effects.L j n ∼ (k r ρ j ) n arises from 1 − J 2 0 and J 0 in equations ( 3) and ( 4), signifying polarization due to FLR effects.Consequently, the GAM/EGAM in equation ( 12) results from the balance and feedback of two radial current mechanisms: curvature drift-induced current and FLR-induced polarization current [1].It is noted that D j n (ω) exists an integral singularity at ω = v ∥ /(qR), representing the particle-wave resonance.For bulk ions with standard Maxwellian distribution, due to ∂ v ∥ F i 0 (v ∥ = qRω) < 0, the bulk ion-wave resonance provides a Landau damping, and consequently conventional GAMs exhibit a damping rate.On the contrary, EGAMs exhibit an unstable branch due to ∂ v ∥ F h 0 (v ∥ = qRω) > 0 and EP-wave resonance provides an inverse Landau damping.There are three terms in square brackets of equation ( 12): the first term is the FOW part corresponding to the poloidal symmetric electrostatic perturbation ϕ 0 , and the elongation effects on it are reflected in G 1 /κ 2 ; the second term is another FOW part related to sidebands ϕ ±1 with elongation modification 1/κ; the last is FLR part with elongation modification coefficient G 2 /κ 2 .Although ϕ ±1 ≪ ϕ 0 , the second term in square brackets of equation ( 12) is still retained due to D j 1 ≫ D j 2 .Based on m = ±1 components of the quasi-neutrality condition, sidebands ϕ ±1 can be obtained: in which n e is the equilibrium number density of electron.
Combining equations ( 12) and ( 14), the EGAM dispersion relation can be formally written as Equation ( 15) is a general EGAM dispersion relation with an elongated cross-section for an arbitrary EP distribution, without the effects of trapped particles.The first two terms in equation ( 15) represent FOW effects related to dominant electrostatic perturbation ϕ 0 and sidebands ϕ ±1 respectively, and the last term corresponds FLR effects related to ϕ 0 .G 1 and G 2 show the influence of elongation on FOW part and FLR part of EGAM dispersion relation, respectively.We stress that the low concentration assumption of EPs, i.e. ε h ≡ n h /n i ≪ 1 (n h and n i stand for EP and bulk ion equilibrium number density), is not utilized in the derivation of equation ( 15), and thus it is valid for an arbitrary ε h .

EP with double-shifted Maxwellian distribution
To model EPs, we first consider the double-shifted Maxwellian distribution, written as Here, v th = 2T h /m h is the thermal speed of EPs with the thermal temperature T h .This distribution has been used for EGAM in analytical and numerical calculations [30][31][32]42], and is adopted in this section to facilitate a comparison with previous simulation of elongation effects on EGAMs [32].
Although the double-shifted Maxwellian is a EP distribution neglecting slowing-down process and adopted for mathematical convenience, it preserves the essential wave-particle resonance and provides analytical tractability in the linear physics of EGAMs, particularly in investigating the relationship between GAMs and EGAMs [42].As mentioned in section 2, the EGAM instability results from the inverse Landau damping of EP-wave resonance at qRω = v ∥ .With proper selection of v 0 , 0 < qRω < v 0 is satisfied, and consequently EP-wave resonance provides the inverse Landau damping, as shown in figure 1.However, it should be pointed out that it can exhibit different behaviors in the nonlinear phase compared to the slowing-down distribution [30,31].In this paper, we focus solely on the linear behaviors of EGAMs, and in next section we will discuss the elongation effects modeling EPs by the slowing-down distribution.From equation ( 16) it follows that so the EP terms in equation ( 15) are integrated as where the subscript D represents 'double-shifted', and Given that the standard Maxwellian distribution is a special case of double-shifted Maxwellian distribution for v 0 = 0, the bulk ion integral terms in equation ( 15) (L i n and D i n ) can be directly reached by following transformation in equation ( 17): v 0 → 0, subscript/superscript h → i and neglecting subscript D, e.g.v ti = 2T i /m i and ζ i = qRω/v ti .
The theoretical lines are according to equations ( 15) and ( 17), and the simulation data are from [32].
When the elongation effects are eliminated, equations ( 15) and ( 17) are reduced to the previous result [42].When the EPs are described by the double-shifted Maxwellian distribution, the EGAM is excited by EPs from an already existing mode [31,42].This already existing mode in the absence of EPs is the initial GAM or an initially highly damped mode.Which mode is driven to be EGAM is determined by the parameters of the EP distribution and the safety factor value.For example, when q = 1.6, T i = T e = T h , v 0 = 2.83v th , the EGAM emerges from the initial GAM, while for q = 3, T i = T e = T h , v 0 = 2.83v th , the EGAM is excited from the damped mode, as demonstrated in figures 1 and 2 of [42].However, when the parameters are not significantly different and EGAMs emerge from the two already existing modes respectively, numerical calculations of equations ( 15) and (17) confirm that the plasma elongation has similar impacts on both types of EGAMs.We only focus on the unstable EGAM in this section by utilizing the same parameters as those in [32], and in this case EGAM is actually generated from the initial GAM.
Figure 2 illustrates the dependence of the EGAM frequency and growth rate on the elongation κ.It is shown that as κ increasing, the EGAM frequency decreases more slowly compared with that of GAM, while the EGAM growth rate decreases rapidly and GAM damping rate increases rapidly.These trends demonstrated in figure 2 agree well with the previous simulation [32].Although the theory derived here is quite consistent with the simulation results on the trend, it should be pointed out that quantitative discrepancy still exists for the growth rate.This quantitative discrepancy mainly results from the finite inverse aspect ratio ε effect (ε = 0.3125 in [32]).As derived in [7], κ 2 +1 ε 2 , the frequency slightly decreases when a finite ε is considered.Consequently the bulk ion damping rate increases sensitively, which could explain that the growth rates of GENE and ORB5 are smaller than our theoretical results as shown in figure 2(b).As mentioned in section 2, with the assumption δ ∼ s δ ∼ ∂ r R 0 ∼ ε ∼ 1/q ≪ 1, the effects of δ, s δ , ∂ r R 0 , ε on the GAMs dispersion relation are all on the order of O(ε 2 ) [12], which means that these parameters should be simultaneously included in analytical or numerical calculations to investigate the impact of shaping on EGAMs in more detail.We look forward to further exploring the impact of these parameters in future work.
To comprehend the diverse elongation effects on GAM and EGAM, as depicted in figure 2, we first focus on the damping/growth rate.Both GAM damping and EGAM instability are attributed to particle-wave resonance at qRω = v ∥ .The resonance strength can be roughly estimated by the value of ∂ v ∥ F j 0 (v ∥ = qRω), as illustrated in figure 3.In figure 3(a) for GAM, an increase in elongation κ significantly reduces the GAM frequency, leading to a substantial increase in the absolute value of ∂ v ∥ F i 0 (v ∥ = qRω) and, consequently, a significant strengthening of the GAM damping rate.For EGAM, as its frequency decreases with increasing κ, the bulk ion-wave resonance strengthens, as shown in figures 3(b)-(d).However, the EP-wave resonance strength exhibits variation with increasing κ: consistently increasing in figure 3(b); initially increasing and then decreasing in figure 3(c); and consistently decreasing in figure 3(d).Although the change in EP-wave resonance strength in figures 3(b)-(d) varies with increasing κ, the EGAM growth rate consistently decreases with increasing κ in all three ε h cases.This is attributed to the dominance of increasing Landau damping induced by bulk ions, given their significantly larger number compared to EPs.
Next, we examine the distinct elongation effects on EGAM frequency and GAM frequency.The results in figure 2 reveal a slow decrease in the unstable EGAM frequency as the elongation κ increases, which differs from the strong dependence of GAM frequency on κ [8].Obviously, the presence of EPs modifies the impact of κ.Considering that L i 2 + L h 2,D = ne T i (k r ρ i ) 2 , we define K D (ω) as the sum of the first two terms normalized by ne T i (k r ρ i ) 2 on the left-hand side of equation (15).Subsequently, we rewrite the EGAM dispersion relation equation (15) as K D (ω) = −G 2 .It is crucial to note that K D (ω) represents the contribution of the first-order FOW effects, while −G 2 corresponds to the first-order FLR effects.When neglecting s κ , K D (ω) is independent of κ, and κ influences the dispersion relation solely through G 2 .To focus exclusively on the frequency, we eliminate the (inverse) Landau damping in the double-shifted case by setting γ = 0.The plot of K D (ω) as a function of ω is presented in figure 4. Additionally, we plot −G 2 for two cases of κ = 1 and κ = 2 in figure 4, which is independent of ω and remains constant as ω varies.The intersections of these two types of lines in the GAM/EGAM region represent the solutions of the dispersion relation equation (15) in the double-shifted Maxwellian case, corresponding to the GAM/EGAM frequency.As shown in figure 4, the FOW part of the EGAM dispersion relation exhibits a more pronounced increase in the region ωR/c s ∈ [1,2] than that of conventional GAM.This suggests that the EGAM FOW part is more sensitive to frequency changes than the FOW part of GAM.As κ varies, changes in EGAM FLR part and GAM FLR part are identical.Consequently, the change in ω EGAM is comparatively smaller than that of ω GAM , as illustrated in figure 4.
From a mathematical standpoint, the transition from GAM to EGAM corresponds to ε h transitioning from zero to a nonzero value in the dispersion relation equation (15).Therefore, it is imperative to conduct an ε h scan to investigate the impact of EP concentration on the elongation effects, as illustrated in figure 5.The trend depicted in figure 5(a) The theoretical points of GAM (ε h = 0) and the theoretical lines of EGAM (ε h > 0) are according to equations ( 15) and (17).
indicates that κ has more pronounced effects on frequency in the smaller ε h range, almost disappearing for ε h > 0.7.The dependence of the growth rate on ε h and κ is relatively intricate, as depicted in figure 5(b).The growth rate as a function of ε h in figure 5(b) initially increases and then decreases, in the ε h ∈ [0.1, 1] range.As ε h increases from 0.1, the frequency rapidly decreases, leading to an increase in Landau damping due to bulk ion-wave resonance.Simultaneously, as ε h increases from 0.1, the EP-wave resonance strength undergoes changes in two aspects: the increasing EP number tends to enhance EP inverse Landau damping, and ∂ v ∥ F h 0 (v ∥ = qRω) initially increases and then decreases due to changes in ω.Consequently, due to these competing factors, the growth rate as a function of ε h in figure 5(b) exhibits a maximum at high ε h (in the approximate region 0.3 < ε h < 0.5).It is noteworthy that as ε h increases from 0 to 5% in figure 5(b), γ experiences a rapid reduction to a negative value and then rapidly increases (except κ = 1.9), with increasing elongation gradually smoothing out this abrupt change.This sharp decrease of γ aligns with previous theoretical and simulated results [31], resulting from the Landau damping of EPwave resonance at v ∥ = qRω.As demonstrated in figure 1, EPs contribute to Landau damping for qRω = v ∥ > v 0 and provide inverse Landau damping (positive growth rate) for 0 < qRω < v 0 .When κ is sufficiently large (as seen in κ = 1.9 in figure 5), the frequency ω is small enough and consistently less than v 0 /(qR).Consequently, the transient rapid decline of the growth rate with ε h (at low ε h ) is replaced by a steady rise.Focusing on the elongation effects, when ε ⩾ the growth rate is consistently decreased by increasing elongation, even in the region of ε h ⩾ 0.7 where frequency exhibits a fairly slight decreases with increasing κ, which means the elongation has a stabilized impact on EGAM for ε h ⩾ 0.1.Nevertheless, for a small amount of EPs (typically, h ≲ 5%), figure 5(b) indicates that the growth rate first increases with increasing elongation and then decreases, which is not specifically highlighted in [32].

EPs with slowing-down beam distribution
In addition to the double-shifted Maxwellian distribution, we also consider the widely used standard slowing-down beam distribution of EPs without pitch-angle scatting, written as where c 0 (r) = ] is the flux function with EPs number density n h = n h (ψ, B), H is the Heaviside function, E is the kinetic energy of EPs, E 0 stands for the injection energy, E c represents the critical energy of the slowing beam ions, is the magnetic moment, λ = µ/E is the pitch angle, λ 0 is the injected pitch angle, and To obtain the dependence of the slowing-down distribution on v ∥ , we must first perform an integral of F h 0 in the v ⊥ velocity space, yielding where Λ = λ 0 B is the normalized injected pitch angle and The dependence of normalized slowing-down distribution Fh 0 on v ∥ is depicted in figure 6, demonstrating a pattern similar to the double-shifted case figure 1.In the slowing-down case, the EPwave resonance at qRω = v ∥ induces inverse Landau damping for 0 Significantly, when qRω/v ti is beyond the upper limitation of v ∥ /v ti in slowing-down case, i.e.E 0 (1 − Λ)/T i , the EP-wave resonance is absent, which is a notable distinction between the double-shifted and slowing-down distributions.
Based on 3 2 and terms of EPs in equation ( 15) are integrated as Here, the subscript S represents 'slowing-down', being the EP transit frequency for initial energy E 0 , and The previous results [26,40] are recovered through assuming τ c → 0, ε h ≪ 1 and setting G 1 = G 2 = 1 to neglect elongation effects in equations ( 15) and (20).When τ c → 0, the integrals I n can be calculated analytically, though this limit is rarely of physical interest: e.g. in [27] E 0 = 170, keV, T e = T i = 4 keV, E c = 59.2 keV and consequently τ c ≈ 0.35.In general, a better choice is to numerically evaluate the dispersion relation based on equations ( 15) and ( 20).
In the case of the standard slowing-down beam, three branches of EGAMs coexist: a marginally stable branch, a damped branch, and an unstable branch.This section specifically focuses on the unstable branch, omitting the other two branches as the EGAM instability holds the utmost interest.The impact of elongation on the unstable EGAM branch in both slowing-down and double-shifted cases is visually depicted in figure 7.With an increasing κ, the unstable EGAM frequency in both cases decreases almost synchronously and much more slowly than that of GAM, and their growth rate decreases also synchronously and rapidly.The similarity in elongation effects on the unstable EGAM between the slowing-down and double-shifted cases can be attributed primarily to the similarity in the distributions in the v ∥ velocity space, as illustrated in figures 1 and 6.
Similar to figure 5, the impact of the EP concentration on the elongation effect is charted in figure 8.The trend observed in figure 8(a) aligns closely with the double-shifted case depicted in figure 5(a): elongation effects on frequency are significantly weakened with increasing ε h .Moreover, as clarified in figure 8(b), increasing elongation consistently and substantially decreases the EGAM growth rate.Notably, the transition from GAM to EGAM, corresponding to the increase of ε h from 0 to 1% in figure 8(b), results in a steady increase in γ instead of a rapid decline, a noteworthy distinction from figure 5(b).This difference arises due to the constant satisfaction of qRω/v ti < 3  √ 2 E c (1 − Λ)/T i in figure 5, where the EP-wave resonance consistently provides inverse Landau damping instead of Landau damping.Figure 8 also illustrates that the unstable EGAM frequency experiences a rapid decrease with increasing ε h for arbitrary values of κ, while the growth rate initially increases, reaching saturation with ε h , and then undergoes a slight decrease.The saturation of the growth rate can be attributed to the interplay of several factors: the enhancement of bulk ion Landau damping, the alteration of EP-wave resonance strength, and the increase in EP number, as discussed in figure 5 in the preceding section.
Another important parameter of the slowing-down model, the normalized pitch angle Λ, also influences the elongation effects, as depicted in figure 9.It is evident that the EGAM frequency experiences a slight initial decrease with increasing shown in figure 6.The lines are according to equations ( 15), ( 17) and (20).Λ, followed by a rapid decrease for Λ > 0.5, while the growth rate consistently and significantly increases with increasing Λ.With an increasing Λ, the diminishing effect of κ on the frequency is weakened and almost disappears for Λ > 0.7.However, the effect of κ on the growth rate remains significant.It should be pointed out that in most cases we can find an  15), ( 17) and (20).
unstable EGAM branch in Λ ∈ [0, 0.8] as shown in figure 9(b), and do not see the instability threshold Λ > 0.4 obtained by Qiu et al [40] in the τ c → 0 limit.Indeed, when τ c assumes a finite value (approximately τ c ≈ 0.35 in figure 9), the instability can occur over a broader range in Λ, consistent with the findings in figure 5 of [30].Finally, it is valuable to explore the influence of s κ on EGAMs, which is disregarded in the previous discussion and figures.As stated in equation ( 1), κ(r) is a function of r.However, the determination of s κ lacks self-consistency within the Miller model, making it challenging to assign a predefined value.Consequently, commonly employed empirical formulas such as s κ = 0 or s κ = (κ − 1)/κ are utilized as a practical approach, as done in [8].According to figure 10, the frequency and growth rate of EGAM unstable branch are both decreased by increasing s κ especially for a larger κ, in the both slowing-down and double-shifted Maxwellian cases.The frequency dependence on s κ is weak, while the growth rate dependence on s κ is relatively strong, which are similar to the effects of elongation κ.

Summary
In the present work, the plasma elongation effects on EGAMs are theoretically investigated based on gyro-kinetic equations and the Miller local equilibrium model.Including an arbitrary elongation κ and a finite s κ , the general EGAM dispersion relation equation ( 15) is obtained for an arbitrary distribution of EPs.The finite inverse aspect ratio ε, Shafranov shift gradient ∂ r R 0 (r), triangularity δ and triangularity radial derivative s δ are all neglected, which is justified by previous analysis on GAMs demonstrating that these parameters only enter the dispersion relation at higher order (than κ and s κ ) [12].Then, we consider the double-shifted Maxwellian distribution and the standard slowing-down beam distribution for EPs, respectively, and exact expressions related to EPs in equation ( 15) are obtained as equations ( 17) and (20).
In both cases, due to the presence of EPs, the frequency of EGAMs decreases more slowly with the increasing elongation than that of GAMs, while the EGAM growth rate decreases rapidly with κ, as shown in figures 2 and 7.These trends indicated by the theoretical results derived here are completely consistent with the previous simulations [32].The similarity in EGAM behavior in both cases is attributed to the analogous dependence of the two distributions on the v ∥ velocity space, as depicted in figures 1 and 6.The particle-wave resonance point of EGAM/GAM, qRω = v ∥ , shifts with increasing elongation, leading to an enhancement of bulk ion Landau damping as plotted in figure 3.This phenomenon is identified as the main reason for the rapid decrease in EGAM growth rate with increasing κ and the significant increase in GAM damping rate in figures 2 and 7. Based on the double-shifted distribution, the FOW part of the EGAM dispersion relation (equation (15) for ε h > 0) is found to increase much more in the EGAM/GAM frequency region than that of GAM (ε h = 0).Consequently, the EGAM frequency exhibits a weaker dependence on κ than that of GAM, as shown in figure 4.
The normalized EP concentration ε h emerges as a key parameter influencing elongation effects on EGAMs, as illustrated in figures 5 and 8. Increasing ε h significantly weakens the elongation effects on EGAM frequency, nearly eliminating these effects for ε h > 0.7.However, the elongation effects on growth rate remain significant throughout ε h ∈ [0, 1], with tendencies differing for various ε h values.For ε h > 0.1, elongation in both cases tends to decrease the EGAM growth rate.However for a low ε h (typically, ε h ≲ 5%), the EGAM growth rate first increases with κ and then decreases in the doubleshifted Maxwellian case as figure 5(a), while in the slowingdown case figure 8(b) it monotonically decreases.This trend was not explored in [32], and is attributable to the impacts of EP-wave resonance induced Landau damping in the doubleshifted case.
The elongation effects on the EGAM frequency can also be weakened by increasing the pitch angle of the slowing-down model as demonstrated in figure 9(a), but the κ effects on the growth rate change are only weakly affected at increasing pitch angle as figure 9(b).In addition, the radial dependence of elongation κ is also included because κ is usually not a constant in the radial direction.It is found s κ can also impact the EGAM frequency and growth rate, especially for a larger κ, and the effects of s κ are similar to the elongation κ effects but weaker in both cases, as demonstrated in figure 10.
The dispersion relations for EGAMs that have been derived here provide a valuable tool for understanding and modeling the impact of elongation on the properties of EGAMs in tokamak experiments.We note, however, that our theory is derived in the case of infinite aspect ratio (ε ≈ 0), and that this must be extended to higher order in order to accommodate finite ε and other higher order shaping effects.To ascertain the order of magnitude of ϕ ±2,±3 , we initially assume ϕ ±2,±3 ∼ ϕ ±1 ∼ O(k r ρ i ϕ 0 + ε h k r ρ h ϕ 0 ) ≪ ϕ 0 .
For the convenience of writing, we replace (k r ρ i + ε h k r ρ h ) with k r ρ, and use k r ρ ≪ 1 as a small parameter due to the long wavelength assumption.Subsequently, in equations (B.1) and (B.2) for m ̸ = 0, due to the presence of ϕ m , we can substitute J 0 (k r ρ j |∇r|) with 1, since our analysis incorporates only first-order FOW and first-order FLR effects, which implies solving equations (B.1) and (B.2) by retaining terms to the order of O (k r ρ) 2 .For m = 0 in equations (B.1) and (B.2), as derived in equations ( 8) and (9), |∇r| only appears in 1 − J 2 0 (k r ρ j |∇r|) and 1 − J 0 (k r ρ j |∇r|), which are both of order O(k 2 r ρ 2 j ).Therefore, even for a large κ where the m = 2 component of |∇r| is dominant, the poloidal coupling between ϕ ±2 and ϕ 0 is still weak enough, and EGAM maintains the nearly poloidally symmetric structure.
In the subsequent analysis, we proceed to determine the true magnitude order of ϕ ±2,±3 .Inserting equation (B.1) into the quasi-neutrality equation of m = ±1, ±2, ±3 poloidal components, we can obtain: Finally, the flux average of quasi-neutrality equation yields the EGAM dispersion relation: Consequently, we can deduce that the incorporation of ϕ ±2,±3 introduces only second-order modifications, O(s 2 κ ), to the EGAM dispersion relation.

(B. 6 ) 2 r
The term enclosed in the box in equation (B.6) indicates the impact of ϕ ±2,±3 on the EGAM dispersion relation.It is noteworthy that ϕ ±2 is absent in equation (B.6) due to the proven calculation:lim sκ→0 ⟨F j 1,±2 ⟩ s s 3 κ q j ϕ ±2n j T j = 0. (B.7)Based on equations (B.5)-(B.7), it is easy to find that the term enclosed in the box in equation (B.6) is on the order O(s 2 κ k ρ2 ), while other three terms are on the order O(k 2 r ρ2 ).