Geodesic Acoustic Mode Oscillation in the Low Frequency Range

In this paper, the Geodesic Acoustic Mode theory is reconsidered with regard to the linear dispersion relation obtained previously by use of the drift kinetic equation. Specifically, kinetics effects in the low frequency range are handled more precisely, suggesting an answer to the question if GAM has only a single frequency of oscillation. It is shown, for a high-aspect-ratio tokamak with circular cross-section, there are two GAM frequencies. The existence of a new solution with a lower frequency is attributed to the unique features of the geodesic response functions defined in this paper. It is also shown that inclusion of kinetic effects renders the geodesic current smaller and narrows the range of GAM existences in the configurationand the plasma parameter-spaces. It is also shown that a system of two mixed helicity (consisting of helical and toroidal ripples) have two modes in a wide range of plasma parameters: The one with the lower frequency mode is a small modification from the tokamak type GAM and the one with higher frequency transits between tokamak GAM to helical GAM depending on the relative size of the curvature of magnetic lines of force. With the transitional plasma parameter, there could be four mode frequencies. 1


Geodesic acoustic mode oscillation in the low frequency range a…
T. Watari, Y. Hamada, T. Notake, N. Takeuchi, and K. Itoh National Institute for Fusion Science, 322-6, Oroshi-cho, Toki-city, Gifu, Japan In order to understand the various appearances of geodesic acoustic modes ͑GAM͒ in experiments, the following specific problems are theoretically addressed: ͑1͒ The asymmetry of the potential field of GAMs, which is enhanced by the coupling with ion acoustic modes.It may affect GAMs in plasmas with electron temperatures higher than those of the ions.͑2͒ The possible existence of GAMs in the lower frequency range: This is discussed in connection with the uniqueness of the kinetic response of the plasma to an external field associated with the geodesic curvature of the magnetic lines of force.͑3͒ The extension of the theory to cover both tokamaks and helical systems: Differences between the helical-type and the tokamak-type GAMs are discussed in terms of their differences in connection length.In a device of mixed helicity, helical natured GAMs are predicted to appear depending on the intensity of the corresponding geodesic curvature and electron temperature.© 2006 American Institute of Physics.͓DOI: 10.1063/1.2206170͔I. INTRODUCTION 1 Since plasma drifts with the velocity E ជ ϫ B ជ , the rotation speed of the plasma is related to the radial electric field E r .The rotating plasma is provided with a restoring force by the geodesic curvature of the magnetic lines of forces, which exist intrinsically in tori.Therefore, the plasmas are subject to oscillatory motions, the existence of which was theoretically suggested by Winsoret al. 1 This oscillation is referred to as the geodesic acoustic mode ͑GAM͒ according to the origin of the restoring force.][7][8][9][10][11][12][13] The sheared flow are shown in turn to regulate transport 14,15 and thus they are attracting more and more attention.][18][19][20][21][22][23][24][25][26] Progress in the physics of flow drive and the regulation of the turbulence have been reviewed, for instance, in Refs.27 and 28.The mechanism of the generation of the radial electric field is in itself an interesting subject of research.0][31] Therefore, they may play a larger role in regulating the turbulence.
Experimental papers [16][17][18][19][20][21][22][23][24][25][26] reported various appearances of GAM oscillations and many questions have arisen: ͑1͒ The GAM oscillations are found only in limited regions of the plasma in some devices, such as the doublet-IIID ͑D-IIID͒, 16 the Axisymmetric Diverter Experiment ͑ASDEX-U͒, 25 and the Japan Atomic Energy Research Institute Fusion Toru Modified ͑JFT-IIM ͒, 21 while GAM is found in a wider domain in the plasmas in the Texas Experimental Tokamak ͑TEXT͒, 17,18 the Japanese Institute of Plasma Physics Tokamak ͑JIPP T-U͒, 19,20 the Tokamak-10 ͑T-10͒, 26 and the compact helical system ͑CHS͒. 22͑2͒ The GAM is found also in the helical system CHS and its theoretical basis has to be developed.͑3͒ An up-down asymmetry of the plasma density is predicted by theory, yet it has not been clearly detected in the experiments.͑4͒ More than two branches of GAM-like oscillations are reported in the experimental papers: GAM-like oscillations are found in two frequencies in ASDEX-U and JIPP-TII U and more than two are found in CHS.
In order to reach a full understanding of the GAM, it is necessary to reconsider as well as to conduct further experimental investigations.Particularly, the prediction of the parameter range where GAMs exist may be useful because it facilitates a comparison between the theory and experimental results.In revising the theory, we will address how the GAM is modified in kinetic description by its coupling with sound waves.The foregoing MHD-based theories suggest that it will cause a frequency shift, If there is such a mode with a high quality factor, it will effectively regulate the drift wave turbulences in the low frequency range. 15Due to the complexity of the response of the plasma in kinetic analyses, there is a possibility that a GAM-like mode is found around the frequency

͑2͒
lower than G,MHD by a factor of 1 / q.It will fall in the low frequency edge of the turbulence, where drift waves condensate due to inverse cascade processes and are finally dissipated.Therefore, the properties of this mode are important in determining the spectrum of the turbulence and thus in determining transport, even if the mode has a low quality factor.
This paper is organized as follows: In Sec.II, the dispersion relation of the GAM oscillation is formulated, clarifying in Sec.II A the approximation used, giving special attention in Sec.II B to the roles of the nonuniform field, and giving in Sec.II C the dispersion relation of the GAM on which the discussions in the following sections are made.In Sec.III, the revised theory is applied to tokamaks.deriving in Sec.III A a reduced form and applying it in Sec.III B to the two typical cases of T e / T i = 1 and T e / T i Ͼ 1.In Sec.III C, the results obtained in Sec.III B are given interpretations from the point of view of the intervention of the sound waves in the GAM mechanism.In Sec.IV, the theory is applied to helical systems, including straight ͑Sec.IV A͒ and toroidal ͑Sec.IV B͒ helical systems.In the latter, a possible coexistence of tokamak-type and helical-type GAMs is noted according to the relative intensity of the ripples in their magnetic field, if T e / T i Ͼ 1.In Sec.V, the damping rates of the modes in the GAM frequency range are studied using the dispersion relation in a full complex analysis.In Sec.VI, the applicability range of results obtained in this paper is discussed, and the problems to be solved are suggested.Conclusions are presented in Sec.VII.

A. Formulation and clarification of assumptions
The extension of GAM theory to helical systems has been addressed in our previous paper, 32 where it was found that the helical geodesic curvature of the magnetic field is the cause of the geodesic acoustic mode.Therefore, the flux coordinate has been introduced with poloidal and toroidal coordinates ͑ , ͒ on which a Fourier expansion of the magnetic field intensity is made:

͑3͒
where is the poloidal flux used as the flux surface label and m , n are the poloidal and toroidal mode numbers of the confining magnetic field.The drift kinetic equation is solved with a drift velocity of the particles ជ D expressed in terms of ␦ m,n ͑͒; the latter is associated with the curvature of the magnetic lines of force: Electrons and ions are not distinguished in Sec.II A, because their motions are subject to the same equation.Based on the straight magnetic field line coordinate ͑ , , ͒ , ជ D in Eq. ͑4͒ is written as follows: and .

͑7͒
Here, B and B are the covariant components of the toroidal and poloidal magnetic field, which are flux functions in Boozer coordinates.ͱ g is the Jacobean.We specify the coordinate to be Boozer coordinates ͑ ͱ g ϰ B −2 ͒, so that ␣ m,n = ␣ m,n ͑͒ is a flux function.We separate the distribution functions into two parts and seek their solutions: f 1 ͑U → ͒, the perturbation caused by the uniform potential field ͗͘ and f 1 ͑NU→ ͒, the part caused by the nonuniform field ͑ − ͗͒͘.͗A͘ is the flux surface average of the quantity A: Here we introduce ˜s, ˜c, and ˜0 by Equations ͑8͒ and ͑9͒ have the operators ͑ ជ ʈ + ជ D ͒ • ٌ ជ f 1 on the left-hand side.These terms are known to create harmonics due to the spatial variation of both ជ ʈ and ជ D .Since they have single-parameter dependences on time t through the spatial dependence on ͑ , ͒, their variations have to be considered simultaneously.A more detailed consideration of this problem is introduced in Ref. 33.For this paper, however, the constancy of ជ ʈ is assumed and ជ D is ignored with respect to the former.The equation is then simplified with the use of the following transformations: where with B and B the contravariant components of the magnetic field.We also assume that K ʈ,m,n is independent of ͑ , ͒, though it is correct only to the first order in ␦ m,n on the Boozer coordinates.Using these assumptions, we obtain where x ʈ = ʈ / T and x Ќ = Ќ / T with T = ͱ 2T / m.The arrow in the bracket ͑U → NU͒ indicates that the induced nonuniform perturbation is due to a uniform potential field.Solutions are obtained to the first order in ͑ T / c,0 k ͒ taking the following forms: Similarly, the nonuniform distribution functions f ˜s͑NU → NU͒ and the distribution functions due to the nonuniform potential field f ˜c͑NU→ NU͒ are obtained from the following equations, which are derived from Eq. ͑9͒:

͑16͒
The solution is obtained in the following way:

B. Determination of the nonuniform field
In determining the nonuniform field, we use quasineutrality.This scheme of calculation has been explored by Lebedev et al. 34 and was extended to helical systems in a recent work by Sugama et al. 35 The sine and cosine components of the density perturbation due to the nonuniform field are obtained by integrating Eqs.͑17͒ and ͑18͒: where Z p ͑ m,n ͒ is the ordinary plasma dispersion function, which is defined by The nonuniform density perturbation caused by the uniform field is obtained by integrating Eq. ͑15͒ over velocity space: Summing up the electric charges from electrons and ions induced by the uniform and nonuniform fields and imposing charge neutrality, we obtain Here, the subscript ͑i , e͒ has been introduced to designate the contributing species; so far the expressions have been applicable both to electrons and ions.The adiabatic approximation, k D,e 2 ͓1+ m,n,e Z p ͑ m,n,e ͔͒ → k D,e 2 , has been made in the calculation of the electron density perturbation due to the nonuniform field.Similarly, only the ion contribution is taken into consideration in calculating the density perturbation due to the uniform field; Z 2 ͑ m,n,e ͒ is small due to the small value of m,n,e .From Eq. ͑24͒, we obtain Here, we have defined for uses in the discussions in the following sections.In similar calculations using ͚ i,e 4e͓n ˜c͑U → NU͒ + n ˜c͑U → NU͔͒ =0, we obtain simultaneously ˜c = 0.It is noted that the only electron-related parameter in Eq. ͑26͒ is T e / T i ϰ ͑k D,e 2 / k D,i 2 ͒ and all other quantities are those of ions.

C. Dispersion relation of the GAM
The radial current across the flux surface due to the uniform field, J͑U͒, is obtained by integrating e ជ D • ٌf ˜s͑U → NU͒ over the flux surface: Substituting Eqs.͑5͒-͑7͒, we obtain Here, we have defined Z 1 ͑͒ by and made an approximation ͐sin 2 ͑m − n͒ ͱ gdd ϳ 1 2 VЈ in the last transformation, f ˜c͑U → NU͒ makes no contributions after integration over velocity space.
Similarly, the current across the flux surface caused by the nonuniform field is obtained as follows by use of Eq. ͑17͒: Substituting ˜s given by Eq. ͑25͒ into Eq.͑30͒, the sum of the radial currents J geo = J͑U͒ + J͑NU͒ is given by The subscript geo indicates that the quantity is due to the geodesic curvature.Similarly, "geodesic current" shall mean current that is due to the geodesic curvature, distinguished from the classical polarization current.We use the following formula of classical polarization current: and integrate it over the flux surface to calculate the total poloidal current J pol .
One of the geometric factors l 2 is defined here by

͑34͒
It is related to the scale factor of the coordinate.However, it may be approximated ͑for easy calculations͒ by the peripheral length of the poloidal cross section of the plasma.Through the discussion above, it was shown that all the mechanisms causing radial currents are proportional to ͑k T / c,0 ͒ 2 , indicating that they are a part of the lowest order finite Larmor radius effect.Therefore, only ion contributions are retained, so that the obvious subscript i is saved; T e appears as a parameter to represent electrons only in m,n .
The dispersion relation is obtained by balancing the classical polarization current with the geodesic currents:

͑35͒
or a simplified formula is obtained by defining the following quantities: Here, ˜is the proportionality constant of the classical polarization current and 5 G 2 is the GAM frequency for a standard tokamak configuration.

˜m,n
2 is the other geometric factor characterizing the confining magnetic configuration.This equation correctly reproduces the results of Ref. 32 as to the radial current due to the uniform field, and a modification to the radial current due to the nonuniform field has been made resulting from the improved model used here.

A. Basic dispersion equation for a simple tokamak
The dispersion relations of the GAM for tokamaks are obtained choosing the n = 0 harmonics selectively; the poloidal mode number m can be retained to incorporate shaping of the plasma cross section, e.g., ellipticity and triangularity.For a tokamak of circular cross section, only the m = 1 term is retained in good approximation to reduce Eq.͑36͒ to a simpler form: where replacements k ʈ,m=1,n=0 → 1/qR 0 → k ʈ , m=1,n=0 → / k ʈ T → , l =2r, and ␦ m=1,n=0 = = r / R 0 were made in Eqs.͑36͒ and ͑37͒ to obtain the relation Equation ͑39͒ is solved in the following sections for , and is calculated by using the relation = / k ʈ T , i.e., and are identified.
We have introduced three kinds of dispersion functions Z p ͑͒, Z 2 ͑͒, and Z 1 ͑͒ defined by Eqs.͑21͒, ͑23͒, and ͑29͒, respectively.It is known that the latter two functions can be rewritten by using Z p ͑͒ ͑see the Appendix for their relations and expressions in series and asymptotic expansion͒.These three functions are different in the multipliers, ␥ 0 , ␥ 1 , and ␥ 2 , in the integrand.They reflect the multiplicity of the intervention of the geodesic curvature in one process.In Fig. 1, they are plotted versus the argument showing different behaviors in the low frequency range.

B. Characteristics of the GAM for two typical parameters: T e / T i = 1 and T e / T i > 1
The formulation in the previous section states that the geodesic current J geo consists of two parts: the part caused by the uniform field, J͑U͒ = q 2 Z 1 ͑͒, and the other part caused by the nonuniform field, J͑NU͒ =−q 2 Z 2 ͑͒.Here, the common factor i ˜• ͑k ʈ T ͒ in Eq. ͑39͒ has been dropped; in this framework the polarization current is expressed as J pol = .These two geodesic terms are calculated and shown in Figs.2͑a͒ and 2͑b͒ assuming T e / T i = 1, manifesting that they are different in the dependences on and are similar in amplitudes.Thus, the uniform and nonuniform fields make similar-sized contributions to the radial current in the standard case of T e / T i =1.
Figures 3͑a͒ and 3͑b͒ show how the GAM frequencies are determined for the two typical cases of T e / T i = 1 and T e / T i =5; q = 2 is assumed common to the two cases.The solid curves give the sum of the geodesic currents J geo = J͑U͒ + J͑NU͒ and the dotted curves give the imaginary parts.The straight lines give the classical polarization current J pol = displayed with its sign reversed: As formulated in the previous section, the GAM frequencies are obtained using the values where the curves and straight lines cross.In Figs.3͑a͒ and 3͑b͒, GAM frequency = GAM is determined to be 3.5 and 5.5, respectively.
These GAM frequencies are interpreted as the ordinary GAM frequencies obtained readily from the MHD equations and classified in this paper, as the first group of solutions.
The second group of solutions is found in the low frequency range: = 1.3 in Fig. 3͑a͒ and = 2.1 in Fig. 3͑b͒.They suggest that there are two solutions in the GAM frequency range if there is one solution in the first group.These solutions of the second group may join the candidates for one of the two frequencies of GAM-like oscillations reported in JIPP-T-IIU 19 and ASDEX U 25 experimental results.It is noted that there is always a solution around ϳ 0, which may be identified as a stationary zonal flow and classified in this paper as the third group of solutions.
To conclude this section, there are two GAM-like frequencies in addition to the stationary zonal flow for both cases of T e / T i = 1 and T e / T i =5.

Ordinary GAM
Equation ͑39͒ is transformed as where we have introduced for simplification a parameter If the term on the right-hand side q 2 Z 2 ͑͒ is small, Eq. ͑41͒ gives the two decoupled dispersion relations and Therefore, Eq. ͑41͒ is viewed as an equation for predicting the coupling of the two modes with q 2 Z 2 ͑͒ regarded as a coupling constant.Equation ͑42͒ gives the standard kinetic GAM frequency treated in Ref. 29, and Eq.͑43͒, which is the dispersion equation of electrostatic waves, has two solutions in the case of e Ͼ 1.The interpretation of results obtained in Sec.III B is simple as to the first group of solutions; we expand asymptotically the three terms composing Eq. ͑41͒ to obtain a dispersion relation

͑44͒
The following solution results from this equation: where accounts for the numerical result in Sec.III B that the GAM frequency is higher if T e / T i is larger.It is noted that the correction terms contain various parallel dynamics of the particles, though they play a minor role in the first group of solutions ͑ordinary GAM͒ if q is sufficiently large.

GAM-like solutions in a lower frequency range
The second group of solutions emerges due to the swing of the geodesic currents, J͑U͒ and J͑NU͒, in the lower frequency range: In order to study the mechanism of these solutions ͑the GAM-like mode in the lower frequency range͒, we to study the properties of J͑U͒ and J͑NU͒ in more detail and consider their relative importance.
In Fig. 4͑a͒, the real and imaginary parts of D e , Re͓D e ͔ and Im͓D e ͔, are plotted versus in the case of e = 1.The real part ͑reactive part͒ varies from 2͑Ϸ1+1/ e ͒ to 1͑Ϸ1/ e ͒ as varies from 0 to 7 ͑ϳ infinity͒ corresponding to the variation of Z p ͑͒ from 0 to −1.This variation indicates that Debye screening of ions vanishes at a large value of while that of electron persists.In general, D e −1 does not show pe- * are shown vs .͑a͒ shows the D e calculated with T e / T i = 1 and ͑b͒ with T e / T i = 5.A critical difference is found between the two cases: In the case of T e / T i = 1 the D e shows a moderate variation from 2 to 1 as varies from 0 to 7. In the case of T e / T i =5, D e crosses the abscissa twice as varies from 0 to 7 ͓see the indication in ͑b͒ made by the arrows with labels A and B͔.Physically, this effect is attributed to the coupling of the GAM with two kinds of sound waves.͑c͒ shows the numerator N u * , which is common to the two cases, being independent of the value of T e / T i .culiar variations but shows monotonous variations for the case of e = 1.The swing of J͑NU͒ in Fig. 3͑a͒ is attributed to the sign change of the numerator N u * = Z 2 • Z 2 as shown in Fig. 4͑c͒; a phase variation in Z 2 ͑͒ by / 2 causes that of N u * by .
In the case of e = 5, the situation is quite different: In Fig. 4͑b͒, Re͓D e ͔ and Im͓D e ͔ are plotted versus .Re͓D e ͔ descends from 1.2= ͑1+1/ e ͒ for the variation of from 0 to 7 and crosses the abscissa twice as indicated by arrows in Fig. 4͑b͒: once at around ϳ 2.0 ͑labeled A͒ and once at ϳ 1.2 ͑labeled B͒.The coincides with the value of ϳ 2.1 in Fig. 3͑b͒, where the geodesic current makes a big swing and crosses the abscissa.The meanings of these values are understood by expanding Re͓D e ͔ asymptotically and in series using the expression given in the Appendix.
We obtain the following two forms suitable respectively for the larger and smaller solutions of Re͓D e ͔ =0: The solutions to Eqs. ͑49͒ and ͑50͒ for e = 5 are = 1.89 and = 0.77, well approximating the corresponding values ϳ 2.0 and ϳ 1.0 obtained in Fig. 4͑b͒.͓The slight disagreements between them are attributed to inaccuracies of the approximations, Eqs.͑49͒ and ͑50͒.͔Since the larger and smaller solutions have asymptotic forms = k ʈ ͱ Z i T e / m i and = k ʈ ͱ 2T i / m i for large e , we shall refer to these solutions in this paper as the electron and the ion branch of sound waves, respectively.The electron branch is known as the ion acoustic wave, having weak damping if e Ͼ 1.The ion branch is less popular and categorized as a class of quasimode having heavier damping.They are inclusively referred to as sound waves.The inversion of J͑NU͒ occurs in the case of e Ͼ 1 at the ion acoustic wave frequency.Though this inversion of the sign occurs also at the sound wave frequency of ion branch, J͑NU͒ does not change sign there in the case of e Ͼ 1, because it coincides with that where numerator Z 2 • Z 2 changes sign.The radial current due to the uniform field J͑U͒ also changes sign in the low frequency range.This is attributed to the uniqueness of the Z 1 ͑͒, where faster and slower groups of particles carry radial currents in opposite directions.The inversion of the direction occurs due to the population change of ions belonging to the two groups in the lower frequency range.
To summarize this subsection, the geodesic radial currents, J͑U͒ and J͑NU͒, eventually have similar frequency dependences, having currents in the sense of polarization current in the lower frequency range; this is a quite interesting feature that MHD equations cannot describe.The GAMlike mode in the low frequency range is thus attributed to this uniqueness of kinetic responses.
It is also pointed out that the coupling to the branches of sound waves occurs under the condition of e Ͼ 3. It is sup-posed that the experimentally observed GAM oscillations under the condition of T e / T i Ͼ 1 may contain modes influenced by the ion acoustic mode.

Nonuniformity of the potential field
The presence of this effect may be checked in the experiments or in the results of computer simulations by diagnosing the up-down asymmetry of the potential.From the discussion in the previous section we obtain

͑52͒
where an approximation = ͑Z i T e / T i ͒Z 2 =−͑Z i T e / T i ͒͑1/͒ has been made, which is derived from Eq. ͑A7͒ for a large value of .
According to recent experimental observations that ͑k r T / c,0 ͒ϳ1/q, ˜s / ˜0 may be of the order of 1 / .Though the latter is regarded as a parameter indicating smallness, ˜s / ˜0 may be detectable in experiments.If the observed GAM is affected by the ion acoustic mode, the nonuniformity is even enhanced due to the increased , making it easier to detect in experiments.
We obtain, for the case of T e / T i = 1, the window in which the GAM can exist as follows: It is expected that GAM may exist in a more limited parameter range than Eq.͑53͒, if the damping is taken into account, and may be practically prevented from existing in the core region of the plasma.On consideration of the possible enhancement of polarization current due to neoclassical effects, 33 the window of the GAM existence may be even narrower.Even under such conditions, the presence of the coupling of GAMs to ion acoustic waves ͑T e / T i ӷ 1͒ may enhance the geodesic current J͑NU͒ and allow the presence of the GAM.

A. Applications to straight helical systems
We first consider a straight helical device with dominant mode numbers ͑m , n͒ = ͑M , N͒, which is determined by the helical pitch numbers of the device.This gives a good reference for the understanding of the GAM in toroidal helical systems.
By using these integral numbers, the dispersion relation may be written as

͑54͒
Since single helicity is assumed, the structure of the equation is the same as that of tokamaks.The variable = / T k ʈ,1,0 used in the previous section is simply replaced by M,N = / T k ʈ,M,N .
Though the formulation has the same structure, the wave numbers are quite different.While k ʈ,1,0 = B / B in tokamaks, k ʈ,M,N ϵ͑MB − NB ͒ / B ϳ −NB / B in helical systems.
For standard linear helical systems, which have large toroidal pitch numbers, the latter wave number takes on much larger tan in tokamaks.Physically, it means that the connection length is shorter in helical systems than in tokamaks.This gives the GAM in helical systems different features from those of tokamaks.Since the pressure asymmetry is more easily canceled along the magnetic lines of force, the geodesic radial current has more difficulty balancing the polarization current.
The condition of the GAM existence is obtained analytically for the case of Since M / N is usually small in standard helical systems, the parameter range of GAM existences will be narrower than that of tokamaks.The previous work, Ref. 32, is based on an asymptotic expansion and therefore caution must be taken in applications to the standard parameter ͑T e / T i =1͒.An exception is the radial domain where MB / NB −1Ϸ 0, in which the connection length becomes larger and this boundary of GAM existence is relaxed.
Figures 5͑a͒ and 5͑b͒ show the determination of GAM frequencies in a straight helical system: T e / T i = 1 and T e / T i = 5.The geodesic current J geo and the polarization currents J pol are plotted versus ͑in place of M,N ͒ to facilitate a smooth connection to the next subsection.Shortage of J geo relative to J pol is evident in Fig. 5͑a͒, where T e / T i = 1 is assumed.Figure 5͑b͒ shows that a helical-type GAM can exist in the cases of T e / T i Ͼ 1.This is partly due to the general enhancement of the geodesic current, but more importantly, due to the coupling with ion acoustic mode.
Returning to Eqs. ͑47͒ and ͑49͒, we may write

͑57͒
Here, 0 2 = 1 2 e + 3 is the frequency of the ion acoustic wave and ␦ =Im͓Z p ͑ 0 ͔͒ is the imaginary part of Z p ͑͒ calculated at = 0 .Equation ͑41͒ is then approximated as follows in the vicinity of = 0 : Here, a ϵ Re͕͓ 0 + q 2 Z 1 ͑ 0 ͔͒ / q 2 ͓Z 2 ͑ 0 ͔͒ 2 ͖ and is regarded to be positive and small in the cases of inferiority of the geodesic currents to the polarization current.We obtain a quadratic equation to which the following two solutions follow: Since the right-hand side is positive independent of the sign, two solutions are predicted to be above the ion acoustic wave frequency: one above 0 by ␦ ϳ e 0 /2a and the other FIG. 5. Determination of the GAM frequency in helical systems.Real parts of J geo ͓͔ and −J pol ͓͔ are plotted vs for three cases: ͑a͒ and ͑b͒ show the calculation in the cases of a straight helical system; T e / T i = 1 is assumed in ͑a͒ and T e / T i =5 in ͑b͒.In the latter case with high T e / T i , geodesic current is enhanced and two solutions emerge in the helical GAM frequency range.͑c͒ is the calculation for a mixed helicity case; the two modes in the different frequency ranges are nearly independent and they affect each other perturbatively.
by ␦ ϳ e a 0 3 ␦ 2 / 2. The former solution ͑ϩ͒ is interpreted as a continuous modification from the ordinary GAM in tokamaks.The latter ͑Ϫ͒ is conjectured to have heavier damping.
With the present assumptions, a Ͼ 1 and ␦ Ӷ 1, neither of the two solutions is far from 0 .
Coupling of GAM with the ion acoustic mode in helical system has more significant effects than in tokamaks: It not only causes the frequency shift of GAM but also enhances the geodesic current to allow its existence.As is shown in Eq. ͑51͒, such GAM will have a larger fraction of nonuniform field s / ˜0 ϰ sin͑M − N͒ than that in the tokamaks s / ˜0 ϰ sin .

B. Applications to toroidal helical systems
Practical helical systems are toroidal and intrinsically have toroidal curvature.Therefore it is necessary to consider at least two kinds of curvatures: For the CHS device ͑m , n͒ = ͑M , N͒ = ͑2,8͒ and ͑m , n͒ = ͑1,0͒.The sizes of the tokamak and helical ripples are similar in size ␦ M,N 2 Ϸ ␦ 1,0 2 at around the r / a ϳ 2 / 3, where r is the averaged minor radius and a is that of the outermost closed flux surface.As estimated in the previous examples, the curvature due to the toroidal ripple plays a more important role due to the longer connection length.From this point of view, toroidal helical systems are expected to have at least tokamak-type GAMs even in the standard range of plasma parameters, T e / T i ϳ 1.They appear in the similar frequency range of tokamak GAM and are classified as a tokomak-type GAMs in toroidal helical systems.
With the coupling of GAM with the ion acoustic mode, helical ripples may play a substantial role.We start with the following model equation:

͑61͒
Figure 5͑c͒ shows the calculations to determine the GAM frequency in the case of T e / T i = 5, where = m=1,n=0 ϳ qN M,N is taken on the abscissa and radial currents on the ordinate.The solid curve shows the sum of the geodesic currents due to the tokamak curvature ͑m , n͒ = ͑1,0͒ and helical curvature ͑m , n͒ = ͑M =2,N =8͒.The straight line shows the classical polarization current with a reversed sign.
˜M,N = 3 is assumed to specify the strength of the helical ripple relative to the tokamak ripple, which is a value representing one third minor radius.
In Fig. 5͑c͒, a pair of solutions is found around = m=1,n=0 = 5.5, where the toroidal ͑m , n͒ = ͑1,0͒ magnetic ripple plays a dominant role.The GAM will then have the features of the tokamak type.Another pair of solutions is found around = m=1,n=0 = 32 in Fig. 5͑c͒, where the radial current due to ͑m , n͒ = ͑M =2,N =8͒ magnetic ripple plays a dominant role.The GAM will thus have features of helical GAM.It is noted that this helical-type GAM is also dependent on ˜M,N , the parameter representing the intensity of the helical geodesic curvatures relative to that of a tokamak.Thus it is predicted that a helical-type GAM emerges as the helical ripple is increased.Such a change of the mode structure may occur in heliotron-type devices spatially, because it is known that the relative intensity of the helical ripples increases with the minor radius.

A. Simple tokamak case
Full complex analyses are required for more exact examinations of the dispersion relation.In order to do this, we formally recover an external driving term 4J ext ͑͒ on the right-hand side of Eq. ͑36͒ and write Here, R͑͒ ϰ D͑͒ −1 is regarded as the response function of the electric field d ˜0͑͒ / d to the external current J ext ͑͒, which is assumed to be caused by turbulence through nonlinear phenomena including Reynolds stress and Stringer mechanisms.and are identified due to the relation = / k ʈ T .In Secs.III and IV, we derived the dispersion relation assuming Re͓D͔ =0; if is real it is equivalent to D H = 0 with D H the Hermitian part of D.
Figure 6͑a͒ shows the calculation made in a tokamak case: the response function ͉R͓͔͉ is shown in a threedimensional ͑3D͒ plot over r =Re͓͔ and i =Im͓͔.Here, parameters q = 2 and e = 5 are used.FIG. 6. Full complex calculation of the response function for a simple tokamak case: Total response function ͉R͉͑͒ is calculated over complex ͑ = m=1,n=0 ϳ qN M,N ͒ plane for ͑T e / T i = 5 and q = 2.0͒: The solutions of GAM are given as poles of the response function.In Fig. 5͑a͒, two poles are found in the fourth quadrant of the plane.The one at Re͓͔ϳ5.5 is the ordinary GAM; this mode is well defined having Im͓͔ϳ0.0.͑b͒ gives the magnified view of ͑a͒, where five satellite poles are found.
In this display, the presence of peaks suggest the possible presence of poles of R͑͒, i.e., the frequency of the GAM can be determined.There are two outstanding peaks found in the fourth quadrant of the plane: the one at around ͑ r = 5.5, i = 0.0͒ corresponds to the ordinary GAM, having a similar value of r that was obtained from Fig. 3͑b͒.The other one at around ͑ r = 0.0, i = 0.0͒ corresponds to the stationary zonal flow.Within the framework of collisionless damping ͑plateau regime͒, the stationary zonal flow has a finite damping rate.Figure 6͑a͒ also has small peaks, suggesting the presence of another group of poles.Figure 6͑b͒ is an expanded view of Fig. 6͑a͒ in the form of a contour plot; each of the peaks is confirmed to be a pole by using Cauchy's theorem.Thus it is found that the GAM-like mode in the lower frequency range, which we discussed in Sec.III, is transformed into modes with substantial damping rate in the full complex analysis.

B. Mixed helicity case
GAM in mixed helicity configurations has been studied using the same full complex analyses as described in Sec.V A. Figures 7͑a͒ and 7͑b͒ give global views of the absolute value of the response function ͉R͓͔͉, shown in a 3D plot over Re͓͔ and Im͓͔.The parameters used in the calculations are commonly q =2, ͑N , M͒ = ͑2,8͒, and =3, and the parameter T e / T i is different between the two cases: Equation ͑61͒ shows that P is a function of via the depen-dences on m=1,n=0 ͑͒ and M,N ͑͒ distinguishing if the cause of their contributions is due to tokamak ripple or helical ripple.
Since they are mutually related by m=1,n=0 is chosen as a single variable .͉R͓͔͉ is then 3D displayed over Re͓͔ and Im͓͔.
In Fig. 7͑a͒, where we assume T e / T i = 1, the helical-type GAM is found at ͑ r , i ͒ = ͑25.0,−8.0͒.It is noted that a solution is given in the full complex analysis under the condition of T e / T i = 1, where no solution was found in Sec.IV ͓see Fig. 5͑a͔͒; the solution has instead a significantly larger damping rate.We can also recognize two satellite poles, which, however, may be more difficult to find in experiments because they are deeper in the lower half plane.
In Fig. 7͑b͒, where we assume T e / T i = 5.0, the helicaltype GAM is found at frequency ͑ r , i ͒ = ͑32.0,ϳ 0.0͒.The real frequency is approximately the same as the one we found in Fig. 5͑c͒, and since i Ϸ 0, the GAM is predicted to be well defined.We can also recognize two satellite poles like the ones we find in Fig. 6͑a͒, which also may be difficult to find in experiments having a large imaginary part.From the comparison of Figs.7͑a͒ and 7͑b͒, it is found that the GAM frequency is higher and the damping rate is lower with increasing T e .
In the lower frequency range, there are tokamak-type GAMs in both Figs.7͑a͒ and 7͑b͒.Since this frequency range is crowded, an expanded view of Fig. 7͑b͒ is shown in Fig. 8, facilitating a direct comparison with Fig. 6͑a͒ with the same e assumed.It is found that both of the two major poles corresponding to a tokamak-type GAM and stationary zonal flow have moved deeper into the lower half plane.The only difference in Fig. 8 from Fig. 6͑a͒ is the presence of an added helical ripple in the former.In the low frequency range the tokamak ripple plays a dominant role and no significant difference is found in the real frequency.A significant difference between the two cases is however found in Im͓͔, which is attributed to the additional damping mechanism due to the helical ripple, which can be treated perturbatively as has been shown in Refs.32, 35, and 36.

VI. DISCUSSIONS
The possible presence of GAM in the lower frequency was suggested in this paper by using the drift kinetic equation.This mode, however, turned out to have a significant damping rate in a full complex analysis, and therefore it is difficult to convincingly predict that it will be observed in experiments.However, it may be noted that these predictions are results of solving the drift kinetic equation in the plateau regime.
Therefore, analyses in a collisionless regime are required in order to clarify the characteristics of GAM-like mode in the lower frequency range.It has been shown in this paper that geodesic currents have a maximum at a certain value of and changes sign in the low frequency range.These results will remain true even with the inclusion of a probable neoclassical polarization current.Therefore, there is a certain possibility that a new low frequency of GAM branch will be found in the collisionless ͑banana͒ regime with a damping rate smaller than that in the plateau regime.

VII. CONCLUSIONS
The geodesic mode was reconsidered in this paper in order to find some clues to understanding experimental observations of the GAM, which have different features depending on the device and operational parameter range: First, the role of the nonuniform potential field associated with the GAM was analyzed and its relation to coupling of GAMs with the two branches of sound waves were discussed.Specifically, coupling of the GAM with the ion acoustic wave was found to have strong effects in experiments carried out under the condition that the electron temperature is higher than that of ions.We therefore propose to measure the asymmetry of the potential for the clarification of this mechanism.In the present analyses, made avoiding the asymptotic expansions, it was found that there are typically two GAM frequencies: The one with higher frequency is the well-known GAM and the one found in the lower frequency range has been given little attention.The existence of the GAM in the low frequency range is attributed to the unique responses of the ions in this frequency range.The analyses with a full complex calculation showed that this mode has substantial damping.It is due to the rather strong dissipative response of ions in the low frequency range.It may, however, emerge as a well-defined mode in the treatment in a banana regime.The analyses developed in this paper allow inclusion of the multihelicity problem; in applications to toroidal helical systems, it was shown that helical devices have at least one tokamaktype GAM.It was found that a GAM of a purely helical nature occurs only under limited conditions ͑T e ӷ 3T i ͒; it emerges as the geodesic current is enhanced by the coupling with ion acoustic wave and has a strong helical asymmetry.In the systems of mixed helicity, the properties of the GAM changes from that of the tokamak type to that of the helical type as the helical geodesic curvature is increased.

APPENDIX: PROPERTIES OF THE DISPERSION FUNCTIONS
Three kinds of the dispersion functions Z 1 ,Z 2 , and Z p have been used in the main text.They differ in terms of the multiplicity with which the particle drift motion played a role: Here, their relationship and expansion formula are tabulated: Z p is the well-known plasma dispersion function.Z 1 and Z 2 are expressed by using it as follows: + ͩ͑͒ 4 + ͑͒ 2 + 1 2 ͪZ p ͑͒, ͑A4͒ Z 2 = + ͩ 2 + 1 2 ͪZ p ͑͒. ͑A5͒

Asymptotic expansion
Asymptotic expansions of Z 1 and Z 2 are obtained from that of the plasma dispersion function Z p :

Series expansions
Series expansions of Z 1 and Z 2 are also obtained from that of the plasma dispersion function Z p : Z 2 Ϸ − 4 3 3 + i ͱ ͩ

͑A11͒
These three dispersion functions are calculated and shown in Fig. 1 versus .The most notable feature is that Z 1 , the ion current caused by the uniform potential field, increases in proportion to ͑ϰ͒ in the sense of polarization current in the low frequency range.This is quite different from the case of Z p , where the current varies with the opposite coefficient of proportionality.It is also noted that Z 2 does not have a term proportional to .These unique features of the geodesic response function originate from the multiplication factors in Eqs.͑23͒ and ͑29͒.

FIG. 1 . 11 GAM,0 2 = ͫ q 2 ͩ 7 4 +
FIG.1.The dependences of the three dispersion functions on ; Z 1 ͑͒, Z 2 ͑͒, and Z p ͑͒ are shown in ͑a͒, ͑b͒, and ͑c͒, respectively.Their definitions are given by Eqs.͑29͒, ͑23͒, and ͑21͒.They show different dependences due to the different weighting functions in their integrals.Particularly, Z 1 ͑͒ is unique having a positive slope around = 0, which can be regarded as a neoclassical polarization current due to passing particles.Z p ͑͒ is the wellknown plasma dispersion function that appears limitedly in ͓see Eq. ͑26͔͒, where geodesic curvature does not play a role.

FIG. 3 .
FIG. 3. Determination of the GAM frequency: ͑a͒ T e / T i =1; ͑b͒ T e / T i =5.The real and imaginary parts of geodesic current J geo = J͑U͒ + J͑NU͒ are shown by solid and broken curves.Polarization currents are shown in ͑a͒ and ͑b͒ by straight lines with their sign reversed.The GAM frequencies are determined as the values where J pol equals J geo = J͑U͒ + J͑NU͒.Comparing ͑a͒ and ͑b͒, it is found that the geodesic current is enhanced in the case of T e Ͼ T i .FIG. 4. The components determining J͑NU͒ =−q 2 N u * / D e .The denominator D e and numerator N u* are shown vs .͑a͒ shows the D e calculated with T e / T i = 1 and ͑b͒ with T e / T i = 5.A critical difference is found between the two cases: In the case of T e / T i = 1 the D e shows a moderate variation from 2 to 1 as varies from 0 to 7. In the case of T e / T i =5, D e crosses the abscissa twice as varies from 0 to 7 ͓see the indication in ͑b͒ made by the arrows with labels A and B͔.Physically, this effect is attributed to the coupling of the GAM with two kinds of sound waves.͑c͒ shows the numerator N u * , which is common to the two cases, being independent of the value of T e / T i .

FIG. 7 .
FIG. 7. Full complex calculation of the response function in a helical system ͑mixed-helicity case͒: Total response function ͉R͉͑͒ is calculated over complex plane; parameter sets ͑T e / T i =1, ˜M,N =3, N =8͒ and ͑T e / T i =5,˜M,N =3, N =8͒ were used in calculating ͑a͒ and ͑b͒, respectively.Two major peaks appear in each figure corresponding to tokamak and helical ripples.In ͑b͒, where T e / T i is larger, the geodesic current is magnified and a helicaltype GAM solution appears with a smaller Im͓͔ value.
We have introduced = N u / D e in Sec.II B with denominator D the external uniform field ˜0 and D e is the shielding factor normalized by k D,i 2 .It is noted that takes a large value for the , satisfying e ϵ͕k D,e 2 / k D,i 2 + ͓1+Z p ͔͖͑͒ and numerator N u = Z 2 ͑͒; is interpreted as the response function of the nonuniform field ˜s to the uniform field ˜0.N u is proportional to the charge directly induced by