Proceedings of ITC/ISHW2007

Geodesic Acoustic Mode with the Existence of a Poloidal Shock
Structure
Naohiro KASUYA and Kimitaka ITOH
National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, JapanIn H-mode plasmas, two-dimensionally steep structures of the potential and density are formed, when a large
poloidal flow exists, and its formation mechanism has been studied for quantitative understanding of the particle
transport in the H-mode transport barriers. Extension of the previous two-dimensional model is carried out to investigate
parallel flow dynamics, when potential and density distributions do not satisfy the Boltzmann relation. The extended
model includes the generation mechanism of a poloidal shock structure and a geodesic acoustic mode, and theircompetitive formation can be studied.
Keywords: poloidal shock structure, geodesic acoustic mode, H mode, poloidal asymmetry, structural formation

1. Introduction
A variety of structures are formed in the toroidal
plasmas, and their formation mechanism is one of the keys tounderstand the transport phenomena. The typical example
appears in the high-confinement mode (H mode) [1]. Thesteep radial electric field plays an important role for turbulent
suppression in H-mode transport barriers [2]. In addition, apoloidally steep structure can be formed, associated with alarge poloidal flow. Theories have predicted that the poloidal
shock can appear in H-mode plasmas [3,4]. The poloidal
shock structure is a steady density or potential jump in thepoloidal direction, resulting from the plasma compressibility
and the inhomogeneity of the magnetic field by the toroidicity,and is important, because it induces radial particle fluxes toaccelerate the density pedestal formation on the L/H transition[ 5 ]. Some experiments have indicated the existence ofpoloidal asymmetry [6,7]. Therefore, it is important tounderstand the formation mechanism of H-modes byanalyzing the two-dimensional (2-D) electric field structure.
We have extended the one-dimensional (1-D) model intokamak H modes to give 2-D structures, taking into
consideration of coupling between different magnetic surfaces
by shear viscosity, and obtained 2-D potential and density
structures in edge transport barriers [5]. Our evaluation
clarifies the non-negligibility of the particle transport arisingfrom poloidal asymmetry, and the self-consistent mechanismof the density pedestal formation on the L/H transition [8].
In this paper, extensions of the previous 2-D model are
carried out to investigate two effects on the structuralformation, i.e., the deviation from the Boltzmann relation andthe parallel flow dynamics. The distribution of the potential
and density different from the Boltzmann relation contributes
to induce a particle pinch [8], and the parallel flow dynamics
is important to obtain the flow pattern, which contributes tothe structural formation [2]. These extensions enable toauthor’s e-mail: kasuya@nifs.ac.jp

analyze the geodesic acoustic mode (GAM) [9], which is theoscillatory zonal flow, caused by compressibility of the E × B
flow in the presence of the geodesic magnetic curvature [10].
The zonal flow nonlinearly interacts with turbulence anddetermines the transport level. Therefore, many experimental
observations have been made to clarify the role of the zonal
flow in plasma confinements [11]. Both the poloidal shockand the geodesic acoustic mode induce density asymmetry inthe magnetic flux surface, so their competition must be
examined to deepen our understanding of the transport barrier
physics.
The paper is organized as follows. The derivation of themodel equations is described in Sec. 2. In Sec. 3, the extreme
cases with a large and small poloidal flow are shown todeduce the formation mechanism of the poloidal shock andthe GAM. The summary is presented in Sec. 4.

2. Set of Model Equations
For analyzing the potential, the density and the flow
velocity, a set of fluid equations consists of the momentum
conservation equation, the continuity equation of the density,
the charge conservation equation, and the Ohm’s law:r td r r r r(1)
mi n Vi = J × B − ∇( pi + pe ) − ∇ ⋅ π i ,
dt
rr
∂n(2)
+ ∇ ⋅ n Vi = 0 ,
∂tr r
(3)
∇⋅ J = 0,r r r(4)
E +V × B = 0 ,rrwhere mi is the ion mass, V is the flow velocity, J is thercurrent, p is the pressure, π is the viscosity. We assume
density n = ni = ne for simplicity, where ni and ne are the ion
and electron density, respectively. We consider a large
aspect ratio tokamak with a circular cross-section and thecoordinates (r, θ, φ) are used (r: radius, θ: poloidal angle, φ:( )Proceedings of ITC/ISHW2007
toroidal angle). The magnetic field is taken to berB=⎛ 0 ⎞⎜⎟
1⎜ Bp0 (r )⎟1 + ε cosθ ⎜⎟⎝ Bφ 0 ⎠ ,(5)where ε is the inverse aspect ration. Using Eq. (4), the flow
velocity can be written to be⎛⎞⎜ − 1 ∂Φ ⎟rB∂θ⎜⎟r rr r E×B ⎜KBp
⎟V = V// +=⎜⎟nB2⎜⎟KB∂Φ1φ⎜⎟−⎜ n
Bp ∂r ⎟⎠ ,⎝(6)(7)

and Φ is the potential. Using Eq. (6), Eq. (2), the poloidal
and parallel component of Eq. (1) are given to be∂n ∂ ⎛ n ∂Φ ⎞ 1 ∂(KB p ) = 0 ,−⎜⎟+∂t ∂r ⎝ rB ∂θ ⎠ r ∂ θ2322

Bp 0 v ti C r ∂M p 01 JBp Bφ∂τmin
B0 rr r tr r t
1 Bp ⋅ ∇ ⋅ π i
1 Bp ⋅ ∇ ⋅ π i−−minmin2whereK ≡ nVp Bp ,Bpshock ordering, which is the perturbations to be O(ε 1/2), isadopted. In the case in which Mp ~ 1, the steep structure in
the poloidal direction is formed, and the perturbations
become larger than O(ε). The set of equations for obtaining
Mp0, Mp1, Φ1, n1 is derived from Eqs. (8-10), assuming Vr /Vp << 1, which is satisfied, even if a strong poloidal shockexists:∂M p1 ,
∂χ(12)= M p 0ε sin θ −∂τ∂θbulk(2⎛⎞ ∂M p 0χ2− ⎜⎜ χ −+ 2ε cosθ ⎟⎟2⎝⎠ ∂τ∂M p1∂χ+− M p0 exp(− χ )∂τ∂τ(9)KBp Bφ ∂ ⎡ B ∂Φ ⎤⎥⎢nr ∂θ ⎢⎣ Bp Bφ ∂r ⎥⎦Bp ∂ ⎛ pe
5 pi 2 3 ⎞n ⎟⎜ ln n +2 n5 3mi r ∂θ ⎝ n⎠1 r r t
1 r r tB ⋅ ∇ ⋅ π i bulk −B ⋅ ∇ ⋅ π i shear−mi nmi n)

× cosθ + 2εM p0 sin θ2∂ ⎛ K ⎞ Bφ ∂ ∂Φn ∂Φ ∂ ⎡ 1 ⎛ KB ⎞ ⎤B2 ⎜ ⎟ −−⎟ ⎥⎢ ⎜∂t ⎝ n ⎠ Bp ∂t ∂r KrB ∂θ ∂r ⎣⎢ 2 ⎝ n ⎠ ⎦⎥−∂M p1)(),(14)∂M p 0∂χ+χ∂τ∂τ∂τB0 1 ∂(M p1 − M p0ε cos θ − χM p0−
Bp0 v ti Cr ∂θ= M p0=−(, (13)2∂2χ
+ D exp(− χ ) 23∂θ∂χ 22 ∂χ+ 1 − M p0+ 2A∂θ∂θ2⎧⎪⎤ ⎫⎪∂M p0
B0 ⎡ 2 ∂ M p0− ε ⎨ D − µˆ+ 4r− 2M p0 ⎥ ⎬
⎢2r
2∂r
∂r
Bp0 ⎢⎣⎪⎩⎥⎦ ⎪⎭22∂ ⎛K⎞n ∂Φ ∂ ⎡ 1 ⎛ KBp ⎞ ⎤ Bp ∂ ⎡ 1 ⎛ KBp ⎞ ⎤⎟⎟ ⎥ +⎟⎟ ⎥⎢ ⎜⎜⎢ ⎜⎜⎜ ⎟−∂t ⎝ n ⎠ KrB ∂θ ∂r ⎢ 2 ⎝ n ⎠ ⎥ r ∂θ ⎢ 2 ⎝ n ⎠ ⎥⎣⎦⎣⎦
5 pi 2 3 ⎞1 JBp Bφ 1 Bp ∂ ⎛ pe=−⎜ ln n + 5 3 n ⎟mi nmi r ∂θ ⎝ n2n⎠r r tr r t
1 ⎛ Bp ⋅ ∇ ⋅ π i ⎞⎟1 ⎛ Bp ⋅ ∇ ⋅ π i ⎞⎟− ⎜− ⎜⎟⎟⎜mi ⎜⎝nmni ⎝,⎠ shear⎠ bulkBp ∂ ⎡ 1 ⎛ KB ⎞ 2 ⎤ Bφ ∂Φ ∂ ⎡ B ∂Φ ⎤⎟ ⎥+⎢ ⎜⎥⎢r ∂θ ⎢⎣ 2 ⎝ n ⎠ ⎥⎦ rB ∂θ ∂r ⎢⎣ Bp Bφ ∂r ⎥⎦shear

B ∂2∂E1{M p0 [exp(− χ ) − 1]}= − µˆ r 2 0∂τBp0 ∂r 2(8)+=− χM p1 +, (10)respectively. Isothermal electrons and adiabatic ions aret
assumed. The viscosity of ions π i is divided into twoterms: the bulk viscosity given by a neoclassical process[12], and shear viscosity given by an anomalous process [2].
The forms of viscosity terms are represented in Ref. [13].
The 2-D structures of the potential, the density and theflow velocity are obtained. These variables are divided into
the average part and the perturbation part: f = f0(r) + f1(r, θ),
where f represents each quantity. The variable K is replacedby Mp, which corresponds to the poloidal Mach number anddefined as
KB0(11)
Mp ≡n v tiCr .
where v ti = 2Ti mi , C r 2 = 5 6 + Te (2Ti ) , Ti and Te is
the ion and electron temperature, respectively. Here, theχ22⎞
M p 0 ⎟⎟⎠−B0 1∂(M p1 − M p0ε cos θ − χM p0 )ε cos θBp0 v ti Cr∂θ−2B0 ⎡ ∂ ⎛5 1 2⎞∂χ ⎤⎜χ +χ ⎟⎟ + ε cos θ ⎥2 ⎢2
⎜
18 Cr
∂θ ⎥⎦
Bp0 ⎢⎣ ∂θ ⎝⎠
2B0 r
1322
Bp0 v ti Cr mir rr r t⎡ JBp Bφ ⎛ Bp ⋅ ∇ ⋅ πti ⎞⎛ Bp ⋅ ∇ ⋅ π i ⎞ ⎤⎜⎟⎜
⎟ ⎥
⎢
−×−⎜⎟
⎜
⎟ ⎥ ,nn⎢⎣ n⎝⎠ bulk ⎝⎠ shear ⎦(15)+whereχ = ln(n n ) ,(16)

∂Φ1 ,
1
E1 =
Bp 0 v ti Cr ∂r(17)

4 π I ps K 0 B0 ,3 n v ti Cr 2(18)D=Proceedings of ITC/ISHW2007
A=M p02+5 1 ,36 Cr 2(19)2istheshearviscosity coefficient, and the form of Ips isµ̂
represented in Eq. (10) of Ref. [4], and depends on Mp andthe collision frequency [14]. The variables n1 and Φ1 are
replaced by χ and E1 in this set, respectively. Time t isnormalized as τ = t t p , wheretp =B0 r
Bp0 v ti C r .

Figure 1 shows the example of the 2-D structure (the detail ofthe simulation parameters is presented in Ref. [8]). With aprescribed inhomogeneous density profile as shown in Fig. 1(a), normal and enhanced radial electric field branches aretaken in the small and large gradient regions (near theboundary r - a = -5, 0 [cm], where r = a is the position of the(20)Eqs. (13) and (14) are the flux surface average and the 2ndorder of Eq. (9), respectively. Current J is obtained from Eq.
(3), which includes polarization current ε 0ε ⊥ ∂Er ∂t , andexternal components as driven by an electrode, orbit losses,etc. [15], where ε0 is the vacuum susceptibility, and ε⊥ is theperpendicular dielectric constant of a toroidal plasma, which
depends on the flow pattern and is of the order of c2 / vA2,
where vA is the Alfvén velocity [2].

3. Formation Mechanisms of Poloidal Asymmetry
A set of model equations consists of Eqs. (12-15). Thisextended model includes the generation mechanism of thepoloidal shock structure and the GAM. Some simplified cases
are described to depict the fundamental mechanisms in thissection.

(c)

3.1. Poloidal Shock Structureand strong toroidal flow damping
Vφ = 0,(22)which makes Mp to be proportional to the radial electric field,
give a simplified set of equations. Equation (14) is given to be
M p0 exp(− χ )B ∂2∂χ{M p0 [exp(− χ ) − 1]}= − µˆ r 2 0
Bp0 ∂r 2
∂τ

()

2
∂2χ
∂χ 22 ∂χD exp(− χ ) 2 + 1 − M p0+ 2A3∂θ∂θ∂θ⎞ ∂M p 0⎛
χ2− ⎜⎜ χ −+ 2ε cos θ ⎟⎟2⎠ ∂τ
⎝+270(r - a = -2cm)

0-70-1400⎧⎪⎤ ⎪⎫∂ 2 M p0
∂M p0B ⎡− ε ⎨ D − µˆ 0 ⎢2r 2+ 4r− 2 M p0 ⎥ ⎬ cos θ2B∂r∂r⎪⎩⎢
p0 ⎣⎦⎥ ⎪⎭+ 2εM p0 sin θ(d)Ep [V/m]The key mechanism for poloidal shock formation is in
the momentum conservation Eq. (1). With the existence of thelarge poloidal flow, nonlinear effects in the convective
derivative and the density gradient generate the poloidal
shock structure by coupling with toroidicity. Assumption ofBoltzmann relationeΦ(21)
n = n exp 1Ti.

(23)
In this case, an iterative process can be taken to obtain thesolution, i.e., time evolution of the 2-D structure is calculated
from Eq. (23) by substituting Mp0 obtained by Eq. (13). Then,using Boltzmann relation (21), the potential profile is obtained.

1θ/π2Fig.1 (a) Radial profiles of the density (n) normalized by
the density at the edge r = a (na) (solid line), and thereciprocal of the density scale length (dashed line).
(b) Radial profiles of the radial electric field at θ = 0(Solutions of the 1-D and 2-D model), and (c) 2-D
structure of the potential perturbation. (d) Poloidal
profile of the poloidal electric field at r - a = -2 [cm]in spontaneous H mode.

Proceedings of ITC/ISHW2007
last closed flux surface), respectively, and a critical layer isformed between these regions (at r - a ~ -1 [cm]). The steep
gradient of the radial electric field is formed in this layer.
The radial electric field profiles in the radial direction are
shown in Fig. 1 (b) by solving the 1-D and 2-D model. The
2-D structure of the potential perturbation is shown in Fig. 1(c). There exists a poloidal structure, and Fig. 1 (d) shows
the poloidal electric field profile. The poloidal shock
structure exists at θ ~ 0.2π - 0.5π in this case.

Acknowledgement
Authors acknowledge discussions with Prof. S.-I.
Itoh, Prof. M. Yagi, Prof. A. Fukuyama and Prof. Y.
Takase. This work is partly supported by the Grant-in-Aidfor Specially-Promoted Research (16002005) and for
Scientific Research (19360418) of MEXT, by Research
Fellowships of the Japan Society for the Promotion ofScience for Young Scientists, and by the collaboration
programsofNIFS(NIFS06KDAD005,NIFS07KOAP017) and of RIAM of Kyushu University.

3.2. Geodesic Acoustic Mode
The GAM exists in the toroidal plasmas by coupling
between the (m, n) = (0, 0) electrostatic potential and (1, 0)
sideband density perturbation, where m and n are the poloidal
and toroidal mode number, respectively [10]. Taking theordering that perturbations have O(ε) with a small poloidal
flow, Eq. (12), flux surface average of Eqs. (13) + (14) and Eq.
(15) with µ̂ = 0 are give to be∂M p1 ,∂χ(24)= M p 0ε sin θ −∂τ∂θ2∂M p0B∂χ ,(25)= − 0 2 ε cos θ∂τ∂θ
Bp0∂M p1∂τ

=−

∂χ ,
∂θ(26)

respectively. This set of equations describes the GAMoscillation. If the density perturbation has a sinθ dependency(27)
χ = χ r (r , t )sin θ ,Eqs. (24-26) gives the dispersion relation
ε 2 B0 22Ω2 −− kθ = 0 ,2 Bp0 2(28)

where Ω is the oscillation frequency and kθ is the wave
number in the poloidal direction. The time is normalized by
tp shown in Eq (20), and the frequency in real unit is givento beω=2Cr v ti ⎛ 2kθ ⎞ ,⎜1 + 2 ⎟q ⎟⎠2 R ⎜⎝(29)

where q is the safety factor.

4. Summary
The extended set of fluid equations, which consists ofthe momentum conservation equation, the continuity equationof the density, the charge conservation equation, and theOhm’s law, is derived to obtain 2-D structures of the potential,
the density and the flow velocity in H-mode transport barriers.
This model includes the generation mechanism of thepoloidal shock structure and the GAM, and theircompetitive formation can be studied. The derivation is thefirst step to analyze the multi-dimensionality of transport in
the toroidal plasmas, which gives quantitative understandingsof the transport barrier physics.

References[1][2]F. Wagner et al., Phys. Rev. Lett. 49, 1408 (1982).
K. Itoh, S.-I. Itoh and A. Fukuyama, Transport andStructural Formation in Plasmas (IOPP, Bristol, 1999).
[3] T. Taniuti, H. Moriguchi, Y. Ishii, K. Watanabe and M.
Wakatani, J. Phys. Soc. Jpn. 61, 568 (1992).
[4] K. C. Shaing, R. D. Hazeltine and H. Sanuki, Phys. Fluids B
4, 404 (1992).
[5] N. Kasuya and K. Itoh, Phys. Rev. Lett. 94, 195002 (2005).
[6] A. Mohri and M. Fujiwara, Nucl. Fusion 14, 67 (1974).
[7] R. J. Taylor, P. Pribyl, G. R. Tynan and B. C. Wells, in Proc.
of 15th International Conference on Plasma Physics andControlled Nuclear Fusion Research, Seville, 1994 (IAEA,Vienna, 1995) Vol. 2, p. 127.
[8] N. Kasuya and K. Itoh, submitted to Nucl. Fusion (2006).
[9] N. Winsor, J. L. Johnson and J. M. Dawson, Phys. Fluids 11,2448 (1968).
[10] For a review, e.g. P. H. Diamond, S.-I. Itoh, K. Itoh and T. S.
Hahm, Plasma Phys. Control. Fusion 47, R35 (2005).
[11] For a review, e.g. A. Fujisawa et al., Nucl. Fusion 47, S718
(2007).
[12] K. C. Shaing, E. C. Crume and W. A. Houlberg, Phys. Fluids
B 2, 1492 (1990).
[13] N. Kasuya and K. Itoh, J. Plasma Fusion Res. 81, 553(2005).
[14] N. Kasuya and K. Itoh, Phys. Plasmas 12, 090905 (2005).
[15] N. Kasuya, K. Itoh and Y. Takase, Nucl. Fusion 43, 244(2003).