Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)581–584
000–000
J. Plasma Fusion

Electromagnetic Effect on Turbulent Transport in Tokamak Basedon Landau Fluid Global Simulation
MIYATO Naoaki1, LI Jiquan1,2 and KISHIMOTO Yasuaki1
1Naka Fusion Research Establishment, JAERI, Naka, 311-0193, Japan
2Southwestern Institute of Physics, Chengdu, P. R. China(Received: 9 December 2003 / Accepted: 20 February 2004)

Abstract
Electromagnetic effect on ion temperature gradient driven turbulence and zonal flow generated from the turbulence is investigated based on global electromagnetic Landau fluid simulation in tokamak plasmas. Turbulent transport
decreases by increasing beta in low beta regime. Two types of zonal flow are observed. One is almost stationary flow
formed in low safety factor (q) region, which is weaker in higher beta. The other is flow oscillating coherently at thegeodesic acoustic mode (GAM) frequency.
Keywords:
ITG turbulence, electromagnetic, finite beta, Landau fluid, zonal flow, geodesic acoustic mode

fluid and an ion temperature equation with HammettePerkins closure [8], and an electron Landau damping model
term is included in Ohm’s law [5]. In the electrostatic limit
with adiabatic electrons the 5-field model reduces to 3-field
ion fluid model like in Refs. [9-11]. Equations for these fields
are the following:

1. Introduction
It is believed that drift wave turbulence such as ion temperature gradient (ITG) driven turbulence and the zonal flow[1] generated from it are responsible for anomalous transport
in tokamak plasmas. Many simulation studies for electrostatic ITG turbulence have been done based on both gyrokinetic[2] and fluid [3] models. Analytical theories for coupled system of drift wave turbulence and zonal flow also have beendeveloped [4]. As well known the ITG instability is linearlystabilized by finite beta effect and the kinetic ballooning
instability is excited above some critical beta [6], where beta
is a ratio of plasma pressure to magnetic pressure. Nonlinear
electromagnetic gyrofluid flux-tube simulation showed
increase of transport with beta unexpected from linear estimation, whose detailed nonlinear mechanism is still unknown[5]. We have developed a global electromagnetic Landau
fluid code and investigated electromagnetic effect on coupledsystem of microturbulence and zonal flow.
This paper is organized as follows. The numerical model
is described in Sec. 2. In Sec. 3, results of numerical calculations are given. Finally the obtained results are summarized
in Sec. 4.

dn=adtdneq
dr∇θ φ − neq∇||v + ∇|| j+ ωd (neqφ − τ Teqn − neqTe) + Dn∇2⊥nd∇2⊥φ = − Teq

dtadneqneqdr(1 + ηi)∇θ ∇2⊥φ +(1)1neq∇|| j⎞⎛T− ωd ⎜Ti + Te + eq (1 + τ )n⎟ + DU∇4⊥φ⎠⎝neqdvdt− β Teqβ

2. Model equations
We use five-field (density n, electrostatic potential φ,parallel component of magnetic vector potential A, parallel
ion velocity v and ion temperature T) Landau fluid equation
system describing the ITG mode at finite β. Compared to theprevious resistive drift-Alfvén model (3-field) [7], the equation system includes an equation of parallel motion for ion∂A∂t

1581neq∇||na dneq
(1 + ηi + τ )∇θ A + Dv∇2⊥v
neq dr= − ∇||φ + τ+Corresponding author’s e-mail: miyaton@fusion.naka.jaeri.go.jp

Teq= − ∇||T − (1 + τ )Teqneq(2)∇||n + βτ Teqadneq

neqdr⎛πmj ⎞τ e |k||| ⎜ v − ⎟ − η j⎝
2mineq⎠(3)∇θ A(4)©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchMiyato N. et al., Electromagnetic Effect on Turbulent Transport in Tokamak Based on Landau Fluid Global SimulationdTdt= Teqadneq

neqdrηi∇θ φ − (Γ − 1)Teq∇||v⎞⎛T+ Teqωd ⎜(Γ − 1)φ + (2Γ − 1)T + (Γ − 1) eq n⎟⎝neq ⎠8Teq− (Γ − 1)

π|k|||T + DT∇⊥T,2(5)parallel current is related with magnetic potential throughAmpère’s law

j = −∇2⊥A,Fig. 1 Linear growth rate and mean heat flux as a functionof beta. They are normalized by the value for β =0.001(6)where, τ = Te0/Ti0, β = (ncTc)/(B20/µ0) is a half of beta value
evaluated on plasma center, ηi = dlnTeq/dlnneq, neq(Teq) is anequilibrium density(ion temperature) normalized by a central
value nc(Tc), B0 is a toroidal magnetic field on a magnetic
axis and Γ = 5/3 is a specific heat. We assume a circular
tokamak geometry (r,θ,ζ), where r is a radius of magnetic
surface, θ and ζ are poloidal and toroidal angles, respectively. Then operators are defined as

Figure 1 shows linear growth rate and heat flux as afunction of beta. The ITG mode is a dominant instability inthe present parameters, although kinetic ballooning mode isdominant for β = 0.006. As in Ref. [5] heat transport decreases with the linear growth rate in a low beta regime.
Figures 2 and 3 show temporal evolutions of E × B∼zonal flow and electrostatic component of heat flux 〈T ∼v Er〉 in
quasi steady state for β = 0.001, respectively, wheredf= ∂t f + [φ, f ], ∇|| f = ∂ζ f − β [A, f ],dt~1 ∂φ∼v Er = − r ∂θ . Two types of zonal flow are observed in Fig.

ω d · f = 2 [r cos θ, f ],[ f, g] =2. In an inner (low q) region zonal flow is almost stationary.
The stationary zonal flow develops around r/a = 0.31 andsuppresses heat transport (box (a) region in Fig. 3). On theother hand, the zonal flow makes coherent oscillation due tothe geodesic curvature in an outer (high q) region. The frequency of oscillation is that of geodesic acoustic mode(GAM) ~ vti/R. It seems that the GAM also regulates heattransport as pointed in Refs. [12,13]. When heat flux increases, subsequently zonal flow becomes strong and suppresses
the heat flux (box (b) in Fig. 3 and box in Fig. 2). Time evolutions of E × B zonal flow for β = 0.003 are shown in Fig. 4.
The stationary zonal flow around r/a = 0.31 is weaker thanthat for β = 0.001. This is due to nonlinear magnetic flutter.
Maxwell stress has little effect on the zonal flow generationeven in β = 0.003 case. When the magnetic nonlinear terms([A, ] term in ∇||) are dropped artificially, stronger zonal flow

1 ⎛ ∂ f ∂ g ∂ f ∂ g⎞−⎜⎟r ⎝ ∂ r ∂θ ∂θ ∂ r ⎠where = a/R is an inverse aspect ratio, a and R are minor
and major radii, respectively. Here the normalizations aretvti/a → t, r/ρi → r, ρi∇⊥ → ∇⊥, a∇|| → ∇||,AT⎞a ⎛ n eφ v,,, ⎟ → (n, φ, v, A, T )⎜ ,⎝ρi nc Tc vti β B0 ρi Tc ⎠where vti = Tc/mi, ρi = vti/ωci, ωci = eB0 /mi. Artificial dissipations (Dn, DU, Dv, DT) are included to damp the small scale
fluctuations.

3. Numerical results
Parameters used in calculations are R/a = 4, ρ i/a =0.0125, τ = 1, neq = 0.8 + 0.2e-2(r/a) , Teq = 0.35 + 0.65(1 −(r/a)2)2, q = 1.05 + 2(r/a)2. Nonlinear calculations have been
done for β = 0.001 and 0.003. All fields are expanded in
Fourier modes in the poloidal and toroidal directions,2f=

Σ f (r, t)exp[i(mθ − nζ )]
mn(7)

m, n

and finite differenciated in the radial direction. The Fouriermodes included in calculations are ones having resonant surfaces between 0.2 < r/a < 0.8 and (m,n) = (0,0), (1,0) components. The number of radial mesh is 256 for β = 0.001 or 512
for β = 0.003. Since higher radial resolution is necessary forhigher beta calculation, computational cost increases withbeta. Analysis for higher beta will be reported in future.
Artificial D’s are set to be ~ 10-7 m4 which is hyperviscositylike and resistivity η = 4 × 10-5.

Fig. 2 Temporal evolution of E × B zonal flow for β = 0.001.

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Miyato N. et al., Electromagnetic Effect on Turbulent Transport in Tokamak Based on Landau Fluid Global Simulation
Table 1 Time averaged zonal flow energy drive due toReynolds stress, Maxwell stress and geodesic curvature effect (GAM drive).

βReynolds drive
Maxwell drive
GAM drive

0.003
6.84
−0.433−5.53

turbulence in tokamaks have been performed based on thereduced 5-field Landau fluid model and finite β effect on turbulent transport and zonal flow has been investigated in thelow beta regime. The turbulent transport decreases with thelinear growth rate of the ITG mode by increasing beta. Twotypes of zonal flow are observed. One is the almost stationary
flow formed in the low q region, which is weaker at higher
beta due to the nonlinear magnetic flutter. The other is theflow oscillating coherently at the GAM frequency. TheReynolds stress is a source of zonal flow energy, while thegeodesic curvature related term acts as a loss channel for thezonal flow. The Maxwell stress has little effect on the zonal
flow in the present parameters. But its contribution increases
with beta and may be important at higher beta.
Further detailed analysis is necessary for understanding
nonlinear dynamics between the ITG turbulence and thezonal flow at finite beta. And nonlinear simulations at higher
beta are in underway.

Fig. 3 Temporal evolution of heat flux for β = 0.001.

is formed like β = 0.001 case as shown in Fig. 4(nomag).
There seems to be no significant change in the GAM oscillation.
Zonal flow energy equation is~∂ 1~~〈vE〉2 = − 〈v~Er Ω 〉〈vE〉 + βˆ 〈B r j 〉〈vE〉∂t 2− ωB〈p sin θ 〉〈vE〉,

0.001
17.1
−0.182−14.7(8)∂φ 0where 〈·〉 denotes flux surface average, 〈vE〉 = ∂ r is the E ×∼∼
B zonal flow, Ω = ∇2⊥φ is vorticity and ωB = 2a/R. The firstterm in the right hand side is a Reynolds stress term, the second is a Maxwell stress term and the final is a geodesic curvature term which shows the coupling with (m,n) = (1,0)
pressure perturbation. Here a viscous term is neglected. The
average contributions to the zonal flow energy from thesethree terms are listed in Table 1 (the final term is labeled as
GAM drive). The Reynolds stress contribution is positive anddrives the zonal flow. The mean GAM drive is negative,
which is a sink for the zonal flow as reported in Ref. [14].
The Maxwell stress contribution is very small compared tothe other contributions, but increases with beta.

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4. Summary
Nonlinear global simulations of electromagnetic ITG

Fig. 4 Temporal evolutions of E × B zonal flow for β = 0.003. The figure labeled “nomag” shows the zonal flow in the case withoutmagnetic nonlinearity.

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Miyato N. et al., Electromagnetic Effect on Turbulent Transport in Tokamak Based on Landau Fluid Global Simulation[8] G.W. Hammett and F.W. Perkins, Phys. Rev. Lett. 64,3019 (1990).
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