Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)570–572
000–000
J. Plasma Fusion

Study of Transport Regimes in Ion Temperature
Gradient Driven Turbulence
TAKEDA Kazuo, BENKADDA Sadruddin1,
HAMAGUCHI Satoshi and WAKATANI Masahiro*
Kyoto University, Uji, 611-0011, Japan
1CNRS URA 773, Universite de Provence, France(Received: 9 December 2003 / Accepted: 23 February 2004)

Abstract
With a low-degree-of-freedom model composed of 18 ordinary differential equations (ODEs), it is shown thatintermittent oscillations are generated by nonlinear interactions between toroidal ion temperature gradient (ITG) driven
modes and self-generated sheared flows. The intermittent oscillations are also observed when higher harmonics areincluded and the number of ODEs is increased up to 56. However, in the cases considered here, if more higher harmonics are included, the saturation level of the kinetic energy becomes lower and the period of intermittency becomesshorter for the same value of the ion pressure gradient.
Keywords:
ITG-driven turbulence, low-dimensional model, transition, intermittency, avalanche, sheared flow,
Nusselt numberequations describing the toroidal ITG mode are given and its
linear stability is analyzed. We also introduce our lowdegree-of-freedom models in Sec. 2. In Sec. 3, our numericalsimulation results are presented.

1. Introduction
The anomalous transport governed by microscopic turbulence is one of the most important issues in the research ofmagnetically confined plasmas [1]. Ion temperature gradient(ITG) driven turbulence [2-5] is responsible for the anomalous ion thermal transport in the core region of tokamaks as
observed in fluid simulations [2,6], gyrofluid simulations[7,8], and gyrokinetic simulations [9].
In order to understand the physics of nonlinear interactions between anomalous transport and sheared plasma flow,
we find it useful to analyze low-degree-of-freedom models
that describe essential dynamics governed by partial differential equations.
It was shown that dynamics of resistive interchange turbulence is well described by extended Lorenz systems, i.e.,
coupled ordinary differential equations (ODEs) for low ordermodes. Takayama et al. [10] studied 5,6 and 7 ODE models
for resistive interchange turbulence and showed that anELM-like behaviour is obtained with the generation ofsheared flows and suppression of the turbulence.
The ITG mode has intrinsically twice the number ofdegrees of freedom compared with the resistive interchange
turbulence since it contains wave phenomena. Hu and Horton[11] gave an 11 ODE model for toroidal ITG turbulence anddiscussed transport barrier dynamics with oscillations in
kinetic energy such as barrier localized modes (BLM).
This article is organized as follows. In Sec. 2, model2. Models
The toroidal ITG-driven turbulence in a slab is describedby the following vorticity and ion pressure equations givenby Horton, Choi and Tang [3]:∂(∇⊥2 φ − φ) + [φ , ∇⊥2 φ ] =∂t(1 − g + K i∇⊥2 )∂φ∂p− g + µ ∇⊥2 ∇⊥2φ ,∂y∂y(1)∂p∂φ+ [φ , p] = − K i + κ∇⊥2 p,∂t∂y(2)∂a ∂b ∂a ∂b−∂x ∂y ∂y ∂x(3)where[a , b] ≡are the Poisson brackets. Here g = 2Ln/R is the effective gravity due to curvature of magnetic field, Ki = Ti/Te(ηi + 1) is theequilibrium ion pressure gradient with ηi = d ln Ti/d ln n, µ isthe viscosity and κ is the thermal conductivity. Here x and ycorrespond to the radial and poloidal directions, respectively.

Corresponding author’s e-mail: takeda@center.iae.kyoto-u.ac.jp
1570©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Takeda K. et al., Study of Transport Regimes in Ion Temperature Gradient Driven Turbulence

The standard drift wave units x ≡ x/ρs, y ≡ y/ρs, t ≡ (Ln/cs)t, p≡ LnTi0 /(ρsPi0Te0)p, φ ≡ eLn/(B0Te0ρs)φ, µ ≡ eLn/(B0Te0ρs)µ,and κ ≡ eLn/(B0Te0ρs)κ are used for normalization, where cs isthe ion sound velocity, ρs = cs /Ωi, and Ωi is the ion cyclotron
frequency.
By linearizing Eqs. (1) and (2), the following dispersion
relation of the toroidal ITG mode is obtained,observed in the 11 ODE model also appears in the 18 ODE
model. The noteworthy difference between the 11 and 18ODEs is that, only in the case of 18 (or larger) ODEs, intermittent bursts (so called avalanches) are observed in the time
evolution of the kinetic energy and the Nusselt number when
Ki is large. This results from the competition of the three factors, i.e., ITG turbulence that generates shared flows, thesheared flows that suppress the ITG turbulence, and viscositythat suppresses the sheared flows.

[− i(1 + k ⊥2 )ω + i k y(1 − g − Ki k ⊥2 ) + µ k ⊥2 ]× (− iω + κ k ⊥2 ) − gKi k y2 = 0.

(4)

3. Numerical results
The kinetic energy due to velocity fluctuations may be
calculated from the vorticity equation (1) as

The dispersion relation (4) gives the linear growth rate as

ω=1{ky(1 − g − Ki k ⊥2 ) − i [κ k ⊥2 (1 + k ⊥2 )2(1 + k ⊥2 )+ µ k ] ± Dk },4⊥∫K = φ 2 + (∇⊥φ)2dV/V = 〈φ 2 + (∇⊥φ)2〉V .

(5)

So the kinetic energy can be calculated separately for shear
flow components (my = 0 mode) and other harmonic modes
as follows

whereDk = {ky(1 − g − Ki k ⊥2 ) + i[κ k ⊥2 (1 + k ⊥2 ) − µ k 4⊥ ]}2− 4(1 + k ⊥2 )gKi k y2.

(11)

Nx

K0 =(6)∑ 4 (1 + m k )(φ ) ,12 2x x

0 2mx(12)mx = 1

From Eq. (5), the most unstable mode is estimated ask x2 << k y2 ∼ k ⊥2 =

1− g.
Ki

1Kmx,my = (1 + m2xkx2 + m2yky2)[(φ cmx,my )2 + (φ smx,my )2 ]. (13)8(7)

To characterize the anomalous thermal transport, we also calculated the Nusselt number, which is the ratio of the totalheat transport including convective or anomalous transport tothe conductive heat transport defined as

Therefore, as the fundamental mode of our numerical simulations, we have selected a mode close to the most unstablemode satisfying

kx = ky / 2(8)∫Nu = 1 + pvxdVwith kx = [(1 − g)/5Ki] and ky = [4(1 − g)/5Ki] .
In order to construct low-degree-of-freedom models,variables φ(t, x, y) and p(t, x, y) are expanded as follows,
1/2

Nx

φ (t, x, y) =1/2

cmx,mydP0(14)iFigures 1 and 2 show simulation results for Ki = 8 from the18 ODE model, which corresponds to Nx = 3 and Ny = 1.
Time evolution of the kinetic energy is shown in Fig. 1.

Ny∑ ∑ [φˆ〈 pvx 〉 V/ ⎛⎜⎝κ dx ⎞⎟⎠ = 1 + κ K .

(t)sin(mxkxx)cos(mykyy)mx = 1 my = 0+ φˆms x,my (t)sin(mxkxx)sin(mykyy)],Nxp(t, x, y) =(9)Ny∑ ∑ [ pˆ

cmx,my(t)sin(mxkxx)cos(mykyy)mx = 1 my = 0+ p̂ ms x,my (t)sin(mxkxx)sin(mykyy)],(10)

where Nx (Ny) is the maximum mode number in the x (y)direction. By substituting Eqs. (9) and (10) into Eqs. (1) and(2), we obtain coupled ordinary differential equations(ODEs) for the harmonics. The 11 ODE model given by Huand Horton [9], which corresponds to Nx = 2 and Ny = 1, generates an L-H like transition and oscillatory behaviour.
However the obtained sheared flow is not strong enough tosuppress the ITG turbulence and therefore the transition
seems to be weak. So we have increased the degrees of freedom from 11 to 18 (Nx = 3 and Ny = 1) [12] and more in orderto analyze the anomalous transport and sheared flows generated by nonlinear ITG modes with higher accuracy. In thesmall Ki region of Ki ≤ 3, a bifurcation process similar to thatFig. 1 Time evolution of (a) total kinetic energy and kinetic
energy of (b) the sheared flow K0, (c) the 1st harmonic
K1,1, (d) the 2nd harmonic K2,1, and (e) the 3rd harmonic K3,1 in the case of Nx = 3 and Ny = 1 for Ki = 8.

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Takeda K. et al., Study of Transport Regimes in Ion Temperature Gradient Driven TurbulenceFig. 2 Time evolution of the Nusselt number in the caseof Nx = 3 and Ny = 1 for Ki = 8.

Fig. 4 Time evolution of the Nusselt number in the case ofNx = 4 and Ny = 3 for Ki = 8.

Intermittent bursts, so called avalanches, are caused by nonlinear interactions between the ITG modes and the shearedflows. As the ITG modes grow, the sheared flows are generated through the Reynolds stress. Sufficiently strong shearedflows can suppress the ITG modes quickly. After the ITG
modes are suppressed, with the lack of a flow source, thesheared flows gradually decay due to viscosity. Time evolution of the Nusselt number Nu is presented in Fig. 2, where
one sees that the Nusselt number bursts at the time when theITG modes grow rapidly.
For comparison, we have also studied the case of Nx = 4and Ny = 3. Simulation results for Nx = 4 and Ny = 3 (56 coupled ODEs) are shown in Figs. 3 and 4 for Ki = 8, i.e., thesame value as that of Figs. 1 and 2. Intermittent bursts of thekinetic energy and the Nusselt number fluctuations are alsoobserved. However the saturation level of the kinetic energy
is lower and the period of bursts is shorter than those in thecase of Nx = 3 and Ny = 1.

In summary, it is verified that the essential nonlinear
behaviour of the system can be at least qualitatively accounted for by nonlinear interactions of several low order harmonics. We also find that such intermittent bursts also occur in
our model of the 24 ODEs (Nx = 4 and Ny = 1), the 30 ODEs
(Nx = 3 and Ny = 2), and the 40 ODEs (Nx = 4 and Ny = 2)
although their saturation levels and the periods of bursts aredifferent. Among these cases, the mode saturation levels andburst periods decrease as the number of higher modes included in the ODE model increases. These quantities may converge to certain values when we increase the degrees of freedom. We believe these models, which contain the mostunstable mode and several low-order harmonic modes, arelikely to capture the essential dynamical effects of the original physical system.
* deceased

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Fig. 3 Time evolution of (a) total kinetic energy and kinetic
energy of (b) the sheared flow K0, (c) K1,1, (d) K2,1, and(e) K3,1 in the case of Nx = 4 and Ny = 3 for Ki = 8.

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