The Runge-Kutta-Gill method is used for the time- integration. The simulation code is well optimized in order to achieve high efficiency for vector and parallel operations.

(1)

§3. Development of a Toroidal ITG Simulation Code for a Flux Tube Geometry

Watanabe, T.-H., Sugama, H. (NIFS)

We have developed the gyrokinetic-Vlasov simu- lation code for the toroidal ITG mode in a flux tube geometry for a tokamak configuration [1]. In the followings, some of the linear benchmark results are briefly described, while nonlinear simulation results will be shown elsewhere.

The coordinate system and the boundary concli- tion are based on the work by Beer [2], where con- centric circular magnetic surfaces with a large as- pect ratio are assumed. 'Ve consider the gyrokinetic equation for ions, the adiabatic electron response, and the quasi-neutrality. As the first trial, we deal with passing ions only, while neglecting the mirror force terrIl. Spatial coordinates in radial (x) and field-line-label (y) directions are discretized by the Fourier expansion, while the parallel (z) derivatives are approximated by the fifth-order upwind finite difference. The parallel velocity (VII) and the mag- netic moment (/1) are chosen for the velocity space coordinates which are discretized by grid points.

The Runge-Kutta-Gill method is used for the time- integration. The simulation code is well optimized in order to achieve high efficiency for vector and parallel operations.

The linear growth rate of the toroidal ITG modes for the Cyclone DIII-D base case parameters [3] are shown in Fig.l, where the solid and dashed lines represent the real and imaginary parts (denoted by

We

and,) of the eigenfrequency obtained by the lin- ear gyrokinetic code [4], respectively. Here, we have employed (84, ±20, ±32, ±64, 32) modes/grid points in the five-dimensional (kx, ky, z, VII, I,)-space, where kx and ky denote the wave numbers in the

x-

and y-directions, respectively. Solid squares and open circles indicate

We

and, given by the gyrokinetic- Vlasov simulation results which agree well with the linear code prediction.

In the absence of the electric field, the initial density perturbation n with the ballooning type mode structure is damped due to the phase mix- ing associated with the toroidal particle drift. Its asymptotic behavior is proportional to r

²

[4], since not only the parallel advection term but also the toroidal magnetic drift terms contribute to genera- tion of fine-scale structures of the distribution func- tion in the phase space. The collisionless damp- ing process can be successfully reproduced by our

simulation as shown in Fig.2, where a finer numer- ical grid for the (vlI,/1)-space is employed, such as (±192,64) grid points, in order to continue the run up to t

= 100L_n

/vti. In lack of the resolution, oth- erwise, n unphysically grows at earlier time. The result demonstrates that, also in a tokamak con- figuration, treatment of the fine-scale structures of the distribution function is one of the key issues for simulating the collisionless damping.

0.15 0.1

"'"

0.05

E-

^O

0-~-0---a

./({j/" ~~Q"

," '0,

.... 0;._ -.--.-.- ... "-.0

,...

~ a

^-0.05-0.1

w,/4 •

r

0

-0.15 -0.2

0 0.1 0.2 0.3 0.4 0.5 0.6,0.7 kePi

Figure 1: Real frequency (we) and linear growth rate (,) of the toroidal ITG modes for the Cyclone DIII-D base case obtained by the gyrokinetic-Vlasov simulation code. Solid and dashes lines indicate

We

and, obtained by the linear gyrokinetic code [4],

.ci ^~_=> 10-4

1::

"

^0. ^10-⁵

'," C 10-⁶ c

"

""

<;.. ₀ 10-⁷

"

""

^10-⁸

.~ 0.

a

10-⁹

...: 10-' 10' 10'

Time (L"Iv,;)

Figure 2: Collisionless damping of the density per- turbation in a tokamak configuration in absence of the electric field (solid), Dashed line represents the asymptotic behavior (oc r2) predicted in Ref.[4],

References

[1] T,-H.Watanabe and H.Sugama, to appear m

J,

The Runge-Kutta-Gill method is used for the time- integration. The simulation code is well optimized in order to achieve high efficiency for vector and parallel operations.

§3. Development of a Toroidal ITG Simulation Code for a Flux Tube Geometry

Watanabe, T.-H., Sugama, H. (NIFS)

We have developed the gyrokinetic-Vlasov simu- lation code for the toroidal ITG mode in a flux tube geometry for a tokamak configuration [1]. In the followings, some of the linear benchmark results are briefly described, while nonlinear simulation results will be shown elsewhere.

The Runge-Kutta-Gill method is used for the time- integration. The simulation code is well optimized in order to achieve high efficiency for vector and parallel operations.

The linear growth rate of the toroidal ITG modes for the Cyclone DIII-D base case parameters [3] are shown in Fig.l, where the solid and dashed lines represent the real and imaginary parts (denoted by

and,) of the eigenfrequency obtained by the lin- ear gyrokinetic code [4], respectively. Here, we have employed (84, ±20, ±32, ±64, 32) modes/grid points in the five-dimensional (kx, ky, z, VII, I,)-space, where kx and ky denote the wave numbers in the

and y-directions, respectively. Solid squares and open circles indicate

and, given by the gyrokinetic- Vlasov simulation results which agree well with the linear code prediction.

In the absence of the electric field, the initial density perturbation n with the ballooning type mode structure is damped due to the phase mix- ing associated with the toroidal particle drift. Its asymptotic behavior is proportional to r

[4], since not only the parallel advection term but also the toroidal magnetic drift terms contribute to genera- tion of fine-scale structures of the distribution func- tion in the phase space. The collisionless damp- ing process can be successfully reproduced by our

simulation as shown in Fig.2, where a finer numer- ical grid for the (vlI,/1)-space is employed, such as (±192,64) grid points, in order to continue the run up to t

/vti. In lack of the resolution, oth- erwise, n unphysically grows at earlier time. The result demonstrates that, also in a tokamak con- figuration, treatment of the fine-scale structures of the distribution function is one of the key issues for simulating the collisionless damping.

"'"

E-

,...

~ a

r

Figure 1: Real frequency (we) and linear growth rate (,) of the toroidal ITG modes for the Cyclone DIII-D base case obtained by the gyrokinetic-Vlasov simulation code. Solid and dashes lines indicate

and, obtained by the linear gyrokinetic code [4],

"

"

""

"

""

a

...: 10-' 10' 10'

Figure 2: Collisionless damping of the density per- turbation in a tokamak configuration in absence of the electric field (solid), Dashed line represents the asymptotic behavior (oc r2) predicted in Ref.[4],

References

[1] T,-H.Watanabe and H.Sugama, to appear m

Plasma Fus, Res, SERIES 6 (2004),

[2] J'v[,A,Beer, S.C,Cowley, and G,W,Hammett, Phys, Plasmas 2, 2687 (1995); J'v[,A,Beer, Ph,D thesis, Princeton University, 1994,

[3] A, J'v[, Dimits, G,Bateman, M, A. Beer, et ai"

Phys, Plasmas 7, 969 (2000).

[4] H,Sugama, Phys, Plasmas 6, 3527 (1999),

327