Finite Larmor radius and finite banana width corrections
Clarisse Bourdelle
03/08/05
The under estimation of the growth rates in Kinezero versus GS2 is due to the simplified way of
accounting for both the finite Larmor radius effects and for bounce average. Therefore it does
not affect the qualitative behaviour of the growth rates, but rather their absolute value and the
shape of the spectra.
[C. Bourdelle, X. Garbet, G. T. Hoang, J. Ongena, R. V.
Budny, Nuclear Fusion 42, 892 (2002)].
As detailed in appendix A.4 of [C. Bourdelle, X. Garbet, G. T. Hoang, J. Ongena, R. V. Budny,
Nuclear Fusion 42, 892 (2002)], the integration in energy of the finite Larmor radius effects is
done separately from the rest for both trapped and passing particles as shown in equation (23)
for trapped particles. Moreover, for the trapped particles, the bounce frequency does not
appear at the denominator of equation (21) with the vertical drift
n ds . It is supposed that the
bounce frequency, as the cyclotron frequency, is higher than the other frequencies
 , n ds .
And the integration with respect to energy is done also separately as shown in equations (23) and
(24).
The physics contained in these effects is the fact that, for wavelength of the unstable modes
below either the Larmor radius or the banana width, the particles concerned do not participate
anymore to the resonances. The Bessel functions integrated in energy do represent this effect,
since they decrease exponentially for either k    s or k r   s .
In GS2 and GYRO the integrals in energy of the finite Larmor radius effect Bessel functions are
done properly, with the rest of the functional. And the energy dependence of the bounce
frequency is kept at the denominator and integrated in energy.
It turns out that the approximation made in Kinezero over estimates the finite Larmor radius
and finite banana width effects.
The Bessel functions arguments are modified arbitrarily in order to be more consistent with the
spectra obtained with GS2.

 becomes Bk  / 20 , for the trapped electrons it becomes
/ 10 . For the passing particles Bk   becomes B k  .
For the trapped ions, B k r  s

B k r  sth
th
r

th
s
th
cs

th
cs
On figure 1, GS2 is compared to both Kinezero with the finite Larmor radius and finite banana
width effects over estimated and modified.
a)
b)
c)
Figure 1: spectra obtained with GS2, red crosses, with Kinezero with previous Larmor radius and
banana width effects, green line, with Kinezero with modified Larmor radius and banana width
effects, magenta circles. The parameters are:
q  2, Te  Ti  4keV , ne  3.1019 m 3 , Z eff  1, R / a  3
r / a  0.5, B  3T , a
with s=0.25 for figure 1.a, s=1 for
 T
 n
 8, a
 2, k  s  0.56
T
n
figure 1.b and s=4 for figure 1.c.
In the previous version of Kinezero, the underestimation of the growth rates was sensible mainly
at higher s, as shown on figure 1.