23P071-P
Anisotropic plasma MHD equilibrium Analysis in LHD
LHDにおける非等方圧力MHD平衡解析
Yoshimitsu Asahi1, Yasuhiro Suzuki1,2, Kiyomasa Watanabe1,2 and Wilfred A. Cooper3
朝日良光1, 鈴木康浩1,2，渡邊清政1,2，W. A. Cooper3
1)

The Graduate University for Advanced Studies, 322-6 Oroshi-cho, Toki 509-5292, Japan
総合研究大学院大学 〒509-5292 岐阜県土岐市下石町322-62)
National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan
核融合科学研究所 〒509-5292 岐阜県土岐市下石町322-63)
Ecole Polytechnique Fédérale de Lausanne, Centre de Recherches en Physique des Plasmas,
Association Euratom-Suisse, CH1015 Lausanne, Switzerland
In anisotropic plasmas, the pressure components parallel and perpendicular to field lines are not flux
surface quantities. The effects on the measured values caused by the pressure distortion from the surface
average are investigated in the Large Helical Device (LHD). The MHD equilibria with the anisotropicpressure are studied using a 3D MHD equilibrium code, ANIMEC, which uses the bi-Maxwellian model as
an anisotropic plasma pressure model in the 3D geometry. We study the position of magnetic axis and themagnetic flux measurements as candidates of characteristic parameter in the MHD equilibrium. As theresults, we find that the position of magnetic axis is not sensitive to the distortion from the flux surface
average, but the magnetic flux is sensitive to the distortion. This suggests that the magnetic diagnostics has
a possibility to estimate the distortion from flux surface average.

1. Introduction
High beta plasmas in LHD experiments aregenerated and maintained only by tangentiallyinjected neutral beams. Because of the long slowing
down time of high energy particles in low density
regimes and the small thermal pressure due to thelow field, the beam pressure cannot be ignoredcompared with the thermal pressure and thereby
expected to cause an anisotropy in the pressure withparallel component along the field lines p|| greaterthan p ⊥ , the perpendicular component[1]. The
MHD equilibrium theory with the anisotropicpressure predicts that the pressure is not the flux
surface quantity. The evaluation of the magnitudeof pressure distortion from the flux surface average
is an important subject to identify the MHD
equilibrium. Recently, a three dimensional MHD
equilibrium analysis code, ANIMEC, wasdeveloped [2], in which a bi-Maxwellian model wasimplemented in the VMEC code to take anisotropicpressure into account. In this paper, the effects ofthe distortion from the flux surface average on themagnetic axis positions and the magnetic flux areinvestigated for p⊥> p|| plasmas.
2. Pressure profile distortion from the flux
surface average
The bi-Maxwellian distribution function(1)

was implemented to energetic ions in the ANIMEC
code. Where s is the radial index, N(s) is the density
amplitude factor, ε is the kinetic energy, mh is themass of the high-energy particles, and T|| and T⊥arethe temperatures of high energy particles in thedirection parallel and perpendicular to magneticfield. The value T⊥/T|| is related to the pressure
anisotropy. Bc is a threshold field strength whichprescribe the magnitude of trapped particles fraction.
The trapped particles are assumed to exist in thesurfaces which include smaller field than Bc,. Fig.1
shows the magnitude of pressure profile distortion
from its surface average[5]. For the case which(a)(b)Fig.1 The maximum absolute value of thedifference between the parallel pressure and its
flux surface average normalized to its averagevalue on axis (a) and the corresponding maximum
absolute value for the perpendicular pressure (b) as
a function of the ratio of <p ⊥ > to <p||>corresponding to the strength of the anisotropy at
the magnetic axis[5].

includes trapped particles, the ratio of the surface
averaged pressure p⊥ to p|| at the axis <p⊥>/<p||>|ρ=0 does not be unity even when T⊥/T|| = 1, and thedifference of pressure from its averaged value
remains considerable, whereas the distortion
decrease as <p ⊥ >/<p||>| ρ =0 decrease in the casewithout trapped particle.
2. Effects of the trapped particles on the axis
position and the magnetic flux
First of all, we define the equilibrium beta
value which is expected to yields theequivalent magnetic axis shift to theisotropic pressure equilibria with the same
beta value. From a previous study[6] with
CGL formula p = p⊥I + (p|| - p⊥)nn and a low
β ordering, a beta value proportional to thePfirsch-Schlüter current can be defined [5] as(2).
Fig.2 shows the axis position analyzed in(a)(b)

Fig.2 The dependence of the magnetic axis
position obtained by a model which trapped
particles does not exist(a) and exist(b). The
gray points correspond to the isotropicpressure plasma axis positions which T⊥/T|| =1 [5].

(a)(b)

Fig.3 The dependence of the magnetic flux
obtained by a model which trapped particles
does not exist(a) and exist(b). The gray points
correspond to the isotropic pressure plasma
axis positions which T⊥/T|| = 1[5].

two models. One is the case without trapped
particles and another is the case withtrapped particles. All of the points in Figs.2(a, b) nearly coincides to the isotropic case.
Fig.3 shows the estimated saddle type loop flux
calculated from equilibrium current. The saddle
loop is placed at the bottom of the LHD torus, andcan detect the vertical field generated by
Pfirsch-Schlüter current. The points housed in red
circle represents the results with T⊥/T|| = 0. The
magnetic flux varies with the trapped particles exist
or not, namely the difference in the shape ofvelocity distribution and the value deviates from theisotropic case. The saddle loop flux seems to detect
the effect of the distortion from the surface average,
and therefore, the flux could be a candidate ofestimating the distortion.
3. Summary
The distortion of parallel and perpendicularpressure from its surface average are estimated withp|| > p⊥. The distortion, particularly for p⊥ wasstrongly affected by the presence of the trapped
particles. To establish the method of identifying thepressure anisotropy and the shape of velocity
distribution, the response of the magnetic axis
position and the magnetic flux were examined in
LHD magnetic field configurations. For themagnetic axis, the position is depending only on βeq,
regardless of the amount of trapped particles or thepressure anisotropy. In contrast, the magnetic flux
vary in whether the trapped particles exist or not, orin other words, the difference in the velocity
distribution prescribed by Bc. This result suggeststhat the saddle loop measurement can detect thedistortion of pressure from its surface average.
Acknowledgments
This study is supported by a project oriented
course budget and a NIFS11ULPP022.
References[1] K. Y. Watanabe, et al. Fusion Science andTechnology 58, 160 (2010).
[2] W. A. Cooper, et al. Comp. Phys. Comm. 180,1524 (2009).
[3] T. Yamaguchi, et al. Plasma Physics andControlled Fusion 48, L73 (2006).
[4] W. A. Cooper, et al. Nuclear Fusion 46,683(2006).
[5] Y. Asahi, et al. Plasma and Fusion Research 6,2403123 (2011).
[6] W. N. G. Hitchon, et al. Nuclear Fusion 23,383 (1983).