H." Sugama

M. Okamoto

S. "Satake

"Satake,  S.

Okamoto,  M.

Sugama,  H."

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§1 O. Formulation of Lagrangian Neoclassical Transport Theory Satake, S. (Grad. Univ. Advanced Studies), Okamoto, M., Sugama, H. Recently, neoclassical transport in the core region of tokamaks again attracts much attention. It is well-known that there appear non-standard guiding-center orbits near the magnetic axis called "potato" orbits[l]. Typical orbit width of potato particles is as large as (q2 p2 RO)1/3, where q is the safety factor, p is the Larmor radius, and Ro is the major radius, respectively. The standard neoclassical transport theory[2,3] constructed in the small-orb it-width (SOW) approximation is not applicable to the near-axis region, and the orbital prop-erties of potato particles should be considered in ana-lyzing transport in this region. Neoclassical transport theory has usually been dis-cussed in Eulerian representation. However, to in-clude orbital properties in the transport theory, La-grangian formulation[4] was found to be suitable for a collision less (banana-regime) plasma. In this approach, transport phenomena are described by a reduced drift-kinetic equation in the space of three constants-of-motion (COM) along a collisionless particle orbit in a tokamak. We apply for the first time Lagrangian formulation to the near-axis region[5] in which the finite-orbit-width (FOW) effect becomes really important. To utilize La-grangian transport theory, we improve the treatment of like-particle collision term by adding a momentum-restoring term to a simple Lorentz operator. This trans-port theory reflects quantitatively the properties of all types of particles appearing near the magnetic axis. The Lagrangian formulation is derived from the drift kinetic equation in an Eulerian representation in the (x, E, fL) space, a . afa ( ) atfa(x, E, fL, t) + X· ax = Cab, 1 where' = d/dt, x is the guiding-center position and Cab is a collision operator. The independent variables in eq. (1) are transformed into three constants of motion in the collisionless limit (Zl,Z2,Z3) = (E,fL, (1jJ)), and the other three variables (Z4, zs, Z6). Here, E is the energy of a particle, fL is the magnetic momentun, and (1jJ) rep-resents the averaged radial position of a particle orbit. The orbit average operator for any function a(z, z) is defined as (a) == -+- f d~ d(d¢ a(z, z), (2) 47r Tp () where (3) is the poloidal period of an particle orbit. Note that we can use Z3 = (r) instead of (1jJ) when it is convenient. By using the set of variables (z, z), eq. (1) is transformed into a .afa atfa(Z, (), t) + () a() = Cab, (4) where the property a / a¢ = a / a( = 0 is used. In the collisionless regime, averaging both sides of eq. (4) yields the reduced drift-kinetic equation in the COM space, al = ~~ . (Jc / az . r(/))) = C, (5) at Jc az \ av where f = f(E, fL, (1jJ), t) and Jc is Jacobian. Equation (5) is expanded in 6b = D..b/ L « 1 and then solved order by order. Here, D..b and L are typical potato width and typical gradient scale, respectively. By taking moments of O(6~) part of eq. (5) with E and fL, one obtains the particle and heat flux equations, (6) (7) where [ ~ ] = [ t: These equations are in the comparable form to those of standard Eulerian representation. The difference arose by using (1jJ) instead of 1jJ can be seen to be factorized by An and Aq . These factors tend to be unity away from the axis and go infinity as (1jJ) --t O. We also emphasize that the transport coefficients Ajk , though we do not have enough space to write the definitions here, have a convenient form to calculate numerically. Then, by using Lagrangian formulation, we can evaluate neoclas-sical transport coefficients near the axis, with including the effect of potato orbits. l)Satake, S., Okamoto, M., Sugama, H., and Wakatani, M., Jounal of Plasma and Fusion Reserch, 77, 573 (2001). 2)F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys. 48, 239 (1976). 3)S. P. Hirshman and D. J. Sigmar, Nuclear Fusion 21, 1079 (1981). 4)Bernstein, 1. B. and Molvig, K., Phys. Fluids 26, 1488 (1983). 5)Satake, S., Okamoto, M., Sugama, Phys. Plasmas (to be published). 185 

Formulation of Lagrangian Neoclassical Transport Theory

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