H.B. "Park

T. Hayashi

T." Sato

Y. Todo

National Institute for Fusion Science (NIFS-Repository)

§4. Phase Space Fluid Simulation of Ion Neoclassical Transport Park, H.B., Todo, Y., Hayashi, T., Sato, T. We have developed a new phase space fluid simula-tion code for investigation of ion neoclassical transport. Separating f into f = fo + h according to small gyro-radius ordering, we can write the drift kinetic equation as and ah at afto + vII' V fo + a· V vfo - C (fo, fo) = 0 (1) + (VII + Vd) . V h + a· V vh + Vd . V fo C (h, fo) - C (fo, fd - C (h, h) = O. (2) A steady state solution to Eq.(I) is a local Maxwellian distribution function (3) where no (r) is the zeroth order density, r minor radius, Vt = Iff the thermal velocity, T (r) the zeroth order temperature and x = ~. Vt In this work we calculate the collision as C ~ C (f{,fM) I27rVllfM ( 2x exp (-x 2 )) - (3 4) 2 er f (x) - Vi 7r + nOVt 7r x / vIIC(f{,fM)d3 v _ ~ fM (er f (x) _ 4x exp (-x 2 )) nOVt x Vi x / v2C(f{,fM)d3 v, (4) where we neglected second order terms. Thus to calcu-late the collision term, we first calculate the flow veloc-ity and determine f{ which in turn is used in Eq.(4). Equation (4) conserves a shifted Maxwellian distribu-tion function almost completely with small numerical error in the calculation of the fluid velocity. In order to verify that our numerical scheme works well for ion neoclassical transport calculation, we test our code for the following case. If we neglect Vd . V h term in Eq.(2), which is of second order but might be important with present day tokamak discharge param-eters in order to take into account finite banana width effect, In the special case of zero ion temperature gradient VT = 0, this equation has the exact steady state so-lution of shifted Maxwellian[l) (6) with the parallel fluid velocity (7) In Fig.I, we present the results of numerical calcu-lation of Eq.(5) with zero temperature gradient. In Fig.I(a) and (b) we can confirm that the numerically calculated radial particle and energy fluxes indeed drop to zero at steady state. We also present the distribution function in Fig.I(c) and (d), where we can recognize that our numerical result agrees well with the analytical solution in Eq.(6) with slight difference in fluid velocity and thus, in the distribution function. ~. r_ (a' yl lv, (e' r .... (b' (v'.v.'}/y,' (<II FIG. 1. Results of the numerical calculation of Eq.(6): (a) particle flux, (b) energy flux, (c) ion distribution function as a function of parallel velocity, and (d) ion distribution function as a function of squared perpen-dicular velocity. References [1] F.L. Hinton and R.D. Hazeltine, Rev. Modern Phys. 48, 239(1976). 313 

Phase Space Fluid Simulation of Ion Neoclassical Transport

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