Approximate expression for beam driven current in tokamak plasmas

Abstract

An approximate expression for the beam driven current in which the trapping effects of fast ions and electrons are included is derived by means of a variational principle, The beam driven current due to arbitrary fast ions can be calculated easily from this expression in an axisymmetric magnetic field with finite aspect ratio. It is shown that in a tokamak with circular cross-section this expression gives a good approximation to the results obtained by numerically solving the Fokker-Planck equation. The beam driven current due to energy diffusion, which is neglected in the approximate expression, is also estimated.

A Laguerre expansion method for the field particle portion in the linearized Coulomb collision operator

“…In this Sonine polynomial expansion procedure, we did not use an assumption of 2T i =m i ( v 2 b ( 2T e =m e that had been frequently used in past analytical theories on the fast ions' slowing down process. 1,2,14,18,20,25 From the viewpoint of the field particle portion C af ðf aM ; f f Þ, the assumption v 2 b ( 2T e =m e previously corresponded to the use of the usual small mass ratio approximation for the electron-ion collisions 9 also for the e-f collision C ef ðf eM ; f f Þ in calculations of the shielding current component in the beam driven currents. 14 For future studies requiring the C af ðf aM ; f f Þ of all thermal particle species a 6 ¼ f, however, these kinds of asymptotic limit approximations giving the e-f and the i-f collision formulas separately will be confusing and inconvenient.…”

“…Since diamagnetic-driven perpendicular and parallel flow velocities u a in toroidal plasmas are determined by (@p a /@r)/n a /Z a of the species a, and this radial gradient scale length j@ ln p a =@rj À1 is common for all species, ju a j $ ju b j $ ju c j $ :::: ( ju f j can be assumed not only for hB f ðl¼1Þ a i in unbalanced tangential NBIs 1,21 but also for the diamagnetic driven flows due to the radial gradients 2,18 @p ?f =@r; @p kf =@r. (Perpendicular and parallel flow velocities driven by ÀcrU Â B=B 2 of the ambipolar electrostatic potential being order of jrUj $ jðrp a Þ=ðe a n a Þj (a 6 ¼ f) are negligible for fast ions' drift motions and thus not calculated in the kinetic equation for the fast ions. 1,2,18,21 ) Therefore, we do not need to retain the Galilean invariant property of the Coulomb collision in Eqs. ( 1) and ( 2) so rigorously in the determination of the f f ðx; vÞ.…”

“…in which the u b are neglected and the non-linear collision operator C ff ðf f ; f f Þ is omitted because of the low-density of fast ions themselves Z 2 f n f ( Z eff n e and the momentum/energy conservation of like-particle collisions, has been widely used in past calculations of the f f ðx; vÞ. 1,18,[21][22][23][24] In contrast to the simultaneous algebraic equation for the thermalized particles, 5,6 the algebraic handling of the Ð d 3 v integrals of the velocity distribution and the collision term is not required in this independent determination without knowing b f a ðx; vÞ of the thermalized particles. Once one has obtained the f f ðx; vÞ, the next step is to solve the kinetic equations for the b f a ðx; vÞ, including the newly added C af ðf aM ; f f Þ.…”

“…Past kinetic theories on NBI (neutral beam injection)heated or burning plasmas from the 1970s into the 1990s were constructed for investigating the beam driven currents 1 and the bootstrap currents. 2 After development of the charge exchange recombination spectroscopy, 3,4 the flow velocities of thermalized ions caused by the existence of the fast ions also are regarded as an important physics issue.…”

The Simulation of Certain Properties of Plasma Affecting Beam-Driven Current in EAST

“…Further developed theories were proposed in succession and all these theories considered some corrections based on Eq. (1) [9][10][11][12].…”