# Kinetic Modelling in Nuclear Fusion

The study of irreversible transport processes in nonuniform plasmas is based on solutions of a collisional plasma kinetic equation

(1)

where f is the test-particle distribution function and z are coordinates in the six-dimensional single-particle phase space. The Vlasov operator

(2)

characterizes the dissipationless time evolution of f, where the Hamiltonian particle orbit z(t;z0) is the solution of the Hamilton equations with the initial condition z0.

In the absence of collisions (C=0), Eq. (1) is the Vlasov equation: df(z, t) / dt=0, whose solution f(z,t) is constant along a Hamiltonian particle orbit. The collision operator

(3)

in Eq. (1) characterizes the dissipative (irreversible) time evolution of f, where the bilinear collision operator C[f ; f'] describes binary collisions between test particles and field particles (with distribution f'); the summation is over all field-particle species (and includes like-particle collisions).

Solutions of the collisional kinetic equation (1) are quite difficult to obtain in general and various approximation schemes must be adopted in order to arrive at useful solutions. One such approximation scheme involves removing fast orbital time scales associated with Hamiltonian particle orbits in six-dimensional phase space. Here, the motion of magnetically confined charged particles exhibits three distinct orbital time scales:

- A fast gyration time scale associated with the gyromotion of charged particles about a magnetic field line;
- An intermediate bounce time scale associated with the periodic parallel motion of charged particle along a magnetic field line;
- A slow drift time scale associated with the perpendicular motion of charged particles across nonuniform magnetic field lines.

After choosing a slow time scale of interest, one can identify all fast time scales which are much less than a time scale of interest. For example, if we are interested in plasma dynamics on the bounce time scale then the gyromotion time scale is considered fast and it can be asymptotically removed. For each such fast orbital time scale, a pair of action-angle variables is assigned: the fast-angle variable and its canonically conjugate action (an adiabatic invariant on a certain time scale, i.e., the fastangle average of action time derivative is zero). The asymptotic elimination of a fast orbital time scale from the Vlasov operator (2) can be carried out to arbitrary order in the chosen small parameter.