Classical Closure Scheme

(2.32) | ||

(2.33) |

For particle, momentum, and energy flows perpendicular to magnetic field-lines, the small (compared to unity) dimensionless expansion parameters upon which the classical asymptotic closure scheme is based are and . Here, we are assuming that the variation length-scales of quantities perpendicular to magnetic field-lines are of order the plasma minor radius, . As is apparent from Table 2.2, the expansion parameters and are indeed small compared to unity in tokamak fusion reactors.

For particle, momentum, and energy flows parallel to magnetic field-lines, the small (compared to unity) expansion parameters upon which the classical asymptotic closure scheme
is based are and . Here, we are assuming that the variation length-scales of quantities parallel to magnetic field-lines are
of order the *connection length*,
, which is the typical distance a magnetic field-line has to travel in order to fully traverse a magnetic-flux surface. As is clear from Table 2.2, the expansion parameters and are actually both large compared to unity in tokamak fusion reactors. It follows that the classical asymptotic closure scheme fails in such reactors (because the confined plasmas are not sufficiently collisional in nature). Nevertheless, in the following, we shall describe the classical asymptotic closure scheme, before attempting to repair it.

According to the classical asymptotic closure scheme [6],

Here, is a unit vector parallel to the magnetic field, and is the plasma current density. Moreover, the
It can be seen that parallel component of the *friction force density*, , is smaller than the perpendicular component by a factor ; this is a consequence of the fact that the collision
frequency decreases with increasing velocity (
), causing the distribution of electrons with large parallel velocities to be more distorted from a Maxwellian
distribution than that of slower electrons [30]. The *thermal force density*, , is also a consequence of the velocity dependence of the collision frequency [18]. (See Section 2.23.)

The electron and ion heat fluxes are written [6]

respectively, where are known as the parallel, cross, perpendicular, and thermal heat fluxes, respectively. Here, theAccording to the previous expressions, the diffusion of heat parallel to magnetic field-lines is characterized by the diffusivities

(2.54) | ||

(2.55) |

(2.56) | ||

(2.57) |

In order to describe the viscosity tensor in a highly magnetized plasma, it is
helpful to define the species- *rate-of-strain tensor*:

(2.58) |

In a highly magnetized plasma, the viscosity tensor is conveniently described as the sum of three component tensors [6]:

where(2.60) | ||

(2.61) | ||

(2.62) |

The tensor
** describes what is known as ***parallel viscosity*; this is a viscosity that controls the diffusion of parallel (to the magnetic field) momentum along magnetic field-lines.
The parallel
viscosity coefficients are given by [6]

(2.63) | ||

(2.64) |

(2.65) | ||

(2.66) |

(2.67) | ||

(2.68) |

According to the previous expressions, the diffusion of parallel momentum parallel to magnetic field-lines is characterized by the diffusivities

(2.69) | ||

(2.70) |

(2.71) | ||

(2.72) |

Table 2.3 shows estimates for the classical heat and momentum diffusivities in a tokamak fusion reactor. It can be seen that the electron parallel diffusivities are much larger than the ion parallel diffusivities, but that both are extremely large. On the other hand, the ion perpendicular diffusivities are much larger than the electron perpendicular diffusivities, but both are much smaller than the experimentally observed perpendicular diffusivities, which are all approximately [52].