Derivation of Neoclassical Fluid Equations
It is helpful to define the socalled magnetohydrodynamical (MHD) velocity [29],

(2.321) 
which is the fluid velocity associated with the ion guiding centers. Here, a guiding center is the imaginary point about
which a charged particle gyrates in the plane perpendicular to the local magnetic field. According to Section 2.11, the
lowestorder ion and electron fluid velocities can be written
respectively,
where

(2.324) 
is the lowestorder electrical current density. Quasineutrality demands that

(2.325) 
The electron and ion continuity equations, (2.26) and (2.29), can be combined with Equations (2.322),
(2.323), and (2.325) to give an electron number density continuity equation of the form [29]:

(2.326) 
If we sum the electron and the ion equations of motion, (2.27) and (2.30), then we obtain
Here, is the total plasma pressure, and use has been made of Equations (2.16), (2.39), and (2.59). In the previous equation, we have neglected electron inertia with respect to ion inertia, because the former is
times smaller than the latter. We have also neglected electron viscosity with respect to ion viscosity, because the former is, at least,
times smaller than the latter. Now, it can be demonstrated that [28,29]:
This important result is known as the gyroviscous cancellation. Furthermore, it is clear from Equations (2.152)
and (2.161) that
acts predominately in the
direction (i.e.,
parallel to
). A model form for
that is consistent with the analysis of
Section 2.18 is
where

(2.330) 
Thus, the neoclassical ion parallel viscosity tensor acts to relax the ion poloidal velocity toward the neoclassical
velocity specified in Equation (2.330) on a timescale that is roughly
.
As has already been mentioned, this effect is known as poloidal flow damping. Equations (2.328) and (2.329)
can be combined with Equation (2.303) to give an MHD equation of motion of the form:
where

(2.332) 
is the poloidal flowdamping time. Of course, the value of the anomalous perpendicular momentum
diffusivity,
, must be taken from experiments.
The electron equation of motion, (2.27), can be combined with Equations (2.59), (2.139) and (2.323) to give

(2.333) 
Here, we have neglected electron inertia and gyroviscosity, because these terms are
times
smaller than the leadingorder terms. We have also neglected anomalous electron perpendicular viscosity because there are no available experimental measurements of this effect in tokamak plasmas. Let us adopt the following
model form for
which is consistent with the analysis of Sections 2.16, 2.19, and 2.20:

(2.334) 
where
Thus, the neoclassical electron parallel viscosity tensor acts to relax the perpendicular current to its lowestorder value [see Equation (2.324)], enhances the parallel electrical resistivity of the plasma, and also generates the noninductive parallel current specified in Equation (2.337). As has already been mentioned, this noninductive current
is known as the bootstrap current. Note that in Equation (2.334) we have neglected the parallel thermal force density, (2.306), in accordance with the
discussion in Section 2.23. We have also neglected the perpendicular component of the thermal
force density, (2.307), because it is
smaller than the similar term in the
previous equation that involves
.
Thus, Equations (2.333) and (2.334) yield the following generalized Ohm's law for the plasma:

(2.338) 
The electron and ion energy conservation equations, (2.28) and (2.31), yield the
following electron and ion energy conservation equations:
where


(2.341) 


(2.342) 


(2.343) 


(2.344) 
and
In writing Equations (2.339) and (2.258), use has been made of Equations (2.139), (2.226)–(2.229), (2.259)–(2.262), (2.322), and (2.323). We have neglected electronion energy exchange (i.e., ), because this is a small effect, due to the smallness of the mass ratio . [See Equation (2.37).] We have also
neglected ohmic heating (i.e.,
), because this effect is
(see Tables 1.2 and 2.1) smaller than the leading order terms. In addition, we have neglected viscous heating, because this effect is, at least,
times smaller than the leadingorder terms.
In accordance with the discussion in Section 2.23, we have neglected the parallel component of the thermal
heat flux, (2.47). We have also neglected the perpendicular component of the thermal heat flux because this term is
the same size as the classical electron perpendicular heat flux (and, therefore, much smaller than the anomalous
electron perpendicular heat flux). The parallel heat “diffusivities”,
and
, are
given the collisionless values specified in Equations (2.319) and (2.320), respectively. Of course, the values of the anomalous perpendicular heat
diffusivities,
and
, must be taken from experiments.
Note, finally, that when combined with the following subset of Maxwell's equations,
our final set of neoclassical fluid equations, (2.326), (2.331), and (2.338)–(2.340), form a complete set.