Neoclassical Transport
Taking the scalar product of Equations (2.163) and (2.164) with
, and
fluxsurface averaging, we obtain
where
and use has been made of Equations (2.130), (2.131), (2.155), (2.161), (2.162), (2.165), and (2.166).
Note that
and
are proportional to the species particle and heat fluxes, respectively, across (i.e., perpendicular to) magnetic fluxsurfaces. Note, further, that the crossfluxsurface components of the particle and
heat flow velocities are
smaller than the influxsurface flow
velocities discussed in Section 2.11.
Now,

(2.271) 
where use has been made of Equations (2.16) and (2.199). It follows that the crossfluxsurface
particle fluxes in tokamaks are automatically ambipolar, as a consequence of axisymmetry, quasineutrality, and collisional momentum conservation [44]. This result is of great significance because it implies that there is no preferred radial
electric field,
, in a tokamak plasma, which means that the plasma can rotate freely
in the toroidal direction. [See Equations (2.228) and (2.231).] The situation is very different
in a nonaxisymmetric magnetic confinement system, such as a stellarator, in which ambipolarity is
only attained at a certain value of the radial electric field. It follows that a stellarator plasma cannot
rotate freely in the toroidal direction [31].
Equations (2.267) and (2.268) can be combined with Equations (2.186), (2.191), and (2.196) to give
where
Making use of Equations (2.193)–(2.195), (2.197), and (2.198), as well
as the large aspectratio approximation (2.230), we obtain

(2.276) 

(2.277) 
Here, use has been made of the results
and
. According to the previous
two equations, the large parallel particle and heat flows present in a lowcollisionality plasma (see Sections 2.18 and 2.19) give rise to a transport of particles and heat across magnetic fluxsurfaces. This transport,
which is far larger than the crossfluxsurface transport predicted by the classical closure scheme (see Section 2.6),
is known as neoclassical transport [22,43].
Now, according to Equations (2.130), (2.131), (2.165), (2.166), (2.181), (2.221), (2.222), (2.243)–(2.250), (2.257), and (2.258),
It follows that
where use has been made of Equations (2.269) and (2.270). Note that we have also made use of the large aspectratio approximation that
. Furthermore, we have neglected a
term that is
smaller than the leadingorder term in Equation (2.282).
In the circular magnetic fluxsurface limit, the previous three expressions reduce to [52]
and

(2.285) 
and
where use has been made of Equations (2.202), (2.209)–(2.211), (2.243), and (2.244).
The first term on the righthand side of Equation (2.284) represents an inward flow of trapped particles driven by the inductive
parallel component of the electric field. This effect is know as the Ware pinch [51]. Consider the toroidal
component of the equation of motion of a particle of species :

(2.287) 
(Note that .) For a passing particle,
is balanced by
. In other words,
the toroidal electric field causes the toroidal velocity of the particle to continuously increase or decrease. This increase or decrease is
ultimately limited by collisions and parallel viscosity. [See Equation (2.175).] The net result is that
passing particles of both species carry a net toroidal current that is driven by the inductive electric field. For a trapped particle, on the other hand, the integral between bounces of the
lefthand side of the previous equation is zero. Thus, we obtain

(2.288) 
It follows that the timeaveraged radial velocity of a trapped particle is

(2.289) 
independent of the particle's mass or charge.
Hence, given that the trapped particle fraction is
, we expect both plasma species to
exhibit a mean radial
flow of the form

(2.290) 
The Ware pinch does indeed have this form (when we take into account the fact that
in a
large aspectratio tokamak).
It is clear from Equation (2.286) that the inward flow of trapped electrons is associated with an outward
electron heat flux.
The remaining terms in Equations (2.284)–(2.286) are all diffusive in nature. According to Fick's
law [16], we expect the diffusive component of the species radial velocity to take the form

(2.291) 
where
is the perpendicular (i.e., crossfluxsurface) particle diffusivity. If we ignore the Ware pinch and temperature gradient terms in Equation (2.284), and make the simplifying assumption that , then we
obtain

(2.292) 
This leads to the following estimate for the perpendicular particle diffusivity (which is the same for
ions and electrons)

(2.293) 
The previous equation can also be written

(2.294) 
where use has been made of Equations (2.102) and (2.202).
The previous formula can be interpreted as follows. The collisions that scatter electrons out of their trapped orbits
displace the electrons across magnetic fluxsurfaces a random distance that is of order the banana width,
.
Such collisions take place at the frequency
. [See Equation (2.94).] Hence, the trapped
electrons have a diffusivity of
[41]. However, trapped electrons only make
up a fraction of the total number of electrons, so the overall electron diffusivity is
[22,32,52].
Note that the ion diffusivity is limited to be the same as the electron diffusivity by the constraint that the
crossfluxsurface particle fluxes be ambipolar.
According to Fick's law [16], we expect the diffusive component of the ion heat flux to
take the form

(2.295) 
We can see that Equation (2.285) does indeed take this form. Our estimate for the perpendicular ion energy diffusivity is thus

(2.296) 
The previous expression can also be written

(2.297) 
We interpret the previous formula as saying that the neoclassical perpendicular ion energy diffusivity corresponds to the randomwalk motions of
trapped ions with stepfrequency
and steplength
[22,32,52].
Note that the neoclassical diffusivity specified in the previous equation is much larger, by a factor
,
that that predicted by the classical closure scheme. (See Section 2.6.)
According to Fick's law [16], we expect the diffusive component of the electron heat flux to
take the form

(2.298) 
If we neglect the Ware pinch, density gradient, and ion temperature gradient terms in Equation (2.286) then
we get

(2.299) 
Hence, our estimate for the electron energy diffusivity becomes

(2.300) 
The previous expression can also be written

(2.301) 
We interpret the previous formula as saying that the neoclassical perpendicular electron energy diffusivity corresponds to the randomwalk motions of
trapped electrons with stepfrequency
and steplength
[22,32,52].
As before, the neoclassical diffusivity specified in the previous equation is much larger, by a factor
,
that that predicted by the classical closure scheme. (See Section 2.6.)
It turns out the the neoclassical crossfluxsurface momentum diffusivities only exceeds the classical
momentum diffusivities by a factor . For example [52],

(2.302) 
Table: 2.5
Neoclassical diffusivities in a lowfield and a highfield tokamak reactor. Here, is the toroidal
magnetic fieldstrength, the perpendicular particle diffusivity,
the perpendicular electron energy diffusivity,
the perpendicular ion energy diffusivity,
and
the perpendicular ion momentum diffusivity.

LowField 
HighField 

5.0 
12.0 













Table 2.5 shows estimates for the neoclassical crossfluxsurface particle, heat, and momentum diffusivities in a tokamak
fusion reactor. Note that, although the diffusivities are much larger than the classical
crossfluxsurface diffusivities shown in Table 2.3, they are still all
much smaller than the experimentally observed crossfluxsurface diffusivities, which are
and
[52].