Trapped and Passing Particles
The motion of a charged particle in the presence of a uniform magnetic field consists of
gyration in the plane perpendicular to the local magnetic field-line combined with a steady drift along the field-line [18].
Suppose, however, that the magnetic field is non-uniform, meaning that there are spatial gradients in its direction and strength.
As is well known, if the characteristic gradient scale-length of the field is much larger than the particle gyro-radius then the motion
still consists of gyration in the plane perpendicular to the local field-line, combined with drift along the field-line (which
is no longer straight). However, the magnetic moment,
|
(2.73) |
of the particle is a conserved quantity during this motion [18,40].
Here,
is the particle's perpendicular (to the magnetic field) velocity, and the local magnetic field-strength. Now, it is clear from Table 2.1
that the gyro-radii of both electrons and ions in a tokamak fusion reactor are much smaller than the dimensions of the reactor. Assuming that the characteristic
gradient scale-length of the reactor's magnetic field is comparable to its dimensions, we conclude that the motions of both ions and electrons
in such a reactor consist of rapid (see Table 2.1) gyration perpendicular to magnetic field-lines, combined with drift along
magnetic field-lines at constant magnetic moment. However, these motions are interrupted by occasional collisions (i.e., scattering events).
Figure 2.1:
Charged particle motion around a magnetic flux-surface of circular poloidal cross-section.
|
Consider the gyro-averaged motion of a charged particle around an idealized magnetic flux-surface of circular poloidal cross-section.
Let us set up a right-handed cylindrical coordinate system, , , , whose axis corresponds to the symmetry axis
of the tokamak. Let and be the major and minor radii of the flux-surface, respectively. As shown in Figure 2.1,
we can specify the location of the charged particle in the poloidal plane in terms of an angular coordinate, , which is
zero on the outboard midplane (i.e., , ). In fact, the particle's coordinates in the poloidal plane are
Now, if we neglect the influence of the comparatively small poloidal currents flowing in the plasma on the toroidal magnetic field then we
expect the overall magnetic field-strength (which is dominated by the toroidal component of the field) to vary as
|
(2.76) |
Here, is the toroidal magnetic field-strength on the magnetic axis (i.e., , ). Note that the overall magnetic field-strength
varies slightly around the flux-surface, being larger at smaller major radii, and vice versa.
Let us suppose that the parallel
(to the magnetic field) electric field-strength,
, is comparatively weak, as is indeed the case in a high-temperature tokamak
plasma (otherwise the field would generate an absurdly large parallel current). (See Section 2.9.) If this is the case then (neglecting collisions, for the moment) our charged particle
drifts around the magnetic flux-surface with a constant kinetic energy (recall that a magnetic field cannot do work on the particle). In other words,
|
(2.77) |
is a constant of the motion. Here,
is the particle's parallel velocity. However, the particle's magnetic moment, which is defined in Equation (2.73),
is also a constant of the motion. Equations (2.73) and (2.77) can be combined to give
|
(2.78) |
Hence, we conclude that the particle is excluded from regions of the flux-surface in which
(because
clearly cannot be imaginary). In fact, if the
particle reaches a so-called bounce point, characterized by
, where
, then its parallel motion must reverse direction (i.e., the sign in the previous equation must flip).
Let
and
be the particle's parallel and perpendicular velocities at the outermost point of the flux-surface (i.e., ), where the
magnetic field is weakest. It follows that and both take the constant values
respectively.
The previous three equations can be combined to give
|
(2.81) |
Finally, Equation (2.76) yields
|
(2.82) |
where
is the inverse aspect-ratio of the flux-surface, and we have made use of the large aspect-ratio approximation
.
The previous equation, combined with the requirement that
not be imaginary, leads to the conclusion that (neglecting collisions) there are two populations of charged particle on a magnetic flux-surface. The first population
satisfies
|
(2.83) |
where
|
(2.84) |
is known as the pitch angle (on the outboard midplane).
This first population of particles are called passing particles because they all have sufficiently large parallel
velocities that they can circulate freely around the flux-surface. The second population satisfies
|
(2.85) |
These so-called trapped particles have small enough parallel velocities that they are trapped on the outer part of the flux-surface, oscillating between
bounce points located at
, where
|
(2.86) |
(See Figure 2.2.)
Assuming that the charged particles at the outermost point on the flux-surface have a Maxwellian velocity distribution, all possible values of
are equally likely. Thus, we conclude that the fraction of the particles on the flux-surface that are trapped is
|
(2.87) |
[A more exact expression for is specified in Equation (2.202).]
Given that
, it is clear that the trapped particle fraction grows as we move from the innermost (i.e., ) to the outermost (i.e., ) magnetic flux-surface in the plasma.
Consider a trapped particle oscillating between bounce points located at
. Because the particle is drifting parallel to the magnetic field, its motion
is characterized by
. [See Equation (1.76).] Let represent path-length along a magnetic field-line. It
follows that
. The particle's parallel equation of motion is
|
(2.88) |
where use has been made of Equations (2.82), (2.84), and (2.86). Hence, we obtain
|
(2.89) |
Taking the derivative of the previous equation with respect to time, and making use of Equations (2.84) and (2.86), we get
|
(2.90) |
where
|
(2.91) |
is known as the bounce frequency [52]. Here, we have
made the large aspect-ratio approximation that, on average,
for a trapped particle. Obviously, Equation (2.90)
has the same form as the equation of motion of a simple pendulum.
It follows that a deeply trapped particle, characterized by
, executes simple harmonic motion in ,
between bounce points located at
, at the angular frequency
.
In this case, the appropriate solution of Equation (2.90) is
|
(2.92) |
A more accurate expression for the bounce frequency, which does not assume that the particle is deeply trapped, is
|
(2.93) |
where
is a complete elliptic integral of the first kind [1].
Note that the bounce frequency decreases with increasing amplitude of the angular motion, eventually approaching zero
logarithmically as
.
Let us now take collisions into account. In the absence of collisions, the value of the pitch angle on the outboard midplane,
, is a constant of a given particle's
motion. Thus, we conclude that if the particle is initially trapped then it remains trapped at all subsequent times. However, collisions cause the
value of
to diffuse in velocity space. (Note that Coulomb collisions in a high temperature plasma are dominated by
small-angle scattering events [18], which means that each collision only causes a small change in
.)
It takes a time of order (where is the collision time) to change
by order unity.
However, it is only necessary to change
by order
in order to de-trap a trapped particle. [See Equation (2.85).] The time
required for collisional de-trapping is thus
|
(2.94) |
Now, if
then a trapped particle is collisionally de-trapped long before it has time to complete an oscillation between its
bounce points. Obviously, in this case, the plasma is sufficiently collisional that it is meaningless to draw a distinction between trapped and passing particles.
On the other hand, if
then collisions are too infrequent to interfere with the oscillations of a trapped particle between
its bounce points.
Table: 2.4
Neoclassical parameters in a low-field and a high-field tokamak reactor. Here, is the toroidal
magnetic field-strength, the fraction of trapped particles,
the electron collisionality parameter,
the ion collisionality parameter,
the electron banana orbit width,
the ion banana orbit width, the electron toroidal magnetization parameter, the
ion toroidal magnetization parameter,
the electron poloidal magnetization parameter,
the ion poloidal magnetization parameter,
the
electron collisionality parameter, and
the ion collisionality parameter. All quantities are calculated with and . (See Table 2.1.)
|
Low-Field |
High-Field |
|
5.0 |
12.0 |
|
0.81 |
0.81 |
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It is helpful to define the dimensionless collisionality parameter [64],
|
(2.95) |
Note that
.
It follows that the criterion for species- trapped particles to exist is
.
Table 2.4 shows that
and
are both much less than unity in a tokamak fusion reactor, implying that populations of
trapped electrons as well as trapped ions exist in such reactors. Moreover, it is clear that the fraction of trapped particles, which is the same for both plasma species, is about on the
outermost magnetic flux-surfaces.
Figure: 2.2
A gyro-averaged trapped particle orbit (solid line) on a magnetic flux-surface (dashed line) of circular poloidal cross-section. The orbit parameters are
,
, and
.
|
Let us examine the motion of a trapped particle slightly more accurately. In an axisymmetric tokamak plasma, the gyro-averaged motion of the particle
takes place at constant toroidal canonical angular momentum:
|
(2.96) |
where denotes magnetic vector potential.
Note that
|
(2.97) |
Suppose that the particle is trapped on a magnetic flux-surface of minor radius . We can write
|
(2.98) |
in the immediate vicinity of the flux-surface, where use has been made of Equation (2.97). Now,
|
(2.99) |
where use has been made of Equations (2.84), (2.86), and (2.88). Thus, employing Equations (2.96) and (2.98),
|
(2.100) |
which implies that the gyro-averaged particle orbit satisfies
where
|
(2.102) |
Here, we have again used the large aspect-ratio approximation that, on average,
for a trapped particle. Note that the signs in Equation (2.101) correspond to motion in the
and
directions, respectively. As is illustrated in Figure 2.2, a trapped particle that oscillates between its
bounce points makes radial excursions from the guiding magnetic flux-surface that are of amplitude
. In fact, the gyro-averaged orbit's poloidal cross-section looks a
little like a banana. Hence, such orbits are known as banana orbits, and
is known as the banana width [52]. [Note that
is the banana width of a barely trapped particle (i.e.,
). Deeply
trapped particles (i.e.,
) have much narrower banana widths.] Finally, as is clear from Tables 2.1 and 2.4, although the electron and
ion banana widths in a tokamak fusion reactor are much greater than the corresponding gyro-radii, they are much smaller than the minor radius of the plasma.