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,

$$\mu_s= \frac{m_s\,v_{\perp\,s}^{\,2}}{2\,B},$$ (2.73)

of the particle is a conserved quantity during this motion [18,40]. Here, ${\bf v}_{\perp\,s}$ is the particle's perpendicular (to the magnetic field) velocity, and $B$ 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., $90^\circ$ 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, $R$, $\varphi$, $Z$, whose axis corresponds to the symmetry axis of the tokamak. Let $R_0$ and $r$ 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, $\theta$, which is zero on the outboard midplane (i.e., $Z=0$, $R>R_0$). In fact, the particle's coordinates in the poloidal plane are

$$R =R_0+r\,\cos\theta,$$ (2.74)

$$Z = r\,\sin\theta.$$ (2.75)

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

$$B(\theta) = B_0\,\frac{R_0}{R} = \frac{B_0}{1+(r/R_0)\,\cos\theta}.$$ (2.76)

Here, $B_0$ is the toroidal magnetic field-strength on the magnetic axis (i.e., $Z=0$, $R=R_0$). 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, $E_\parallel$, 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,

$$K_s = \frac{1}{2}\,m_s\,v_{\parallel\,s}^{\,2}+ \frac{1}{2}\,m_s\,v_{\perp\,s}^{\,2}$$ (2.77)

is a constant of the motion. Here, $v_{\parallel\,s}\,{\bf b}$ 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

$$v_{\parallel \,s}(\theta)= \pm \left(\frac{2}{m_s}\left[K_s-\mu_s\,B(\theta)\right]\right)^{1/2}.$$ (2.78)

Hence, we conclude that the particle is excluded from regions of the flux-surface in which $K_s<\mu_s\,B(\theta)$ (because $v_{\parallel\,s}$ clearly cannot be imaginary). In fact, if the particle reaches a so-called bounce point, characterized by $\theta=\theta_b$, where $K_s=\mu_s\,B(\theta_b)$, then its parallel motion must reverse direction (i.e., the sign in the previous equation must flip).

Let $v_{\parallel\,0\,s}$ and $v_{\perp\,0\,s}$ be the particle's parallel and perpendicular velocities at the outermost point of the flux-surface (i.e., $\theta=0$), where the magnetic field is weakest. It follows that $\mu_s$ and $K_s$ both take the constant values

$$\mu_s = \frac{m_s\,v_{\perp\,0\,s}^{\,2}}{2\,B(0)},$$ (2.79)

$$K_s = \frac{1}{2}\,m_s\,v_{\parallel\,0\,s}^{\,2} + \frac{1}{2}\,m_s\,v_{\perp\,0\,s}^{\,2},$$ (2.80)

respectively. The previous three equations can be combined to give

$$\frac{v_{\parallel\,s}(\theta)}{\vert v_{\parallel\,0\,s}\vert} =... ...{v_{\parallel\,0\,s}^{\,2}}\left[1 -\frac{B(\theta)}{B(0)}\right]\right)^{1/2}.$$ (2.81)

Finally, Equation (2.76) yields

$$\frac{v_{\parallel\,s}(\theta)}{\vert v_{\parallel\,0\,s}\vert}\s... ...l\,0\,s}^{\,2}}\,2\,\epsilon\,\sin^2\left(\frac{\theta}{2}\right)\right]^{1/2},$$ (2.82)

where $\epsilon = r/R_0$ is the inverse aspect-ratio of the flux-surface, and we have made use of the large aspect-ratio approximation $\epsilon \ll 1$.

The previous equation, combined with the requirement that $v_{\parallel\,s}(\theta)$ 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

$$\tan^{-1}\left(\!\sqrt{2\,\epsilon}\right) < \xi_{0\,s} < \frac{\pi}{2},$$ (2.83)

where

$$\xi_{0\,s} = \tan^{-1}\left(\frac{\vert v_{\parallel\,0\,s}\vert}{\vert v_{\perp\,0\,s}\vert}\right)$$ (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

$$0 < \xi_{0\,s}< \tan^{-1}\left(\!\sqrt{2\,\epsilon}\right).$$ (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 $\theta=\pm\theta_b$, where

$$\sin \left(\frac{\theta_b}{2}\right) = \frac{\tan\xi_{0\,s}}{\sqrt{2\,\epsilon}}.$$ (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 $\xi_{0\,s}$ are equally likely. Thus, we conclude that the fraction of the particles on the flux-surface that are trapped is

$$f_t = \sqrt{2\,\epsilon}.$$ (2.87)

[A more exact expression for $f_t$ is specified in Equation (2.202).] Given that $\epsilon\propto r$, it is clear that the trapped particle fraction grows as we move from the innermost (i.e., $r=0$) to the outermost (i.e., $r=a$) magnetic flux-surface in the plasma.

Consider a trapped particle oscillating between bounce points located at $\theta=\pm\theta_b$. Because the particle is drifting parallel to the magnetic field, its motion is characterized by $d\varphi/d\theta = q$. [See Equation (1.76).] Let $s$ represent path-length along a magnetic field-line. It follows that $ds \simeq R_0\,d\varphi= q\,R_0\,d\theta$. The particle's parallel equation of motion is

$$\frac{ds}{dt} = v_{\parallel\,s} = \pm\vert v_{\parallel\,0\,s}\vert\left[1- \frac{\sin^2(\theta/2)}{\sin^2(\theta_b/2)}\right]^{1/2},$$ (2.88)

where use has been made of Equations (2.82), (2.84), and (2.86). Hence, we obtain

$$\frac{d\theta}{dt} =\pm \frac{\vert v_{\parallel\,0\,s}\vert}{q\,R_0} \left[1- \frac{\sin^2(\theta/2)}{\sin^2(\theta_b/2)}\right]^{1/2}.$$ (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

$$\frac{d^2\theta}{dt^2} = -\omega_{b\,s}^{\,2}\,\sin\theta,$$ (2.90)

where

$$\omega_{b\,s} = \frac{v_{t\,s}}{q\,R_0}\!\sqrt{\frac{\epsilon}{2}}$$ (2.91)

is known as the bounce frequency [52]. Here, we have made the large aspect-ratio approximation that, on average, $\vert v_{\perp\,0\,s}\vert\simeq v_{t\,s}$ 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 $\theta_b\ll 1$, executes simple harmonic motion in $\theta$, between bounce points located at $\theta=\pm\theta_b$, at the angular frequency $\omega_{b\,s}$. In this case, the appropriate solution of Equation (2.90) is

$$\theta\simeq\theta_b\,\sin(\omega_{b\,s}\,t).$$ (2.92)

A more accurate expression for the bounce frequency, which does not assume that the particle is deeply trapped, is

$$\omega_{b\,s}(\theta_b) = \frac{\omega_{b\,s}}{(2/\pi)\,K(\sin[\theta_b/2])},$$ (2.93)

where $K(k)=\int_0^{\pi/2}\left(1-k^2\,\sin^2\theta\right)^{-1/2}\!\!d\theta$ 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 $\theta_b\rightarrow\pi$.

Let us now take collisions into account. In the absence of collisions, the value of the pitch angle on the outboard midplane, $\xi_{0\,s}$, 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 $\xi_{0\,s}$ 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 $\xi_{0\,s}$.) It takes a time of order $\tau _s$ (where $\tau _s$ is the $90^\circ$ collision time) to change $\xi_{0\,s}$ by order unity. However, it is only necessary to change $\xi_{0\,s}$ by order $\sqrt{\epsilon}$ in order to de-trap a trapped particle. [See Equation (2.85).] The time required for collisional de-trapping is thus

$$\tau_{d\,s} = \left(\sqrt{\epsilon}\right)^2\,\tau_s = \epsilon\,\tau_s.$$ (2.94)

Now, if $\omega_{b\,s}\,\tau_{d\,s}\ll 1$ 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 $\omega_{b\,s}\,\tau_{d\,s}\gg 1$ 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, $B$ is the toroidal magnetic field-strength, $f_t$ the fraction of trapped particles, $\nu_{\ast\,e}$ the electron collisionality parameter, $\nu_{\ast\,i}$ the ion collisionality parameter, $\rho_{b\,e}$ the electron banana orbit width, $\rho_{b\,i}$ the ion banana orbit width, $\delta _e$ the electron toroidal magnetization parameter, $\delta _i$ the ion toroidal magnetization parameter, $\delta_{\theta\,e}$ the electron poloidal magnetization parameter, $\delta_{\theta\,i}$ the ion poloidal magnetization parameter, ${\mit \Delta }_e$ the electron collisionality parameter, and ${\mit \Delta }_i$ the ion collisionality parameter. All quantities are calculated with $q=3$ and $r=a$. (See Table 2.1.)

	Low-Field	High-Field
$B({\rm T})$	5.0	12.0
$f_t$	0.81	0.81
$\nu_{\ast\,e}$	$1.7\times 10^{-2}$	$4.0\times 10^{-2}$
$\nu_{\ast\,i}$	$1.2\times 10^{-2}$	$2.8\times 10^{-2}$
$\rho_{b\,e}({\rm m})$	$4.1\times 10^{-4}$	$1.7\times 10^{-4}$
$\rho_{b\,i}({\rm m})$	$2.8\times 10^{-2}$	$1.2\times 10^{-2}$
$\delta _e$	$2.2\times 10^{-5}$	$2.2\times 10^{-5}$
$\delta _i$	$1.5\times 10^{-3}$	$1.5\times 10^{-3}$
$\delta_{\theta\,e}$	$2.0\times 10^{-4}$	$2.0\times 10^{-4}$
$\delta_{\theta\,i}$	$1.4\times 10^{-2}$	$1.4\times 10^{-2}$
${\mit \Delta }_e$	$3.2\times 10^{-3}$	$7.7\times 10^{-3}$
${\mit \Delta }_i$	$2.3\times 10^{-3}$	$5.5\times 10^{-3}$

It is helpful to define the dimensionless collisionality parameter [64],

$$\nu_{\ast\,s} = \frac{q\,R_0\,\nu_s}{\epsilon^{\,3/2}\,v_{t\,s}}.$$ (2.95)

Note that $\nu_{\ast\,s}\simeq (\omega_{b\,s}\,\tau_{d\,s})^{-1}$. It follows that the criterion for species-$s$ trapped particles to exist is $\nu_{\ast\,s}\ll 1$. Table 2.4 shows that $\nu_{\ast\,e}$ and $\nu_{\ast\,i}$ 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 $80\%$ 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 $\theta _b= 60^\circ$, $r_b= 0.5\,R_0$, and $\rho_{b\,s}=0.1\,R_0$.

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:

$$L_{\varphi\,s} \equiv m_s\,R\,v_{\varphi\,s} +e_s\,R\,A_\varphi = {\rm constant},$$ (2.96)

where ${\bf A}$ denotes magnetic vector potential. Note that

$$B_\theta = -\frac{dA_\varphi}{dr}.$$ (2.97)

Suppose that the particle is trapped on a magnetic flux-surface of minor radius $r_b$. We can write

$$A_\varphi(r) \simeq A_\varphi(r_b) - B_\theta(r_b)\,(r-r_b)$$ (2.98)

in the immediate vicinity of the flux-surface, where use has been made of Equation (2.97). Now,

$$v_{\varphi\,s}\simeq v_{\parallel\,s} = \pm\sqrt{2\,\epsilon}\,\v... ...eft(\frac{\theta_b}{2}\right)-\sin^2\left(\frac{\theta}{2}\right)\right]^{1/2},$$ (2.99)

where use has been made of Equations (2.84), (2.86), and (2.88). Thus, employing Equations (2.96) and (2.98),

$$\pm m_s\,R\sqrt{2\,\epsilon}\,\vert v_{\perp\,0\,s}\vert\left[\si... .../2} + e_s\,R\,A_\varphi(r_b) - e_s\,R\,B_\theta(r_b)\,(r-r_b) = {\rm constant},$$ (2.100)

which implies that the gyro-averaged particle orbit satisfies

$$r= r_b \pm{\rm sgn}(e_s)\, \rho_{b\,s}\left[\sin^2\left(\frac{\theta_b}{2}\right)-\sin^2\left(\frac{\theta}{2}\right)\right]^{1/2},$$ (2.101)

where

$$\rho_{b\,s} = \frac{\sqrt{2}\,q}{\epsilon^{\,1/2}}\,\rho_s.$$ (2.102)

Here, we have again used the large aspect-ratio approximation that, on average, $\vert v_{\perp\,0\,s}\vert\simeq v_{t\,s}$ for a trapped particle. Note that the $\pm$ signs in Equation (2.101) correspond to motion in the $\pm \theta$ and $\pm\varphi$ 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 $\rho_{b\,s}\,\sin(\theta_b/2)$. 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 $\rho_{b\,s}$ is known as the banana width [52]. [Note that $\rho_{b\,s}$ is the banana width of a barely trapped particle (i.e., $\theta_b\simeq \pi$). Deeply trapped particles (i.e., $\theta_b\ll 1$) 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.