The axi-symmetric toroidal magnetic configuration has both radial gradient and curvature of the magnetic field, therefore, a charged particle motion is initially determined by the super-position of the magnetic drifts. However, in the absence of a poloidal magnetic field (no toroidal plasma current), the charge-dependent magnetic drifts give rise to a charge separation resulting in a vertical electric field. An outward ExB drift of both electrons and ions follows, expelling the whole plasma out of the high field side of the torus to the outer wall. Figure 2.4 shows the directions of the magnetic and ExB drifts of electrons and ions in a cross-section of a toroidal axi-symmetric configuration.

To avoid the outward electric drift of plasma, a toroidal plasma current is generated in the toka- mak via the transformer action technique. This toroidal current induces poloidal magnetic field, which generates a twist in the magnetic field lines. Following the twisted field lines, the charged particles move through the entire poloidal cross-section, thus generating a “shortcut” between the separated charges and avoiding the electric drift expelling the plasma.

Trapped and Passing Particle Orbits

Charged particle motion in tokamaks can be classified by two main groups: particles passing around the torus and particles trapped in magnetic wells formed by the difference in the magnetic field

Guiding centre Vfi and curvature drift velocities of electrons and ions in a toroidal magnetic configuration

FIGURE 2.4 Guiding centre Vfi and curvature drift velocities of electrons and ions in a toroidal magnetic configuration (no poloidal magnetic field) result in vertical electric field and in outward electric drift of both electrons and ions.

strength between the inboard and outboard sides of the torus. The trapping of the particles depends on the ratio between particle velocity along the magnetic field and perpendicular to the magnetic field, and can be understood by considering (2.35) as the force on the magnetic moment p of the particle orbit. The magnitude of the magnetic field along a particle trajectory in the torus has its lowest

value, fimin = B01 I--, in the median plane. Denoting the velocity at this point by a subscript

V Ro)

zero and using the condition that pis almost constant during the particle motion, we obtain

If there is a bounce point along the particle orbit, vM = 0, the energy conservation implies that the perpendicular velocity at the bounce point satisfies v = v|0 + v,p. Substituting this expression into (2.41) gives the value of the magnetic field at the bounce point as

Thus, the pitch-angle determines whether the particle is passing or trapped, as well as how


short is the trajectory of the trapped particle between bounces. The boundary between trapped and passing particles is determined by the condition that the highest value of the magnetic field on the inner side of the torus is equal to the value of the “bounce” magnetic field,

Thus, using (2.41), the requirement for the particle trapping, Bbounce < £max, is

Using this critical condition for an isotropic distribution of particles, the fraction of the trapped particles as a function of minor radius r can be found to be approximately

The bounce orbits can be calculated using (2.35) for strongly trapped particles, which have their bounce points in the plasma cross-section (at some toroidal angle) within a narrow poloidal angle,

i? «: 1, and — «1. By writing the major radius coordinate Kn

so that the magnetic field takes the form and its parallel gradient, dZ? / ds is

/**d ^

The equation of the field line is-= — so that t? = -——. Thus, combining (2.47) and (2.35), the

, , , , ds В rB

equation of motion takes the form

where the bounce frequency of a trapped ion between the bounce points is


and the so-called “safety factor” is q = ——. We multiply both sides of (2.48) by ds/dt and inte-

grate to obtain the equation of the particle motion along the magnetic field Because 8 =•= s, the S-component of motion is given by and the equation of motion along the poloidal angle is

The drift surface on which trapped particle orbit lies could be obtained by considering the radial component of the vertical drift due to the toroidal field gradient, vd = j(ejB,R), according

to the following equation

By combining (2.52) and (2.53), we find

By integrating (2.54) we find the equation for the drift surface:

This surface has the shape of a banana, as shown in Figure 2.5, and the half-width of the drift orbit

• . iVa

is A о —-•


The guiding centre drift orbits of passing particles form trajectories similar to the magnetic flux surfaces, but shifted from the flux surfaces by the drift orbit size, as shown in Figure 2.5. Some regions of the particle phase space also form “non-standard” orbits, such as stagnation and potato orbits (not shown in Figure 2.5).

Examples of Fat and Non-Standard Orbits of Highly Energetic Ions in JET

For highly energetic charged fusion products, such as fusion-generated alpha particles, the particle drift orbits may not be thin banana orbits, but fat orbits comparable to the minor radius of

Guiding centre drift orbits of charged particles in a tokamak

FIGURE 2.5 Guiding centre drift orbits of charged particles in a tokamak: trapped particle orbit (“banana” broken line) and passing particle orbit (circle broken line) shifted from the magnetic flux surface (circle line).

Examples of trapped orbits of He ions in the MeV energy range in JET tokamak

FIGURE 2.6 Examples of trapped orbits of He ions in the MeV energy range in JET tokamak.

the plasma. Figure 2.6 illustrates this point by showing orbits of trapped He ions accelerated with ICRH on JET up to the MeV energy range (experiment mimicking trapped fusion alpha particles). Furthermore, one can see here that Larmor radii are comparable to the drift orbits, and so full orbit description may be essential in solving some problems, such as interpreting fast ion losses to the first wall.

Next, some tokamak scenarios develop magnetic field topology significantly different than the one considered for discharges with toroidal current peaked at the plasma centre. To underline the role of the “twist” in magnetic field lines in the formation of drift particle orbits, we illustrate how energetic particle orbits appear in JET with strongly reversed magnetic shear equilibria, described in more detail in Chapters 4, 9, and 10. The reversed magnetic shear is formed in JET if the plasma current profile is hollow, that is, significantly reduced near the plasma centre, sometimes down to the zero value (so the safety factor q(r) —> ~ in the plasma core). Such magnetic configurations are of interest for “advanced tokamak” scenarios aiming at triggering internal transport barriers. Fast ion distribution function resulting from on-axis ICRH satisfies

The trajectories of hydrogen ions launched at the same energy 500 keV in the reversed-shear equilibrium (pulse #49382 discussed in more detail in Chapter 9) are shown in Figure 2.7. This distribution function consists of some trapped particles with bounce points, Vf, = 0 at the В contour through the magnetic axis, B=B0. However, it also consists of non-standard ion orbits, for which V[, does not change sign. Because these orbits never bounce, that is, ф>0, its toroidal drift frequency is larger than the poloidal frequency. We see that many ions within qmin have orbits of the non-standard type, so some unusual equilibria could result in non-standard orbits deviating significantly from the usual trapped and passing types and should be inspected individually.

Drift orbits of ICRH-accelerated H-minority ions in JET pulse #49382 with reversed magnetic shear, q=3.95

FIGURE 2.7 Drift orbits of ICRH-accelerated H-minority ions in JET pulse #49382 with reversed magnetic shear, qmin=3.95, цВо/Е = 1 corresponding to on-axis ICRH.

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