Constants of Motion Phase Space

By defining each particle by its CoM, a three-dimensional CoM phase space can be considered (with a fourth dimensionσdrequired to distinguish between the direction

of travel in degenerate regions where more than one orbit exists with a certain E,

Pζ, µ). This phase space can subsequently be divided by topological boundaries

into different types of orbits. The boundaries are surfaces in the three dimensional phase space, whereby orbits on each side fall under different orbit classifications. These are crucial, and the crossing of these boundaries represents the transition to a different orbit type or even to a region where they will no longer be confined by the plasma and are lost altogether, as defined by the second type of fast particle transport explained in Section 2.7.4.

Region Orbits A Trapped B Co-Passing C Counter-Passing D Counter-Passing (Lost) E Trapped (Lost) F Co-Passing (Lost) G Stagnation H Potato

I Co-Passing & Counter-Passing Table 4.1: Fast particle orbit classifications.

A slice at constant energy of such a CoM phase-space diagram is shown in Figure 4.2. Regions can be seen where the fourth dimensionσdis required to resolve

degeneracy, for example the trapped region where at any point in time the particles could be travelling along the direction of the plasma current Ip or in the opposite

direction to it depending upon their location within a bounce.

Trapped 1 P_ζ / ψp(a) Lost Counter- passing Co-passing Stagnation (G₎ Potato (H₎ Λ -1 0 0 A C B D E D F I

Figure 4.2: CoM phase space schematic. The constants of motion are (E,Pζ,µ), but

to simplify the figure it is normalised such that the y-axis is Λ =µB0/E. The third dimension is E, hence this figure is a ‘slice’ through the phase space at a constant energy. Orbit classifications are separated by the coloured lines.

The different regions describe the type of orbit phenomenology of the parti- cles contained within it, separated by the coloured topological boundaries described below. The energy for any particle orbit may be written in the form

E= 1 2mv 2 ₌ 1 2mv 2 k+ 1 2mv 2 ⊥ = 1 2mρ 2 kω2c +µB = (Pζ+ψp) 2_B2_e2 2mg2 +µB

which is left in SI units in order to show what the energy would be like for particles of different species. In the plasma physics units used in this thesis (m=e= 1) it is re-written as

E = (Pζ+ψp) 2_B2

2g2 +µB . (4.23)

a particle is transported from one of the confined regions such that its orbit now lies on this line, it will be lost from the plasma. At the point the particle exits the plasma,B=Bmin,ψp =ψp(a) andθ=π. Putting these into Equation 4.23 defines

the equation for the line as

(Pζ+ψp(a))2Bmin2 2g2_(ψ

t(a))

+µBmin− E = 0 . (4.24) The red line in Figure 4.2 represents the LFS boundary of the plasma. Orbits represented by a point on this line will be lost from the plasma, for example that shown by the cyan line in Figure 4.1. At this point, B = Bmax, ψp = ψp(a) and θ= 0. Putting these into Equation 4.23 gives the equation for the line as

(Pζ+ψp(a))2Bmax2 2g2_(ψ

t(a))

+µBmax− E = 0 . (4.25) For an arbitrary distribution function, there may be particles whose unperturbed CoM lie in the lost particle region between the green and red lines. If they are immediately lost within one poloidal orbit, they are referred to as ‘prompt loss’ particles.

The magnetic axis of the plasma equilibrium is shown in Figure 4.2 by the magenta line. Orbits that at some point go through the magnetic axis lie along this line. Figure 4.2 shows that co-passing, counter-passing, trapped, stagnation and lost orbits are all capable of passing through the magnetic axis at some point. The yellow orbit in Figure 4.1 shows the boundary between potato and trapped orbits, and passes through the magnetic axis. On the line itself, B = B0 and ψp = 0, so

the line is given by the equation

P_ζ2B₀2

2g2₍₀₎ +µB0− E = 0 . (4.26) The orange boundary in Figure 4.2 discriminates between trapped orbits that have a bounce point and those which do not, completing full passing or- bits. This occurs (to leading order) when ρk = 0. From the unperturbed toroidal

canonical angular momentum (Equations 2.57b) and 4.23 this line is defined by

µB(ψp, θ) =E; Pζ =−ψp ,

at θ= 0 and θ=π, and is subsequently referred to as the trapped-passing bound- ary. The region encased by the orange line is therefore the trapped particle region.

Where this region coincides with the lost region, trapped particles are lost from the plasma, and where it coincides with the passing region is where the potato orbits lie. A small region remains, encased by the black line. Within this region are passing orbits that do not encircle the magnetic axis - the stagnation orbits.