# Appendix A: for Chapter 3

## А.1 COLLISIONAL RELAXATION OF ANISOTROPIC ENERGETIC BEAM IONS

Consider a temporal evolution of energetic beam ion distribution function, *f(v _{x},v_{y},v_{z}y* in the velocity space due to Coulomb collisions with plasma electrons and ions. The plasma considered is homogeneous, and an axial symmetry of the beam distribution function is assumed during the evolution:

The beam evolution can be described by the Fokker-Planck equation for hot ion distribution function /(v,r>) with initial velocity *V _{0}* for the typical range of hot ion velocities between ion and electron thermal velocities, v, « V

_{0}«

*v*Neglecting self-collisions between hot ions, the linear Fokker-Planck equation can be written as [A.1,A.2]:

_{e}.

where

*L* is the Coulomb logarithm, *m _{0}* is the beam ion mass,

*E*are the initial energy and speed of fast ions, respectively, and the last two terms in the right-hand side of (A.2) represent sink and source of the energetic ions, respectively. The source term satisfies the normalisation

_{0}, V_{0}

A comparison of the terms proportional to *a, b,* and *c* shows that for the highest energy range of the beam ions, *v ~V _{0},* the dominant relaxation effect is the slowing down of the beam ions due to Coulomb collisions with thermal electrons and ions. This effect is determined by the term proportional to

*b(v).*The effect of beam scattering in the pitch angle

*i)*is the next in the ordering. This effect proportional to c(v) is mostly determined by beam collisions with thermal ions, while the electron contribution is small as

*V*Finally, the velocity diffusion term proportional to a(v) is the weakest, with a small factor

_{0}/v_{e}.*T*in front of it.

_{e}/E_{0}To analyse (А.2), we drop the small term proportional to *a(v)* and represent both the source and the distribution function of the energetic ions via a series of Legendre polynomials [A.2]:

Here, *Pi* (cos$) are the Legendre polynomials satisfying

so that (A.2) can be represented as set of one-dimensional first-order differential equations for the functions *к, = b(y)f*

resulting in characteristic relations

The evolution of the beam ion speed in time can be found from (A.9): resulting in time evolution where

Expression (A.l 1) gives v = *V _{0}* at

*t=0*and the following simple expression The beam slows down to

*v~ V,*at

The /-th component of the velocity distribution function can be obtained from (A.9). After straightforward lengthy calculations, the following general form of k> can be found for arbitrary source and sink:

where

and

satisfies W(v, *t)* > v,

Here,

**REFERENCES TO APPENDIX A**

[A.l] Yu.N. Dnestrovkii and D.P. Kostomarov, *Mathematical modelling of plasma,* Nauka, Moscow (1993) (in Russian).

[A.2] H.L. Berk et al., *Nucl. Fusion* 15 (1975 ) 819.

**Appendix В for Chapter 4**

# Appendix B: for Chapter 4: Curvilinear Coordinates in Toroidal Geometry

## В.1 COORDINATE TRANSFORMS

We begin with Cartesian coordinates (x, *y, z)* with the basis of unit vectors (V, , *e _{y}, e_{z})*

In this case, representation of a vector has the form:

with components of a vector:

If one transforms to another coordinate system *(a, /5,* y), then
with

Let us generalise the maths above to curvilinear coordinates (§’, £^{2}, with the covariant basis

(V§ V^{2} + The

covariant components of the vector take the form:

where Jacobian is given by *J* =[(V§‘ x Vg^{2} )■ V^^{3} J By substituting the Jacobian, one obtains:

If one transforms to another coordinate system then

with

We can also introduce a contravariant basis (,/(V^{3}j, ^{x}^^^{2}))

with contravariant components: *A' = A-* V^^{1}, *A ^{1} = A-* V<^

^{2},

*A*V£

^{}}= A■^{3 }The contravariant representation of a vector is then obtained in the form:

Metric tensor is a relationship between covariant and contravariant components:

where the contravariant components of tensor *g* are given by

and covariant components of tensor *g* are given by the inverse matrix:

Volume element in the general geometry case takes the form: and the length element takes the form:

In vector calculus, divergence, which is a vector operator that operates on a vector field producing a scalar field giving the quantity of the vector field’s source at each point, becomes:

while the curl, which is a vector operator describing the infinitesimal rotation of a vector field in three-dimensional Euclidean space, takes the form:

where *e _{ijk}* is an anti-symmetric unit matrix whose only non-zero components are

## B.2 SHAFRANOV COORDINATES FOR TOROIDAL PLASMA

We start from the coordinate system employed by Shafranov for analysing plasma equilibrium in toroidal geometry (Figure B.l).

The following expressions relate the different sets of coordinates: with the following inversed relations:

FIGURE B.1 Cartesian coordinates (.v, *y,* Z), polar coordinates(/?, *p,* Z), and toroidal coordinates (r, *0, <; =* -чр)

used for describing toroidal plasma equilibrium. Here, *R _{0}* is the major radius of the magnetic axis, and Д is the Shafranov shift (the distance between

*R*and the geometric major radius).

_{0}The 3x3 matrix for transformations from *(x,* y,Z)to<*' =r,<*^{2} = i?,£^{3} = £ has the following elements:

so that the covariant components of the metric tensor
are calculated easily, for example, for *g _{n}: *
to give the following matrix:

with the following Jacobian relevant to Shafranov coordinates:

Here, we used the large aspect ratio ordering

The contravariant metric elements can be found from the general expression:

These contravariant elements are:

## В.З FLUX-TYPE COORDINATES IN TORUS

For further investigations of plasma stability and waves, we need to introduce a safety factor

which plays a major role. A significant simplification of the maths associated with plasma stability and waves can be achieved if the safety factor is *independent* of the poloidal angle variable <9 As the density of magnetic field lines is higher at the inner side of the torus (where equilibrium magnetic field is higher), a different poloidal coordinate than the usual poloidal angle used in Shafranov coordinates is required. Due to the explicit connection to the flux of equilibrium magnetic field, the coordinates with such improved poloidal angle variable can be called flux-type coordinates. One can search for a desired set of flux-type coordinates (/>;r9,•;£/•) related to Shafranov coordinates

(r_{s};r>s;Cs) via

where the expression for can be found from the condition *dq/ddf* = 0. For this, we split *В* into poloidal and toroidal parts,

and obtain

where *J _{f} =* l/(V/y x • V£j-) is the

*flux-type Jacobian.*We now explicitly require and obtain

with the ^-dependence involved only in *R ^{2}.* The ratio of the Jacobians derived above gives a differential equation for the relevant poloidal angle variables:

One can then find the flux-type poloidal angle variable via Shafranov angle:

Here, C=0 as must change by 2*л* whenever *r> _{s}* does. Thus,

Inverting the expression for *t> _{f} (/y*,

*i9*

_{s}) gives and we can write using Taylor expansions:

where

We can now rewrite Shafranov coordinates to obtain the flux-type coordinates:

From this set of coordinates, we obtain the relevant flux-type Jacobian

and for the flux-type coordinates we obtain the covariant matrix (put #/ = i? here):

The contravariant metric elements are:

**Appendix C for Chapter 6**