ELECTROSTATIC TRIVELPIECE-GOULD MODES IN A TORUS

(1)

HAL Id: jpa-00219281

https://hal.archives-ouvertes.fr/jpa-00219281

Submitted on 1 Jan 1979

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ELECTROSTATIC TRIVELPIECE-GOULD MODES IN A TORUS

F. Stössel

To cite this version:

F. Stössel. ELECTROSTATIC TRIVELPIECE-GOULD MODES IN A TORUS. Journal de Physique

Colloques, 1979, 40 (C7), pp.C7-603-C7-604. �10.1051/jphyscol:19797292�. �jpa-00219281�

(2)

JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au n07, Tome 40, JuiZlet 1979, page C7- 603

ELECTROSTATIC TRIVELPLCE-GOWD MODES

IN

A TORUS

F.P. Stksel.

I n s t i t u t e for l'heoreticaZ Physics, Universi.ty- o f Innsbruck, Innrain 52, A-6020 Innsbruck/Austria, Europe.

Abstract: Electron plasma waves are treated in a V ~ Y = 4x en 1 torus of square cross section with an infinitely

aE

-

(k,~/m)"~ is the velocity of sound. This strong azimuthal magnetic field. Eigenfrequencies

leads to the following equation for the electrosta- and eigenfunctions are calculated numerically.

tic potential:

Although many experiments were performed in to- -v2y

a

- ^{a2 12}E o v ~ +y^lo2e a2 ~ r = 0 y roidal devices a well-developed theory of toroidal

a

t2 r2 a@2 r a@2 waves is not available as yet. As a step in that

Variahles can be separeted by the following ansatz:

direction electron plasma waves are investigated in

iwt iknz im@

a toroidal cavity of square cross section, i.e. the Y = e e e Tl(r) same configuration as used by Swanson (1) in dis.-.

cussing ion cyclotron and fast waves. Cylindrical coordinates r,

9,

z are used. The torus has a major radius R and a square of lateral length 2ro

.

The cavity is completely filled with a homogeneous plasma, and the walls are ideal conductors. In con- trast to the original paper by Trivelpiece and Gould (2), where the magnetic field has only a component in the z-direction, the magnetic field here lies in the $-direction. A linearized fluid descrip- tion is used (3).

The uniqueness-condition restricts the mode numbers m to integers; m=O describes the steady-state solu- tion. The perfectly conducting walls require that Y vanishes at the walls, i.e. Y(r, z=0) =

= Y(r, z=2r ) = 0, which results in a sinusoidal z-dependence (u sin k z with k = n?r/2ro) and in

n n

the conditions y(r=R-ro, z) = Y(r=R+ro, z) = 0 whence we obtain boundary conditions for the radial eigenfunctions T (r) which obey the following dif-

1 ferential equation:

The magnetic field is assumed to be infinitely 1 m2 w2

TI''

+ 7

TI' + ( pe

- - -

m2 ^k:) ^.TI^{= 0} strong, so that the r- and z-components of the ve- r2U2 - m2ai r

locity can be neglected. Thus the system of line- Primes denote derivatives with respect to r. This arized equations consists of the equation of con- eigenvalue problem with w as the eigenfrequency was ti nu it^, the $-.component of the equation of motion, solved numerically by a shooting method. Numerical and Poisson's equation: values typical of large present-day experiments

were chosen. E.g,, the Debye-length was set equal to 7.43xl0-' x R. The figures below show some I a examples of the eigenfunctions T, for an inverse

a

= - a 2 L ? L n l + e n --y

"0

TF

$ E r a$ rn era$

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797292

(3)

aspect ratio a = 0.3. The eigenfunctions are nor- malized by setting dT/dr = I at r = R

-

r The

0'

first six modes in 1 (1 = 1

-

6 ) are plotted one over another in each figure for mdl, n=l in Fig.1, for m=3, n=3 in Fig.2,and for m=5, n=5 in Fig.3.

Fig. 1

The eigenfrequencies w / w for these cases are:

Pe

From the figures one can clearly see the deviation of the radial eigenfunctions from symmetry about r = Rwhich should obtain for linear geometries.

For the appropriate linear geometry, a parallelepi- ped of length ZrR, and side-length 2ro, the eigen- functions are sinusoidal. The deviation increases with the inverse aspect ratio, i.e. with the cur- vature of the torus. However, as can already be suspected from the three examples given here, it turns out that the mode number n has much more in- fluence on the shape of the eigenfunctions than the aspect ratio. On the other hand variation of m changes the eigenfrequencies but does not alter the shape of the eigenfunctions.

The eigenfrequencies for linear geometry are generally somewhat smaller than those for the to- xus, For the above-mentioned examples, the deviation lies within 1

-

4 % for most of the modes and reaches a maximum of 23 % for the 1=1, m=5, n=5 mode.

This work bas been supported by the Gsterreichi- scher Ponds zur Farderung der Wissenschaftlichen Forscfiung Grant Nr. 2781/S.

References

(1) Swanson B.G., Phys.Fluids

2,

1974, 2241 (2) Trivelpiece W., Gould R.W., J.Appl.Phys.

30

1959, 1784

(3) Cap F.F., Handbook on Plasma Instabilities, Academic Press London, 1978, vo1.2, p.817