Equations of charged particle motion in toroidal (tokamak) magnetic …eld F. Spineanu and M. Vlad and Plasma Theory team Bucharest, INFLPR Abstract A large number of versions of derivation and formulation of the equations of motion of particles is reviewed. This rather strange approach (instead of a simple and concise presentation) is useful when there are applications with particular requests of representation of the particle motion or, to simplify the lecture of the large number of papers. This is one of the Lectures in the Work Session of Plasma Theory. The text is never …nal.

1

1.1

Equations for the particle’s motion and for the variation of the velocities along the orbits Notes

After derivation of the equations of motion for the guiding center (by averaging over the gyromotion) we should stop making confusions between the guiding center and the charged particle. For example if we have an electric …eld and a GC particle we should not expect the GC to move in the sense of the electric …eld. All intuitions about the charged particle are NOT valid for GC. In the paper on intrinsic rotation DIIID. Creation of a charged particle with only perpendicular velocity will lead to an average parallel ‡ow. 1

The particle is created with only v ? B. This means that there will be gyration. It is su¢ cient that the particle has gyration, without any other initial displacement for the particle to feel the space variation of the magnetic …eld magnitude rB. Then the drift exists b n

rB

and this is perpendicular to the magnetic surface. It exists, for the guiding center a velocity with radial component vr . From this it results vr

B ! motion in toroidal direction

The invariant is Rmvk + eRA' = const and means that the fact that the particle traverses the magnetic surfaces (equivalently, it changes A' ) induces parallel velocity vk . The invariants refer to the motion of the particle itself, not only the guiding center.

1.2 1.2.1

Expressions for the velocities Neoclassical drift of the guiding centers

A formula for the drift of particles vk B+r B vk b+ r = vk n B

v =

where

k

=

vk

eB m

=

kB kB

vk c

We have b + vD v = vk n

e r kB m k e = r B+r m k k e 2 ( j) = m k 0 eB b r k n m k

vD =

2

k

B

Ignoring for the moment the term depending on the current density and only retaining the second term, we have vD =

b vk n

r

vk

which is one of the expression used frequently (Rosenbluth Hinton alphas). The radial part in the neoclassical drift vD r = vk

B r @ B @

k

B (r r ) B r

This calculation can be further developed. We take into account that in the case of circular surface B r @ B @

B 1 @ BT r @ 1 @ = qR @ @ ' @z =

which makes these factors to combine in vk

1 @ B r @ = vk B @ qR @

from here we retain

1 qR The other factor of the full expression requires the calculation of b r = n

b (RB' ) ( b B (r r ) n er ) (1=r) b e = b r B r n 1 1 b b = RB' n e' 1= (qR) r 1 B = (RB' ) (qR) r B' = RB' (check however the signs) and k

B (r r ) = B r 3

k

(RB' )

Now we return to the initial expression of the projection of the drift velocity on the radial direction and we obtain = vk

vD r

1 @ qR @ k RB'

since (this will be denoted I) RB' = R0 B0' = const I it will get out of the derivation to , vD r = vk

1 @ RB' qR @

vk

Introduce the expressions I

RB'

@ 1 @ b r =n qR @ @ We note that this formula consists of the usual projection from the parallel to the poloidal direction 1 @ qR @

RB 1 @ B @ = rBT R @ BT r@ = rk =

The expression can be re-written b r vD r = Ivk n

@ @

vk

and this is the expression used by Rosenbluth Hinton for alphas. It is useful because allows to average explicitely the terms in the expansion of the distribution function over bounce. We take into account that in the case of circular surface B r @ B @

B 1 @ BT r @ 1 @ = qR @ @ ' @z =

4

we obtain vk @ k RB' Rq @ vk 1 @ = Ivk qR @ ci

vD;r =

Separately I RB' = rB' = RB = jr j qR R RB then jr j = We can take as variables

I qR

= the total particle energy =

2 mvk2 mv? + +e 2 2

= the magnetic moment =

2 mv? 2B

and then, for IONS 1=2

2 ( vk = m

B

@ 1 @ @B vk = jej @ mvk @ @ 1 "B0 = sin mvk (1 + " cos )2 2 1 v? " sin = vk 2

jej

@ @

The variation of the parallel velocity vk is due to the variation of the modulus of the magnetic …eld along the magnetic …eld line as it turns from exterior to the interior of the torus. The derivative of the magnetic …eld is @B @ = @ @

B0 1 + " cos "B0 sin = (1 + " cos )2 5

It results the known formula: 2 =2 + vk 1 v? sin R i

ions = vD;r

The component of the neoclassical drift projected on the toroidal angle ' direction vD r' =

vk

B r @ B @

k

B (r r') B r

or, the toroidal drift, vD r' =

vk

b (r n

1 @ qR @

1 qR

r')

!

1 b If the vectorial product r r' is rR er then the scalar product is zero and there are no component of the drift velocity on the direction of the gradient r'. We must conclude that this vector-gradient is not transported along the line up to the current point. This toroidal component of the drift velocity should represent the toroidal precession of the particles.

e m (slow temporal variation of the radial electric …eld). And vk b r vD = vk n vD =

1

rB + vk2 (n r) n +

n

A formula written in general terms d

1;2

dt

=

1

r

@J

g33 @ g H

2;1

d vk

where 1;2 are the coordinates de…ned by the magnetic line, g33 is the longitudinal component of the metric tensor, whose determinant is g. As coordinates, one can choose (r; '0 ). A simple expression

vD?

1 = eB0 =

mvk2 R0

2 mv? + 2R0

!

2 2 1 v? =2 + vk it is almost vertical R0

6

where the formula is used in [?]. This formula combines the grad-B drift and the curvature drift from the general expression of the drift velocity. The direction is approximately vertical which means parallel with the torus main axis of symmetry. The drift of particles in the RADIAL (r) direction, averaged on the bounce is ZERO. For trapped particles one can use vk

v?

which simpli…es the expression for the curvature drift trapped vD? =

2 =2 1 v? R0

with e < 0 for electrons. This velocity is directed vertically, almost parallel with the torus main symmetry axis. The projection on the radial direction, for the electrons 2 v? trapped sin vD;r = 2R0 j e j electrons This is often used as an equation of motion in the radial direction, for the trapped electrons 2 v? trapped r = vD;r sin (1) = 2R0 j e j 2 For exemple, for the ions at the temperature Ti , v? =

2Ti mi

.

Parallel drift motion The equation (valid for passing and trapped particles) of balance of forces acting upon the particle m

dv = e (E + v dt

B)

rB

then the parallel component is m

dvk = eEk dt

@B @

(where is a coordinate along the magnetic …eld, obtained from the toroidal angle) and, for B = B0 (1 " cos (k0 )) with k0 = 1=qR0 , and taking Ek = 0 we have vk =

2 v? "k0 sin 2

7

(2)

(Note that this suggests (poloidal angle)

k0 =

lk qR0 B = lk BT B = BT B = BT =

r r lk dl dlk

qR0

or

which is approximately correct). Comparing (1) and (2) it results r = r0 +

vk

or r = r0 + as the radial displacement of the particles, due to the drift. The parallel velocity vk in this formula is oscillatory for both passing and trapped particles. But for trapped it changes the sign.

1.3

Trapped particles

Banana regime Condition for the banana regime =

ii

!b

1

where ii is the ion-ion collision and ! b is the bounce frequency. This is the small parameter of the banana regime. The bounce of the trapped particle is much faster than the collisions. Collisional di¤usion Frieman 1970. Rosenbluth Frieman Hazeltine. Shaing trapping detrapping.

8

Region in the space of parameters w=

v2 = 2 =

or

e m w

2 v? 1 2 v B (x) Note that in some other articles it is taken

=

B (x) = B0 h and ! end then, since here

=

2 v? 1 = 2 v h

2 v? 1 v 2 B(x)

we have p vk = 2w (1

We de…ne m

0

B)

1 B (x) = the largest for which the function f (x; ; w) is de…ned =

and c

Bmax

=

1

Bmax where = the maximum of jBj along a …eld line

and c is the critical for trapping. The trapped region is c
a the fast ion will intercept the limiter and it is considered to be in the loss region. Fix a spatial position, for exemple: on the equatorial plane, at the distance x = 0:3a Fix the energy of the particles, for example " = 10 keV for fast injected ions. The loss region boundaries dependes now on the pitch angle and are de…ned by: The pitch angle (nearest to 180 ) which correspond to the untrapped-trapped particle boundary. (Note: a pitch angle greater than =2 means that the parallel component of the velocity is opposite to the direction of the magnetic …eld). The smallest pitch angle at which the banana width is large enough so that the orbit intersects the limiter. Note: small pitch angle means that the direction of the particle velocity is close to the direction of the magnetic …eld and only very small transversal velocity is left (small magnetic moment). This particle has small chance to be captured but, if it is, it has very large radial deviation, since iy is not far from the barely trapped particles, which are known to have largest radial deviation. Comming from small values of the pitch angle toward larger values, a particle will be captured at a certain value of and, precisely in this region it will have largest radial deviations. That is why one limit of the loss region must be searched for at small values of the pitch angle. The banana width decreases with decreasing the particle energy. Then the loss region will vanish as the parameter: qv P = a 2 a mv = 0 Ze R I where I is the total plasma current (with which we calculate the poloidal magnetic …eld at the plasma border) and 0 is the vacuum permeability. 12

To determine the loss-region boundaries. The fast-ion is produced at a point with the major radius Rs . Then the magnetic …eld is R0 B B0 Rs We calculate the invariants of the particle motion. Magnetic moment: 2 2 mv? mv? Rs = = 2B 2R0 B0 2 = v 2 sin2 . where v? Energy (in the absence of the radial electric …eld):

"=

m 2 v + vk2 2 ?

Canonical angular momentum Ze + mRv' = const where is the poloidal magnetic ‡ux which in the circular surface approximation is = R0 A' where A' is the magnetic potential due to the plasma current. Taking a model for the current density h r n ip j = j0 1 a

we obtain the expression for the magnetic potential A' =

0I

2

F (r)

where F (r) is a sum of powers of r. This expression allows to represent =

R0

0I

2

as

F (r)

and to re-write the expression of the canonical angular momentum F (r)

P R R

Rs sin2

The choice of the signs is:

13

1=2

= const

(3)

the minus sign must be used when v J>0 the plus sign must be used when v J