Stabilization of trapped-ion instability in two-component torus

Volume 50A, number 1

PHYSICS LETTERS

4 November 1974

STABILIZATION OF TRAPPED-ION INSTABILITY IN TWO-COMPONENT TORUS* Hi. BERK Lawrence Livermore Laboratory, University of California, Livermore, California 94550, USA Received 2 August 1974 The trapped-ion instability in a two-component torus reported by Ohkawa and Bhadra is stabilized with an extremely small spread in the drift velocity.

A recent note by Ohkawa and Bhadra [1] suggested that in the two-component torus (TCT) concept, the notion component can excite a mode at the gradient-B drift frequency, and this is destabilized by the collisions of trapped particles in the plasma background. Here we point out that the model used by ref. [1] applies only if there is no dispersion in the drift frequency, and that the mode is stabilizedfor very small spreads in the drift frequency, namely, (1)

l&A)DH/WDHI~>(flH/flo)(Te/TH),

where WDH is the gradient-B drift frequency and AL~DHits spread, ~H the hot-beam density, n0 the background electron density, Te the background electron energy, TH the beam energy. In practice one can expect ~ ‘~‘DH’hence eq. (1) is highly stable for TCT parameters where typically nH/n0 ‘~005,Te1TH 0.05. To illustrate our point we use the simplified Kadomtsev and Pogutse [2] trapped-particle model to describe the background plasma and treat the beam ions with a distribution function nH(r)FH(E) (r is the radial position). The dispersion relation then has the form ‘~i

a

~H

~H

fl0J

~ ~,—•w’~ ~i ~ ~—e’ +~ fl0

—FH(E=0)—c’ aE

l+—r——T ..i3 ~

r’r

.

(m.,+w”) - + 1V~/E (A)

(~ 1?

=0,

~DH(V) +

(2)

where w~ kycTe/eBr~r = T~/T1,v~is the collision frequency for the/th species, e, i, and H denotes the electron, ion and hot-ion component respectively, e is the aspect ratio, and 1c~is the wavenumber perpendicular to magnetic field and density gradient. Now, c~DHis the mean drift frequency of a trapped hot-ion and thus is a function of pitch angle and energy [2] and it has the opposite sign of w’~’for badly drifting particles, (In general in a tokamak geometry one can find particles with both signs of ‘~‘DH-) The last term of eq. (2) differes from eq. (1) of ref. [1] in that they treated FH in the numerator as if it is Maxwellian with energy TH and, (A)DH(V) in the denominator as constant. Now, if c~ WDH 1CYCEH/BR (where EH is the particle energy) and ~F/8E FlEH then -~

ETe(aFIaE)w]/w*F~C,

(3)

and we can neglect the term Te(aFH/aE). To illustrate the nature of spread in can be expressed as an integral over ~ alone and for FH we take

WDH,

we suppose that the u integral

2+ ~~WDH]},

(4)

FH(O~DH) (~wDHIlx)fll[(~.~DHCODH) where ~~DH = (kYEHc)/eBR. Work performed at PrincetonUniversity, Plasma Physics Laboratory Princeton, New Jersey 08540.

‘~

65

Volume SOA, number I

PHYSICS LETTERS

4 November 1974

For the last term in eq. (2) we then have (neglecting ~H) WDH)]

=

If we then search for a root near ~

[1/(w

“DH + ‘I~DHDI.

—

and assume i~DH~

-~‘~

(5)

~

fl

11r

(A)

=

W~FJ II&)DH~+ e”2

—~

~°

/ WD_huIDHI+

\

e (1+

‘1H (A)W —

r)2 ~o

*

1/2~/ej{T/~DH[(v

L (1 + r1 l6

T[(v/e)+L~w

I

~ ~H’~O’~

1, we find

})

+ 2 ~WDH + (A)~~]

1/e)+ IL~WDH1/ [(v~/e)

(6)

_______________

[(~

2 I- ‘~‘~~H + ~DH] 1/e)

We see that when W*~DH<0 (the criterion for instability when &A.DH

> (1/2) (~H/~

2]

0), stability is guaranteed if

.

0) (Te/EH)[T/(l + r)

This is essentially the result we stated at the beginning of this paper. The author wishes to acknowledge discussions with Dr. Harold P. Furth. This work was supported by U.S. Atomic Energy Commission Contract AT(1 l-1)-3073.

References (1] T. Ohkawa and D.K. Bhadra, Phys. Lett. 48A (1974) 140. [21 B.B. Kadontsev and O.P. Pogutse, Nuci. Fus. 11(1971)67.

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