5.7 Destabilization of Energetic Particle Modes by Localized Particle Sources (in collaboration
with University of California, Irvine)
Using the same approach of Ref. [1] and the formalism adopted in section 1.3.4, it is possible to
derive the dispersion relation for the low frequency branch of Energetic Particle Modes (EPM)
excited by ICRF induced fast minority ion tails:
2i
    2

2s (1  2DI )  E  ,
 A 4 | s | 
(1)
where DI is the Mercier index and E is the same as that of Eq. (3) of section 1.3.4, i.e.,
 3 /4
x x  3/ 4
8 b 
F(k LE , x)  xG(k  LE , x)
exp( x)dx ,
E    E 
3
(3/
4)

x


/

A 
0
 2
 2
2
2 J0 ( E )
2 J 0 ( E )J 0 ( BE )
F(k  LE , x)  
dy , G(k  LE ,x)  
dy ,
 0 1  y 2 
 0
1  y 2
(2)
Nothe that, Eq.(2) reduces to the same expression given by Eq.(2) of section 1.3.4 in the limit
 / A 1/ 2 .
Further novel features if EPM excitations emerge when their destabilization by highly localized
particle sources is considered. Usual approaches assume that, even for small magnetic shear, the
mode structures for high- n modes are contained within a distance of order 1/ nq' from the mode
rational surface. This means that k|| sets the radial extent of the mode, since equilibrium is fixed on
such scales. By localized particle sources we mean fast particle populations characterized by sharp
gradients and varying on distances shorter than 1/ nq' in very weak shear equilibria.
Here, we assume, e.g., the case of a low-frequency EPM excited by a strong energetic particle
source (ICRF-induced), characterized mainly by spatial density gradients. The latter assumption in
made only for the sake of simplifying algebra and does not limit the generality of the present
discussion. Assume also k LE ,k BE 1 and that the particle source is localized near r0 , where
|  /  A || nq(r0 )  m | . This is crucial to minimize continuum damping (cf. later).
In this case, it is readily shown that the eigenmode equation is given by
 E0 
r02  2
(r  r0 )2 

 2 2   1 
1
  0 ,
2


m r

  A0 

(3)
2
2
where  0 A   2 / 2A  nq(r0 )  m , a spatial dependence in the form E   E0 1 (r  r0 ) /  
has been assumed with (r  r0 )2 / 2 1 for consistency and
2


4
x7 / 4
exp(x)
 b E0 
1  
E0 
dx  .
 2  3 (3 / 4) x   / 

dE
0
Equation (3) is easily solved and it yields the dispersion relation:
(4)
r


E0,l   A0 1 0 1  2l  ,
 m

(5)
where l is the radial mode number. Consistency conditions are satisfied provided that l is moderate
and:
r02  2 (r  r0 )2
r
 0  1 ,
2
2 
2
m r

m
which is compatible with | nq'  |1 if | s |1 . Analogously, consistency conditions on the
starting assumption of negligible continuum damping are obtained using the form of the
eigenfunction
 r  r 2 
0
  exp 
 ,
2



D
(6)
with 2D  2(r0 / m) . Since the interaction of the mode with the continuous spectrum takes place at
(r  r0 )  (r  r0 )C  ( /  A )(r0 / ms) , continuum damping is exponentially small provided that
(r  r0 )2C
2D
 1 
 2 r0
r0
2 2A s2
.

1

1


2 2
2 As m
m
2
Under these conditions, we have, thus, shown that there exist radially localized EPM instabilities 
see Eq. (6)  driven by localized particle sources, according to Eq. (5), which are affected by
negligible continuum damping, i.e., which should have a negligible excitation threshold.
[1] S.T. Tsai and L. Chen, Phys. Fluids B 5, 3284, (1993).