NOVEMBER 1979
PPPL-1609
TWO-DIMENSIONAL ANALYSIS OF
TRAPPED-ION EIGENMODES
BY
R. MARCHAND, W, M. TANG/
AND G. REWOLDT
wsrai
PLASMA PHYSICS
LABORATORY
15TH1BUTOH DF TIIS DOCUMENT IS UKLIKITEO
PRINCETON
UNIVERSITY
P R I N C E T O N , NEW JERSEY
This work was supported by the TJ. S. Department of Energy
Contract Ho. Ey-76-C-O2-3073. Reproduction, translation,
publication, use and disposal, in whole or la part, by ot
for the United States Government is permitted.
Two-Dimensional Analysis of Trapped-Ion Eigenmodes
R. Marchand,
W.M. Tang, and G. Rewoldt
Plasma Physics Laboratory, Princeton University,
Princeton, New Jersey
08544
We present the first fully two-dimensional
e: genmode analysis of the trapped-ion instability
in an axisymmetric toroidal geometry.
Although
attention if. focused on this particular instability,
the calculations also takes into accovmt the basic
dynamics associated with other low frequency modes
such as the trapped-electron instability and the iontemperature-gradient instability.
The poloidal struc­
ture of the mode is taken into account by Fourier
expanding the perturbed electrostatic potential, <t> ,
in e .
Assuming that the perturbation varies mildly
over a typical ion-banana-width, p
, each poloidal
harmonic is then expressed as a truncated Taylor
series in the minor radius to account for the radial
structure.
The resulting set of coupled ordinary
second-order differential equations is solved by the
method of finite elements after prescribing the appro­
priate boundary conditions.
Our governing equations
involve summations over ion bounce and transit harmonics
so that they are valid for mode frequencies Less than,
and of the order of the ion bonnce and transit freauencies
BTSTftlBimOK OP THIS MCUMEIET IS u m r t U T H j t V
-2-
(JtjjJ < J!
~w ) .
Our formalism is also applicable to
the familiar radially local and one-dimensional radial
approximations.
Results obtained in these limits are
presented and found to be in reasonable agreement with
previous calculations.
In low shear plasmas, the full
two-dimensional calculation is in qualitative agreement
with the one-dimensional radial approximation.
However,
for higher shear (which gives rise to multiple rational
surfaces in the region where the mode is localized) it
yields a somewhat different picture.
Specifically, the
radial structure of the trapped-ion mode is found to
differ considerably from that previously obtained in
the one-dimensional approximation.
Here, the analysis
presented is limited to long wavelength (k p, .<l,k
r
r
being a typical radial wave number) and electrostatic
perturbations.
Present Address: University of Maryland, College Park, MD 20742.
-3-
1.
INTRODUCTION
The relevance of microinstabilities to tokamaks is associated
with the well-known fact that, even when the plasma is stable
against the most dangerous MHD .modes, microturbulence in the range
of frequency
|u)| - |i" l
Ae
(w* being the electron diamagnetic drift
e
frequency defined later) can cause anomalous transport.
A thorough
review ot the present status of microinstability theory in tokamaks
has recently been given.
In this paper we concentrate on instabil­
ities which occur in the range of frequency
| u | ~ |u>* | 5
e
w
h l
~
a )
t i
'
w, . and u . being the typical ion bounce and transit frequencies.
In addition, we will require that the plasma be sufficiently collisionless to fall in the so-called "banana regime" (where v ,.
. < u. . ,
with v £ * and u. . being the effective collision frequency for
ef
trapped particles and the typical bounce frequency for species j ) .
Specifically, we present an eiyenmode analysis of the trappedion instability which can be used to calculate the full twodimensional structure of low frequency microxnstabilities in an
axisymmetric toroidal system.
In addition to the usual effects
associated with the trapped-ion mode, our analysis also includes
important dynamics commonly associated with specific instabilities
such as the trapped-elgctron mode, and the ion-temperature-gradient
mode.
The resultant calculations are, therefore, valid over a
wider range in parameter space than previous analyses dealing with
each of these instabilities on an individual basis.
Moreover, our
governing equations include summations over ion bounce and transit
harmonics so that they are valid for mode frequencies of the order
of or smaller than the typical ion bounce and transit frequencies.
-4-
The trapped-ion mode (TIM) was discovered by Kadomtsev and
Pogutse.
in this first simple calculation, the authors showed
that when both the electrons and the ions were in the banana regime
Ca situation expected for thermonuclear plasmas) new low frequency
instabilities could exist in the plasma.
Considering a low 6
plasma {|3 being the ratio between the plasma pressure and the mag­
netic pressure) in a large aspect-ratio tokamak and ignoring tempera­
ture gradients, Kadomtsev and Pogutse used a simple Krook collision
operator to generate an eigenmode equation in the electrostatic
limit.
In tF»e radially local limit, they solved the resultant equa­
tion and found two types cf unstable modes.
limit (v
In the collisionless
, v- •*0) , they obtained a purely growing "trapped-particle
interchange niode" and in the dissipati-vie limit (V../E << | o) | << v ./£ ,
t; being the local inverse aspect ratio), they found a drift-type
1
mode propagating in the electron diamagnetic direction lii,.!!)^) ,
For the dissipative mode, the electron and ion collision frequencies
were found tc! be destabilizing and stabilizing, respectively
discovery of
a
This
potentially dangerous unstable mode in a parameter
range of relevance to thermonuclear plasmas stimulated much sub­
sequent research directed at better understanding the stability
criteria and possible effects of such modes on plasma confinementWith regard to analyzing the spatial structure of these eigePmodes, several contributions have been made by numerous authors in
3
the past several years.
For example, Rosenbluth et al.
used a var­
iational method to study the poloidal structure in the radially lc?cal
limit.
The radial structure was subsequently treated by Gladd and
Ross in a one~dimensioral model.
In this approximation the ^cituibed
potential v<as assumed to be nearly constant along the field lines.
-5-
The analysis presented in this paper is the first to account for
the full two-dimensional (poloidal and radial) structure of the
trapped-ion instability in toroidal geometry.
The outline of this paper is as follows.
In Sec. 2 we derive
the basic eigenmode equation, point out the various approximations
made as well as the corresponding limitations, and emphasize the
new features.
Section 3 deals with the radially local limit.
Specifically, we concentrate on the dissipative limit in which an
analytic dispersion relation can be obtained.
Comparison of the
analytic results obtained with previous calculations provides us
with a good test of our physical model.
In Sec. 4 we focus on the
one-dimensional approximation of Gladd and Ross
bility is assumed to be nearly flutelike.
full two-dimensional calculation.
in which the insta­
In Sec. 5 we present our
The results are compared with
those obtained in the one-dimensional approximation.
Also, we
present results obtained using idealized TFTR parameters.
A brief
summary of the results together with concluding comments are given
in Sec. 6.
2.
DERIVATION OF THE EIGENMODE EQUATION
In this section we derive a linear eiger.mode equation which
describes the trapped-ion instability in a large aspect ratio toroidal
system.
The analysis is appropriate for a low g plasma, and it is
assumed that the radial wavelengths are long compared with to a
typical ion banana width.
The general procedure consists of solving
the linearized drift-kinetic equation and using the resultant per­
turbed distribution function to calculate the perturbed density
-6-
response for each plasma species.
In the familiar limit of suffi­
ciently long wavelengths (kX << 1) and sufficiently low frequencies
D
(|(D| << u) .] , the quasineutrality condition can be applied to generate
a set of coupled second-order ordinary differential equations which
governs the pe-turbed electrostatic potential.
Together with the
appropriate boundary conditions, these equations determine the
1
spatial structure anc eigenfrequency of the instability.
Details of
these calculations will now be presented.
2.1
The Drift-Kinetic Equation
The form of the linearized drift-kinetic equation governing the
perturbed distribution, f , is given by
^ T + (v
| | +
v )-?
D
= - ^.VF
f l
v
M
+ e
Cv
| | +
v W<^-v(E,A>(f
D
1 +
!ji-F j ,
M
(1)
where
2
v., = ° n " C T r ) r i - A / h o ) ]
V
i
=
D
X
ryVB +v^n-Vn]
_
B x V$,
v^-c—j-L
V!
,
(2)
,
(3)
m
v j E r A )
(4,
_ ^4±«i^
Y
1
,
y
Vi |l-e-Ai
h
,
(5)
'
|l-E-A|
h(9) = 1 + e cos 6
,
(7)
1
4irn_e ' lnX„.
a i
'
h
(2T )*Vll
(8)
'
(continued)
-7-
Y = E/T
e
H ( x )
,
= *P^>
vTx
<
+ (l--V)erf(X)
2X
.
9 )
CIO)
In these equations the subscript 1 is used to label perturbed
quantities.
F
is a local Maxwellian in which the density and
temperature only depend on the minor radius, r ; n is a unit vector
parallel to B ; SI is the gyrofrequency; E is the ratio of the minor
radius to the major radius, e = r/R ; In A
is the usual Coulomb
logarithm ,• 3nd erf is the standard error function.
Here, the
spatial coordinates used are the usual toroidal variables, r , 8 * £ /
(see Fig. 1 ) , and the velocity space variables areff.., E , and A /
with o..
representing the sign of the parallel velocity along the
field line, E being the total kinetic energy of the particle, afd A
being the pitch-angle variable defined by
where \i is the magnetic moment and B
magnetic field on axis.
is the magnitude of the
In Eq. (1), note that collisions are modeled
by a simple energy and pitch-angle dependent Krook collision operator.
An interesting fact about this operator is that it automatically
accounts for the enhancement in the effective collision frequency of
trapped and barely circulating particles.
As will be discussed later,
it is inadequate in the small collision frequency limit (v. ./E << |w|)
but becomes quite accurate in the large collisionality limit.
Also,
it should be noted that for a low $ plasma in a large aspect ratio
tci1f.a5!J.a.fc, It is. 3.^9-c^ici«,te. to CQ^ai'd^x -cix'CMl.a'E wsws«rA.Txc: IVcnt strcfaces and a safety factor of order unity; i.e..
-8rB
q = g-^- - 1
o 6
•
(12)
Also, consistent with the large aspect ratio assumption, we shall
keep all quantities of orders
of order
1 and /E
and some selected quantities
E , when the lowest-order terms vanish.
ignore corrections of order
We will generally
E to quantities of order
1.
Finally,
note that in the equations above, we assume that the plasma consists
of a single ion species of charge unity.
Equation (1) can be solved by the method of characteristics.
After multiplying by the integration factor e
d ,,
( £
dt >
vt .
e
H
d ,. vf .
e
(
»= ^Fldt *
e
-+(1) „„
i vt I
' - ^ F
1
e
it can be rewritten
V
)" E
'
V P
M
vt
e
,,,,
( 1 3 )
'
where
dT - £
+
<V*D>'
V
( 1 4 )
represents the total time derivative along the unperturbed guidingcenter trajectory.
t' =-»
t' = -=>
Integrating each side of Eq. (13) from time
to t' = t , and assuming that the perturbation vanishes at
we find
+
*.<« = - V - * M l _ .
d t ,
1 )
7 p
[-!T-T5F-'i - iiHp
I v < t ,
t
1
- > -
(15>
Note that in taking dF /dt = 0 , it is assumed that typical radial
excursions along the unperturbed trajectory are small compared with
the equilibrium scale-length.
-9-
Proceeding with the analysis, the integrant! in Eq. fl5) is
written as a function of time so that the integral can be performed
explicitly.
Since our problem is linear and stationary in time, ai.d
since we assume an axisymmetric geometry, the time and azimuthal
dependences of all perturbed quantities can be assumed to be of the
form exp[i(tc-ut)] .
Since the r and
6 dependences of the per­
turbed potential are not known, we are left with a two-dimensional
problem in configuration space.
In order to account for the poloidal
structure of the mode, we expand
<tj as a Fourier series in 8 ; i.e.,
>(r,6,5,t) = I <f (r)exp[i Ue-me-wt) ]
ra
Substituting
f,
= -
eO,
P
F
F
M
this
+
i n Eq.
(15)
eeFF
rt
-Y^expt-ime)
m
(IS)
gives
d
t
'
(-iu+i" ;",?)*,, ( r ' )
1
* — DO
x e x p { i U ( ? ' - C ) - m ( O ' - B ) - <u + i v ) ( t ' - t ) ]} ,
(17)
where
exp[-iU?-uit) ]
(m) _ ,,(m)
u*
[1 + n ( Y - / ) ]
m>
J
2
*r
fm) _ _ m cT 1 dii
*
r eB n d r
o
n n
This
yields
an integral
(18)
,
(19)
(20)
'
d(lnT)/dr
d(lnn)/dr
equation
(21)
for
the
Fourier
coefficients
<p (r)
m
In the following we shall make the simplifying assumption that each
-in­
harmonic varies only weakly over a typical orbit radial excursion.
The form of the solution given in Eq. (17) will, then, lead to a
set of coupled differential equations which proves to be tractable
to numerical and analytical procedures.
If the condition
k f, '1
r
is well satisfied, where k
and <,
hi
r
I'i
arc the typical radial wave number and ion banana width respectively,
the potential can be approximately written in the form,
i fr') * 1> (r) i -(r'-r)*' (r) + '/• (r'-r)V'fr) , (22)
ram
m
m
so '"hat Eq. (17) becomes:
dt'[-i. + UrJ)[^
L
f = - -TT r' + -=•-}'exp(-iinn)
M
)i
(r) + (r'-r^'tr!
' - '"
in
+ 7 <r'-r)Vm (r) ]exp{i [P.U'-O - m (f> '-9) - (u+i v)( t' -t) ] • .
?
(23)
Since the unknown functions, 1> , can now be factored out, the
m
integral in Eq. (23) can, in principle, be evaluated explicitly
because it only involves known equilibrium quantities.
In practice,
this is done by judiciously expanding some of the periodic equilibrium
quantities as Fourier scries in time.
Specifically, for the radial
excursions of trapped particles, we take:
r - r (0)
where
r
y
=
r
(n)
exp(ina). t) ,
(24)
is the orbit averaged value of r
r
( 0 )
= ±- H t r
,
(25)
b
w
b
and
T, being t h e bounce -.requency and p e r i o d , r e s p e c t i v e l y .
b
a r e defined by:
They
-11-
M '°*
= 4 .-_ — J < K - U - H / h ( 6 ).-'/,
f "
/Vv., ^0
th
G
b
(26)
and
"b =
Here,
2
T
/
•
\
(
2
?
)
f'„ is thf? angle of the trapped-particle turning points,
" = arccos [ (.'.-D /' 1 , and v , is the thermal velocity defined by:
t>
The corresponriinq quantities for circulating (passing or untrapped)
particles are the transit period,
dO[l-A/h(G)l
/vv
t h
(29)
Jo
and the transit frequency,
u. = 2 r / T
t
Here, the conve-ition is that
t
.
(30)
to, and w
are always positive.
In the
following analysis, we will restrict our attention to radial excur­
sions of trapped ions only.
Those of circulating ions and trapped
electrons are smaller by factors of order
/z
2
and (r: /M-) ,
e
respectively, and will be reglected.
The exponent appearing in Eq. (23) can be written in the form:
*{?'-;) -m(9'-6) = ! at"\l-&t
I- dt
J
-m-^-]
dt
•
(31)
-12-
33)
b,t
;
(0>
(m)
5 ( r ) = J-q(r) - m
,'.S = ?lq(r> - q ( r
,
( 0 J
)]
W(t) = -AS(0' -':) +
(35)
,
< 3fi)
dt Aw
Jo
(37)
u
•.v'• i.- jrj rcwrile I-',;j. (51) as :
U -,'---,) - m("'-f<) = (a)
( 0)
D
+ H 0 S w ) (t'-t) + W* - W + S(f)'-O)
n
t
- HO Sw (t'-t)
where
H = H(1-c-A)
,
(38)
vanishes for trapped particles and is equal to
one for circulating particles.
In £.j. (38) we omitted the arguments, r , and the superscript, m
We shall continue to do so unless they are needed to avoid ambiguity.
Also in Eq. (38), W
1
and w stand for W(t')
and W ( t ) , respectively.
From Eq, (38), the exponential appearing in Eq. (23) can be written
as:
e
l f 3
= exp[-i(a)- J^
- Hff S(o + iv) ft'-t)Jexp{-i [S (e '-6) - Ha Sw (t'-t)
((
(/
£
+ w' - W ] } .
The second exponential is a periodic function of time and can be
expressed as:
(39)
-13-
e
1 M
- j a 9xp[in(o.
n
(t'-t) ]
.
(40)
D. t
when Eqs. (24), (39), and (40) are substituted in Eq. (23), the
time integration can be explicitly performed to yield the following
solution for f i "•
f = -^-F
'
"
~
ZaxpC-imOMu-u
m
[G(n+n
) - G ( n ) ] + 'A *"
M
*
+
)£a [* G(n) + C; I r ^ i ' e x p d n ^ t l
r,
n,
>
r '"' r
( n ? )
e x p [i (n , + n ) ^
7
t]
iij n „
x [ G ( n t r i j + rij) - 2 G ( n + n , ) + G ( n ) ]
/
(41)
where we define
G(n) = [w - ^
0 )
- (n+HO S)oa
|(
b t
+ iv]~
.
(42)
3
Explicit expressions for the coefficients r
anc. a
can be
n
derived straightforwardly .6 The resr^.s are
n
.1*1 -
/Y /if I
D
\
0
,
n even
,
(4 3)
and
a
nl
= e x p [ - i ( S S - n o 3 t + W)3 ^
J
b
trapped
a_j
,
n r a
(44)
b
= e x p { - i [ s 8 - (n+a S)m t + W ] } A
'..irculating
t
L
I de ' [1-A/h (9 ' ) ]
t ^0
x cos[Se' - (n+o^S)u t' + W'l
J / i
t
(45)
with the definitions:
-14-
P
G
=
I
V
t h J
=
n
( 4 6 )
'
dfl c o s ( n i i i , t )
,
(4 7)
h
J
= | °d. | l - A / h ( n ) ] -
nm
] / 2
2
[cos (^)cosfs
( m )
e ) c o s ( n a . t + w)
b
J
+ sin (^}sin(s
L
b
L
=
lk \ " d e i l - A / h t e ) ] "
= £
f
' -
ao[i-i\/h(en"
7 2
I / 2
( r a )
ej in(n
s
U h
t + w)] ,
(48)
,
( )
4 9
.
f 5 0
)
IT
Here, t is a function of 9 and a., and has a one-to-one corre­
spondence with each point of any single orbit.
R q
f
6
It is defined by
,.
d9'[l-A/h(6') ]~
h
i
t(9,a„) = '
T./2 - t(6,+l) I
,
!I
(51)
=
for trapped particles and by:
R
B
t
^
.
.
i
t(e,a ) = — —
o.. f de' [l-A/h(e'}]~
^v
/Yv..
"%
Jn
k
M
11
t h
for circulating particles.
Note that the expression given in Eq.
(41) provides a complete solution for the perturbed distribution
function, f , in terms of an arbitrary electrostatic potential <P
t
(52)
-15-
2.2
The Perturbed Density and the Quasineutrality Condition
The next step consists of calculating the perturbed density
response, n, , by integrating over velocity space.
Making use of
the fact that
,
r=°
d v - 2-
f
00
3
r
0
dv
Vj
_d
Vl
«
r
,h(6)
rh(6)
0 0
„ m /VJ
2
= i(4r)
d
I
Y
/
ww-i
!
M
dAfhte) [ i - A / h ( e ) ] ' }
(53)
we find that
n^r.e)
n
has the form:
= [ e x p ( - i m B ) [P (r,6)*U (r> +Q (r,e)iJ> J + R. (r,0)» ] . ( 5 4 )
m
n i
l
ni
i
i
T 1
] ] l
This can now be substituted in the quasinputrality condition to
yield a system of equations of the form:
Zexp(-ime)(P *; Q * ; R * ) = 0
m
m
+
nl
i
+
ra
m
.
(55)
These equations, however, have the disadvantage that they involve
the two variables r and 8 .
It is convenient to remove the explicit
dependence on 0 by taking the Fourier transform of Eq. (55) with
respect to this variable.
Hence, operating on Eq. (55) with
_T de exp (ipe)/2TT , reduces it to a set of coupled ordinary differen­
tial equations of the form
+B
+C
( 5 6
I<V*» pm*™ p m V = ° '
>
m
which is independent of 8 .
Equation (56) determines the Fourier
coefficients of the perturbed electrostatic potential.
Using the
definitions in Eqs. (47), (48) and defining
K
n „ » = f 3e[i-A/h(e)]-
1/2
(p)
C0S
[s e- [
( m )
n + s
Kt]
,
( 5 7
)
-16-
a
we find, after performing the
summation, that the matrix elements
can be expressed as:
i + e
A . ^ L fl> ^
r d v ^ - M - - „- - - t - ^ - h-t L, Jo
M
y
Di
J
J
l n+n +n ,p n,m ~
]
Tl p
=
pm
fl+C
"
v/
r
J
a
Y
Y
e
Y
bi
I
r J
>'
odd
n + n , ,p n - n ,m '' n,P ti-n,-n ,m J ,(58)
J
J
?
„
dYYe~
?
til-HI .
m
7
Z
*
T i
*
y
Z n.
n,
.i
odd
T
(0)
n OJ-UI„. - no>, ,+iv.,
Di
bi
i
J
•n '' l-c
2J
?
,-<"
j.
1A
?
L, 0
b
> 1.1
e
1
b
1
i
J
J
-J ,
J
J
n,p n-n^m
n+iij.p n,ra'
1
(59)
and
^
4]
h
V
J
i
an
J,,,,
tr *»
L
-Y
*Te
De
J
l-c b 0
7?
+ I "
,
L,
L
dA
r
.
T
e
L
H
v
y
^
-Y
(0)
M
y
""-%Ti "
T
+ 1 V
i
n,p n.n
J
M)
-
where
, .
(l + T)<S
pm
,
is the electror-to-ion temperature ratiy defined by
The only contribution to the
A
and
K
K
V n,p,ra n,m,m
(60)
r = T /T .
H
B matrices comes from the trap­
ped ions, because radial excursions of particles other than trapped
ions have been neglected.
a)
Here, we ignore the average drift frequency,
, for circulating particles because it is smaller than its trap­
ped particle counterpart and typically much smaller than | to | (see
-17-
Appendix A of Ref. 7). Also, it should be noted that, the C matrix
has no contribution from the nonadiabatic circulating electron
response and the trapped electron non-time-averaged response.
The
reason is that, in most of the plasma cross section, these contribu­
tions would be smaller than their circulating ion counterparts by a
factor of order M /M-^ and can, therefore, be ignored.
e
In the following calculations, we will ignore the phase factor,
W , which appears in the definition of J
n ra
given in Eq. (48). This
approximation is appropriate because inclusion of this factor only
leads to corrections of order
p 74r or u
/ ^ M ' with Ar being
the distance between adjacent rational surfaces.
For the parameters
of interest, these corrections are always small.
As a practical
point, by ignoring this factor, the numerical evaluation of the
matrix elements can be considerably simplified; i.e., without the
W term , the angle integrals, J
2.i
, are now independent of energy.
Boundary Conditions
To obtain a well-posed eigenmode problem, we nee-1 to supplement
Eqs. (56) with appropriate boundary conditions.
To do so we assume
that the plasma is in contact with an axisymmetric conducting limiter
at a minor radius, a , at which the tangential electric field must
vanish.
We also require that, on the axis, the electrostatic energy
be integrable and that the perturbed electric field does not cause
any accumulation of charge there.
These conditions yield:
-18-
<f (a) = 0
for all
t (0) = 0
,
m
m j- 0
m
,
(61)
,
(62)
d
*o 1
7TF"
= 0
.
(63)
'r-0
The system of equations (56) along with the boundary conditions
given in Eqs. (61) to (63) determine the basic form of the trappedior. eigenmode problem which will be studied.
2.4
General Remarks
The system of eigenmode equations just derived contains some
interesting features, and it is appropriate, at this point, to make
the following comments.
A.
First Derivative Terms
An interesting fact about Eq. (56) is the presence of first
derivatives of the perturbed potential (associated with the B macrix).
This might seem surprising, since such terms were absent in previous
4
eigenmode calculations.
However, since the presence of these terms
is generated by ballooning effects (neglected in the earlier studies),
our basic equation is in fact consistent with those appearing in
previous studies.
Specifically, when the phase factor, W , is neglected
in the definition of J
given in Eq. (48), it can be shown that the
n,m
^
3
diagonal elements of the B matrix vanish identically.
Thus, when
the coupling between the various poloidal harmonics is ignored, it
follows that no first-order derivative terms appear in the eigenmode
equations.
This is indeed the case in the earlier calculations,
where the mode is assumed to be flutelike; i.e., approximately
-In­
constant along the field line.
B.
Truncation
In order to solve our eigenmode problem, in practice, it is
clearly necessary to truncate the number of unknown functions and
equations to a finite number; i.e., to introduce lower and upper
bounds, M 4„ and M „„ , for the indices p and m in Eq. (56). In
m
ra
doing so we must make certain assumptions concerning the structure
of the instability.
In what follows we choose to limit our atten­
tion to modes which nre not very strongly ballooning and take
M
™
>^
{65)
'
m a x
The influence of this truncation on the final result can be investi­
gated by varying M
m i n
and M
x
.
In the results
which will be
presented, this truncation error is always less than 10%.
C.
Reduced Intervals
Finally, in practical calculations, it proves convenient to
solve the eigenmode problem in the subinterval [x,,x ], instead of
2
[0,1] (0 _< x, , x <_ 1) , where x is the normalized radial coordinate,
2
x = r/a .
This is necessary to avoid certain divergences at x- 0
and, as a practical point, it can also be exploited to make the
numerical solution of the problem more efficient.
The next task is
to ciioose the appropriate boundary conditions to apply on this
reduced interval.
In what follows we will always use the boundary
conditions given in Eqs. (61) to (63) and will treat the left and
right boundaries of the interval as if they were located at x = 0
-20-
and x = l .
'This proves to be adequate, provided the solution is
sufficiently well localized within [ x x ] .
](
points, x, #nd x
2
2
Specifically, the
must fall sufficiently far into the evanescent
regions of the solution.
3.
RADIALLY LOCAL LIMIT
In the limit when the ion banana width is much smaller than
the equilibrium scale length, derivatives rith respect to r can be
ignored, and the system of equations (56) reduces to a simple set
of algebraic equations of the form,
[C
m
p m
*
m
= 0
.
(66)
This radially local eigenmode equation has the virtue of being
considerably simpler than its full two-dimensional counterpart.
It is readily amenable to numerical solution by standard methods
and, in some limiting cases, to a completely analytic solution.
In this section we consider the radially local limit in some
detail.
First, we find an analytic dispersion relation valid in
the dissipati
ve
limit. Then, we compare our results with those of
3
Rosenbluth et al., who developed a variational principle that
enabled them to study analytically the mode structure along the field
line and to treat collisions with the more accurate Lox*entz model.
This comparison serves as a good test oE our sir.ipie Krook colli­
sion operator-
-21-
The approximations involved are that (i) the plasma is in the
banana regime,
U
5
« b|
;
( 6 7 )
4
(ii) the aspect ratio is large,
-2- << l
and
.
(68)
(iii) the familiar dissipative li.iit (within the banana regime)
is appropriate,
v..
v .
J i << | | << -ll
.
u
In a d d i t i o n ,
the following simplifying
\ui\
<< i L
a s s u m p t i o n s a r e made:
,
bi
S.q = m + V
(69)
2
(TO)
,
(71)
and
«(8,e)
= *„ e x p [ i ( i c - m 6 ) ] + *
m
+
1
e x p { i U c - ( m + U &J >
•
(72)
Consistent with Eq.
(70) we shall treat terms in
(nonvanishing) order.
Note that Eq.
<»/\-
only to lowest
(71) corresponds to a mode which
is localized halfway between rational surfaces.
This assumption
(which facilitates the evaluation of several integrals) is suggested
by the strong circulating-ion Landau damping that should occur close to
rational surfaces.
Ref. 3.
Finally, Eq.
(72) is suggested by the results of
It corresponds to a weakly-ballooning mode for which the
Fourier series in
6 can be truncated to only two poloidal harmonics.
-22-
It should also be noted that, consistent with Eqs. (68) and (69),
we will neglect u
compared with
D
|u| and treat the circulating
ions as being collisionless.
Let us now evaluate the elements of the C matrix and solve
Eq. (66) analytically.
In doing so, it is convenient to distinguish
between the various contributions to C and to evaluate each one
separately.
In practice, given the above assumptions, this is done
by rewriting Eq. (60) as:
r
-DID
l+E
Y
TT
^w-^C^i>^-"
Si [ aY/7e" " ""'
5/
l-£
i
b
J
0
v
Vi
. l
M
-Vi)
2r/e
l-e
2
2
2
Tip
Tim
0
1+E
dA ("
(i)((,i-.^
dA
"t
f
y
dY/Y e
Y
=
_.)
*Tl
r
m)
n (D - [ ( n + s f K
JQ
T K
K
Tipm niran
2
2
( 1 + T)
pm
]
(73)
The various terms are the trapped-electron, thp r-^llisionless and
collisional time-averaged trapped-ion, the (nj£ 0) non-time-averaged
trapped-ion, the circulating-ion, and the adiabatic contributions.
After some algebra,
it follows that the C matrix is of the form:
C =
a + a.
(5 + a + a,
(74)
-23-
where 6 , a , and a
represent the adiabafcic, the time-averaged
trapped particle and the non-time-averaged (nf 0) trapped
circulating-ion resonances, respectively.
be solved perturbativelyv
and
Equation (66) can now
As usual, for the trapped-ion instability,
the adiabatic and ,ttie time-averaged collisionless trapped-ion terms
are taken to be of order unity.
The other contributions are all
assumed'to be small perturbations of the same order.
To lowest order, we have:
6
( 0 )
+ 2a<°> = 0
,
(75)
which yields the lowest order mode frequency:
lU
o; ' = 0.577
,^ *
1+ T
e
.
(76)
Here, the superscript indicates the order oZ the various terms in
the small expansion parameter.
It is clear from Eqs. (71), (72),
(74), and (75) that t'.ie lowest order eigenfunction is of the form
4(0,C) °= cos(6/2)exp[i£(c - q6) ]
;
(77)
i.e., the mode is ballooning on the outer side of the torus.
To first order, we have
( 1
2a >
+
a^>
+
a™
= 0
.
(78)
Here, it is convenient to define
J
where Y , 1 , Y
e
t
T L
l )
, Y
= ifYe + Yi+Y^+Ya,)
C L
,
(79)
represent the collisional trappad-electron
and trapped-ion growth rates and the trapped and circulating-ion
-24-
Landau growth r a t e s , r e s p e c t i v e l y .
order
-
Then, Eq. (78) y i e l d s t h e
first-
corrections:
= 0.798 . ^
(1)
T
(1+1.41n
)
(80)
1 + T
o
Y
i
1 +
V
T
— = - 4 . 6 1 ^-±-^(1 -
r
ii
E
In I n ( 1 +
1.0S5r\.)^~*e
u.
T)V
1.06
(81)
ii
a
Y
•>
TL
3
U J
,
f *e l
(82)
and
^
(83)
- 3 . 2 0 ( ^ 1 U - ' A n ^ hID p
where
w
i s d e f i n e d by
u
v
t =
t
A
(84)
q
A very similar dispersion relation was obtained by Rosenbluth
et al.
3
Assuming the same ordering as in Eqs. (67) to (71), they
found
- o - O . S B ^ ^ .
,
(85)
3/
/2
£
2w
2
' *e
= 0.84(1+ 1 . 4 1 n ) 7 e
rT
(86)
ei
Y,
— = - 0.31(1 - 0.57r
(87)
(88)
-25-
with the corresponding lowest order mode structure
*(e,;) « (cos 9/2 + 0.14 cos 3e/2 - 0.02 cos 59/2) exp [i (£5 - q8) ] .
(89)
We recall that this result was derived using a variational principle
and that collisional effects were calculated using the Lorentz
operator.
Comparing these results with the ones found with our
Krook operator, we see that, except for the ion collisional growth
rates, the two are in basic agreement.
The most important point in this comparison is that it provides
a direct test of our simple Krook collision operator in two collisionality regimes.
Indeed, since the electrons are assumed to be
strongly collisional (v> ,/z >> \ 101) , and the ions, nearly collisionless (v. ./c << |ui|) , we see that a comparison of the electron and ion
collisional growth rates obtained in both cases will show how
accurate the simple Krook model is in both of these limits.
Thus, by inspection of the above dispersion relation, it appears
that the Krook collision operator is quite accurate in the large
collision frequency limit, but that it fails to predict the correct
asymptotic behavior of Y- in the opposite limit.
This lack of
agreement in the small collision frequency limit is a consequence
of the fact that collisions take place predominantly in a narrow
layer at the boundary between trapped and circulating particles and
cannot be reproduced accurately with a simple Krook collision operator.
Mote, however, that in the code, we do include ion-ion collisions
using the simple Krook model.
wil
1
The reason is that, in practice, we
consider cases for which the ions are not far into the colli-
sionless regime.
In such cases, the asymptotic limit corresponding
-26-
to Eqs. (81) and (87) would not be strictly applicable.
It should be noted, however, that for the case of unstable modes
far into the collisionless regime, the use of an exact form of the
ion collision operator would be inconsequential since both Eqs. (81)
and (87) would give a vanishingly small contribution to the growth
rate in this instance.
Finally, we obberve that it woul3 be possible
to obtain better agreement in the scaling of y
with
j ..
and
-?
n . given in Eq. (87), if we choose [A - 1 + c|
-i
instead of ;/ - 1 + t.
as the A dependence of v in Eq. (6), This modified
A dependence
of the collision frequency yields an' ion collisional growth rate
scaling predominantly as (v..)V? . Additionally, the r dependence
here is precisely the same as that given in Eq. (87). However, this
modified collision operator fails to give the proper results in the
large collision frequency limit.
Specifically, the scaling of the
electron collisional growth rate with
i: is incorrect.
This is the
reason for our use of the more standard collision operator, given in
Eqs.
(5) and (6), for both electrons and ions.
4.
ONE-DIMENSIONAL RADIAL ANALYSIS
In this section, we deal with the one-dimensional radial analysis.
Following the approximation of Gladd and Ross,
assumed to be nearly flutelike.
4
the instability is
Then, the governing eigenmode equa­
tion has the form:
2
pf-^-4 + 0(r)<J> = 0
,
(90)
dr
where
p . = /e Q . , w i t h t h e b o u n d a r y
L
bi
n
tfi
conditions
$(0)
= 0
,
(91)
0(a)
= 0
.
(92)
-27-
The radial potential, Q , is given by
for arbitrary m.
Here, A,^ and C
Jtm
have the same form as given
in Eqs. (58) and (60) except that: (i) S = Cq - m is artifically set
equal to zero everywhere in the trapped-particle responses.
the circulating-ion response, we take S = S.g - m
the closest integer to lq .
factor
S
, with m
being
Note that with this prescription, the
only vanishes exactly at the rational surfaces.
The factor m appearing in the definition of io
t
replaced by
(ii) In
(iii)
in Eq. (20) is
>.q . Vfe note that the particular form of the eigenmode
Eq. (90) suggests that the radial potential, Q , can provide a rough
measure of k P.. , since
<*r
»M
2
> -
1. d
dr
(94)
2
Thus, our assumption, k p . < 1 , requires that |Q| be smaller than
r
one, at least in the region where the mode has most of its amplitude.
The parametrization of the equilibrium and the values of the
parameters used are given in the Appendix.
Except for the density,
they correspond almost exactly to those of Gladd and Ross.
reason for using a greatly reduced density fn (0) = 5 x10
13
of
5 * 10 cm
The
instead
— 3
] is that without the destabilizing effect of balloon­
ing introduced ari.j.1 ically in Ref. 4 through the n
e
and
n
i
correc-
tion factors multiplying the collision frequencies, the modes con­
sidered here would be stable.
-28-
The results are presented in Figs. 2 to 6.
In the calculations
it proves convenient to turn on and off the contribution to Q from
the nonadiabatic circulating-ion response.
This is implemented i
the computer code jy multiplying this contribution by an artificial
weight factor, w . , which can, then, be varied gradually between
0 and 1,
The results presented below are divided into three parts:
(i) the influence of circulating ions on the radially local spec­
trum; (ii) the influence of circulating ions on the radially non­
local spectrum; and (iii) the radial structure of the instability.
The radially nonlocal spectrum mentioned above is obtained by solving
the radial eigenmode Eq. (90) subject to the boundary conditions
in. Eqs. (91) and (92), The result yields a discrete (quantized)
frequency spectrum.
As discussed in Sec. 3, to obtain the radially
local spectrum, Eq. (90) is solved in the appropriate limit; the
radially local equation,
Q(r) = 0
is solved.
,
.'«)
Since solutions to this equation can be obtained at
arbitrary points in the plasma, and since the mode frequency depends
continuously on the minor radius at which it is evaluated, the
resulting spectrum is continuous.
The primary goal in the present
section is, of course, to gain an understanding of the radially
nonlocal spectrum and the associated eigenmodes, 4>(r) . Neverthe­
less, the radially local spectrum is still of interest since it
provides a rough initial estimate of the eigenvalues for the non­
local calculations.
Before presenting the results in more detail,
it is worth mentioning that the following calculations were also
carried out using the form of the radial eigenmode equation derived
-29-
in Ref. 4.
This led to results which were in qualitative agreement
with the ones presented
(see Ref. 6 for more details).
results corresponding to Eq.
A.
Only the
(93) will now be considered.
Influence of the Circulating Ions on the Radially Local Spectrum
The results of interest are displayed in Figs. 2 and 3.
Figure 2 shows the radially local spectrum found in the calculations
when the nonadiabatic circulating-ion response is ignored
(w =C) •
c i
Figure 3 shows the corresponding spectrum when the full circulatingion dynamics is taken into account.
ions can be summarized as follows.
The influence of circulating
(i) Only one mode corresponding
to the conventional trapped-ion mode (TIM) is found to be unstable
when the nonadiabatic circulating-ion response is ignored.
(ii)
When the full circulating-ion dynamics is taken into account, the
radially local spectrum has two unstable branches.
One corresponds
to the conventional TIM, and the other to a new low frequency
(!-'",l 5 ^' ) drift-type trapped-electron mode
e
i
ti
(TEM) .
(iii) Circu­
lating ions have a strong stabilizing effect on the TIM close to
rational surfaces.
Accordingly, when the full circulating-ion
dynamics is considered, this instability tends to be localized
between rational surfaces.
In a radially local sense, this mode
will grow away from the rational surfaces and will be damped close
to them.
(iv) In contrast, the TEM tends to be localized at the
rational surfaces.
For this case, the radially local eigenvalues
indicate growth at the rational surfaces and damping far away from
thfc:Tl.
The preceding observations, (iii) and
(iv), can be understood
by recalling the fact that, when the mode freguency is smaller than
-30-
a typical ion transit frequency (|tu| "^ ",..,), the nonadiabatic cir­
culating ion contribution is only important in narrow layers at
rational surfaces where
|w/(sijj
) | >1 .
Away from these layers,
the nonadiabatic circulating-ion response is small (of order
and has little influence on the TIM.
\H;\'/K'.)
However, close to rational
surfaces, this contribution becomes important and causes the strong
Landau damping of the conventional TIM.
Also, it is the fact thac
the circulating-ion response is nearly hydromagnetic close to
rational surfaces, which allows for the new TEM with frequency
- (U.
HI
r
B.
*e
Influence of the Circulating "*,< ns on the Radially Nonlocal
Spectrum
As the full circulating-ion dynamics is gradually turned on
(w . is increased from 0 to 1 ) , we observe that the various radial
eigenmodes behave in two different characteristic ways.
This
suggests the classification of those modes as (i) radial trappedion
ides (TIM), and (ii) radial trapped-electron modes (TEM).
Th' ;e are associated respectively with the radially local trappedion and trapped-electron branches.
Figure 4 shows that the nonadiabatic circulating-ion response
has a stronger influence on the TEM than on the TIM.
Indeed, as w
c j
is artifically increased from 0 to 1, the real part of the frequency
for mode #1 (TEM) rapidly increases from | ui | < [o^J , to |<* | ~ | u
i e
The frequency of mode #2 (TIM), however, is observed to be fairly
insensitive to w J .
function of w
±
This characteristic behavior of u as a
can be best understood by examining the radial struc­
ture of the instabilities.
I •
-31-
C.
The Radial Structure
The radial structure of the trapped-electron and trapped-ion
eigenmodes along with the corresponding radial potential, Q , are
shown in F^qs, 5 and 6.
In particular, the radial TEM (mode tl)
is mainly localized a_t a particular rational surface, while the
radial TIM (mode #2) is mainly localized between rational surfaces.
It is o? interest to note that there is a fairly sharp struc­
ture in thr radial potential, Q , located at rational surfaces
fr/a - 0.74 and 0.94).
ion response.
aiven by
This comes from the nonadiabatio circulating-
As stated, the width of this structure is roughly
.. /(Jq'v .) .
Specifically, for the (lower frequency)
TIM, it is quite narrow and its primary influence on the eigenfunction is to cause a rapid variation of the slope at rational sur­
faces.
On the other hand, for the (higher frequency! TEM, the
structure in Q associated with the nonadiabatic circulating-ion
response is sonewhat broader.
Its width is comparable to that of
the mode itself, and in this case, it acts as a potential well in
which the TEM is analogous to a bound state.
Hence, it follows
that at low frequencies ([to^J < to .) , the TEM is localised at a
particular rational surface because of the potential well produced
by the circulating ions.
We note that the circulating-ion response
is roughly hydromagnetic close to rational surfaces.
-^^5.
TWO-DIMENSIONAL RESULTS
In this section we present results obtained when the full
two-dimensional structure of the instability is taken into account.
In short the problem is that of solving a set of coupled ordinary
differential equations with prescribed boundary conditions.
This
calculation is carried out numerically by the method of finite
i
4. f>,9,10
elements.
'
In analysing the results below, it proves convenient to pro­
vide a quantitative measure of how well the basic assumption,
k p, . < 1 , is satisfied.
This is done by plotting k p,.
r
as a func;-
tion of the minor radius, for each poloidal harmonic, and by •com­
puting the average value of k p.
r
<k f), • > = I [
r'bi
L
I
according to:
d x <J>*k n , * \ l [
d x $*<!> ]
« r ' f c i HI M
.
(96)
ram
The radial v/ave number used is defined for each poloidal harmonic
by:
Roughly speaking, our calculation will be consistent provided that
<k rT)i
a. > is smaller than one, and also, orovided
that k r p,bi. is
"
smaller than °ne in the regions where the corresponding poloidal
harmonic has significant amplitude.
5.1
Comparison with the One-Dimensional Approximation
First, we consider the same parameters as in Sec. 4 and compare
our results with the ones obtained there.
The results are displayed
iu Eiqs. 1 t,Q L5 and thsir presentation parallels that of Sec. A,
-33-
The influence of the nonadiabatic circulating-ion response on the
radially (non) local spectra is shown in Figs. 7 to 9.
The spatial
structure of the eigenmodes, along with typical plots of <t and
m
the corresponding value for k p
r
, are displayed in Figs. 10 to 15.
The perturbed potential shown in Figs. 10 to 13 is the potential
along the field line and is defined by
0(r,e) = l * (r)exp[iUq - m) G]
m
.
(98)
in
A comparison of these results with the corresponding ones
obtained in Sec. 4 shows that, for the parameters considered, the
basic characteristics found in the one-dimensional radial calcula­
tion still persist in the two-dimensional calculation.
Specifically,
in the radially local limit, the nonadiabatic circulating-ion
response contributes strongly to tho destabilization of a low
frequency (|u
I < u . ) type of TEM close to rational surfaces.
Additionally, it accounts for the strong Landau damping of the con­
ventional TIM near such surfaces.
Away from rational surfaces, the
nonadiabatic circulating-ion response is small and has no significant
influence on the TIM.
In carrying out the nonlocal calculation, we
still find two characteristic types of eigenmodes.
One is localized
a_t a particular rational surface and corresponds to a nonlocal TEM.
The other is primarily localized between rational surfaces and corre­
sponds to a two-dimensional TIM.
These radial characteristics are
consistent with the fact that the nonadiabatic circulating-ion
response (which, we recall, is important only close to rational sur­
faces) has a stronger influence on the nonlocal TEM than on the TIM
(see Fig. 9).
-34-
Despite the gualitative agreement mentioned above, it is
important to emphasize that the two-dimensional calculation does
yield new information about the instability, which cannot be obtained
from the one-dimensional radial calculation.
In particular, it
allows us to test the validity of the basic assumption made in the
one-dimensional analysis concerning the nearly flutelike poloidal
structure of the instability.
10 and 13.
This feature is exhibited in Figs.
The TEM exhibits a weaker ballooning than the TIM and
is, therefore, closer to the flutelike structure assumed in the
ont-dimensional radial calculation.
balloon strongly to the outside.
However, the TIM is seen to
This suggests that, in the present
calculation, the one-dimensional approximation will lead to results
that aie more reliable in the case of the TEM than in the case of
the TIM.
For the real part of the eigenfrequency, this behavior
tends to be supported by the following comparison of eigenfrequencies
obtained in both calculations.
In units of 10 sec
these fre-
quencies are (with the notation ('J,Y) for w_ + i\] :
1-Dimensional
2-Dimensional
TEM
< - 4.71 , 0.65)
( - 4.35 , 2.28)
TIM
( ~ 0.50
( - 1.56 , 2.04)
, 0.41)
However, it should be noted that ballooning, which is taken into
account only in the two-dimensional calculation, has a sizable
destabilizing influence on both the TIM and TEM.
Finally, we observe
that the TIM has some finite structure close to rational surfaces
where the mode is nearly flutelike.
The main localization of the
-35-
itiode, however, occurs between rational surfaces where, as stated
earlier, the mode is ballooning to the outside.
When the expression given in Eq.
that
'k p
(96) is calculated, we find
> = 0.4 2 and 0.37 for the TEM and the TIM, respectively.
f" hi
This tends to indicate that our basic assumption,
justified.
k
r
P
1 > is
<
b i
This is further supported by Figs. 12 and 15 which
show a typical poloidal harmonic and the corresponding
for both modes.
The important point to note is that
in the regions where
also, that
k n
r
i>
k (r)p
r
k P
T
b i
has some significant amplitude.
becomes quite large at the boundaries.
consequence of the boundary conditions
here, and of the definition of
k
[* fx,) = *
k p, . should go to infinity at the boundaries.
r
We note,
This is a
(x,) = 0 ]
given in Eq. (97).
is small
imposed
Mathematically,
Numerically, it is
large but finite.
5.2
TFTR Parameters
Let us consider the TFTR parameters listed in Table 1.
correspond roughly to a high current discharge
idealized profiles.
only one ion species.
These
with somewhat
For simplicity, we assume that the plasma has
Besides being more relevant to a real machine
than the Gladd-Ross parameters just considered, these parameters
have the advantage of corresponding to a somewhat smaller value of
P /a .
b l
Hence, our basic assumption,
k p
r
< 1 , is more likely to
be satisfied here.
We stai-t by considering a case with artifically low shear so
that for a chosen toroidal mode number,
1= 5 , the plasma contains
-36-
only two rational surfaces.
Furthermore, we choose the q profile,
such that the rational surfaces are located at r/a=0.5 and
(where
S,q = 12 and 13) .
0.8
These radial positions are away from the
reqion where the mode is localized.
The reason is that we first
want to consider a case where the presence of rational surfaces (and
the associated strong circulating-ion Landau damping) does not have
a strong influence on the solution.
The influence of increasing
shear will be considered later.
The procedure followed here is very similar to that in the
previous subsection.
In carrying out the calculation, we first
find the radially local spectrum neglecting the nonadiabatic circu­
lating ion contribution.
These eigenvalues are, then, used as the
initial guesses for the eigenvalues in the corresponding twodimensional calculation.
Next, focusing only on the most unstable
eigenmode, we gradually turn on the full circulating-ion response.
The spatial characteristics of the mode aie displayed in Figs.
16 and 17.
In particular, we observe that the instability is
strongly ballooning to the outside and tr.at it .is well localized
radially.
Specifically, the mode is wel] localized between rational
surfaces.
Because the rational surfaces are located away from
where the mode is localized, they have only a weak influence on the
mode frequency and spatial structure.
In Pig. 18 we plot various relevant frequencies as a function
of the minor radius.
Comparing the mode frequency with these, it
is interesting to note that the electrons are collisional (JM; << v /e)
el
but that the ions are not really collisionless (v /c 5 jw[) .
ii
Hence,
in contrast to the dissipative limit, effects associated with finiteion collision frequency might prove to be important here.
This is
-37-
noted because, as discussed in Sec. 3, the model collision operator
used in our calculations may not be appropriate in this collisionality
regime, v^./t < \ u | .
We next examine the influence of increasing shear on the eigenmode just discussed.
This is done by varying uj on axis and at the
limiter according to:
q(0) = 2.2718 - 0.3718w
q(a) = 2.78-16 + 0.1154w
g
,
g
,
(99)
(100)
where q(r) =q(0) + [q (a) - q (0) ] (r/a) , a'id where w
weight factor which is varied from 0 to 3.5 .
is an artificial
Of course, w_ is
limited by the requirement that q(0) _> 1 for MHD stability.
As
w
s
is increased, we follow the evolution of the mode frequency as well
as the spatial structure of the instability and the value <k p, .> .
r
The results are shown in Figs. 19 to 22.
From Fig. 19 we see that the real part of the frequency to ,
increases monotonically with shear while the growth rate, y , is
fairly constant over a wide range of w
shows <k p >
1
value of
as a function of w
,P
Part (b) of this figure
The tendency is that the
S
Dl
< k
.
.
>
v
generally increases with shear.
This is expected
since stronger shear implies more rational surfaces in the region
where the mode is localized, and hence, more rapid radial variations.
We note that <k r o.bi.> seems to have a maximum at ws = 3 . However,
'
the numerical evaluation of <Jc,p,.>
at this maximum is somewhat
~ bi
inaccurate since the estimates obtained for N = 300 and N = 350
(N being the number of grid points used in the finite element method)
differ by as much as 15%.
This is in contrast to the corresponding
-38-
evaluation of the mode frequency which differs by less than 1% f „
these two values of N .
It is generally true that the evaluation
of the mode frequency is more accurate than that of ^k n, > .
r
The
bi
reason is that the latter involves integrations of absolute values
of derivatives, which are usually difficult to accurately nvaluate
numerically.
Let us now look at the eigenfunction in more detail for a
selected value of w
w
(w > 0).
Specifically, let us consider
=1.5 which corresponds to five rational surfaces in the region of
interest ( 0. 4 <_ r/a ^_ 1. 0) .
shown in Figs. 20 and 21.
The spatial structure of the mode is
Comparing these with Figs. 16 and 17,
we see that stronger shear does not influence the global shape of
the mode.
It merely causes some rapid variations of the eigenfunci- ion
close to rational surfaces.
This is due to the strong contribution
from the nonadiabatic circulating-ion response there.
The value of
<k p, ,:• computed in this case is 'k r. .> - 0.25 .
r bi
r bi
This is consistent with the long wavelength approximation, k n
r
)
>-'l.
Finally, Fig. 22 shows a typical plot of diagonal elements of
the C and A matrices as a function of r .
This figure is of interest
because it shows the sharp structure in the C matrix localized at
rational surfaces.
This structure comes from the nonadiabatic
circulating ion contribution and it is analogous to that in the radial
potential Q introduced in Sec. 4.
In part (b) we note that the
diagonal elements of A are localized at the corresponding rational
surface (for each poloidal harmonic) and that the characteristic
width of each element is of the order of the distance between adjacent
rational surfaces.
This suggests that, for the case considered, only
the coupling between the first few adjacent poloidal harmonics might
-39-
be important,
6.
CONCLUSION
The new formalism developed in this paper was tested by
applying familiar approximations and, then, comparing our results
with those of previous calculations.
In the radially local dissi-
pative limit, we found that our analytic dispersion relation agreed
with that of Ref. 3, except for the ion collisional growth rate.
We
concluded that our Krook collision model was accurate in the large
collision frequency limit (|u| << M/K) of primary interest.
However,
it cannot be adequately applied on the small collision frequency
limit (v/r '<• |w|), and its validity for intermediate collision fre­
quencies (V/L •• I to I ) is, as yet, untested.
In the one-dimensional radial approximation, our results were
in reasonable agreement with those obtained from the eigenmode equa­
tions of Ref. 4.
Specifically, we found that two types of modes
could be unstable; one being localized at a particular rational sur­
face (a low frequency form of the "trapped-electron mode", or TEM)
and another, primarily localized between rational surfaces (the
"trapped-ion mode", or TIM). Note that no unstable TEM was
reported in Ref. 4.
The existence of such unstable modes here is
probrtbly a consequence of the artificially low density used in the
calculations of Sec. 4.
When applied to the Gladd-Ross parameters of Sec. 4, the twodimensional calculation produced results which were in qualitative
agreement with those obtained in the one-dimensional approximation.
Here
again we found both a low frequency ((J - in
r
< u . ) TEM
-40-
localized a_t a rational surface, as veil as a TIM, mainly localized
between rational surfaces.
The TEM was observed to exhibit weak
ballooning and was, therefore, more consistent with the flutolike
assumption made in the one-dimensional radial calculations.
The two-dimensional formalism was also applied to moro realistic
TFTR parameters and the influence of shear was considered.
For the
particular eigenmodn considered, wo observed that the real part of
the frequency increased monotonically with shear, but that the
growth rate was relatively insensitive to this effect.
We also
noted that, aside from causing rapid radial variations in their
neighborhood, multiple rational surfaces did not change the global
structure of the instability.
In particular, it is important, to
emphasize that, in contrast to the picture given in Ref. 4, wo founr?
that the mode could have finite amplitude at a rational surface
(and still regain unstable) and that the radial structure seemed
to be determined primarily by the equilibri-m variations rather
than by the position of the rational surfaces. In addition to toroidal
1
coupling effects, absent in the one-dimension,.: radial calculations,
this may also be due in part to the fact that t e equilibrium pro­
files used here are generally sharper than those of Ref. 4.
Also, we did attempt to reproduce a mode with the same radial
characteristics as found by Gladd and Ross
ple rational surface's.
in the presence of multi­
These results were not included in Sec. 5
but will be briefly described here.
In particular, we used the same
parameters (including the density) as for the 4 = 5
case of Ref. 4.
We first found an unstable two-dimensional TIM neglecting the nonadiabatic circulating-ion response.
full circulating-ion response.
Then, we gradually turned on the
The result was a marginally stable
mode exhibiting many of the characteristics of a TEM (e.g., large
-41-
amplitude at rational surfaces and w ~ u ).
r
te
Because of the computa­
tional difficulty of this exercise, we had to limit our attempt to
one mode only.
Of course, the negative result found here does not
definitively rule out the possible existence of a two-dimensional
multiple rational surface TIM with similar radial characteristics to
that found in Ref. 4.
It is appropriate to recall the main limitations of our
analysis, and indicate some possible future developments.
serious limitation
The most
here is that associated with the long-wavelength
assumption, k p, . < 1 .
Even though this inequality was roughly
satisfied in the cases discussed in this paper, we found that the
computed value of <k p, .>
was generally not much smaller than one.
Furthermore, it seems likely that, when considering more realistic
parameters corresponding to cases with multiple rational surfaces in
the plasma
and with sharp equilibrium profiles, this basic approxi­
mation would break down.
It
then
follows that a more < _r.eral,
integral formulation of the problem, valid for arbitrary values of
k p
, would be necessary.
As noted earlier, the Krook collision model used in this calcu­
lation is inaccurate in the low collision frequency limit and its
validity is uncertain for intermediate collision frequencies,
v/e — f to I
This regime of collisionality may be of interest in
practice (as for the TFTR parameters of Sec. 5). Because, in this
instance, the collisional terms might have a sizeable effect, it
would be desirable to improve our collision operator in this regime
of collisionality.
This could be achieved by solving the gyro-
kinetic equation using the Lorentz collision operator.
Unfortunately,
-42-
even with present computers, this multi-dimensional calculation in
configuration and velocity space is prohibitively time consuming.
The inclusion of other effects, even though not essential to
the conventional analysis of trapped particle instabilities, could
nevertheless lead to changes in the numerical results by factors of
order unity.
For example, noncircular flux surfaces or anisotvopy
in the equilibrium distribution could considerably affect the thres­
hold conditions and even introduce new branches (beam-driven modes)
ia the dispersion relation.
It would thus be of interest to consider
generalizing the present approach to treat more general
[numerically
generated) equilibria.
Finally, it is worth pointing out that, although it does
represent the first fully two-dimensional analysis of low frequency
instabilities in the trapped-ion regime, the present formulation of
this problem is nevertheless somewhat inefficient numerically.
Specifically, when dealing with multiple rational surfaces, several
poloidal harmonics need to be included in the calculation.
In such
cases, the numerical solution of Eq. (56) takes several minutes of
computer time and very accurate initial guesses are needed (1% or
better) for rapid convergence.
This puts a rather stringent limita­
tion on the applicability of our approach to cases of practical
interest.
This difficulty could be alleviated by developing a
ballooning formulation of the problem, similar to the one already
applied to MHD modes 12and, more recently, to the trapped-electron
mode.
-43-
ACKNOWLEDGMENTS
The authors are grateful to Dr. D.W. Ross for helpful discus­
sions and suggestions.
Helpful comments by Dr. W.H. Miner on
numerical methods are, also, appreciated.
One of us (R.M.) wishes to thank the National Research Council
of Canada for their financial support through a graduate scholarship.
This work was supported, in part, by the United States Department of
Energy Contract No. EY-76-C-02-3073.
-44-
APPENDIX
PARAMETRIZATION
We define the parametrization of the equilibrium and list the
explicit values of the parameters used in each calculation.
The
parameters are assumed to determine the equilibrium through the
model profiles
a
n(x) = n(Q)exp[-(x/x ) "]
,
Ul)
] ,
(A2)
n
3
T (x) = T (0)exp[-(x/x )
e
T ix)
1
e
Te
Te
= T (0)expI-(x/x )
i
Ti
T i
] ,
(A3)
,
(A4)
q(x) = q(0) + [q(a) - q(0)}x*
where x
is the normalized minor radius, x=r/a .
The specific
values of these parameters, used in each calculation, are listed
in Table 1.
- 4 5TASLE
1
Gladd
& Ross
TFTR
R (cm)
80
24B
a (cm)
23
85
B
2.8
5.2
n(0)(10 cm" )
0. 5
6.0
x
n
1.0
0.75
\
3.5
1.5
2.0
5.0
1.0
0 . 75
3.5
2.5
T ^ O ) (keV)
1.0
5.0
x
1.0
0.75
T i
3. 5
2.5
q(0)
1.14
variable
q(a)
2.1
variable
M.(amu)
1
2
2
I
1
1
h.
1
1
3
5
(Tesla)
0
l !
T (0)(keV)
e
X
Te
a
!
T e
Ti
a
I
I
-46-
REFERENCES
•'•W.M. Tang, Nucl. Fusion 18:, 1089 (1978) .
2
B.B. Kadomtsev and O.P. Pogutse, Sov. Phys. JETP 2£, 1178 (1967).
3
M.N. Rosenbluth, D.W. Ross, and D.P. Kostorr.arov, Nucl. Fusion 12^
3 (1972) .
4
N.T. Gladd and D.W. Ross, Phys. Fluids 16_, 1706 (1973).
5
D.L. Book, NRL Memorandum Report No. 3332 (1978).
6
R.Marchand, Ph.D. Dissertation, Princeton University (1979).
B. Coppi and G. Rewoldt, in Advances in Plasma Physics (Wiley,
New York, 1976), Vol. 6, p. 421.
8
W.M. Tang, Nucl. Fusion 1_3, 883 (1973).
9
G.Strang
and G. Fix, in An Analysis of the Finite Element
Method(Prentice-Hall, 1973).
10
W.H. Miner, Report FRC #138, University of Texas (1977).
11
R.Bergman
e t a l . , Princeton Plasma Physics Laboratory Report
PPPL-1275 (1976).
12
M.S. Chance et al., in Plasma Physics and Controlled Nuclear
Fusion Research, (Proc. Conf. Innsbruck, 1978) , (IAEA, Vienna,
1979), Vol. I, p. 677.
13
E.A. Frieman, G. Rewoldt, W.M. Tang, and A.H. Glasser, Princeton
Plasma Physics Laboratory Report PPPL-1560 (1979).
-4 7-
Fig. 1.
<t
792231
Toroidal coordinates.
-48-
2.0
1.5
x=I.O
O
1.0
0.5
1.0
•0.5
0
w (l0
r
4
-1
sec )
792292
Fig. 2. Radially local (solid line) and radially non-local
(dots) spectra found neglecting the non-adiabatic circulatingion response.
-19-
0.94*0.95
Radially Local
Spectrum, TEM
Branch
Mode* 2
Radially Local
Spectrum T I M ,
Branch
oj {\0
r
x=I.O
1
sec" )
792293
Fig. 3. Radially local (solid line) and radially non-local
(dots) spectra found with the full circulating-ioti response.
-505.0
4.0
-ID, (Mode#n
3.0
o
"a
2.0
y (Mode#2)
05
792247
Fig. 4, Variation of the mode frequencies with the artificial
weight factor, w j , multiplying th& non-adiabatic circulatingion response.
c
-51-
0.6
r/a
792495
Fig. 5. Radial structure of the eigenfunction, 0, and of the
corresponding radial potential, Q, for mode *1 (TEM). Here,
1 = 3 and w = (-4.71,0.65) x lo* sec" .
1
-52-
792497
Fig. 6. Radial structure of the eigenfunet ion, 4>, and of
the corresponding radial potential, Q, for mode #2 (TIM).
Here, Jl = 3 and w= (-0.50,0.41) x Wsec
.
1
-53-
I
1
-0.95
x = I.O I
0.9 T
^ - M o d e #1
—
3—
0.5"
\
o
.Mode # 2
2 2—
V*\
Radially Local
Spectrum
0.7 \
—
0.6^
0
1
1
0
4
_ l
w, ( I 0 s e c )
792499
Fig. 7. Radially local (solid line) and radially non­
local (dots) spectra obtained with the Gladd-Ross para­
meters, when the non-adiabatic circulating-ion response is
neglected.
1
1
1
1
1
1
1
1
1
1
5—
C.94
- 4 -
Rodiolly Locol
Spectrum,
/ T I M Bronch
0.95
\o.96
_
0.93 +
o 3 —
Mode # 1
.0.97
Z —
—
/ /
V.
11
x=0.92-/
P*
/
/ l
1
TI
V\
Radially Locol
- Spectrum,
TEM Branch
V ,-7 ,
A
//0.9
f
f0.80
^-Mode #2
T
\ \\ v V x.
l
<
: ;
1
1
-5
-4
-3
w (10* sec" )
K
r~ s o-65
^^ ^i ^S<_ ^Sv
0 6
J
1
r
792244
Fig, 8. Radially local (solid line) and radially non-local spectra obtained with
the Gladd-Ross parameters, including the full circulating-ion response.
-55-
oi (Mode # I)
r
y (Mode #1)
0.5
792287
Fig. 9. variation of the mode frequency with the artificial weight
factor, w , multiplying the non-adiabitic circulating-ion response
for modes #1 (TEM) and #2 (TIM), obtained with the Gladd-Ross parameters.
o i
-56
792239
Fig. 10. Two-dimensional structure of the trapped-electron mode (mode
#1) obtained with the Gladd-Ross parameters. Here, 2. = 3, w= (-4.35,2.28)
x 10" s e c , and < k P > = 0.42 .
- 1
r
bi
-57-
702290
Fig. 11. Radial structure, along 9 = 0 , of the two-dimensional
trapped-el^ctron mole (mode #1) found with the Gladd-Ross parameters.
-58-
im
M^i\
r/a
792498
Fig. 12. A typical poloidal harmonic and the corresponding
k p. for the trapped-electron mode.
ir
/~
Im{4>(r,e}}
792238
Fig, 13. Two-dim«nsiona.l s t r u c t u r e of the trapped-ion mode (mode #2)
obtained with the Gladd-Hoss parameters. Here, £ = 3, to = ( - 1 . 5 6 , 2 . 0 4 )
0.37.
and < k P >
x 10" sec
_ 1
r
M
.60-
Fig. 14. Radial structure, along 8 = 0, of the two-dimensional
ion mode (mode #2) found with the Gladd-Rcss parameters.
792291
trapped-
-61-
r/a
Fig. 15. A typical poloidal harmonic and the corresponding k^p
for
the trapped-ion mode.
*r
* bi
H
h1
-02-
0,6
08
0.6
O.i
r/a
.0
792240
Fig. 16. Two-dimensional structure of mode *1 obtained with the TFTR
parameters, including the_full circulating-ion response. Here, £= 5,
u = (0.55,2.23) x 10* sec 'and < k P > = 0,12.
r
b1
-03-
Re{*(r,0 = O)}
0
lm{${r,8:0)}
0.4
.0
7023136
Fig. 17. Radial structure, along 6 = 0. of mode #1 found with
TFTR parameters, including the full circulating-ion response.
-64-
sec
r/a
792347
Fig, 18. Various frequencies corresponding to the TFTR
parameters, flere, w =0.
-65-
o
(/>
ro
O
0.6
1
l
-.Al
f
• N=250
o N=300
0.4 -h A N= 350
l
i
\
l
-
V-G
%
0.2 h
0
i
i
i
0
w.
792496
Fig. 19. Mode frequency and the averaged value of k a
as a
function of w .
r bi
H
- 66-
Lm ($(r,0)}
•~tr
0G
0.4
0.3
r/o
r/a
792367
Fig. 20. Two-dimensional s t r u c t u r e o( the i n s t a b i l i t y for w =1.5.
tere.w = ( 1 . 6 2 , 2 . 5 3 ) x 10 sec
, and " V > = 0.25.
s
l
:
h i
-67-
Re{<&(r,0 = O)}
0
-*ss
r
Im{*(r,0 = O ) } - ^ \
\
i
\
i
0.4
0.6
\
I
\
i
0.8
r/Q
792366
Fig. 21. Radial structure of th« instability, along 0 = 0,
for w = 1.5.
s
1.0
urt-
'a)
/"
y
..^
14
m
0.6
0.8
0
r/a
Re
A
{ mm}
792363
Fig. 22. Typical diagonal matrix elements us a function of
the minor radius, for \v =1.5.
E X T E R N A L DISTRIBUTION IN A D D I T I O N TO TIC
A L L CATEGORIES
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C e n t r a l Res. Inst, lor Physics, Hungary
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R. Shingal, Moor n College, India
A . K . Stindaram, Phys. Res. Lab., India
M. Naroghi, A t o m i c Energy O r g . of Iran
B i b l i o t e c a . Frasc a t i . Italy
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G. Rostagni, U i m . Di F'udova, Padova, I t a l y
Preprint Library, Inst, de Fisica, Pisa, Italy
L i b r a r y , Plasma Physics Lab., Gokasho, U j i , Japan
S. M o r i , Japan A t o m i c Energy Res. Insl.,Tokai-.Mura
Research I n f o r m a t i o n C e n t e r , Nagoya U n i v . , Japan
S. Shioda. Tokyo Inst. Qf Tecfi.,Japan
Inst, ot Space St ,.\ero. Sci., U n i v . of Tokyo
T. Uchida, U n i v . of Tokyo, Japan
H. Yarnato, Toshiba R. A: D. C e n t e r , Japan
M. Yoshikawa, J " \ E R I . Tokai Res. Est., Japan
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Minerales, Spain
L i b r a r y , Royal Institute of Technology, Sweden
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Librarian, Fom-Instituut
Voor
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V. E. G o l a n t , A . F . Ioff«s Physical-Tech. Inst.,USSR
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USSR
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M a x - P l a n c k - l n s t . fur Plasmaphysik, W. Germany
N u c l . Res. Estab., J u l i c h , West G e r m a n y
K. Schindler, Inst. Fur Theo. Physik, W. G e r m a h y
EXPERIMENTAL
THEORETICAL
M. H. Brennan, Flinders U n i v . A u s t r a l i a
H . B a r n a r d , U n i v . of B r i t i s h C o l u m b i a , Canada
S. 5creenivasan, U n i v . of C a l g a r y , Canada
J . Radet, C . E . N . - B . P . , F o n t e n a y - a u x - R o s c s , F r a n c e
Prof. S c h a t z m a n , Observatoire de N i c e , France
S. C. Sharma, U n i v . o i Cape C o a s t , Ghana
R, N, A i y e r , Laser Section, India
B. B u t i , Physical Res. Lab., India
L. K. Chavda, ,S. G u j a r a t Univ., India
l - M . Las Das, Banaras Hindu U n i v . , India
S. Cuperman, T e l A v i v Univ., fsraef
E. Greenspan, N u c . Res, C e n t e r , Israel
P. Re.-., .lau, Israel Inst, of T e c h . , Israel
I n t ' l . Center for Theo. Physics, T r i e s t e , i t a l y
I. K a w a k a m i , Nihon U n i v e r s i t y , Japan
T . N a k a y a m a , R i t s u m e i k a n U n i v . , Japan
S. Nagao, Tohoku Univ., Japan
J . I . Sakai, Toyama U n i v . , Japan
S. T j o t t a , U n i v . 1 Bergen, N o r w a y
M . A . H e l l b e r g , U n i v . of N a t a l , South A f r i c a
H. Wilhelmson, Chalmers Univ. of T e c h . , Sweden
A s t r o . Inst., Sonnenborgh Obs.,The Netherlands
N.G. Tsintsadze, Academy of Sci GSSR, USSR
T. J . Boyd, U n i v . College of N o r t h Wales
K. Hubner, U n i v . H e i d e l b e r g , W.Germany
H. J . Kaeppelcr, Univ. o l S t u t t g a r t , West Germany
K. H . Spatschek, U n i v . Essen, West G e r m a n y
EXPERIMENTAL
ENGINEERING
B. G r c k , Univ. du Quebec, Canada
P. Lukac, Komenskeho U n i v . , C z e c h o s l o v a k i a
G . ffon'kosni', W a f f Lab for High Energy Physics,
Tsukuba-Gun, Japan
V. A . G l u k h i k h , D.V. E f r e m o v Sci.
Res. I n s t i t . o f E l e c t . App., USSR
EXPERIMENTAL
F. J . Paoloni, Univ. of Wollongong, A u s t r a l i a
J. K i s t e m a k e r , Forn Inst, tor A t o m i c
& Molec. Physics, The Netherlands
THEORETICAL
F. Verhccst, Inst. Vor Theo. M e c h . , Belgium
J . Teichmann, U n i v . of M o n t r e a l , Canada
T . Kahan, U n i v . Paris VII, France
R. K. C h h a j l a n i , India
S. K. Trehan, Panjab U n i v . , India
T. N a m i k a w a , Osaka C i t y U n i v . , Japan
H. N a r u m i , U n i v . of H i r o s h i m a , Japan
Korea A t o m i c Energy Res, Inst., Korea
E. T. Karlson, Uppsala Univ., Sweden
L. Stenflo, U n i v . of U M E A , Sweden
3. R. Saraf, New U n i v . , U n i t e d K i n g d o m