PFC/JA-90-28
Nonlinear Collisional Impurity Transport Including
Bifurcations, and Coupling to Rippling Mode
D. J. Sigmar, G. S. Lee, C. T. Hsu and K. W. Wenzel
August 1990
Massachusetts Institute of Technology
Plasma Fusion Center
Cambridge, MA, USA
Paper presented at: Joint Varenna-Lausanne International Workshop on Theory of Fusion
Plasmas, Varenna, Italy, August 27-31, 1990.
I
Abstract and Summary
Since the discovery of neoclassical-like impurity peaking in Alcator C and other
tokamaks1' 2 after pellet injection and the discovery of strong [0(E)] up/down asymmetries of the impurity densities in the flux surface', 4 the need for a more comprehensive
impurity transport theory containing neoclassical as well as anomalous effects has become
apparent. In fact, both kinds of effects may co-exist since neoclassical transport is driven
by parallel (friction) dynamics which can remain near neoclassical' even if the ion cross
field transport is dominated by B x V
fluctuations. Here, we report on progress in two
areas.
(i) The observed poloidal asymmetry of the impurity density can be understood from
a nonlinear extension of neoclassical theory (occurring when A = (pi0/r)(1)
where Ppi is the poloidal ion Larmor radius, r, the impurity density scale, wLt the main
ion transit frequency). When A Z 1, ("strong ordering theory"') the poloidal modulation
n, - no(r) = n' sin8 becomes 0(r/R,) [rather then the standard small O(ep, /rn)] and
the ensuing nonlinearities produce a bifurcation of the poloidal ion flow velocities and
Recent results 4 on injected impurity density
abrupt changes of the diffusion fluxes Fi, Fr.
asymmetries obtained in the Texas Tokamak (TEXT) exhibit several features in agreement
with the strong ordering theory, as will be shown.
(ii) Concomitantly, on TEXT, a marked up/down (poloidal) asymmetry of the turbulent density fluctuation spectrum has been observed which, moreover, flips from the top
7
7r/2) to the bottom (0 = -7r/2) when the plasma current is reversed. We propose
8 9
a link with the VZeff -driven rippling mode theory, ' specifically retaining the poloidal
(9
=
modulation Zff - Z,'ff(r) = Z.ff sin0 corresponding to the observed poloidal impurity
density modulation. The effect of B - V71,q
#
0 (with 7,, = Zef f *-lspitzer)
and its depen-
dence on the sign of the equilibrium current density j through Ohm's law is worked out.
The flipping of the fluctuation asymmetry is theoretically confirmed as a consequence of
the neoclassical "strong ordering" poloidal impurity variation.
1
References to Abstract
1 R. Petrasso, D. Sigmar, K. Wenzel et al, Phys. Rev. Lett. 57 (1986) 707.
2 K. Wenzel, D. Sigmar, Nucl. Fusion 30 (1990) 1117.
3 P. Smeulders, Nucl. Fusion 26 (1986) 267.
4 K. Wenzel, Bull. Am. Phys. Soc. 34 (1989) 2153, paper 8R5.
5 K. Shaing, Phys. Fluids 31 (1988) 2249.
6 C. T. Hsu, D. Sigmar, Plas. Phys. Contr. Fusion (July 1990), in press.
7 D. Brower et al, Phys. Rev. Lett. 54 (1985) 689.
8 P. Rutherford, in "Physics of Plasmas Close to Thermonuclear Conditions," Proceedings Varenna 1979, EUR FU BRU/X11/476/80.
9 T. Hahm, P. Diamond, P. Terry et al, Phys. Fluids 30 (1987) 1452.
2
II Neoclassical Theory of Strong Poloidal (Up/Down) Asymmetry of Impuri-
ties
II.1
Introduction
Experimentally, strong variations of the impurity density ni(r,0) with respect to the
~
poloidal angle 0 have been known since 1977 [2.1]. By "strong" we mean that ni,/n1
8
[2.2]. Here, e is the inverse aspect
0(e) rather than nl,/nr ~ O(e6,), the "standard" result
ratio and we represent
ni(r,9) = n10 (r)+niecos9+n, sin 9 +...
(2.1)
6, = pp/rn is the poloidal Larmor radius parameter with
1 dn
pp = vth/(eB,/mc) and rn = (nd)
.
(2.2)
A novel first principles theory adopting in this strong ordering has been developed by
us [2.3] recently. It is shown that nrI/nm ~ O(e) if
2 >1iZ2Vii
A =
(2.3)
which is easily realized in the outer region of the plasma where even the main ions i can be
in the Pfirsch-Schliiter regime (v;i > wt) and, foremost, the smallness of standard poloidal
Larmor radius parameter bpi is overpowered by the square of the impurity charge number,
Z 2 . This parameter arises as the balance of two terms. The first is the parallel ion impurity
friction
BR11
-
-ciminivii(Bvi
- Bvjii)
-
c2 niB - VTi
(2.4)
where vili (the Pfirsch-Schliter "return flow" of species i) is driven by Vip/B,. Secondly,
this friction is balanced by a parallel pressure gradient B - Vpj in the fluid momentum
balance, thus giving rise to the parameter A as was recognized early [2.4a, 2.4b] but not
pursued by those authors to its nonlinear consequences as was done in Ref. [2.3]. (Essential
features of this work will be summarized below.)
Experimentally, up/down impurity asymmetries were observed in Alcator A [2.1],
PLT [2.5], PDX [2.6], ASDEX [2.7] and most recently in TEXT [2.8]. Unfortunately so
3
far, Alcator A seems to be the only machine in which the toroidal field direction was
reversed causing the poloidal excess impurity density to flip from the top to the bottom.
This is precisely the neoclassical expectation due to the fact that the drift velocity
2
VDi =
B x VIBI
B2
+
which is responsible for the (diffusive) upward migration of the impurity ions changes sign
when BT
II.2
-+
-BT
(but VIBI remains the same).
Brief Summary of Conventional vs. Strong Ordering Theory
Physically, one recalls that any poloidal variation of the ion pressure gives rise to a
force x B drift in the radial direction. On flux average, one can write
Fr=b
where b = B/B, f = eB/mc, f,
x
VenT er).br
(2.5)
VnT
= eB,/mc, with B the total and Bp the poloidal
field. Thus, to determine the radial particle transport, the reasons for
87
f 0 must be
investigated. (We note that pj = p3 (0) is not incompatible with ptotal = E, pj (r,6) being
a function of r only since
0 = -B - Vpj - Bejnj
1 -V
+ BR1j;
j = i, I,e
(2.6a)
yields E, B - Vpj = 0 when charge neutrality and collisional momentum conservation is
invoked, in the absence of strong plasma rotation.) On flux averaging, (2.6a) becomes
0 = -(ejn
1B
- VD) + (BR 11 ).
The steps in (2.5) follow from Vp = Vf 9 + V09, and B - V =
(2.6b)
where
=
V4 x Vo - VO and B = V4 x VO + IV4. Here, I = RB, = I(,0). In (2.6a) the parallel
friction has the form given in Eq. (2.4) (cf. [2.2], [2.3]) and from the lowest order (i.e.
zeroth order in collision frequency) continuity equation
V -nv = 0
4
and lowest order perpendicular momentum balance
env = B x (Vp* + enVq5*)/B
(2.7a)
2
there results
(2.7b)
v =K(#) B + wR 2 Vq.
n
Here,
.
a(D
-O=-+ (e
(2.7c)
The superscript * indicates that p0 = p*(b), 4 = -V(4b). Any collision driven 0-depencence
enters in higher order.
8,0 -
1 -8r
RB,
is the poloidal flux derivative and the integration
constant K(tk) = nv,/Bp measures the yet to be determined poloidal flow velocity v,.
' = 0' is the plasma potential gradient which will be discussed later. For now, it suffices
to observe that it drops out of R11. The pressure gradients in (2.7b) are assumed given.
Thus, to obtain Fr,, BR 1i, must be determined. We start with
(vil - vill)B =
K -
ni
\ni
B2
-
(W,
-
w)I
(2.8a)
needed in the first terms of (2.4). For the purpose of the present section, we will neglect
the second therm, the thermal friction oc B - VT, i.e. we assume T = T(O) (but the
0-dependence will be kept in the final results of this paper). To determine the poloidal
flows Kj (,0) it is necessary to consider the small Larmor radius expansion underlying all of
transport theory and it is at this point that the conventional neoclassical impurity theory
will be replaced by the "strong ordering theory".
Returning to (2.1) for nj(r, 0) and an analogous representation for k(r, 0), the standard
~ e5,.
ordering is f-~ e6,,
(Here the tilde quantities denote the cos0 and sine
components.) Consequently, the poloidal gradient of 4 in (2.6b) is ordered out being of
O(52) and there remains (RIIB) =0. This can be used with the help of (2.8) to express
-
\(~
n
n B
/
1)
/
\
Kn(4) _ K
fIG
5
)i
2 = (wi - wu)I(ip)
((B')
(B
(2.8b)
if and only if we neglect the 9-dependence of nI, ni and vii as small in bp. Then (2.6a)
becomes
TrB - Vn 1 = -minivii(wi - wi)I (
/B2
(2.9)
-
where the last factor is the well known Pfirsch-Schliiter term ~ O(E). Consequently
fi(,
( 0)) ~
s/2-6 , Z2 V,
Enio(o)
,0)
~fi
ni
(2.10a)
--
enio(0)
where a =
(A)
Wti
(2.10b)
Z0 a
is the impurity strength parameter. The functions f; and fi are linear
for Az < 1 but turn over like (1 + A
2
)
1
for A > 1, as will be shown. Thus fiI and i do
not increase monotonically with A. Upon inserting (2.9) in (2.5) one obtains the standard
Pfirsch-Schlilter result for the impurity flux r (cf. [2.2]). Note that an explicit solution
for the poloidal flows Ki, K 1 was not required in this linearized theory and Eq
-
-B - V<D
was ordered out. S! dropped out in the friction term wi -wI, of course. A posteriori then,
the neglect of the 9-dependence in n 1 , ni and viI assumed above and leading to (2.9) was
predicated on the assumption
A ~ O(6 ,1) < 1.
(2.11)
This is the "standard ordering" in impurity transport theory, first implicitly assumed in
[2.2]. On inspection of (2.10a) one recalls that !-
= e 3/ 2 v.;, where v.i is the main ion
collisionality parameter such that v.i < 1 is the banana regime and v* > E-3/2 is the
Pfirsch-Schlilter regime. Consequently, near the Pfirsch-Schlfiter plasma edge, and with
Z2 <
(102) even for carbon and oxygen, it will be very difficult to justify retaining the
standard ordering (2.11). Rather, the appropriate "strong ordering" is
(2.12)
> 1.
V
/-8,-Z-2
Wti
However, if this ordering is adopted all 0-dependences of nj, n; and vi, become 0(e) rather
A =
than 0(6,) and must be retained in (2.8) and elsewhere rather than linearized out. Thus,
the neoclassical impurity fluid momentum (and other) equations become nonlinear in i
and fi1 . The consequences are the subject of [2.3] and the present paper and lead to several
explanations of hitherto baffling experimental observations.
6
11.3
Outline of Nonlinear Theory
When Eqs. (2.7) are inserted in the coupled momentum balance Eqs. (2.6a) for the
main ions i and impurity ions I one obtains
B - Vn T; + B - V
i + eniB -V4 = c 1minivij (BIII - BvIj)
B - Vn1 T + B -V - II + Zen 1 B -V. = -ciminivi 1 (Bvy;;
-
c2 njB - VT.
(2.13a)
- BvI) + c2 njB -VT;. (2.13b)
where from (2.7b)
j = i,I.
B2 + Iwi(o) ;
BVI= K,()
Here we have re-introduced the parallel viscous forces, important for 9 Ir even in the PfirschSchlinter regime. If T; = TY is assumed to be independent of 0, these equations are a coupled
nonlinear set for the unknown functions i (0, 9), K,(40). If B -VT
$
0 is allowed, the set
has to be extended to include the parallel heat balance equations, to be included later.
The intrinsic nonlinearity of Eqs. (2.13) becomes readily transparent when a simplified
form of (2.13) is rewritten in terms of the dimensionless variables
Y
i 1 (4, 9)
Enjo (0'
U,=
-
Y (0) cos
_-
K1 B o
+ Y,(ik) sin0
(2.14a)
(2.14b)
Bo
These variables are normalized such that Y and U1 are of 0(1) in typical cases, as will
be shown. As shown in (2.10), the radial dependence of the poloidal amplitudes Y, Y,
is driven by A = A(r), see (2.10a), and we will show below that A can be a strongly
increasing function of r in the outer regions of the plasma. Further, defining an inverse
gradient scale length (but normalized to the plasma radius a)
Ad
=
-
-
ln(niT)
2 [Or
ZT1
ln(n1 0 Ti)
I
(2.14c)
Equation (2.13b) takes on the explicitly nonlinear form (balancing B - Vp against BRIJ)
-6
~ A(UiY + 2Ad cos 0)
7
(2.15)
(B - VT was dropped for brevity.)
To close the system, one further invokes the flux
averaged equations (2.13) after dividing them by njZ:
(B)
- V -Iii(2.16a)
ni
ni
(BV -z)
((2.16b)
Znj
Znj
(The parallel gradient of the scalar pressure and the potential annihilate in an axisymmetric
system where (B - VA) = 0 for any A.)
Using the constitutive relation of [2.10; 2.15,
Eq. (A10)] or Eq. (2.10)
(B -V - I95) OC pig opj +
=
/1259j
,
(2.16c)
where 'v,, qp are the poloidal particle and heat flow. The parallel viscosity coefficients are
given in [2.9], their ratio
/'2/A1
is shown as a function of collisionality and aspect ratio in
Fig. 2.1. (It determines the magnitude of vj,.) Equations (2.16) are algebraic equations for
the flux functions Kj(0), coupling to the poloidal density variations A (0, 9) in Eqs. (2.13).
For impurities in the collisional regime (III oc v-1) the left hand side of (2.16b) can be
neglected, to lowest order, for the determination of v1,.
To include the poloidal heat flow qp and B . VT requires extending the set of equations
to include the energy balance, as mentioned. Another extension we kept in [2.3] is arbitrary
plasma rotation (or flow velocity) v which - as is apparent from Eq. (2.7a) - is controllable
by an externally maintained pressure and ambipolar potential gradient. The latter can be
set by imposing electrodes [2.12] or via neutral beam induced toroidal and poloidal rotation
[2.13a,b]. From (2.7a)
Vthi
~O
o
- )+o(
a
Vthi
showing that the first term (due to Vp) remains bounded
<
1, but the second term can be
chosen to dominate. In this "strong rotation" case, the inertial term Bx (m,/e,)(v.V)v/B 2
has to be retained on the right hand side of (2.7b). This has been done in [2.3]. However,
we used a limiting ordering, namely
vo/Vthi
< Z-'/ 2 and consequently U1 < 1 (as defined
8
in [2.14b]) which means VI./vthi
DIII-D for v,
1
<
(B,/B)6,p. The recent experimental results [2.13b] on
very near (or even outside) the separatrix show a larger poloidal rotation,
not compatible with our ordering which is suitable further inside. In fact, the DIII-D
results for vr are so large as to question the validity of the (pi/a)expansion.
II.4 Some Results of the Nonlinear (Strong Ordering) Theory
11.4.1 Explicit solution for the poloidal variation of the impurity density
When the full nonlinear equations (2.13) and (2.16) are expanded in the smallness
of e < 1 and Z-' < 1 and taking the limit of small toroidal rotation typical for ohmic
plasmas they reduce to (as shown in [2.3]) two simplified equations, i.e. neglecting the
impurity viscosity oc v,,
0 yields
(Bpii)=
(2.17a)
U1 - U, = Ad
and TIB - V In nr = -BRIli, yields
-
a&UY = 2aaAd cos
(2.17b)
0.
To obtain a second equation for U1 , U , we use the main ion equation
B.ii
=
B
and finally find
=2a&Ad
1 +a2 A2
ni,
(2.17c)
2 D(a)
(2.18a)
o
Here,
a
T +T.
Ti
where
576 + 488 v/a + 128a 2
2
576 + 1208v" a + 434a
is related to the friction coefficient cl of Rutherford, see [2.4] and we recall a
n 1 Z 2 /ni.
Now, from (2.17) and (2.18) our previous remark is quantified which stated that the
sin 9-amplitude fii,
of the impurity density first rises with increasing A and then decreases
again for large A. The reason for this turnover lies in the nonlinear coupling term -aAUIY
9
in (2.17b) which came from the friction between the poloidal flow components oc K of
(2.8a), while the right hand side of (2.17b) came from the diamagnetic flow components
oc Wj of (2.8a).
Equation (2.17c) is the main analytic result for the up/down asymmetry of the impurity density in the flux surface and will be used for experimental comparisons in the next
section.
11.4.2 Natural appearance of bifurcated solutions
Because of the nonlinearities shown, e.g. in (2.17), the plasma flows in the flux surface
and the transport fluxes across the flux surface can have more than one linear solution.
One finds (numerically) from Eq. (2.16), i.e. the balance of parallel friction and viscous
force [in conjunction with (2.13)] the results shown in Fig. 2.2a, 2.2b. Where the curves
intersect determines the self-consistent roots for U1 , i.e. essentially the impurity poloidal
flow velocity (cf. (2.14b)), and once this is known, the up/down asymmetry Y (cf. (2.14a)).
Note in Fig. 2.2b that as (pressure gradient x A) steepens, three roots for U1 appear! The
increasing structure of these curves vs. U1 comes from
f(A),
see (2.10a), entering the
friction, as well as the viscous force. The potential connection between this bifurcation
which leads to a corresponding low and high radial particle transport and the H-mode will
be addressed elsewhere [2.11].
11.5
Observation of Up-Down Asymmetry of Impurities in Alcator A and TEXT
The first experimental evidence of poloidal impurity asymmetry was discovered in
Alcator A [2.1] where the Oxygen VI brightness showed a marked up/down asymmetry,
as shown in Fig. 2.3. The magnitude of the asymmetry agreed well with the neoclassical
model of parallel ion diffusion (in the Pfirsch-Schliter regime) combined with B x VIBI
updrift, and moreover, the asymmetry flipped from the top to the bottom upon reversal
of the toroidal B-field, indeed.
Recently, carbon, aluminum, titanium and iron were injected into similar discharges
in TEXT, and - O(e) poloidal up/down asymmetries of these impurity densities were
10
studied (using a horizontal x-ray imaging array). While it was not possible to reverse the
toroidal B-field to see if this would flip the excess accumulation from the top to the bottom
a comparison with the "strong ordering theory" of the previous sections will be carried
out with a particular focus on the predicted Z-dependence of these asymmetries. When
the plasma current is reversed the neoclassical impurity asymmetry remains unchanged
(which is in accord with neoclassical theory) but the also observed turbulent density
fluctuations flip from the top to the bottom of the tokamak. This will be the subject of
Section III of this paper. A brief description of the TEXT observations is given next.
(Typical parameters of this tokamak are B = 2.5 T, I, = 300 kA, i, = 3 x 10
13
cm 3 ,
T,(o) = 1 keV, a = 26 cm, R0 = 100 cm.) Defining the x-ray brightness of a viewing chord
through the top as I+, and another chord through the bottom as I-, let
ACP=I+ _II++ Iwhich corresponds to n1 ,/n
of aluminum.
of Eq. (2.1).
0
For example, Fig. 2.4 shows the asymmetry
Figure 2.5 shows A. 1 , for injection aluminum at three different plasma
radii (viewing chord impact parameters) and one observes an increase of the up/down
asymmetry with increasing radius.
Figure 2.6 shows measured and calculated values of the sin 9-component
5
(up/down
asymmetry of various impurity densities as a function of Z, namely carbon, aluminum,
titanium, and iron. The parameter
Wy
=a)(
(2.19)
Ad
whose factors were defined in Eqs. (2.18a), (2.3), (2.14c) independent of Z and is evaluated
at a fixed radius r = 17.7 cm. (TEXT has a = 26, R, = 100 cm, thus E = .18 at r = 18
cm.) The theoretical expectations for the asymmetry are those of Eq. (2.17c) which can
be rewritten as
hi_ = 2fqktZ 2
nio ~1 + p2Z4
(2.20)
Thus, the Z-dependence of the asymmetry is nonmonotonic and peaks at pit
ax =
1 giving
a maximum asymmetry of
(i)
at
=~E
11
Z7 ,2 =
-1
(2.21)
independent of all the plasma parameters contained in the quantity p. In TEXT typically
I $ .01 yielding Zm,., > 10. Thus the injected aluminum should show a larger asymmetry
than the intrinsic impurities (C and Ti in TEXT) which is borne out experimentally in
Fig. 2.6. By magnitude, the experimental asymmetry is indeed very close to E (= .18 at the
location of measurement).
Theoretically, this large up/down asymmetry can only come
from employing the strong ordering nonlinear impurity theory and, indeed, the strong
ordering parameter A (defined in (2.3)) equals 1.35 for iron, .73 for aluminum and .22 for
carbon.
In Fig. 2.6 the Z-dependence of ii
5,/n, (n,, and (n) are almost identical) is shown for
three different values of p, reflecting the range of gradient scale lengths entering Ad. The
gradients at r = 17.7 cm are somewhat difficult to pin down but p = .01 is a good fit.
The vertical error bars are due to statistical scatter from the brightness averaged over a
sawtooth period. Not shown are horizontal error bars on Z which has been assumed equal
to the coronal equilibrium charge states at the prevailing Ta-profile. Finally, the chord
average at an impact parameter r = 17.7 cm is not exactly equal to the brightness at
a plasma minor radius r. These measurements were repeated after reversing the plasma
current but no clear current direction dependence was found.
These difficulties notwithstanding the unambiguous experimental features of aluminum
injection, namely (i) a clear O(E) up/down asymmetry (Fig. 2.4); (ii) which is radialy
increasing (Fig. 2.5); (iii) showing the Z-dependence of Fig. 2.6; (iv) showing smaller
asymmetries for the intrinsic impurities which have either Z (carbon) < Zma. (Al) or Z
(Fe), Z (Ti) > Zm.a2
(Al); and (v) showing a lack of dependence on the direction of the
current (poloidal B-field), all of these features are in agreement with the nonlinear impurity transport theory in the "strong ordering limit". That these (nonlinear) collisional
transport effects should be sustained in the outer region of the plasma which appear dominated (as we shall see in the next section) by resistive turbulence cross field transport
can be understood recalling that the up/down asymmetry results from collisional parallel
Pfirsch-Schlilter-like diffusion combined with the upward VB drift and K. C. Shaing has
demonstrated [2.15] that generic low frequency turbulence hardly affects parallel transport.
12
References
[2.1] J. L. Terry, E. S. Marmar et al., Phys. Rev. Lett. 39 (1977) 1615.
[2.2] P. H. Rutherford, Phys. Fluids 17 (1974) 1782.
[2.3] C. T. Hsu, D. J. Sigmar, Plas. Phys. Contr. Fusion 32 (1990) 499.
[2.4a] C. S. Chang, R. D. Hazeltine, Nucl. Fus. 20 (1980) 1397.
[2.4b] K. H. Burrell, S. K. Wong, Nucl. Fus. 19 (1979) 1571.
[2.5] S. Suckewer, E. Hinnov et al., PPPL Report 1430 (1978).
[2.6] K. Brau, S. Suckewer, S. K. Wong, Nucl. Fus. 23 (1983) 1657.
[2.7] P. Smeulders, Nucl. Fus. 26 (1986) 267.
[2.8] K. Wenzel, Bull. Am. Phys. Soc. 34 (1989) 2153, paper 8R5.
[2.9] S. P. Hirshman, D. J.Sigmar, Nucl. Fus. 21 (1981) 1079.
[2.11] C. T. Hsu, "Strong Ordering and Bifurcated Poloidal Equilibria Including VT Effects,"
in preparation, 1990.
[2.12] R. Taylor, M. L. Brown et al., Phys. Rev. Lett. 63 (1990) 2365.
[2.13a] M. Murakami, P. H. Edmonds, G. A. Hallock, R. C. Isler, E. A. Lazarus et al., in
Plasma Phys. and Contr. Nucl. Fus. Res. (Proc. 10th International Conference,
London, 1984) (IAEA, Vienna, 1985) Vol. I, p. 87.
[2.13b] R. J. Groebner, K. H. Burrell et al., Phys. Rev. Lett. 64 (1990) 3015.
[2.14] K. W. Wenzel, R. D. Petrasso, W. L. Rowan et al., "Observation of Up-Down Asymmetric Impurity Densities in TEXT using a Horizontal X-ray Imaging Array," presented at the 8th Topical Conference on High Temperature Diagnostics, Hyannis, MA,
May 9, 1990.
[2.15] K. C. Shaing, Phys. Fluids 31 (1988) 2249.
13
Figure Captions
Fig. 2.1:
Ratio of viscosity coefficients (introduced in Eq. (2.16c)) as a function of ion
u
collisionality v.;
Roqvjj/e3/2vthi, for various aspect ratios E = r/R. This ratio de-
[N+(1+
termines the parallel ion flow velocity (V'jiB) = -I'-I(Ti/ei)
2
i/-t)
]
where the prime denotes '.
Fig. 2.2:
In Figs. 2.2a and b normalized viscous and frictional forces are plotted vs
normalized impurity poloidal flow UI.
Here, Z = 8, Ad = 1, a = 1, A = 1.5 in
Fig. 2a; and A = 6 in Fig. 2b.
Fig. 2.3:
0 VI emission profiles for each toroidal field direction. The solid line indicates
the emission profile with the field direction for which the ion drift is upward; the
dashed line shows emission with the field and drift directions reversed.
case, the deuterium plasma conditions were jie = 2.5 x
1014
In each
cm-3, BT = 50 kG,
q(a) = 4.8. Alcator A.
Fig. 2.4:
Aluminum signal in TEXT (subject to sawtoothing). The upper viewing chord
is at r = 17.7 cm, the lower chord at -17.7 cm. The difference is a measure of the
impurity up/down asymmetry. Aluminum is injected from the bottom at t = 300
Ms.
Fig. 2.5:
Aluminum brightness asymmetry Ax = (I+ - I-)/(I++ I) at three different
radial positions. Note the increasing asymmetry toward the plasma edge.
Fig. 2.6:
Impurity poloidal asymmetry (essentially equal to Ax of Fig. 2.5) for carbon,
aluminum, titanium and iron. Experimental data points are shown as error bars.
Theoretical predictions using approximate analytical formula of Eq. (2.20), with
parameter IL defined in Eq. (2.19). jp depends mostly on ni and nj radial profile
gradients.
14
0f
II
N
-
0
-
1~
-0
-o
1~
-o
1~
N
0
I
0
I
LO
0
LO
1
1*~
0
0
0
Fig.
15
2.1
I
LO
0
0
IO
K(0)
12
/
-+
a
I
Viac.
Ada&
I
=
-
1.1
Pric.
0
-0
-12
-
I
I
12
.
-0
Ul
4
2
4
Ib)
*--.-- // Visc.
- -
Pric.
a
0
-
-6
-4
-2
-
0
U'
Fig.
-I
2
2.2
16
I.
4
COLUMN ORIGMTNESS
Aft UNITS
I
0
7
6
6. WWL
0
oA
* *04
I
I
I
I
-10
I
I
-g
-6
-4
-2
0
2
4
COLUMN MEIGHT lam)
Fig. 2.3
17
6
a
10
Aluminum Injection
600
,
I
5I
U
'I
'
I
I
500
+17.7
0 "0 4
E
300
C
.2
200
100
ni
290
-17.B
-
I
I
310
I
I
I
I
I
350
330
time (ms)
Fig.
2.4
18
b I
370
--
I
--
390
Aluminum A,
35.0
17.7 cm
30.0
25.0
20.0
15.0
10.0
5.0
0.0
290
310
350
330
time (ms)
Fig.
2.5
19
370
390
0.30
I
II I I
I
I
I
I
I
I
I
I
I
I
I
I
I
I rr 7
)p=0.0025
,a=0.0075
25
p=0.01
0.25 H-----
I
0.20 F-
A
/
C
/
0.15
/
/
0.10 -
\
,-
/
N
I
'I
/
I
I
I..
'I
5%
I
/
/
I
0.05 I-
I
I
I
I
II
/
/
- - - -
I
0.00
0
5
15
10
z
Fig. 2.6
20
I
I
I
I
20
25
Effects of Poloidally Asymmetric Impurity Concentrations on Rippling
III
Instability in Tokamak Edge Plasma
III.1
Introduction
Detailed experimental characterization of the ubiquitous broadband fluctuations from
microturbulence, especially for the tokamak edge, has become increasingly available. Results have indicated a high level of electrostatic potential (e-i/T), density (u/n0 ), and
electron temperature ('e/Teo) fluctuations (10% - 50%
port fluxes
[3.1].
),
as well as the associated trans-
Furthermore, these fluctuations exhibit a surprisingly asymmetrical na-
ture in their spatial as well as spectral distribution. In particular on TEXT, an up/down
(poloidal) asymmetry of the turbulent density fluctuation has been observed which, moreover, flips from the top (9 = 7r/2) to the bottom (9 = -7r/2)
when the plasma current is
reversed (at fixed BT) [3.2]. For the wavenumber-frequency spectrum S(k, w), it is also
observed that the mean parallel wavenumber (k1j) of turbulent fluctuations exceeds the
nominal estimate of the equilibrium connection length (1/qR0 ) by almost an order of magnitude [3.3]. This finite mean parallel wavenumber represents the asymmetrical nature of
the turbulent spectral distribution.
In order to understand these experimental observations, extensive studies of instabilities and resultant turbulence due to various equilibrium gradient free energy sources were
carried out including collisional drift modes and resistivity gradient driven modes.
For edge plasma parameters of "TEXT-class" tokamaks, the most widely accepted
theoretical model of edge turbulence is resistivity gradient driven turbulence, which evolves
from linear rippling mode instability [3.4]. However, the original model of this instability is
not capable of predicting some of the observed characteristics of tokamak edge turbulence
[3.5]. To remedy this deficiency of the model, several additional effects, namely the role
of impurity density gradient (VZf f) [3.6-3.7], impurity line radiation [3.8], equilibrium
poloidal flow (or equivalently, E,) [3.9], were considered but still the observed asymmetries
in the turbulent fluctuations could not be explained even qualitatively.
21
Here, we present a theory of the impurity density gradient driven rippling instability
[3.10],
including the poloidal asymmetry of Zff (r,0)
corresponding to the observed strong
[O(e)] poloidal impurity density modulation which we derived in the previous section.
Consequently, we are led to a possible link between the instability and asymmetries in the
observed density and potential fluctuations.
111.2
Theoretical Model
We employ the following simplified set of reduced resistive MHD equations in the
electrostatic approximation including the evolution equations for resistivity fluctuations
which are driven by thermal fluctuations ( ,,) and impurity density fluctuations (Z.ff),
in cylindrical tokamak geometry [3.4,3.7]
-B.V (1)4 =#J-- +77-J
P
(3.1)
,
(3.2)
BzV(O)j.
dv2
the Spitzer resistivity evolution equation,
- XT
t
*1 ,, = -V
(3.3)
dr
and for the single, low-Z impurity species case, the impurity density fluctuation evolution
equation as originally proposed by Rutherford [3.6],
Zeff - Xz. "
Here,
4 is
Zeff =
[v±E x
i -V±Z'f]
.
(3.4)
the fluid stream function (4= -4/B., where 4 is the electrostatic potential),
J. is the parallel current density, po is the mass density. In Eqs. (3.2)-(3.4), the convective
derivative is
d
_8
d-= 8 + (V J X i) -V j ,(3.5)
dt
Ot
and the derivative with superscript (o) indicates the derivative taken parallel to the equilibrium magnetic field. This set of equations can be closed by noting the relationship
17 = Zef fap
22
with the average quantities 17 0 = Z'f77*,, and the perturbed ones (with tilde),
+ 17 Zef
i=Z *f sp5
(3.6)
.
Here, a new physical mechanism will be introduced corresponding to the observed strong
poloidal impurity density modulation, that plays the role of additional instability drive as
well as localizing the mode structure with respect to poloidal angle
Ze*f(r,0) = Z' f(r) + E(Z'ff(r) cos 0 + Z'ff(r) sin0)
The schematic drawing of the tokamak geometry is given in Fig. 1.
affect the evolution of
Zff
(3.7)
.
This new term will
fluctuations through the right-hand side of Eq. (3.4).
To illuminate the new physical effects, it suffices to choose an isothermal electron
response
(
,, = ''
= 0) corresponding to the high XT parallel electron heat conductivity
regime appropriate even for the edge plasma of contemporary tokamaks [3.6].
However,
X,/Zv;
<XT.
the parallel impurity ion diffusivity, XZvi kept finite, since X< =
Neglecting toroidal curvature effect (low-0), the basic instability can be represented by
an "effective" finite length cylinder possessing a definite sense of top/bottom with respect
to the impurity density modulation. This is due to the unique ion B x VB-drift direction
at fixed BT (which is independent of the equilibrium current direction, as will become
important later).
For simplification of the analysis, the cylindrical coordinates (r, 9, z) are chosen as in
Fig. 1, rewriting (3.7) as
Z'ff(r, 0) = Z' f(r) + EZ1 f(r) cos E
where
Zeff(r) = [(Zef f (r))2 + (Z:ff(
))2
and
o = (0 + 0.with
0, = cos~ 1 [Z:ff(r)/Z'ff(r)]
23
,
(3.8)
(In our notation here, we made a distinction between a geometrical angle 9 and a newly
defined relative angle 9 in the poloidal direction as in Fig. la.)
Using the equilibrium model and eliminating dependent variables from Eqs. (3.1) (3.5), we obtain the following nonlinear evolution equations
aV
~
-
~Vid+V71 4x s - ±Vj1
JzoBzV(o)
B2
Po007o)
(o
PfV9i
VPo)2Z
f
1
dZ9
SdZ
Vek
dr
(3.9)
11Vu
POO-
drcosS] -V,
['Z,
fsin0
,
(3.10)
To avoid the resistive decay of the equilibrium magnetic flux and to be consistent with
equilibrium force balance, (tg,) = 7(r)Zyf f(r) was used in Eq. (3.9). These two nonlinear
equations for
and Zf f will serve as our basic model for the study of edge turbulence
driven primarily due to a poloidal impurity density gradient Oz *ff' # 0.
111.3
Linear Analysis
In this subsection, the linear rippling instability including the effects of poloidal modulation of the impurity concentration described by Eqs (3.9) - (3.10) is studied in detail.
All the perturbations can be represented as a Fourier expansion in E and C = z/Ro
0,
( 5,
elt*emnt
t) =
n
Ze
m
e?"_mn(r)e(me-nC)
ff(r, 9, C, t) = E
ni
(r)ei(me-ne)
m
Here,
Ok(r) =
k(r )
m n(r)
imn (r)
24
= 0in,-n(r)
= 'm,---nr)
guarantees the real-valuedness of the perturbed fields. The linearized system of equations
for a given toroidal mode number n becomes
E7/n [V
J..B] (n-
2/q
1~n~i~
smn(r)e
ime
m/q)2
(n - R.
R
.
+
(3.11)
,
mn(r)e'mE)
eff
___
[-
m/q) m(r)eime
(n -
B
o70Z
+
eim®
.mn(r)
mn(r) -
m.m(r )e'm* - EZeeff sin 0
+ d*co
e
td. 12)
It reduces to a single second order radial eigenvalue equation for an (m,n) Fourier coefficient
v2m
^In [V2mn(r) -
-2
PO2
=-
x
{m
dZ 'f
B.O
$mn(r)
J
(n -m/q)
2r
+jZ11
dr'[
R
-
_-M/q)2
o~z(n
dZ'Of
dZ
_ m/q)2_
2
(nn()
1(n
o7
nrjLolpeffJ
nr
rdr
12
gn+
xz
R2
[(rZ
m + 1)$m+1,n + (m - 1)
KVrem+1,n)
- (Vrdm-,n
m-1,n
m1n
.
(3.13)
Here,
(B
-
1
RV)B
8 +BpRo
Br
0
50
-
Ro
0
8
For later discussions of plasma current reversal Jz, --+ -Jzo
1
q(r) 80)
it is important to note here
that q(r) carries the sign of Bp(r)!
For the given equilibrium, the second order poloidally coupled eigenvalue equations
can be solved by numerical integration, for a given n number. However, further analytic
25
progress is possible by ignoring radial nonuniformity of equilibrium quantities for radially
localized eigenmodes and expanding in the small inverse aspect ratio e.
The reduced
equation can be solved in a perturbative approach (weak coupling approximation or strong
coupling approximation) based on the strength of the mode coupling terms with respect
to the conventional driving term in Eq. (3.13).
As a first step we recall standard VZeii-driven problem in the absence of mode
coupling (i.e., for E = 0). Then the eigenvalue and eigenmode can be obtained by following
the path of previous analyses
d2
14
Az'
2_
(})=
(3.14)
nn
4x4
dx 2
where the superscript (') signifies the lowest order solution in the E expansion. Here we
have introduced the relevant characteristic length scales in units of minor radius a such as
the resistive layer width (with Z:ff)
XR =
1
Po7 0,Z*O
no
r 22
1/4
/a,
ff2L
p
the displacement of the mode from the rational surface
La
= [Jzoi oZ,*ii
L
fnBz
and the parallel diffusion layer width
_f
Jzor7 Z,*5 1 r 2 L 31/ 3
SP
B.
B
XZ m 2 L
/a,
a
with the shear length L, = (q 2 Ro/rq'), the impurity gradient scale length Lz
2
[Z,*f /(dZ,*f /dr)],
and the normalized radial coordinate x = (r - r,)/a which represents the distance from
mode rational surface r,. The eigenvalue of Eq. (3.14) can be obtained using the standard
WKB-method which leads to the following dispersion relation
..
Z( ' X
]
o ([
[ ++ if x x2)
_ X2)1/22
)1/
dx = (21 + 1)7rx , ,
26
(3.15)
where the turning point x. is the solution of
A;,
and £ is the radial mode number.
For the two limiting cases corresponding to the small parallel diffusivity limit (xz - 0)
and the large parallel diffusivity limit (X, -+ oo), the dispersion relation can be solved
analytically and the two known growth rate results can be recovered. The linear growth
rate for (A. /
) < 1(x. -
0) is given by
I..
(*) -
po~z~o
a
2L,
(-o:/)
B.
4
)
Ma- 2/5
1-/
ma
-1
/
RO L2 r
h
where S = rR/rhp is the Lundquist number with the resistive diffusion time Ra = (psoa 2 /i',pZeff)
and the "poloidal Alfv6n time" -r7, =
mode
(i
RO/VA.
The linear eigenmode for the fastest growing
(O)exp
{(
= 0) is given by
((x)
=
= i()
-()
mn (X'
_z)24
m dZ~rf) exp
r. dr(X-A
2
)2 /42
R
(3.16)
which is a radially shifted Gaussian function. In the opposite limit (AZ/C') >> 1(xz -0),
the linear growth rate is given by
a2 ma (AoJzoRo
_ [fLL2 r,
Bz
L
R- X
R2*
4/ :_
-'S
Thp*, 7
where the normalized parallel diffusivity is
3
= XZ/(R2/TR)-
Again, the eigenmode has a shifted Gaussian shape with progressively narrower width for
higher 5,
value. The numerical solution for the linear eigenmode with three different 5.
values in cylindrical geometry is given in Fig. 2 for m/n = 22/11 with S = 105.
27
The next step consists of using the eigenmodes for the case without poloidal modulation of the impurity density (e = 0) as basis functions, whereupon the poloidally coupled
Eq. (3.13) can be solved with perturbation theory. Again ignoring the radial nonuniformity
of equilibrium quantities in Eq. (3.13), (as for the derivation of Eq. (3.14)), the reduced
eigenmode equation for the toroidal mode number n can be constructed by defining a "reference" rational surface r* such that q(r,) = m 0 /n. (This defines the "dominant" poloidal
mode number m0 (m0
m0 +
j
>
1). The neighboring surfaces with the same n but poloidal number
are labelled by j = ±1, ±2,...)
The Fourier expansion of the perturbed field can be rewritten as
0(r, 9, Ct)
)e0'e~nC
(x,
= Ln
n
where the n-th component can be, in turn, represented by
CO
$(x)e'(-.+j)e
On(X, O) = =-oo
with the normalized radial coordinate x = (r -r')/a. From Eq. (3.13), a system of coupled
equations for
dX2
'j (x)
)-
is then obtained
k2(x)] -
n 2
4
-
-
OjX
(AMi)
\+
e
x{L~1
+1(x)
A[ (X - jAn)
4 [1 + (-")(X -a)2]
))
- jAn)
(
)(x - jAn)2 ]
j+(x) -
-1(x)] + 1
+ (
where the poloidal wavenumber is kj = (M
+
di-
1(X)]
(3.17)
j)a/r,, the normalized rational surface
spacing at fixed n is An = (ki.)-- with shear strength
.
=
(rq'/q), the radial scale of
1 and the coupling coefficient is M = [.
ZOff(r) is Lo = [Z' f/(dZOff/dr)]a~
ef
eff
a
which can be ~ 0(1). For m,,
>
ZOff
]
1, the system of coupled equations admits a general class
of solutions due to translational invariance in periodic potential functions such as
d0)(x - jAn)
= e"'I
-j(x)
28
(3.18)
where 0 < r.< 27r is a phase angle which will be determined. Here 4
is a shifted mode
function which satisfies the differential-difference equation (representing mode coupling
with left-shifted and right-shifted mode functions),
k22
d2
dx2
P-()
dx24
z2
k ±)+44-
AzX
+ (EM)
x[1 + (-A)X2
2
+~X
x {L
[eikhgI)(x
in ( (+ _)(x
4x4[1+ ()x2]
zn)
-
-
-
5~(x +
+ ei
d(
ee-')
)
= 0.
)(x + A,)
(3.19)
The solution of this differential-difference equation determines the mode structure in the
(r,() plane as well as modified dispersion relation, driven by the poloidal modulation of
the impurity density.
Although an exact analytic solution of Eq. (3.19) is not possible, perturbation theory
can be applied to obtain order by order solutions.
In the limit of sufficiently small E, and retaining only a triplet of poloidal harmonics
centered about the dominant mode number mo
$(.
= I
)(x -)
*)(X + An(X),
the weak coupling approximation can be used to obtain the first order correction to the
eigenmode, and the dispersion relation. Then, the general mode solution with toroidal
mode number n can be constructed by superposition of the triplets. The two first order
equations are
dx 2
-[k
t An)]
( ±
[dX2
2 + (X ,A~)2
0
_ (EM)
4
2 ) 4X4[1 + (
44I1
'.)(X ± An)
()X ± An)1
4 [1 + (_')(X ± A.)2]
± A9)eti(
29
)
±zdxAn
{kf7(
1)(X ± An)2]
0
T
(3.20)
<
We first apply the weak coupling approximation to the (</)
1 regime. Using the
zeroth order eigenmode in Eq. (3.16), the dispersion relation with the first order correction
becomes
0o
A(o) + EA)
where
tz2
1
-k
+
16x 4
\XR
Mx R
A(O) -
2
2)(3.21)
/
is the zeroth order dispersion relation for the fastest growing
P)
=
V-XR
(3.22)
4n
4(-
x-L.
= 0) eigenmode and
x
+ eI
e
e
(i
dx (
e
X
e
After collecting all terms from Gaussian integrals, we finally obtain
A = M
2
_
-AZR
\4
/
+cAn
14'
LR(
e - ..
Cos
(
-i
[
cosK+iLnsinK]
4)
sinK
(3.23)
.
The real-valuedness of the dispersion relation constrains the phase angle n to 0 or ir. These
two choices will give two different eigenvalues and corresponding eigenmodes which can
be designated "dominant" and "subdominant".
The final expression for the dispersion
relation is
[
4i2R
16xi
k
+_
2
e^)J
4X4
LW- + gJ
4x 2
cos
= 0.
(3.24)
After straightforward root finding of Eq. (3.24), the two linear growth rates can be written
transparently as
)
7(o)(1 + EA cos K)
30
(3.25)
where -y() is the zeroth order linear growth rate and the correction term
a
16
=MAe
3R
16
3
2
!,2
80
"
L
-A
r
[L
+
YY(O)
2
+
All parameters in Eq. (3.25) were defined previously and A is -evaluated at -y =
<0)
Experimentally, from the tomographic measurement of edge impurity radiation intensity
as well as the strong ordering theory which we discussed briefly in the previous section, it
is clear that the asymmetry of the impurity density is increasing toward the edge region of
the plasma, in accordance with the fact that LO (defined after Eq. (3.17)) is positive in most
experimental situations. However, even for the Lo -0+ 0 limit corresponding to radially
uniform Z3ff, the two different eigenmodes with distinct growth rates persist. In this case,
the "dominant" root with eigenvalue -y(+) occurs for the phase angle
K
= 0. Furthermore,
the enhancement factor in the growth rate is a function of the coupling constant M as
well as the coupling strength 'R =
(Al /A2)
which is the ratio of the mode rational surface
spacing An and the mode width A,. ~ (A,/2) where A. measures the mode displacement
from the rational surface.
The formal perturbation theory employing the small
E
expansion has allowed us to
obtain the first order analytic correction due to the poloidal mode coupling but the new
physical effects of poloidal impurity density modulation on the impurity gradient driven
rippling instability are far more important than the correction term in the linear growth
rate in Eq. (3.25). The most important consequence is the emergence of a two dimensional
mode structure in the (r, 0)-plane. Its envelope function emerges from the general solution
for the toroidal mode number n which is constructed by superposition of the triplets.
In the next section we will discuss in detail the mode structure which results in
poloidally asymmetric fluctuations, and the mean parallel wavenumber (k 1 )mcan
which
exceeds the nominal value of (1/qR0 ).
It is also important to notice that the radial length scale LO can contribute substantially to driving the instability when the impurity density gradient length L. becomes too
31
large, i.e. when I
>(
Lo).
)I>
In this case, the resulting instability is an intrinsically
toroidal perturbation, not directly related to the basic impurity gradient driven rippling
mode with the growth rate given in Eq. (3.16).
Either this case (L, -* oo limit) or the
high m 0 microinstability case which implies R > 1 (R oc (mo)6/5), requires the limiting
analysis often referred to as strong coupling approximation [3.11]. Although the radially
shifted mode structure of the basic rippling instability, (which is essential to the instability
mechanism and cannot be ignored), prevents us from following conventional analysis techniques employing the Fourier transformation to the k, space (equivalent to replacing the
discrete j indices by a continuous variable), the perturbation theory based on expanding
M degenerated harmonics expansion can be used to calculate the modified dispersion relation. The preliminary results from this approach indicate that the qualitative behavior
of the mode structure as well as two distinct growth rates for n = 0 and r =
7r
persist.
However, these results are not essential to our discussions in the next subsection and will
be reported in a future publication.
III.4
Characteristics of Asymmetric Fluctuations
Previously, it was postulated that the source of up/down asymmetry in measured fluctuations might be connected to the saturated resistivity gradient driven turbulence [3.2].
The more detailed analysis showed that the previous theory of resistivity gradient driven
turbulence [3.4,3.7] can explain neither the poloidal modulation of fluctuation amplitude,
nor its flipping with reversal of plasma current, even though the rippling mode instability
has explicit plasma current dependence in its growth rate. There were attempts to explain
the experimental observation via existence of a poloidal "ring" limiter, which could provide breaking of axisymmetry of the tokamak equilibrium near the edge. However, this
explanation still remains speculative because the physical effects due to existence of a conducting poloidal limiter, especially line-typing effects, are shown to cause the reduction of
the growth rate in the scrape-off layer (SOL) region of the plasma which leaves unexplained
the large radial extent of the observed asymmetry inside the limiter radius [3.2]. There is
second, yet puzzling experimental measurement on the edge plasma of the TEXT tokamak
32
using a toroidally separated Langmuir probe set: The measured value of the mean parallel
wavenumber (kI)mea (not rms value!) exceeds the nominal value (1/qR) by almost an
order of magnitude [3.3].
In this subsection, we will describe the mode structure of toroidally coupled shifted
Gaussian functions in a (r, 0) plane. Using intrinsic asymmetries of this toroidal eigenmode, we will explain the origin of finite (kii )mean and its link to the poloidal modulation
of fluctuation amplitude. Finally a plausible explanation of the observed flipping of fluctuation asymmetry with the reversal of plasma current will be presented on the basis of the
linear theory shown here. Ultimately, of course, the measured fluctuation spectrum reflects
the saturated turbulent state which evolves from the linear instability. However, based on
previous numerical results of saturated resistivity gradient driven turbulence (with C = 0),
the important linear mode feature of the shifted Gaussian shape persist throughout the
saturated state with the radial correlation length AK replacing the linear mode width A,.
Hence, the qualitative features of the following discussion can be extended to the saturated
turbulence regime with corresponding adjustments, which can be found in previous works
[3.4]. Before giving the full description of new toroidal eigenmode structure including the
radial displacement in its zeroth order solution necessary for instability, we first consider
the usual case with a radially symmetric Gaussian function. This highlights the role of the
phase angle . in determining the two dimensional mode structure and helps to visualize
the poloidal envelope function. For the given toroidal mode number n, the two distinct
"toroidal" eigenmodes corresponding to different eigenvalues in Eq. (3.25) can be constructed by the superposition of all radial components with index j around the dominant
mode with mode number m.
The simple symmetric Gaussian function is given by
00
Mx.4n+j)9
Oj (X, 0) =
j=-O
with
(x) = doeki exp[-(x
-
jan)2 /2z ]
where A, is the mode width of the zeroth order radial solution. Schematic drawings of the
superposition of all j's with phase angle choices n = 0 and . = 7r are given in Figs. 3a and
33
+), the mode amplitude at
3b, respectively. For the r. = 0 eigenmode with growth rate
the rational surface r = r, is given by
)r
n+)(x
= o1 + 2e
= 0, 0) ,)=o1+ecos0+
(3A)
+2e
cos 20
2
2&3co
+2 2e+r
2&2co
2
2Ar
cos30+...j ei'
0
(3.26)
This series represents the poloidal envelope function of fluctuations whose maximum is
located at 9 = 0. (In our case this produces upward "ballooning".)
The n = 7r eigenmode with growth rate -y(-) has the mode amplitude
&
- 2e
(=0,
6) =
cos 0 + 2e
2r
-(34,)3
-2e
21r
2
2.(2&n~)
cos 39+..
2
cos 20
. e "'*
(3.27)
This series has the opposite poloidal modulation to the previous series of Eq. (3.26). (In
this case downward "ballooning" results.) The simple examples which we just described,
help to visualize the envelope function of fluctuations and to explain the role of phase angle
re. The solution with the greater growth rate will dominate, experimentally.
As emphasized before, an unstable rippling mode solution requires a shifted radial
mode shape. For the zeroth order solution with index j, the shifted Gaussian radial function
is
-
Oj (x) = SOeI exp[-(x -
jLn)/2A r]
Schematic drawings of the superposition of all j's with phase angle choice n = 0 and
K = ir are given in Figs. 3c and 3d, respectively. The radial displacement A 2 was defined
previously and is a function of (J,0 L,/L,). For normal tokamak shear and our convention,
the sign of the radial shift with respect to the rational surface r, depends only on sign
of L,. In Figs. 2 and 3, we used the outward radial shift corresponding to the L, > 0.
Following the same procedure for the previous examples, the eigenmode with growth rate
-i+ can be written as
-1)22,
L&2i!
2i
+ [e
iee(-
+An)
-"
r2An
±e 2 1e
+
34
2
2A2
r:
+..emoe
.
(3.28)
At the rational surface x = 0 (or r = r,), the series is given by
n
;+)(x
=0, 6) =~
+ e-
j, e
(
2
&-F (1-2
0'
+
8
(
Le*2e
I
(3.29)
+ .. .}e'm*
r
e
This series, which differs from the previous example, represents the poloidal envelope
function of fluctuations in complex form. It turns out to be a "helical" envelope function
in a (9, C) plane for r = r,. The size of modulation and the pitch of helical envelope is a
function of the coupling strength R = (A2 /A2) as previously defined. In the R
-+
0 limit,
2
&A
+
,e
SAem[1+ 2 cos Eo+ 2 cos 29 +
]
The second eigenmode with growth rate -y(-) has the mode amplitude
(X = 0, 0) =
_2 e®e-
eSAe-42
+ e-
e
r+
12-2
+ ...
eimeo
.
(3.30)
This series differs only with respect to the "helical" pitch of the mode envelope which is
opposite to the one in Eq. (3.29). That is, the handedness of the helical envelope function
with respect to the handedness of the equilibrium magnetic field for a given plasma current
direction (at fixed BT) could decide the preferred state between the two eigenmodes. This
is analogous to the instability criterion of the current convective instability in a shearless
magnetic field. In that case, the unstable mode chooses the mode pitch that has the same
sign as the equilibrium magnetic field pitch.
Next, using the functional form of Eq. (3.28), the spectral structure of the toroidal
eigenmode can be examined by applying the V( *-operator. The mean parallel wavenumber of a given toroidal mode n on the rational surface r = r, can be considered to represent
what was measured using a toroidally separated set of probes in TEXT [3.3]. For the radially symmetric Gaussian shape as in Figs. 3a and 3b, the j = 0 mode has no contribution
to (kjj)mean due to the definition of q(r,*) = '- and the contributions from the sidebands
35
with positive j indices are cancelled by the contributions from the sidebands with negative
j indices.
Consequently, the mean value (k)mean is bounded by the nominal value of
(1/qR0 ). On the other hand, for the realistic case with shifted radial mode structure in
Eq. (3.28) (also in Fig. 3c) exact cancellation between contributions from the sidebands
cannot occur so that the mean parallel wavenumber acquires the "positive-definite" and
finite value (k1 )mean ~ F(7Z)/q(r,*)Ro where F(R) is the function of the coupling strength
parameter
. (Here, the positive definite value comes from our definition of the safety
factor q(r) and the outward shift of the radial mode structure.) Even in the multi-helicity
turbulence which evolves from the new eigenmode, the (k
)mean
will retain a similar func-
tional form provided that the radial correlation length replaces the linear radial width in
the definition of 7.
Therefore, the radial extent of poloidal modulation of the fluctuations
is intimately related to the value (k1 )mean through the function F(R). To illustrate the
physics of modes with finite parallel wavenumber such that (k)mean > (1/qR,), the extended (9, () plane at r = r, (produced by flattening the cylindrical surface with length
of 7rq(r,)Ro in the C direction), is plotted in Fig. 4 with magnetic field lines for the two
different current directions (but at fixed BT). The sinusoidal wave forms on the field line
represent the wave front with (k
)mezn
in spatially standing form along the field line. Then,
it is clear that current reversal will result in flipping the maximum amplitude position in
9 by 1800 at the fixed toroidal position ( = C, which represents the viewing chord position
from the reference angle C = 0. This is a direct consequence of a change of state (with a
definite handedness of the envelope function) with changes in current direction.
To answer the question on the specific phase of the helical modulation seen through the
viewing chord, the reference angle C= 0 is fixed by the existence of a poloidal limiter which
does not play any dynamical role in this calculation. Thus, through the theory we just
presented and its extension to the nonlinear turbulent state, the two puzzling experimental
observations of asymmetries in fluctuations can be explained and the relationship between
spatial and spectral asymmetries can be understood. The nonlinear extension of our theory
is in progress. The resultant saturation amplitude and transport consequences will be
reported in a future publication.
In this paper, we focussed on the impact of poloidal impurity density modulation on
36
the instability and its resultant turbulence. As we have seen in Section II, the poloidal
modulation of impurity radiation emissivity is not uncommon near the plasma edge, and
MARFEs [3.12] can also play a role. The nonlinear impact of these impurity asymmetries
on the edge turbulence problem will be studied by extending the linear method employed
in this paper.
Acknowledgment
One of the authors (G. S. Lee) appreciates numerous clarifying and stimulating discussions with Dr. S. Wolfe at MIT. This work is supported by the US Department of
Energy Contract DE-AC02-78ET-51013.
References
[3.1] C. P. Ritz, D. L. Brower, T. L. Rhodes, R. D. Bengtson, S. J. Levinson, N. C. Luhmann, W. A. Peebles and E. J. Powers, Nucl. Fusion 27, 1125 (1987).
[3.2] D. L. Brower, W. A. Peebles, N. C. Luhmann and R. L. Savage, Phys. Rev. Lett. 54,
689 (1985).
[3.3] C. P. Ritz, E. J. Powers, T. L. Rhodes, R. D. Bengtson et al, Rev. Sci. Instrum. 59,
1739 (1988).
[3.4] L. Garcia, P. H. Diamond, B. A. Carreras and J. D. Callen, Phys. Fluids 28, 2147
(1985).
[3.5] A. J. Wootton et al., Plasma Physics and Controlled Fusion 30, 1479 (1988).
[3.6] P. H. Rutherford, in Physics of Plasma Close to Thermonuclear Conditions, edited by
B. Coppi (Pergamon, New York, 1981) Vol. I, p. 143.
[3.7] T. S. Hahm, P. H. Diamond, P. W. Terry, L. Garcia and B. A. Carreras, Phys. Fluids
30, 1452 (1987).
[3.8] D. R. Thayer and P. H. Diamond, Phys. Fluids 30, 3724 (1987).
37
[3.9] K. C. Shaing, G. S. Lee, B. A. Carreras, W. A. Houlberg, E. C. Crume, in Plasma
Physics and Controlled Nuclear Fusion Research, Nice, 1988 (IAEA, Vienna, 1989)
Vol. 2, p. 13.
[3.10] K. W. Wenzel, Bull Am. Phys. Soc. 34, 2153 (1989).
[3.11] J. B. Taylor, in Plasma Physics and Controlled Nuclear Fusion Research, Bertchesgaden, 1976 (IAEA, Vienna, 1977) Vol. 2, p. 323.
[3.12] B. Lipschultz, B. LaBombard, E. S. Marmar, M. M. Pickrell, J. L. Terry, R. Watterson
and S. M. Wolfe, Nucl. Fusion 24, 977 (1984).
38
Figure Captions
Fig. 3.1:
Schematic representation of tokamak configuration (a) cross-sectional view with
definitions of the geometrical poloidal angle 0, and the relative poloidal angle 9.
(A sense of top/bottom is determined by the ion VB-drift direction.) (b) side view
with definitions of the reference toroidal angle C = 0 (the location of the "ring"
limiter) and the relative toroidal angle
C = C0 (the location
of the viewing chord of
FIR-scattering).
Fig. 3.2:
The structures of the eigenfunctions
with (a) 51 = 0, (b) 5 = 6000, and (c)
Fig. 3.3:
;jj
(
m$%and
Z7')) in
the radial direction
= 18000.
Schematic drawings of poloidal harmonics
4j (x)
(a) symmetric modes with
phase angle r. = 0, (b) symmetric modes with phase angle r = ir, (c) shifted modes
with phase angle r = 0, and (d) shifted modes with phase angle K =
Fig. 3.4:
7r.
Schematic drawing of the extended and flattened (9, C)-plane at r = r, with
two different signs of poloidal magnetic field. The sinusoidal wave forms on the field
line represent the wave front with (k1i) >
39
'-.
9=0
Ion VB - drift
9
I
Re
C=
(a)
A
c(b
(b)
Cwc
Fig.
40
3.1
Constantplane
A(o)
mn
4
0
3
-2
2
-3
I I I Ii I I I I
4
(a)
-4
-
-
-
-
-
-
0
0
4
3
2
-2
4
-3
-(b)
0
0
2'
-t
-2
[
I
I I I IFI I
I
I I
-3
-4
0.35
0
0.40
0.45
0.35
r/a
Fig.
41
3.2
0.40
0.45
j=-2 j=-l j=
0
j=+lj=+2
(a)
I
j=-2
j= 0
(b)
|
I
Ir
j=+2
I
I
r
'S
j
=
-1
Figs.
j= +1
3.3a and 3.3b
42
j=-2 j=, lj=O
(C)
I
I
I
|
j= +1j= +2
|
r
Ag
An,
~i
j=+2
j=0
j=-2
(d)
.
r
A'
A,,
ZI
j=+1
j=-1
Figs.
3.3c and 3.3d
43
II
~1~~
II
II
1/7
II
T
I
av
Fig.
3.4
44