PFC/RR-87-8
DOE/ET-51013-225
UC20 G
Evolution of Poloidal Variation of Impurity
Density and Ambipolar Potential
in Rotating Tokamak Plasma, Part I
Sigmar, D.J.; Zanino, R.t; Hsu, C.T.
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
tDipartimento di Energetica, Politecnico di Torino, Italy
September 1987
This work was supported by the U. S. Department of Energy Contract No. DE-AC0278ET51013. Reproduction, translation, publication, use and disposal, in whole or in part
by or for the United States government is permitted.
Evolution of Poloidal Variation of Impurity Density and
Ambipolar Potential in Rotating Tokamak Plasma, Part I
D. J. Sigmarl
R. Zanino2
C. T. Hsu
1 Massachusetts Institute of
2
Technology, Cambridge, MA, USA
Dipartimentodi Energetica,Politecnico di Torino, Italy
Abstract
When the (beam driven) toroidal plasma rotation (Ve1 ) reaches thermal velocity (vik,)
levels such that
Vthz < VZ ~ Vei~Vthi
where i denotes the hydrogenic and z the impurity ions, the particle density n3 and the
ambipolar potential ( can develop strong poloidal variations in the flux surface.
The
rapid time evolution of nj(r,9), V4(r, 0), and the plasma flow velocity components in the
flux surface V11 , V 1i is calculated in Part I. In a forthcoming Part II these results will
be evaluated numerically using a spectral code to describe the strong poloidal couplings.
Results will be compared to data from the ASDEX tokamak.
I. Introduction
The connection between neutral beam induced plasma rotation and impurity transport has
been the subject of numerous investigations [1,2,3], mainly focusing on the radial (across
the flux surface) transport of impurities in experiments with neutral beam heating. In
most of these works, the steady state beam-plasma equilibrium was analyzed describing
the potential variations of all quantities A, in the simplest form
A(r,0) =
X + Ac cos 0 +
1
A" sin0
where A ''/~j was ordered
-
e = r/Ro. However, using soft X-ray imaging techniques
Smeulders [4] obtained results from the ASDEX tokamak of the form
2
n, (r,6) =
' cos 6 + n' sin M8)
+
t=1
with 0(1), not 0(e) - Fourier amplitudes. This necessitates a theoretical approach different
from that used by previous authors. This is the subject of the present paper which will
be restricted to the Pfirsch-Schliiter regime of collisionality for all species, and will assume
the impurity strength parameters a = nZ 2 /ni to be of 0(1). The transient time evolution
of the particle densities, flow-velocities and ambipolar potential is treated. The nonlinear
couplings introduced by the inertia and other terms in the momentum balance equations
are fully retained and contribute essentially to the final steady state poloidal impurity
distribution. A complete description results of the fast time evolutions of the poloidal
rotation and the ambipolar electric field (first described in a simplified model in Ref. [5]).
On the slow time scale of the toroidal evolution, the neutral beam momentum source Mj
is explicitly kept, balanced by the Braginskii viscous force modelled by a phenomenological
momentum drag term - mjnjvd, Vj where vd,j is taken from experimental observations.
(A first principle explanation of this drag as a gyroviscous force was given previously in
Ref. [6]. Its fully selfconsistent theory is still under investigation [7].) The slow (diffusion)
time scale evolution of toroidal rotation (driven by the radial flow) is not treated in this
report.
In Section II, the underlying fluid equations are formulated. Section III contains a summary. Appendices A and B contain details on the parallel viscous force and inertia, Appendix C gives a simplified physical model for the time evolution of the radial ambipolar
field, E,, and Appendix D justifies the neglect of the radial flow veloctiy on the fast time
scale of interest here.
2
II. Fluid Equations
We adopt the usual (#,9,<p) flux coordinate system of an axisymmetric tokamak. The
particle conservation equation for the ion species j = i, z is
i+ V - njj = 0
(1a)
For the purpose of this paper, namely to calculate the evolution in the flux surfaces where
the slower radial diffusion flow -is frozen in, the component V - VO is ordered out (cf.
Appendix D) and (1a) becomes
9n -
1 a
+ --
J'90 JnfVj . VO = 0
(1b)
where
J- 1 =
- VO
is the inverse Jacobian of the equilibrium. The momentum conservation equation is
ejnj
-V + -' x B
+ R j + Mi - mnjvd-i
V
c
(2)
where Rj is the collisional friction, Mj the beam momentum source and IIj the Braginskii
viscous stress tensor. The last term models an effective drag force needed to describe those
experimental observations where the observed rotation damping is faster than the classical
perpendicular viscosity. (Cf. Ref. [6, 8] for a discussion.)
The system is closed invoking charge neutrality
E
ejnj(4,0) = 0
j=e,i,z
and the Vp component of Ampere's law
c
- .c
+ - -E - Vi = VO - V x B= 0
cO
3
(3a)
where the last step requires axisymmetry. Using
c
dV)
Ampere's law becomes
5wIV&Ik2
4(-1) is discussed below.
= 47rcj- VO
(3b)
From parallel electron momentum balance and assuming all
temperatures are constant in the flux surface
T = Tj()
(4)
O = TB -Vn.+ enB - V
(5a)
one obtains
Combining Eqs (3) and (5) one can express B.V4 in terms of the ionic density variations
in the flux surface:
ejnjB -V4/T. = ZjnjB - Vln(n, + Zn,)
(5b)
For the parallel friction we simply use
R1i = -mnjviz(Vjj - VII.),
Ri1 = -Rj
(6)
and the only stress tensor element needed explicitly is given in detail in App. A, Eq. A.4.
The required ordering V
~ vt1, is incorporated by noting in Eq. (2) that to order 61[6 =
p/a, the Larmor radius expansion parameter,}
-VO(-0 + '(0)
A.
V
=(-'0
or
x
A/c
=0
40) = 0(-)(0)
(7a)
(7b)
Then,
(0)= c 8(_
VII
b -8
4
) b x VO
B
)
with the rotation frequency
w
9
(9)
which can be assumed given at t = 0.
Here, I = RBW = I(M), b
B/B, Vp = e ,/R, and the magnetic field is B = VW x VO +
IVW.
-(0)
The total lowest order flow velocity V
V(0)
=
can then be written as
V 1 -+
(kg-
= (iiI+cw
R2
b- WR2
(10a)
whence
4 +w
0
=
-(o
R 2 VW.
=V
--
B,
=
wR 2
1
- R2
)2 /R 2 B 2
(10b)
(BB)
(10c)
(lvl V+ + 'i WI) = 0
(11)
I)
(
Using (10), the continuity Eq. (1b) is rewritten as
+n
Using (9), also the parallel inertial term (V -VV) - b can now be worked out and one finds
the result given in App. B, Eq. (B. 1). Combining these equations (including Eq. A. 4)
there results for the parallel momentum balance Eq. (2) of ion species j
mi VIj + m,
- B/2(b-.
= -3/ Z2b-=
-V
B
[(Vi
-wR2] + Tj$-Vn;+mingva'Vix
+w)
2V)-vv+
+ Z VnIj-
b.VB
' B
I
I --
-B
b.MVBI I
flj ZjTeb VlIn(ni + Zn,.) - mjnjv3 3 , (VIj1 - Vj1jj) + M111
5
J
(12)
To obtain the evolution of w, we take the
ix
V-
projection of Eq. (2), i.e. the radial
component of the perpendicular momentum balance, and sum over species which yields
the radial current
-ic=
V#b- x
+
min
++
i
- II + minvdjdf}-Ia
(13a)
Here, the lowest order 11j
of Eq. (10a) can be used for V. Before summing over species,
in tokamak flux coordinates, this equation is
j
eC V -V0 = -mjni
-m
[i ,
jn
(I
IN 2 b- VB
V1\+
+
B
w
(T1i. Vnj + ejnjb - V)
vo B-n2
IVik120-
"'
+ 2
(Mi)
4[
\-.2
B2
2B
/
j-R
- V (II 11 /B3 )
+ (R
(13b)
)
On summing over species the b- V0 term and the friction term R(
will annihilate. Equa-
tion (13b) can also be obtained by subtracting the toroidal (R 2 VW) component of the
momentum balance from the parallel component given in Eq. (12) as can be seen from the
identity (used in Eq. (8) before)
x -.
= -
(R
A
where A represents the momentum balance equation. This reveals the meaning of (MI)
as the perpendicular component (in the - b x V0 direction) of the momentum source
term, projected onto the toroidal (p) direction. We then utilize Ampere's law Eq. (3b)
and take the flux average of Eq. (13b) to obtain
(E(mn) + 42
at+
b-VR
+2 +2Z~n~nR
(mnV
R
6
12 Z(mnVgb.
(mnvd)w
VB)
+ii (mnb -VR)
2
V01
(b . VB)
IHII)
(13c)
B2R2
Here R 2 B 2
where we dropped the species index for brevity and E =
= 12
+
IVfI 2
was used (and for a low 3-equilibrium IVik is a flux surface quantity.) We note that using
B 2 /47rmnc 2
=
c2 /c 2
<
1 the term (B2)/47rc2 is negligible. (13c) is an equation for the
time evolution of w(-O) oc E, of the form
8w
+ CjW2 + C2W + C3 = 0
C1
(14a)
( mn. VRj
BT
(14b)
C2 =mnV
VB
+
(o
-
D IBP2R
E (77o
((.
VB)(2b. VVI +V
D =
III.
B
lmV2b.
mn I.bVB
IIB
BT
C3
(b- VB)
1(mn) +
B)
B
(14c)
2
-
+-( 2
(14d)
M/R)
vi~o
BB
(B 2 )/47rc2
(14e)
Summary
We have obtained a closed set of Eqs. (11) for
A~ni(O, B),
(12) for
AVj1 ia,
(
) and (14) for
A-w(O)(j = i, Z) which describes the evolution of nj and VjIj in the domain 0 < 0 < 27r,
on a fixed surface
4,
and the evolution of w oc E, as an initial value problem.
The evolution of nj(,0, 0) is driven by the poloidal flow V = (V +
to steady state such that 2"
!w)BE
and will relax
becomes a flux function. The evolution of V and thus
V, is naturally driven by the parallel component of the momentum source M11 , which
7
simultaneously couples to the poloidal variation of the density via terms involving b.- Vnj
in Eq. (12). For nwit is interesting to note that in addition to the explicit drive by the
usually small toroidal projection of M 1 , w is driven indirectly by M11 through VII as shown
in the first term of Eq. (14d). That is, even for purely parallel injection, a large radial
electric field can be built up when V reaches a level VI ~
Vthi
Concerning the damping mechanisms of the system, a simplified physical discussion of the
poloidal damping of V is given in Appendix C. Here we only remark that the parallel
viscosity involving ?7o provides a damping mechanism for both w and VII on the time scale
r,, while the momentum drag term provides the toroidal damping mechanism on the time
scale 1rT. However, it is worth mentioning that in most tokamak experiments the time scales
for 7, and 1rT are not as widely separated as is usually assumed theoretically, involving the
small classical perpendicular viscosity [9] q7
-ri
(experimental) ~ (50 - 100)(i71/a
2
~ io(liri)- 2 . As mentioned after Eq. (2),
) where a is the plasma radius, cf. Ref. [8]. After
the transients A {n,, Vj,w} have vanished, Eqs. (11), (12) and (14) yield a steady state
system with nontrivial solutions for Vig and V,, = BpK j (Ob)/nj maintained by the finite
momentum sources. Note that the quadratic form of Eq. (14a) implies two possible roots
for w
-
E, thus permitting a bifurcation of the steady state plasma.
8
Appendix A on B - V -I
From Braginskii [9] the 0(8*) - part of the viscous stress tensor is
21 1(bb- 3I
S
with
IiiI = -7OWii
,
and
VY+(VY)T -
W1 = bb:
)]
1(V-
(Al)
- 3V-V
=2b-VVII -2b-Vb-Y1
where 7o corresponds to i in Eq. (C5) such that 77o = -1. In the Pfirsch-Schliter regime,
77o = .96p/v.
Using the lowest order Eq. (10) for V gives either
WI1 = 2
[
VV +
VB)V
B1(
(A2a)
wb-VB]
--
or
W, ~ 2 tb V
-
(A2b)
2(b.- VB)]
depending on the decomposition
V = V+b+V1 =V
RVp ,
V =VI+Vp
Using
2
(B.
b+ (b-Vb)rI -
VHII
one obtains
B -V - 1
= B 3 /2 B -V(I/B
(A)
3 /2
)
and using W11 from (A.1) in the (V,, V,) representation one obtains Eq. (17) of Ref. [6] for
-
-(0)
B -V -H
. Using (A.2a) gives
B-V -
=-
B
-BV)[B
( 2b-VVI +
9
B
IW -vB]
B2
-V)]
(A4)
Appendix B on the inertia term
Using Eq. (10) for the lowest order V one has
(V
VY)$=
=
+ ) (-VV
bR
- b-Vb V-1
+')(g. 1( b I\b
(vi
/Vi
-.
R
VS- R2V)
2
b-VR
2
V.
(Bl)
- 2R2
(V +w
=
W~)-!2g
w2
1+W B
b
S +W
2 V.
-
For the flux averaged equation determining E, ~ w (see Eq. (9)) the expression
G,=(nm2VW. (-
. V) g-
is needed. It is worth mentioning that if '9 5 0 in Eq. (1a), G,, does not vanish, even for
axisymmetric systems. This is readily seen if one uses the relation
a
,(nV)+
av
V - (nVv) = n-
+n(V -V)V
which is obtained invoking
On
&+
V -(nV) = 0
Then,
(R 2 V
- n(V -V)V) = ((V -R2V)
) +
-V' - (R 2 VW -nVV- V4)
(B2)
The last term was obtained from the general flux averaged divergence formula
(V. -
V'A- VO)
=
1)
and from,
nVV:
VR 2 VW = 0
since VV is a symmetric and VR 2 VW an antisymmetric tensor (but only if
last term in (B.2) vanished since V.VO = 0 (see Eq. (10)). Thus, from (B.2),
(R 2 VW -n(V -V)V) 5 0
if
On
10
=
0). The
Appendix C on the evolution of E, for a simplified model
In an early paper focusing on a physical explanation, Hirshman
[5] constructed
a simplified
model problem to show the evolution of Er, V and V., leaving out inertia, variations of
n3 in the flux surface and assuming a phenomenological toroidal damping time rT such
that r,
<< rT.
Here r
is the poloidal rotation damping time and the mechanism for rT
can be a perpendicular viscous force.
From the toroidal component of (2), and summing over species and using Ampere's law
(3b), Hirshman obtained rigorously
P(
mjnjR2V
g+
-
(E - V4i)/47rc
=0 + 0(-ri)
(Cl)
Thus, P. is a constant of the evolution on the fast r, time scale. Using Eq. (10a) for Vj,
Eq. (C.1) can be seen to link the time evolution of Vi1 (or V,) with E, in accordance with
our more general result, Eq. (13c).
It is instructive to develop this link in more detail. From Eqs. (C.1), (10a) and cE.Vk =
-W(VO)2 one gets
a
&8
K
47rmnc 2 rn22
B2
IB
,
-
wR 2 B
j~
-
2 =0
(C2)
where the last term can be neglected since
4imnc2 /B
2 =
c2 /C2
Equation (C2) implies that
mn
Vijj
(mnR2)
=
Next, from parallel momentum balance
(nrmjBV1 y) = -
-V. -I)
=
-
The last step was taken from Ref. [5]:
(B - V - I) = 3((b . VB) 2 )ji 1 (V/B,)
11
B p1(C4)
(C3)
pi = RqnmVthv./[(1 + V.)(1 + -- /2v.)]
(C5)
where for concentric circular flux surfaces
((b - VB) 2 )/(B
2
2q 2 R 2
Thus, in the Pfirsch-Schlnter regime (e3/2V. > 1)Aj = n&mfvo2 /v,. Thenfrom Eqs. (lob),
(C3)
BpB
P
V
+
~E
TiVI
(C6)
Eq. (C4) thus becomes
&
~>jminjj
mjnjVj ~- -( Y
Vpj
m-j~
with the poloidal damping frequency given by
= 2
Vi
Finally, owing to Eq. (C6) the radial electric field, E,. = RBpw/c will relax to a quasisteady
state at the same rate r;
1
as the poloidal flow.
12
Appendix D on neglecting
V()-
V.
We neglected this term in the continutity equation (1b).
inertial term ~
2
V0 2
Scaling V
through the
in Eq. (13b), and Ve = V, from Eq. (10c), one obtains readily
V )/V ~ 8,, the Larmor radius parameters - in the poloidal field. The diffusion driven
radial flow TV
is yet smaller than V,(
by the factor (ni7r-i)
rj1 is the collision frequency.
13
in an impure plasma, where
Appendix E on alternate form for w evolution
The purpose of this appendix is to obtain an alternative form of Eq. (13c) which explicitly contains information of toroidal angular momentum conservation which is useful for
numerical computation.
The flux-averaged equation, Eq. (13c) resulted from forcing the constraint w = W(V)
(defined in Eq. (9)) on the unaveraged equation (13b). However, one always has the choice
to multiply by any -varying function before flux-averaging and thus to arrive at different
forms of the resulting equation. In Eq. (13c), we chose to annihilate the term involving
Tb. Vn by determining E(B2 -Eq. (13b) ).
2 VC -Eq. (2)
By taking E(R
), one finds
the toroidal momentum conservation equation
[Mn (V
+ (d
=(C
B
VB
B2 W
(
- R2 C)
VO) +
(El)
Therefore, instead of annihilating Tb . Vn, we preserve the form E
determining
Z(
A (m
n
Eq. (13b) ) which yields
aw+C'w2 +C2w+C1=0
(E2)
where
+ 2 12
+31
1
E-B
Bn
2
Bm(iVi
-
B
(-
2
355(1
14
)I
2
2
|VB
--
1
=
nm(j..V IV
(
C,=
C3~
w) by
vB) 2
2 )]
B2)+mnvd
B
+
.
B4
.VB
B2
2b.-VVII +
b.VB
1
B
)I + (RM± )
+T( B
D' = (V1
2
4-irc 2
+
B
By straightforwardly calculating D'- Eq. (E2), E(IVVJ
Z(V1
m Eq. (11)
),
V
2)
mW Eq. (11)
),
y
Eq. (12)
),
one can reconstitute the toroidal momentum conservation equation
(El).
15
References
[1] K. H. Burrell, T. Ohkawa, S. K. Wong, Phys. Rev. Lett. 47, 511 (1981).
[2] C. S. Chang, Phys. Fluids, 26, 2140 (1983).
[3] W. M. Stacey, D. J. Sigmar, Phys. Fluids 27, 2076 (1984).
[4] P. Smeulders, Nucl. Fusion 26 (1986)267.
[5] S. P. Hirshman, Nucl. Fusion 18 (1978)917.
[6] W. M. Stacey, D. J. Sigmar, Phys. Fluids 28 (1985)2800.
[7] C. T. Hsu, D. J. Sigmar, Sherwood Fusion Theory Conf. 1987 (San Diego) Paper
2C-6.
[8] W. M. Stacey, M. A. Malik, Nucl. Fusion 26 (1986) 293.
[9] Braginskii, Revs. Plasma Physics Vol. I, p. 250.
16