PFC/JA-94-28
Thermal Bifurcation of SOL Plasma
and Divertor Detachment
S. I. Krasheninnikovl, D. J. Sigmar,
T. K. Soboleva 1 ,2 , P. J. Catto
MIT Plasma Fusion Center
Cambridge, Massachusetts 02139 USA
September 1994
1Kurchatov Institute of
2 Instituto de Ciencias
Atomic Energy, Moscow, Russia
Nucleares, UNAM, Mexico DF, Mexico
This work was supported by the US Department of Energy under contract DE-FG02-91ER-54109.
Reproduction, translation, publication, use, and disposal, in whole or in part, by or for the US
Government is permitted.
Submitted for publication in: Physics of Plasmas
Thermal Bifurcation of SOL Plasma and Divertor
Detachment
S. I. Krasheninnikov a), D.J. Sigmar, T.K. Soboleva a,b),
and P. J. Catto
MIT Plasma Fusion Center, Cambridge, MA 02139, USA
a) Kurchatov Institute of Atomic Energy, Moscow, Russia
b) Instituto de Ciencias Nucleares, UNAM, Mexico D.F., Mexico
Abstract
Models to investigate the main features of plasma neutral interaction in
the recycling region are developed for the two opposite extremes of fluid
and Knudsen neutrals. Both neutral models show that a reduction of the
heat flux into the hydrogen recycling region below some critical value
leads to bifurcation of the plasma parameters near the target. This
bifurcation causes behavior in the SOL which is in agreement with all of
the main features of detached divertor regimes in current tokamak
experiments: i) decrease of the plasma temperature near the target to
about 1 eV, ii) plasma pressure drop in the recycling region, and iii) strong
decrease of the target heat load and plasma flux onto the target. It is also
shown that in the Knudsen limit, the neutral density in the divertor region
can not exceed some maximum density, which is of the order of 1-2 x 1013
for current experiments.
1
I. Introduction
Recent experiments on most diverted tokamaks have demonstrated
so called detached divertor regimes (See Ref. 1 and the references therein).
These regimes are characterized by 1) high energy radiation losses from
the scrape-off layer (SOL) region; 2) low plasma temperature near the
divertor plates; 3) strong decrease of the plasma energy and particle fluxes
onto the plates; and 4) strong plasma pressure drop along magnetic field
lines in the divertor volume.
One of the main attractions of these regimes from the ITER divertor
design point of view is the very low heat loads on the divertor plates.
However the physics of these regimes is not yet understood and, therefore,
it is not clear that detached divertor regimes can be considered as a basis
for ITER divertor design. In this work we improve detachment modelling
by removing the constant plasma pressure assumption and retaining the
neutral energy flux into the target.
To get a rough estimate of the dependence of the plasma
temperature in front of the target, Td, on the heat flux entering the
recycling region, q ,, for a high recycling divertor one can use the global
energy balance equation of the recycling region which for sufficiently high
temperature can be written in the form (see for example Ref. 2, 3)
grc - ndCd sin V (yTd
+
E)
,
(1)
Here y is the heat transmission factor, nd is the target plasma density,
Cd =(Td /M)1/ 2 is the sound speed at the target plasma temperature, M is
the ion mass, V is the angle between the target and magnetic field line,
and El is the ionization cost of the incident charged particles due to
2
ionization of the neutral outflux and hydrogen radiation energy losses (E 1 >
I, with I = 13.6 eV is the hydrogen ionization potential). The factor id =
ndCd sin V on the right hand side of Eq. (1) describes plasma specific flux
on the target, which for the high recycling regime equals the neutral
outflux from the target. The first term in the parenthesis describes the
heat flux transferred to the target due to kinetic energy of the charged
particles, while the second one describes the energy loss due to ionization
of and radiation by the neutral outflux.
The total heat flux reaching the target, qd, (qd s qrc) must account
for the energy I released by recombinating ion and electron and the
portion 6EI of the radiated energy reaching the target. It can be written as
qd - ndCd sin V(yTd + I + 6E 1 )
(2)
,
where 6EI the part of the radiated energy coming to the target.
Re-writing the Eqs. (1), ( 2) in terms of target plasma pressure,
Pd=ndTd, one gets
CrC-
Pd sin V (yTd + El)
qd - Pd sin V
(yTd
MTd1/2
U
PdGrc (Td),
+ I + bE)/ (MTd1/2 )
PdGd (Td) -
(3)
(4)
It is easy to see that for these approximate forms the functions G(...)( Td)
have a minima, G(...), min , at Td=Tm-5 eV increasing at lower and higher
temperatures, but it is known that at small Td < 5 eV these forms must be
replaced with more accurate expressions. Before deriving
3
these
expressions we discuss why they are needed.
If one simply assumes that the plasma pressure near the target Pd
can be balanced by upstream plasma pressure, Pu , then for Pu=Pd Eq. (3)
can only be satisfied if
qrc z qmin - PuGrc,min
(5)
.
If Eq. (3) is satisfied then according to relation (4)
qd 2 PuGm, d ,
(6)
and the asymptotic form of jd for Td -+ 0 is
jd - Pu/ M Td/2
cx Td-1/2
(7)
Note, that supersonic plasma flow at the target does not change the
sense of inequality (5). In this case one has to replace temperature Td in
Eq. (3) by MV 2 , where V is plasma flow velocity along magnetic field line
at the target, and use the momentum balance Pu - nd w
2.
Sometimes, the inequalities (5), (6) are interpreted as restrictions on
the values of the heat fluxes grc and qd at the prescribed upstream plasma
pressure. But these interpretations are not correct. Indeed, let us consider
the global energy balance equations (1), (3) of the recycling region. For the
stability of this balance the energy sink (right hand sides of these
equations) as a function of temperature has to have positive slope so it will
decrease with the decreasing target plasma temperature. That means that
4
the solutions (3), (4) are not stable or accessible for Td<Tm 2, 3. To get
stable solutions in low temperature region one has to consider more
sophisticated models of the SOL plasma- neutral interaction 2, 3 to be
discussed in the next chapter (section).
In this book we will develop physical model of tokamak SOL and
show that the reduction of the heat flux into the hydrogen recycling region
below the critical value, qcrit - qmin, automatically leads to a decrease in
the plasma particle flux onto the target and plasma pressure near the
target, increase of the neutral gas density in the divertor volume, and
expansion of the recycling region. The physical mechanisms responsible for
the decrease of the plasma flux onto the target and plasma pressure drop
along magnetic field lines in the divertor volume (depending on the SOL
plasma parameters and divertor geometry) are neutral gas viscosity and
pressure influence on plasma flow 2, 3, and friction between the plasma
flowing onto the target and the neutral gas scattered by the sidewalls 4.
Both the increase of the neutral gas density in the divertor volume and
expansion of the recycling region can be the reasons for the increase of the
impurity radiation losses from the SOL region and the decrease of the heat
flux into hydrogen recycling region. Taken together all these can lead to
thermal bifurcation of the SOL plasma parameters and detachment of
divertor plasma.
The simplified physical model and geometry of the problem are
considered in Section II. Plasma/neutral recycling for the different limits
of neutral gas flow in the recycling region are considered in Section III
(fluid approximation) and in Section IV (Knudsen approximation). The
results of the analysis of these limits are discussed in Section V. The main
conclusions are summarized in Section VI.
5
II. Model of tokamak SOL plasma
We will consider simple slab divertor geometry (see Fig. 1) where
the axis x, y and z are the "radial", "poloidal" and "toroidal" coordinates
respectively. We will assume that magnetic field is in the y, z plane and
b=By/B<<1, where By and B are the poloidal and total magnetic fields
strengths respectively.
We consider regimes with high recycling of the plasma in the
divertor e. g. Fd >> Is , where rd is the particle fluxes on the divertor
plates and Is is the particle flux from the bulk plasma to the (SOL) and
will assume that the particle flux on the first wall is negligible. In this case
it is possible to distinguish two main regions in tokamak edge plasma (see
Fig. 2). The first one, labelled the H-region, is the region of neutral
hydrogen recycling. It characterized by relatively low plasma temperature,
T, and high neutral gas density, NH, which can even become comparable
to the plasma density, nH. The second region is the remainder of the edge
plasma and denoted as the E-region. The plasma temperature there is
higher than in the H-region and the neutral density NE is much lower than
the plasma density nE .
The main physical processes in the E-region are the energy transport
across and along the magnetic field lines, and the energy radiation losses
mainly due to the impurity radiation. Thus the processes in the E-region
determine the energy flux coming to the hydrogen recycling region, Qrc,
and the width of the SOL Ap (where Qrc = Sqrc , S = 4nRAp, and R is the
tokamak major radius). To simplify the problem we will assume that
plasma/neutral parameters only depend on the poloidal coordinate and the
radial scale length, Ap , of the plasma/neutral parameters is fixed or can
be found from scaling laws.
6
Assuming that plasma convection does not effect the plasma
momentum and heat flux (a good approximation for the high recycling
divertor plasma), in the one dimensional (D) case one can write down the
momentum and energy balance equations in the E-region in the form
(8)
P(y) - Pu - const. ,
b2
dy
e d -T
dy
Wrad,
(9)
where P is the total plasma pressure,
ie
is the electron heat conduction
coefficient along the magnetic field lines, T is the plasma temperature (we
assume that electron and ion temperatures are equal), and Wrad is the
specific energy radiation losses due to impurities. Note that in spite of a
relatively low neutral density NE < nE, neutrals can strongly affect energy
balance in the E-region due to a) an increase of the impurity radiation
from hydrogen-impurity charge-exchange; b) an increase in wall
sputtering by charge-exchange neutrals; and c) an increase in hydrogen
charge-exchange energy losses. The most significant impact of the neutrals
on the energy balance is due to a) and c) which one expects to occur
between the H-region and X-point and in the vicinity of the X-point where
the neutral density is still high enough and plasma temperature is too low.
The main physical processes in the H-region are plasma/neutral
recycling and energy losses due to hydrogen radiation, charge exchange,
and elastic collisions. Since we consider regimes with Fd >> Fs , the
plasma flux on the target Fd as well as plasma/neutral parameters near
the target are determined by plasma - neutral interaction in H-region. We
7
will see below that the E-region affects the flux
rd
only by the plasma
pressure Pu and heat flux Qrc.
We will assume that the plasma flow can be treated by a fluid
approximation (e.g. mean free path of the charged particles, Xe, i, smaller
than the longitudinal scale length, 1). Then, there are two limits in which
the description of neutral gas flow can be drastically simplified: a) the
short mean free path (with respect to neutral-ion collision, we assume that
neutral density is low enough that neutral-neutral collisions can be
ignored) neutral fluid approximation 2, 3, 7, and b) the Knudsen
approximation of long mean free path neutral gas flow. The fluid limit can
be applied for relatively high plasma density in the H-region, when neutral
mean free path, XN, is smaller than AP, while the Knudsen approximation
describes the opposite extreme XN >> ApIn the following sections/chapters we consider plasma/neutral
recycling for these two limits of neutral particle dynamics assuming 100%
particle recycling at the target and Fs=0. We will also assume that plasma
parameters in the E-region and the heat flux Qrc into recycling region are
prescribed, and that the plasma-neutral mixture consist of only one kind of
isotope.
III. Neutral recycling (fluid approximation)
When both the plasma and neutrals can be treated in a fluid
approximation it is convenient to consider the equations for the averaged
plasma-neutral velocity, i;,
temperature, T (we will assume here that the
electron, ion and neutral temperatures equal), and relative velocities of
plasma and neutral species. For simplicity, we assume here that Ap = A.
Then (see for example Ref. 5 and the references therein) one has
8
N)VyVii)+ b dy
- (M(n
y
dqy_dy
dy
...
where i
V-
+
(V.
-EIKI(T)nN,
=
NVy
(10)
(11)
(12)
- K, (T)n N,
Vi
)-o,
(d9/dy)
V(NT)
MNnKiN (T)
+
(13)
n + N),
(14)
nVi, y ,
(nVP +
N9N
n (Vp) and N (VN) are the plasma and neutral densities (velocities), P is
the total pressure, P - (2n + N)T; H is the viscosity tensor, qy is the
total heat flux along y coordinate, (. ..) I is the parallel component of (...)
vector, and M is the ion/neutral mass. The ionization, hydrogen
radiation/ionization energy loss, and ion-neutral collision (charge exchange
and elastic) rate constants are denoted by KR(T), KI(T), and Kiv(T),
respectively, and EI(T), ERKR/KI+I is the neutral ionization cost with ER the
characteristic energy loss. We omit here the influence of the thermal force
on the relative velocities of plasma and neutral species and the heat
transport.
Since we consider complete particle recycling at the target and F s =0,
9
the net mass flow along y coordinate equals to zero everywhere in the SOL
(15)
Vy= 0 ,
and therefore Eqs. (10), (13) can be written as
dP
b - + (v - fi) I
0,
(16a)
-v(NT)
MNnKiN
(16b)
dy
VN - Vi
Assuming that plasma velocity Vp is directed along the magnetic
field, from Eq. (14) one has
(n + N)Vy
-
b nVi, 1 + N(bVN,11 + b'VN, y') m 0,
(17)
where the y' coordinate lies in the y, z plane and is directed perpendicular
to the magnetic field, and b' - (1 - b2)
. From Eqs. (15), (16b) and
(17) one finds
j
bnVi, I 'M(N
d(NT.
+ n), N dy
(18)
Since there is no mass flow in the y direction, the heat flux qy is only
determined by plasma and neutral conduction
10
NT
qy=-(Ke(T)b2+
dT
dT
_
MnNyN
where Ke (T) is the electron heat conduction coefficient along the magnetic
field, c, is the formfactor depending on ion-neutral collision cross section
features and K is the global heat conduction coefficient.
Substituting expressions (18) and (19) into Eqs. (11) and (12) one has
d
dy
1
d(N
-K ()nN()
K(T)nN
M(N + n)KiN dy(NT)
d
NT
+ cX MN
dy IMnVjN)
d{(Ke(T)b2
dTl
I
dy
- EK(T)n N,
(20)
(21)
which, together with Eq. (16a), describe the plasma-neutral interaction in
the fluid approximation.
To close system of the equations (16a), (20) and (21) we impose the
following boundary conditions
qy (y -- 0) - -Crc
(22)
,
qy (y -- 0) - -Td -Ypbnd(Vi, II)d
N(y -
o)
YNNdCd}
(23)
(24)
- 0 ,
Vi, I (Y - 0) - -C
+
(25)
,
11
where (...)d is the (...) value near the target (y=O), Cd - (Td/M) 1 / 2 , a s 1,
Yp and YN are the plasma and neutral gas heat transmission coefficients,
and qrc is the specific heat flux coming to the H-region. Notice that the
heat flux on target, Eq. (23) accounts for the neutral energy flux to the
target.
It is possible to show that the influence of the viscosity (mainly
neutral viscosity) on the total pressure variation is relatively weak
(oc a2 s 1) and does not make s significant difference. Therefore from Eq.
(16a) we have
(2n
+
N) T
=
(26)
Pu.
To simplify analytic solution of Eqs. (20), (21), (26) we replace the
exact temperature dependence of the neutral ionization cost E,
and
ionization K, and ion-neutral collision KiN rate constants by the following
dependencies
KI(T) -
i (T/Tk) 2 0(T - TI) ,
KiN (T) = KiN (T/Tk)
(27)
,
El(T) . El - const.
where e(x) - 1 for x>0 and e(x) - 0 for x<O; K, and KiN are
normalization constants, and Tk and T, are characteristic temperatures
(Tk -10 eV>> T - few eV).
12
Let us consider first of all the case Td
>
TI. Then, from Eq. (20), (26)
and (27) one gets
.d_
2
dpN.
dy 1 + pN dy
p (1 - pN)
(28)
/AF
where pN - NT/PU, and
AF
-
MKiN, d KI, d
2Td I/Pu2
(29)
KiN, d = KiN (Td), KI, d a KI (Td)- Multiplying Eq. (20) by El, subtracting
it from Eq. (21) and introducing (19) one finds
dT
dY
2E, Td
dpN
M(1 + PN )KIN,d dy
qrc ,
(30)
where boundary conditions (22) and (24) are taken into account. The
physical sense of the Eq. (30) becomes more pronounced when (18) and
(19) are employed to rewrite it in terms of heat and particle fluxes
-(qy
+ E, ji) - qr
(31)
.
One can see that Eq. (30) or (31) describe energy
balance where
"internal" energy transport owing to radiation/ionization is taken into
account.
Returning to Eq. (28), multiplying by (2/(1 + pN ))(dpN /dy), and
13
integrating once using boundary condition (24) one finds for the root of
interest that
dpN
=
-
AF
dy
(32)
F(PN)
where
F(pN
=
{2[pN - ln(1
+
pN
-
PN 2/2
1/2
(33)
As a result, (18) becomes
(34)
ji - - Pu Cd d(Td)F(PN )/Td,
here Od (Td ) = (2K (Td )/KiN (Td ))1/2
1
= jd/Tk)
2KI
iN 11/2
Using boundary condition (25) one can rewrite expression (18) for
the plasma flux onto the target in terms of Td and pN to find
ji, d - ab (1 - PN, d)Pu Cd
C/2Td) .
(35)
Then, from Eqs. (23), (31), (34) and (35) one gets the algebraic system of
the equations determining Td and PN, d
(1 - PN,d)/F(pN,d) - 2cd(Td)/(ab),
(36)
qrc - Pu Cd ab(yp
(37)
+
EI/Td)( 1 - PN,d)/2 + YN PN,dt -
14
Notice that (37) reduces to (1) only if PN,d
=
0 (and a
=
1). The left hand
side of Eq. (36) is the monotonicaly decreasing function of PN, d . It varies
from oo to 0, when PN, d changes from 0 to 1, so that the Eq. (36) always
has the solution. Therefore, Eq. (36) can be solved for pN, d as a function of
Td and the result inserted into Eqs. (35) and(37) one can rewrite the right
hand sides of the resultant equations as the functions of Td , that is
.
ji,d - -Pu J(Td)
qrc - Pu G(Td),
(38)
One can get the approximate solution of Eq. (36) analytically, by
replacing the function F(pNd) by the first term of its Taylor's expansion
to find
PN, d
(39)
{1 + I0d(Td )ab).
Then
G(Td) - Cd
J(Td) -
7
Yp + j
Cd Gd(Td)
1
1j+
1 +
+ YN
Od(Td)P
.
'
Gd T
(40)
(41)
Let us consider the behavior of Eqs. (39), (40) and (41). The ratio of
ionization to ion-neutral collision rate constants is of the order of unity for
15
the high temperatures and very small for low plasma temperatures. Then,
for high temperatures Td , such that Gd (Td )/(a b) >> 1 one has
G(Td) - Cd YP
+
J(Td) - Cd
'
,
(42)
PN,d - (ab)/[od(Td)] << 1.
For low temperatures Td , bit with Td > E, Od (Td )
G(Td) -EICd
Gd(Td
EIJ(Td)
-
const.
a b) << 1 one has
PN, d - 1
(43)
2Td
where (27) is employed to find ad x Td 1 / 2 .
From the expressions (42) and(43) one can see that the relations (3)
and (7) can only be applied when plasma temperature near the target is
relatively high and neutral pressure does not significantly affect plasma
flow. Only in the high temperature constant plasma pressure limit will the
function G have a parabolic dependence on plasma temperature Td with a
plasma flux on the target that increases with decreasing Td .
When temperature Td becomes low (Ti < Td < Tm) the relations (3), (7)
and (42) can no longer be applied. From Eq. (43) one then sees that for
low Td both functions G(Td) and J(Td) are essentially constants and the
neutral pressure near the target approximately equals upstream plasma
pressure.
Next, we consider now the qualitative behavior of the functions
G(Td), J(Td), and PN, d(Td) for very low plasma temperatures near the
16
target, such that Td < TI.
For these
very low temperatures
(T
<
T1 )
the
value
of
plasma/neutral flux on the target i,d is not determined by neutral
ionization, but rather is determined by neutral diffusion into the region
with high plasma temperature (T
>
T1 ) where neutral ionization can take
place. To estimate the size of jid as PN,d - 1 (recall Eqs. (35) and (43)) we
return to (18) and write
Ji, d - -Tk 2
yI
(44)
MKiN),
by taking yj = - [d ln(NT)/dy]-l as the width of this very low temperature
layer (0
<
y < yI) as determined by the relation T(y) - T1 . The
magnitude of yi has to be found from the energy balance equation (26),
which in the region (0
<
y
<
y 1 ), where there are no radiation losses
gives
qy (y) = -
dT
-
dy
(45)
= qy, d
To analyze the global heat conduction coefficient dependence on the
y coordinate we employ
=
b2
e
(T)+ c,
2
PN
Tk
MKiN ( 1 - PN)
(46)
The first term on the right hand side of Eq. (46) increases with increasing
temperature (and therefore for increasing y),
17
while the second one
increases as T decreases becausepN -1.
Therefore, the function
K
(y) has a
minimum, cmin , in the region 0 < y < yI. Then, from Eqs. (23), (25) and
(45) one can get the estimate
Kmin
yi
(47)
" YN Pu Cd
PNd
-
(48)
i d Cd
-2M
abPU
Finally, from Eqs. (31), (38), (44), (47) and (48) one obtains
1 - PN,d '-
J(Td) - Cd
G(Td) m YN
d2
2 YN
ab min KiN T(
YN
ocTd ,(49)
Td1 / 2 ,
N Tk
(50)
M Kmin fiN T,
El
Tk
id ~
+
=
EIJ(Td) + YN Cd
1/2
Td11 . (51)
M G)min yiN (I
Thus, for very low temperatures T < TI, both G(Td) and
J(Td) decrease
with decreasing Td , while PN, d (Td) approaches unity.
Comparing Eqs. (36) and (49) one can piece together an expression
for an arbitrary temperature dependence of the rate constants. Consider
the following equation for pN, d:
18
1 - PN,d
F(pN, d)
2 (Xd(Td )+
Cd
T
(52)
F (1)Kmin KiN T
ab
where Id (Td) - (2KI (Td )/KiN (Td ))1/2 contains the ratio of exact
ionization and ion-neutral collision rate constants. It is easy to see that for
high plasma temperatures, Id (Td )/(a b) >> 1, Eq. (52) fits relations (42),
since G(Td) and J(Td) do not depend on the quantities inside the
parentheses of (52). For low temperatures, Id (Td )/(cc b) << 1, (52) fits
relations (49)-(51). A reasonable estimate of the value of Kmin from Eq.
(46) as Kmin . (2cKTk )/(MKiN) one can simplify Eq. (52) to read
1 - PN,d
2
F(pN,d)
ab
d(Td) +
Td
YN
Td
2 F(1)q, T)
(53)
The form of G(Td) and J(Td), and pN,d(Td) obtained from numerical
solutions of Eqs. (37), (53) are shown in Fig. 3. Summarizing the results of
analysis of the functions G(Td), J(Td) and PN, d(Td) one can see the
following: PN, d (Td) monotonically increases (approaching unity) with
decreasing temperature,
J(Td) increases with the decreasing Td for the
high temperatures and then decreases for lower Td, and G(Td) can have
an N-like dependence, with a positive slope for high and low temperatures
and a negative slope for intermediate ones. These results are in agreement
with the results of Ref. 2, 3 where simple zero dimensional and 1D
numerical models of the recycling region were employed.
19
IV. Neutral recycling (Knudsen flow approximation)
In this section/chapter we consider the plasma-neutral interaction in
the recycling region for the case X N >> A p (Knudsen flow approximation).
We will assume, however, that plasma still can be described by fluid
equations consisting of the plasma continuity equation (12) and
d
2
, ( MnVi, 1
dq
+
P - -MnNKiN (Vi,
(54
1 - VN(54)
- -EnNKiN - EjnNKj .
(55)
d111
Here, P=2nT
~'
qp,
(11'
2
+ 5T nVj, j+qeJ1 ,
lii is the coordinate along the magnetic field with
parallel vector component,
(56)
...
)i indicating the
d(...)/dll, - bd(...)/dy,
F
is
the
characteristic energy loss due to ion-neutral collisions, and &p and &e are
the total plasma heat flux and the heat flux due to electron heat
conduction.
We assume random scattering of the neutrals by the sidewalls, N
=
N(y) only, and describe neutral transport by a diffusion equation that
neglects the effects of neutrals traveling nearly parallel to the
y
coordinate. Balancing the divergence of the channel - DNVdN/dy with the
ionization in the SOL (bKinN) gives
20
d
6 --
dNDN -
=
where 6 - A/Ap Z 1,
(57)
KnN,
DN - CD A V0 ,O is characteristic velocity of the
neutrals, and CD is the numerical coefficient of the order of unity.
We impose boundary conditions (24), (25) and
qp, iI(Y ~
00)
= - qrc /b ,
qP, II y
0)
=
-~
-y pdd
(58)
(Vi, II)d.
(59)
where the neutrals transfer their energy to the sidewalls and so do not
contribute to (59).
Since the net mass flow is zero (i. e. ANVN, y + ApbnVi, 11
=
0) and
there is no neutral toroidal velocity one has
Vi,
- VN,I1 - Vi, 11 1 +
N.
(60)
Considering subsonic plasma flow and assuming b2/6 < N/n , Eqs. (54) and
(60) gives the y component of the flux to be
b2
dP
.
ji - - MNKiN --2
dy
(61)
Therefore, using NVN,y =-DNaN/ay, the zero mass flow condition can be
written as
21
b2
dN
dP
-
+ DN
-MNK.
(62)
- ~ = 0.
N Y
iMN
Equations (55), (57), (62) and the boundary conditions describe
plasma - neutral interactions for the large neutral mean free path
XN >> Ap . Let us analyze them assuming the following dependencies of
the rate constants on plasma temperature
K1(T) - K, (T/Tk)e(T - TI) ,
KiN (T) - KiN r const. ,
(63)
E1(T) m E1 - const. .
E(T) - T Te(T - TO),
where E is the numerical coefficient of the order of unity, and To is the
effective temperature of the neutrals, (To < TI). Then from Eqs. (24) and
(62) one obtains
P
-
=Pu
)
(64)
,
where i - N/Nmax and
N
2
-
2b 2 P
N
(65)
22
Multiplying plasma continuity equation on E1, adding it to the energy
balance equation (55), and integrating over all y, and employing the
boundary conditions gives
(b Pu)
I
Y ~d+
Cic
+
ji, d
-
ji,d
ctbPuCd (1 _ d2
ddn
1(6
E
0
PU J(Td),
/Jid
(67)
where relation (64), 'Id = Nd/Nmax, and dy =- DNbdN/i are employed. The
main difference between the global energy balance equation (66) and its
short mean free path counterpart (37) is the presence of a volumetric
energy sink due to ion neutral collisions (second term on the right hand
side).
For the case Td > T the
'j
dependence of
ji can be evaluated. Then
from Eqs. (57) and (64) one obtains
1
(68)
2
AK
dy
2)1/
2
where
2
2
AKI
Using ji,d
Tk DN
(69)
u
= -
8DNNmaxdq/dyly=O along with Eqs. (65), (67), (68) and (69)
gives
23
.
Td
Tk
C2
2
2 2
'q
iN
i
(70)
Tld 2 1 - 'qd2/'
Inserting expression (68) into Eq. (66) to eliminate ji,d
= -
6DNNDdi/dy and
using the definitions for Nnd Ak and ji,d gives
G(Td) - b2
(
YP + E
1 -
'9d )
1/2
1/2j
d
+Mf
1 -,2
_ 2/2),
(71)
To simplify expressions (67) and (71) wer first consider high plasma
temperatures, Td >> Tk (a/2 )2 (KiN
2
(C2
Wl KiN
K,
G(Td) - abCd
I ). From Eq. (70) one finds
Tk
(72)
Td<1
(Yp
+ E Tk KiNKI
+ EI
+
Td
)
(73)
so that (67) and (71) reduce to
J(Td)
J~dJ
0'2b Cd
Td
(74)
24
2
For lower plasma temperatures, T 1 s Td << Tk (a/2 )2 (KiN /Kl), id
-1
in Eq. (70) giving
ld
d
G(Td)
1
1
-b
(75)
Td KI 1/2
Tk KiN/
cc
Tk
M
/2
El
K,
2 Tk
KiN
+ 0. 35T
J(Td)
J~dj
2b
Tk
/
iN
-
const. ,
. const. .
(LZ
M KiN) 1/2
(76)
(77)
The exact solutions of Eqs. (67), (70) and (71) are shown in Fig. 4. In
the plasma temperature range Td a TI, the functions G(Td) and J(Td)
saturate at low temperatures and lid approaches unity, Note that
depending on the values of F, KiN/KI, etc. the function G(Td) may
increase at low temperatures.
Analysis of Eqs. (57) and (66) shows that function G(Td) does not
decrease in the temperature range T1 2 Td 2 To
.
We omit the details
here because of the complicated algebra. However, qualitatively one can
see directly from Eq. (66) that G(Td) does not decrease. Notice from Eq.
(66) that G(Td) has the following structure
25
d + f dy
(..)j,
G
G(Td)
(78)
''
0
di,d
where Iji/ji,di < 1. Therefore, G(Td) can decrease with decreasing Td only
when both i, d and id decrease, but this would contradict relation (67)
and so it is not possible.
This contradiction can only be removed when Td decreases below
the value To. Indeed in this case the structure of equation (66) becomes
Crc
=
b
(79)
0
Ji, d
d + fcdi
(--
(79)
.
where T10 corresponds to the coordinate yo such that T (y) = To for
y s yo. Since ii, d is determined by the value of r1d, it is now possible to
decrease the right hand side of Eq. (66) by decreasing both qo and ji, d.
General analysis of this case is very cumbersome, therefore we
consider here the limit of very small
rc (qrc << Pu G(T d a To)).
First of all let us determine the energy losses within the region
yo 5 y ! yl, where yi corresponds to the temperature T(yj) - T1 . The
energy balance equation in this region, where there is no neutral
ionization, can be written in the form
dJ 5 T ji d + b
dy
'
e(T)
-
Pu Nmax KiN
2
dy
Eq. (80) can be simplified for small
(80)
rc . Indeed, in this case the ion flux
ji, d is small and the energy heat flux is mainly transported by electron
26
heat conduction. Moreover, we will see that the neutral density
variation in the region yo z y s Yi is strong and
YiI
Then, imposing the boundary condition (dT/dy)
=
(I
<<
1)
=n (Y I) << i
=
1.
0 and using the
linear dependence of the neutral density on the coordinate y (see Eq. (57)),
from Eq. (80) one obtains the length 110 - y - Yo (110 << yo), and the
expression for the heat flux qO dissipated in the region yo r y z yj
1/2
-1/2
CI
11/2
-
24 b2 Ce(TI)TI
7 r PU Nmax KiN
(3/14)b2 E Pu Nmax KiN ICe(TI )TI 1/
We assume here that e(TI)TI >>
2
(82)
e(To )To.
The expressions for the ion flux ji, d and the heat flux qj dissipated
in the region y a yj can be found from Eq. (55) and the plasma continuity
equation (rid - 1):
ji, d
q,
b bP
-
Tk
djd
M
Ei + E Tk
1/2
-
DN Nmax
6110
rio
-
DN Nmax
,
(83)
6 YI
(84)
-
The values of qi, qi, and yi can be found from global energy balance
in the recycling region
27
(85)
qrc - q 0 +q.
From Eqs. (82)-(84) one sees that for low grc values (rj1
qrc
<
<
«<
1)
{(3/14) T b2 Pu Nmax KiN Ice (TI )TI 1/2
3 4
cx(b2 PU) /
1/2 A)1/2
(86)
so that practically all the energy flux is dissipated in the region
yo r Y s Yi due to ion-neutral collisions (and the part related to
ionization losses in the region y a yi is small). The ion flux on the target is
3
fr
rP
i d
o
yDNa
iN
3b2 ice(T ) Ti
c qrc
Pu
62
Ap b 4
(87)
The estimates for 1, 0 , yl, and ql
-1
-3
110 , qrc
<< Y1cqrC,
0
2
3
Il
Crc
(88)
<<110
q rc << 1,
(89)
obtained from Eqs. (81) and (83), show that (as assumed above) for low
qrc , neutral density variation in the region yo r Y r Yi is strong for
TII << 710.
Thus, for the case XN >> Ap (Knudsen flow approximation) G(Td)
28
(J(Td)) decrease (increases) with the decreasing of plasma temperature
and saturates at low temperatures Td
>
To, while 'qd approaches unity (as
we noted above function G(Td) may even increase at low temperatures).
When plasma temperature Td decreases below To, the plasma flux on the
target, length of the recycling region, etc. are directly determined by the
heat flux qrc (see for example Eq. (87)). Therefore, in Fig. 4 functions
G(T d ) and
J(Td ) have vertical lines at Td =To.
V. Bifurcations in the SOL plasmas and divertor detachment
We have shown in the previous Chapters that generally speaking functions
G(Td) and J(Td) have more or less similar dependencies in the opposite
extremes of the fluid and Knudsen limits. Therefore, it seems reasonable to
assume that in the general case the functions G(Td) and J(Td) have the
same type of behavior: G(Td) increases with increasing temperature for
low (G(O)=0) and high Td values and decreases for intermediate Td ; and
J (Td ) increases
for low temperatures (J(O)=0) and decreases at high ones,
reaching the maximum at intermediate Td .
Consider now the global behavior of the SOL plasma assuming G(Td)
has an N-like dependence with the local minimum Gmin corresponding to
the temperature Td - Tm (one can see from Figs. 3 and 4 that for both
the fluid and Knudsen approximations Tm - 5 eV). Then, the global
energy balance of the SOL plasma can be written in the following way
Pu
G(Td )S
- Qrc R Qheating - Qrad, bulk - Qrad, SOL ,
.
(90)
where S = 4nRAP , R is the tokamak major radius, Qheating is the total
tokamak heating power, Qrad, bulk is the radiation losses in the bulk
29
plasma, and Qrad, SOL is the energy radiation losses in the SOL plasma (Eregion). Note that the value of Pu determined by the upstream SOL plasma
density and temperature depends the edge plasma transport and on the
heat flux QSOL
Qheating - Qrad, bulk entering the SOL.
Q
Assume that QSOL , Pu and S are constants. If radiation losses in the
SOL are low and QSOL is high enough, then Eq. (90) gives QSOL
>
Qbif,
where
Qbif = Pu Gmin S -
(91)
As a result, one finds regimes with high plasma temperature near the
target Td
>
Tm, balanced plasma pressure Pd - Pu, and high plasma
( d - Pu J(Td ) S) and heat flux on the divertor target (Qid) , which can
be esmitted from Eqs. (1), (2) to be
+SOL
Qd, d " QSOL
Fd
(92)
Q
YPTd + El
(93)
QSOL
YPTd + El
With the increase of radiation losses in the SOL, the heat flux Qrc will
decrease causing the decrease of Td (Td
>
Tm) and the increase of plasma
density near the target, nd - Pu/Td, and plasma flux rd. When radiation
losses
from
the
SOL
plasma
becomes
high
enough
that
QSOL - Qrad, SOL < Qbif , bifurcation of solution of Eq. (90) occurs and
the temperature Td drops well below Tm (according to the Figs. 3 and 4,
30
the temperature Td after bifurcation to the low temperature branch of
G(Td) is smaller than 1 eV). Simultaneously, there is a strong decrease of
plasma pressure near the target. As far as the heat load on the divertor
target is concerned, there is a big difference between fluid and Knudsen
neutrals.
Indeed, in the fluid approximation the heat flux qrc going to the
divertor plate can only be dispersed to the side walls by radiation if the
length of the recycling region, 4c - Yi + AF, becomes less than the
plasma column width AP . For the case 4c
target
<
Ap the heat flux on the
Qx, d cannot be smaller than Qrc/2 (which corresponds to half of
the radiated photons hitting the plate). Therefore, for fluid neutrals a
strong decrease in the heat flux on the target is only possible for the case
when Lrc
>
Ap
.
From Eqs. (29) (47), (51) one can see that AF x Pu 1 and
yi ( Qrc~1. Thus, for fluid neutrals one can only expect to have a strong
decrease of
Q7, d for low Qrc
Qrc < Qcrit, F (Ap ) ,
(94)
In the Knudsen approximation energy can easily be dispersed to the
whole target surface and to the sidewalls by the neutrals if the length of
recycling region is high enough. This is exactly corresponds to the situation
after the bifurcation to the low temperature branch of the curve G(Td),
when Td =To (see Fig. 4). Moreover, for the case Td =To the plasma flux
on the target Id is proportional to Qrc3 (see Eq. (87)) one has the
following scaling for
Qx, d O I Fd :
31
QId
(95)
Qrc
Q 3.
Thus, the specific heat flux on the target and sidewalls decreases much
more rapidly then the heat flux into recycling region.
Let us now show qualitatively one effect of the influence of the
radiation losses from the bulk plasma on the divertor plasma parameters
assuming Qrad, SOL= 0 . As one can see from Eq. (91) Qbif is proportional to
the upstream plasma pressure. But for a high recycling divertor, the
upstream plasma temperature Tu is determined by the heat conduction
equation (see Eq. (9)) and hence by QSOL. (Note that the effect of radiation
loses from the SOL plasma on the upstream plasma temperature Tu is
much less pronounced, especially if the radiation region is localized in the
divertor.) An increase in Qrad, bulk causes a decrease in both QSOL and
upstream plasma temperature Tu (assuming fixed upstream plasma
density). In this case Pu . Pu (QSOL) MPu (Qheating - Qrad, bulk) and
Qbif will decrease with increasing Qrad, bulk. Therefore, the transition to
the low temperature branch, which is accompanied by a decrease in Td ,
plasma pressure drop, etc., can only occur for higher radiation losses then
for the case of SOL radiation.
So far we have assumed that the radiation losses in the SOL (E region) are not affected by plasma parameter variation in the recycling
region. However, as was noticed in Chapter II, even a small fraction of
neutrals can significantly influence the impurity radiation. Then, taking
into account that the neutral density is a function of the heat flux Qrc
(N a N(Qrc )), Eq. (90) can be written in the form
32
Pu G(Td)S
=
Qheating
-
Qrad,SOL(G(Td)) ,
(96)
where we put Qrad, bulk =0. Assuming that the radiation loses increase
with increasing of neutral density one can see the following: 1) bifurcation
of divertor plasma parameters may occur for Qrc higher
than
Qbif - Pu Gmin S and 2) after bifurcation to the low temperature branch,
where the neutral density is much higher than on the high temperature
branch, the magnitude of Qrc , plasma temperature, and particle and heat
fluxes on the target (even for the fluid neutrals) may drop more strongly
than they would have had we not accounted for this effect. A detailed
analysis of the influence of the neutrals on impurity radiation and SOL
plasma stability will be considered in a separate paper.
One can see from the preceding that the plasma/neutral parameters
in the divertor and the target heat load corresponding to the low
temperature branch of the curve G(Td) (especially for Knudsen neutrals)
fit all of the following main features of detached divertor regimes :1) high
energy radiation losses from the SOL region, 2) low plasma temperature
near divertor plates, 3) strong decrease of the plasma energy and particle
fluxes onto the plates, 4) strong plasma pressure drop along magnetic field
lines in the divertor volume.
We are now ina position to make some estimates. First we estimate
Qbif for current tokamaks and ITER. For a typical value of
b - Bp/B - 0.05
, and
upstream
plasma
pressure
Pu=
Po - 10 14 cm-3
MW/m
2
x
100eV one finds from Figs. 3 and 4 that qbif - 6
. The typical S values for current tokamaks (DIII-D, ASDEX-U, JET,
JT-60-U)
having
a
single
33
null
divertor
is
S
-
4a
x 0. 01im x 2 m - 0. 25m 2 with an upstream pressure of about 0.1+0.3
PO . Consequently, for current tokamaks Qbif - 0.15 + 0.5 MW. For C-MOD
where R=0.67 m but Pu -0.5
assuming Pu -5
PO, one gets Qbif - 0.5 MW. For ITER,
PO, R - 8 m and A P - 0. 02 m one has Qbif - 60 MW.
All these numbers look quite reasonable.
Next we make some estimates for the neutrals. The tokamaks DIII-D,
ASDEX-U, JET, and JT-60-U are characterized by quite moderate plasma
density near the target, nd si 014 cm-3 when XN a Ap. Therefore, neutral
transport in these machines is closer to the Knudsen regime. For C-MOD
where plasma density is higher, the neutrals are closer to the fluid regime
(for the attached case). However, in detached case in which the plasma
density near the target drops, neutral transport near the target may be in
the Knudsen regime, while at some distance from the target it still can be
in the fluid regime.
We can estimate the magnitude of neutral density Nma reached at
low temperature branch of the curve G(Td ). For b-0.05, cD -1, A -5 cm,
6-5, KiN-10- 8 cm- 3 , VO -5105 cm/s , and Pu-0.3+0.5 PO one has
Nma
-
1+2 1013 cm73 . Again, these values are close to experimental
observations. For the ITER parameters (A -10 cm, Pu -5
Nmax " 4 10
13
PO ) one finds
cm-3 .
V. Conclusions
We have developed models to investigate the main features of plasma
neutral interaction in the recycling region for the two opposite extremes of
fluid and Knudsen neutrals. We have shown that for both neutral models
the reduction of the heat flux into the hydrogen recycling region below the
critical value Qbif leads to bifurcation of the plasma parameters near the
34
target. This bifurcation causes a decrease of the plasma temperature near
the target to about 1 eV, plasma pressure drop in the recycling region,
increase of the neutral gas density in the divertor volume, and expansion
of the recycling region. For the Knudsen neutrals bifurcation also causes a
strong decrease of the target heat load. This behavior in the SOL is in
agreement with all the main features of detached divertor regimes in
current tokamak experiments. We also find that in the Knudsen limit, the
neutral density in the divertor region can not exceed some maximum
density, which is of the order of 1-2
x
1013 for current experiments.
The physical mechanisms responsible for the decrease in the plasma
flux onto the target and plasma pressure drop along magnetic field lines in
the recycling region depend on neutral gas regime of interest. For fluid
neutrals, the neutral pressure influence on the plasma flow is responsible,
while for Knudsen neutrals, friction between the plasma flow onto the
target and the neutral gas scattered by the sidewalls causes these
reductions.
Acknowledgments
This work was performed under DOE grant DE-FG02-91-ER-54109.
35
References
1 I.H. Hutchinson, R. Boivin, F. Bombarda, P. Bonoli, S. Fairfax, C. Fiore,
J.
Goetz, S. Golovato, R. Granetz, M. Greenwald, S. Home, A. Hubbard, J. Irby,
B. LaBombard, B. Lipschultz, E. Marmar, G. McCracken, M. Porkolab, J. Rice,
J. Snipes, Y. Takase, J. Terry, S. Wolfe, C. Christensen, D. Gamier, M. Graf, T.
Hsu, T. Luke, M. May, A. Nemczewski, G. Tinios, J. Schachter, and J. Urban,
Physics of Plasmas 1, 1511 (1994).
2 S. I. Krasheninnikov, A.S. Kukushkin, V.I. Pistunovich, and V.A. Pozharov,
Pis'ma Zh. Techn. Fiz. 11, 1061 (1985). 12th Europ. Conf. on Plasma Phys.,
Budapest, 1985.
3 S. I. Krasheninnikov, A.S. Kukushkin, V.I. Pistunovich, and V.A. Pozharov,
Nuclear Fusion 27, 1805 (1987).
4 P.C. Stangeby, Nuclear Fusion 33, 1695 (1993).
5 P. Helander, S. I. Krasheninnikov, P.J. Catto, will appear in Physics of
Plasmas 1, October (1994).
36
Figure Caption
Fig.1. Geometry of the problem.
Fig. 2. Different regions in the tokamak SOL plasmas.
Fig. 3. Function G(Td) [kW/cm 2 ], J(Td) [1020 cm- 2 s- 1 ], and relative
plasma pressure near the target p(Td) - 2ndTd/Pu for the fluid neutrals
(T 1
=
J (Td)
3 eV, a=0.5, CN= 4 , Yp=6, Y N=0.2, and b=0.05). Functions G(Td) and
are multiplied on the pressure Po
=
104 cm-3 x 100 eV.
Fig. 4. Function G(Td) [kW/cm 2 ], J(Td) [1020 cm- 2 s- 1 ], and relative
plasma pressure near the target p(Td) - 2ndTd/Pu for the Knudsen
neutrals (a=0.5, y p=6, E=1, and b=0.05). Functions G(Td ) and J(Td ) are
multiplied on the pressure P0 = 10
37
14
cm
3
x 100 eV.
*
14
-
X-point
SOL
core
side
wall
side
Lx
wall
'z
x
A
--
p.
'
0
I
1
.
.
.
divertor
Fig. 1.
38
.
plate
*
X-point
energy
transport
side
side
wall
wall
impurity
radiation
4
hydrogen
recycling
Lrc
A
I__
I
Divertor plate
Fig. 2.
39
Fluid Neutrals
1-.0
- - -G
T
/p (Td)
.8
|
4
-I
.2
- .......
0
.
5
10
Td, eV
Fig. 3.
40
J(Td )
.
15
20
Knudsen Neutrals
1.00
p(Td)
.75
-
.7I
-
~
-
.
-
-
-- ~G(Td---
.50
.25
--
J(Td)
.
0
0
5
10
Td, eV
Fig. 4.
41
15
20