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Divertor Detachment Basic Analysis
I H Hutchinson, MIT PSFC.
Summary: Detachment of the divertor plasma may be essential to reactor target plate
longevity. Non-linear analysis of a thermal front[1], representing detachment, predicted the
rather narrow density-window over which detachment is localized in the divertor (rather than
at the x-point) subsequently observed in experiments. Poloidal flux expansion lengthens the
field-line producing minor benefit, but the analysis predicts a much larger operational window
for divertors with total-field-variation Bx > Bplate . This is an important topic to investigate
experimentally. A figure of merit for detachment localization is qk L. Consequently, the
physics of a localized detachment thermal front cannot be accessed at the lower qk and
shorter field-line length typical of linear “divertor simulators”.
When radiative energy loss rises steeply as the temperature decreases, as is the case for
light impurities (and hydrogen) in the scrape-off-layer, the radiation tends to be localized
to a region called a thermal front, which is a free-standing localized transition from a high
temperature Th to a low temperature Tc . It may be essential for divertor survival that it be
detached. Detachment is equivalent to saying that the divertor plate must be on the cold
side of a thermal front. Generally the upstream SOL is at a temperature corresponding to
the hot side, where radiative loss is small. The balance between parallel thermal conduction
with conductivity κk and volumetric energy input H (H = S − E where E is the radiative
(emissive) loss, and S some a heat source such as divergence of perpendicular heat flux) can
be written
B B
κk .∇T
H = ∇.
B B
!
B
1 B
= B .∇
κk .∇T
B
B B
!
= B∇k
1
κk ∇k T
B
d
=B
dl
κk dT
B dl
!
,
where l is the length along the field line.
In reference [1] two parallel power flux densities, qf , the dissipation by the front, and qi ,
the incoming flux density arising from upstream input were analysed. Thermal equilibrium
of the front consists of the two fluxes being equal qi = qf . If the equilibrium is violated then
the front moves: upstream (towards the hot region) if qi < qf or downstream if qi > qf . An
equilibrium is stable against such motion only if dld (qi − qf ) ≤ 0.
These qf and qi are the power fluxes per unit area (A0 ) that the flux-tube would have at
the place where the total magnetic field is B = B0 (a constant reference field). Local qk is
the power flux per unit local flux-tube area (A), which varies with position ∝ 1/B. One
can simplify the analysis by rescaling the parallel distance to a new coordinate
that
z such
B02 dT
d
d
dT
B/dl = B0 /dz. The energy conservation equation then becomes: dz κk B 2 dz = dz κ dz =
H, where κ ≡ κk B02 /B 2 is a scaled conductivity and contains all the B-variation. The
relation between the rescaled quantities and the local quantities is
Quantity Significance
Local Dependence Axisymmetric
qf
Front dissipation
(B0 /B)qk
(R/R0 )qk
qi
Heat inflow
(B0 /B)qk
(R/R0 )qk
z
Parallel coordinate dz = (B0 /B)dl
dz = (B/Bφ )R0 dφ
2
κ
Scaled conductivity (B0 /B) κk
(R/R0 )2 κk
1
The total radiative loss in the front (ignoring S there) is shown to be
dT
qf ≡ − κ
dz
s
!
=
h
−
Z Th
Tc
B0
2κHdT =
B
sZ T
h
Tc
q
2κk EdT = KI fI pu
B0
B
where KI is a constant of proportionality determined by the radiation function of the impurity I whose radiation predominates; fI is the fractional density of that impurity; and pu is
the electron pressure in the upstream (hot) region.
The analysis [1] predicted that, when B-variation in the divertor is small, the fractional window in midplane SOL density (or SOL power) for which the thermal front remains in the divertor is rather small: depending on field-line lengths approximately as
1 − (Lmain chamber /L)−4/7 , typically 20% or so. Experiments subsequently confirmed[2] the
density-window to be quite narrow, observing that the detachment all-too-quickly became
an x-point marfe.
Poloidal flux expansion increases the flux-tube length, but according to this analysis
does nothing else to improve the situation (other second order effects might actually be
unfavorable, because of reducing perpendicular neutral penetration). Divertor legs which
extend to substantially larger major radius than the x-point have, by contrast, substantial
total flux expansion (A ∝ 1/B). Then this analysis[3] predicts the fractional density-window
opens up rapidly: a much stronger effect than the simple contact area scaling.
Many important questions about the achievement and robustness of detachment remain
unanswered, and undoubtedly include multidimensional effects, convection, neutrals, momentum loss, and possibly unsteadiness. These must be explored in relevant experiments.
The parallel-extent of the thermal front Lf is inversely proportional to qf . Quantitatively
Lf ≈ κk T /qf evaluated at the temperature of peak radiation, which for coronal carbon (and
nitrogen is similar) is ∼ 17eV; so Lf ∼ 40/qf [MW,m]. The figure
√ of merit predicting the
relative localization for a field line length L is therefore qk L ∝ fI pu . Consequently longfield-line divertors with high parallel heat fluxes experience localized fronts. The thermal
front dynamics at the much lower qk and substantially shorter field-line-lengths of linear
“divertor simulators” is not localized in the same way. Although the plasma near the plate
in a linear simulator might experience some pressure loss, it is being controlled by localized
plate recycling, not by the upstream radiative detachment dynamics that must be favorable
for a reactor. It therefore appears essential that experiments to understand and control
detachment be carried out in toroidal devices. High magnetic field and hence high qk is
optimal, because at fixed temperature (needed to satisfy atomic physics), if nu ∝ B/R (e.g.
by Greenwald scaling), then qk L ∝ BL/R.
References
[1] I H Hutchinson Thermal Front Analysis of Detached Divertors and MARFES, Nuclear
Fusion, 34 1337 (1994)
[2] B. Lipschultz, B. Labombard, J. L. Terry, C. Boswell, and I. H. Hutchinson Divertor
Physics Research On Alcator C-Mod, Fusion Science and Technology, 51, 369 (2007).
[3] B Lipschultz et al, in preparation (2015).
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