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SUMMARY:Generalization of the Heuristic Drift Model for Finite Collisional
ity and Implications for ITER
DTSTART;VALUE=DATE-TIME:20210512T162500Z
DTEND;VALUE=DATE-TIME:20210512T164500Z
DTSTAMP;VALUE=DATE-TIME:20210414T210525Z
UID:indico-contribution-17382@conferences.iaea.org
DESCRIPTION:Speakers: Robert Goldston (Princeton Plasma Physics Laboratory
)\nExperimental results from tokamaks with low gas puff rates have shown g
ood agreement with the Heuristic Drift (HD) model (1) for the power scrape
-off width. However\, H-mode results with high gas puffs on ASDEX-Upgrade
(2\,3) show significant radial spreading compared with the HD model\, whic
h was explicitly developed for low gas puff conditions. Here we present an
analysis based on the physics of the HD model\, generalized to take into
account the longer energy confinement time in the scrape-off layer (SOL) a
ssociated with Spitzer thermal resistance\, as well as power loss near the
target. This new generalized HD (GHD) result is in qualitative agreement
with ASDEX-Upgrade data and will be compared with the latest results from
both ASDEX-Upgrade and JET to be presented at this meeting. While there ar
e indications of regions of parameter space where the SOL width increases\
, and so heat fluxes to divertor targets can be reduced\, these are not ac
cessible to ITER. However\, ITER’s low collisionality at high density ma
y allow easier H-mode access and high H-mode confinement.\n\nIn the HD mod
el it is assumed that heat flows to the divertor plate at a speed proporti
onal to the ion sound velocity. Here we take into account the thermal resi
stance of the SOL plasma at finite collisionality. Using the two-point mod
el\, we define the SOL energy confinement time as $\\tau_E \\equiv 3 n_u e
T_u L_{\\parallel}/q_{\\parallel}$. If we assume\, as in the HD model\, t
hat the power scrape-off width scales as the drift velocity multiplied by
the confinement time\, we can derive an implicit formula for the dimension
less enhancement factor\, $\\lambda_q^{\\prime}$\, over the collisionless
result $\\lambda_q \\approx 1.6(a/R)\\rho_{pol}$ (4). We include a factor\
, $f_{power}$\, the fraction of the upstream heat flux that is radiated ne
ar the divertor target\, and also take into account the scale lengths of $
p$ and $T$ in a radially finite flux tube that is wider than either one. T
his gives\n\n$\\lambda_q^{\\prime}\\left[1-\\left(\\frac{1-f_{power}}{\\la
mbda_q^{\\prime}}\\right)^7\\right] = 7.39 \\cdot 10^{-2}\\left(\\frac{7/3
}{1+2\\lambda_T/3\\lambda_n}\\right)\\left(\\frac{1+\\overline{Z}}{\\overl
ine{A}}\\right)^{1/2}f(Z_{eff})\\nu^*_{SOL\,Z=1}$\n\nwhere $f(Z_{eff}) = 0
.672 + 0.076Z_{eff}^{1/2}+0.252Z_{eff}$ (5) and $\\nu^*_{SOL\,Z=1} = 10^{-
16}n_uL_{\\parallel}T_u^2$ evaluated at the separatrix. The solution of th
is equation is shown in Figure [1]. Low gas puff experimental results resi
de in the region along the $x$ axis from about 5 to 15\, explaining their
weak dependence on collisionality\, while high gas puff results inhabit th
e range roughly from 30 to 40.![Enhancement factor over HD model SOL width
\, $\\lambda_q \\approx 1.6(a/R)\\rho_{pol}$\, vs. collisionality.][1]\n\n
\nThe HD model reflects the low collisionality limit of the GHD model\, bu
t expressed in terms of operational parameters such as $P_{SOL}$ and $R$ t
he HD model employs the upstream temperature derived on the basis of colli
sional thermal conduction. This gives rise to its weak\, but finite\, $P_{
SOL}^{1/8}$ dependence. It is reasonable therefore to ask over what range
in collisionality this mixed approach can be justified. Applying the two-p
oint model just to a narrow flux tube near the separatrix\, we can solve f
or the enhancement to the separatrix poloidal gyro-radius due to the sheat
h thermal resistance\, $\\rho^{\\prime}$\, a factor that is neglected in t
he HD model. As shown in figure [2]\, this is a small effect in the region
of low gas puff data\, justifying the use of the mixed HD model for these
conditions. Indeed\, the small variation in $\\rho^{\\prime}$ in figure [
2] over the range of low gas puff data serves to counteract the small rise
of $\\lambda_q^{\\prime}$ vs. collisionality in this range in figure [1].
![Enhancement factor for $\\rho_{pol}$ due to sheath thermal resistance vs
. collisionality.][2]\n\nThe collisional broadening of the SOL relative to
the HD model shown in figure [1] is not likely to benefit ITER either by
spreading the heat flux or allowing a larger radiating volume\, since ITER
is projected to operate at low collisionality. Furthermore\, experimental
ly operation at higher collisionality correlates with reduction in confine
ment (3). To take advantage of the effect shown in figure [2] is also prob
lematic\, since large values of $\\rho^{\\prime}$ imply that the divertor
target temperature approaches the upstream temperature\, which would resul
t in unacceptable sputtering of tungsten.\n\nThe confinement reduction obs
erved experimentally at high collisionality may be due\, at least in part\
, to reduction in flow shear near the separatrix. On the core side of the
separatrix\, the drift is approximately given by $v_{E\\times B} = -T_i/(e
B\\lambda_{p_i})$ (6)\, where the minus sign indicates flow in the directi
on opposite to the ion diamagnetic drift. On the SOL side of the separatri
x\, in the absence of parallel currents\, the $E \\times B$ drift is appro
ximately given by $v_{E\\times B} = (0.71T_u + 2.29 T_t)/(EB\\lambda_{T_e}
)$. This implies an arbitrarily high shearing rate just at the separatrix\
, mitigated only by cross-field viscosity. Thus the combination of high he
ating power\, leading to high separatrix ion and electron temperatures\, i
n combination with a collisionless edge\, implying a narrow SOL at high $f
_{power}$ and low collisionality (figure [1])\, should be conducive to tur
bulence suppression by sheared flow near the separatrix. This could ease t
ransition to the H-mode and provide high H-mode confinement properties\, p
otentially favorable implications for ITER’s low collisionality\, high d
ensity operating point.\n\nFinally\, we note that the anomalous radial the
rmal diffusivity required to produce a SOL width greater than that of the
HD model scales as gyro-Bohm (7)\, so projections indicating that the HD m
odel gives significantly too narrow a SOL for ITER require transport near
the separatrix to scale less favorably than implied by the observed H-mode
gyro-Bohm confinement trend.\n\n$\\textit{This work supported by US DOE C
ontract No. DE-AC01-09CH11466}$\n\n(1) R.J. Goldston\, Nucl. Fusion **52**
0130088 (2012)\n(2) H.J. Sun et al.\, Plasma Phys. Control. Fusion **57**
125011 (2015)\n(3) T. Eich et al.\,2020 in press\, Nuclear Fusion\, https
://doi.org/10.1088/1741-4326/ab7a66\, and R.J. Goldston et al.\, EPS-DPP M
ilan\, 2019\n(4) R.J. Goldston\, Journal of Nuclear Materials **463** 397
(2015)\n(5) R.J. Goldston\, M.L. Reinke\, and J.A. Schwartz\, Plasma Phys.
Control. Fusion **59** (2017) 055015\n(6) E. Viezzer\, Ph.D. Thesis\, 201
2\n(7) X.Q. Xu et al\, Nucl. Fusion 59 (2019) 126039\n\n [1]: https://nst
x.pppl.gov/DragNDrop/Scientific_Conferences/IAEA/IAEA_2020/Synopses/Figure
s/Goldston_IAEA2020_Figure1.jpg\n [2]: https://nstx.pppl.gov/DragNDrop/Sc
ientific_Conferences/IAEA/IAEA_2020/Synopses/Figures/Goldston_IAEA2020_Fig
ure2.jpg\n\nhttps://conferences.iaea.org/event/214/contributions/17382/
LOCATION:Virtual Event
URL:https://conferences.iaea.org/event/214/contributions/17382/
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