Effect of Scrape-Off-Layer Current on Reconstructed Tokamak Equilibrium

3.9 MB·

by  J. R. King

Effect of Scrape-Off-Layer Current on Reconstructed Tokamak Equilibrium
J. R. King,1 S. E. Kruger,1 R. J. Groebner,2 J. D. Hanson,3 J. D. Hebert,3 E. D. Held,4 and J. R. Jepson4
1Tech-X Corporation, 5621 Arapahoe Ave. Boulder, CO 80303, USA
2General Atomics, PO Box 85608, San Diego, CA 92186–5608, USA
3Auburn University, 206 Allison Labs, Auburn, AL 36849, USA
4Department of Physics, Utah State University,
4415 Old Main Hill, Logan, UT 84322, USA
(Dated: January 3rd, 2017)
7 Methodsaredescribedthatextendfieldsfromreconstructedequilibriatoincludescrape-off-layer
1 current through extrapolated parametrized and experimental fits. The extrapolation includes both
0
the effects of the toroidal-field and pressure gradients which produce scrape-off-layer current after
2
recomputation of the Grad-Shafranov solution. To quantify the degree that inclusion of scrape-

n off-layer current modifies the equilibrium, the χ-squared goodness-of-fit parameter is calculated for
a caseswithandwithoutscrape-off-layercurrent. Thechangeinχ-squaredisfoundtobeminorwhen
J scrape-off-layer current is included however flux surfaces are shifted by up to 3cm. The impact on

1 edge modes of these scrape-off-layer modifications is also found to be small and the importance of
3 thesemethodstononlinearcomputationisdiscussed. Publishedversion: Phys. Plasmas24,012504
(2017) [http://dx.doi.org/10.1063/1.4972822]

]
h
PACSnumbers: 52.30.-q,52.35.-g,52.55.Tn,52.40.Hf,02.60.Lj,52.35.Vd
p
Keywords: Grad-Shafranovequilibrium,scrape-off-layer,macroscopicstability,peeling-balooningmodes
-
m
s I. INTRODUCTION orders of magnitude below the equilibrium stored
a
magnetic energy [3], and ion-temperature-gradient-
l
p driven turbulence produces density fluctuations less
. Understanding of tokamak plasmas has greatly
s thanonepercentoftheequilibriumdensity[4]. The
c benefited from the separation of time scales inher- small-fluctuation hierarchy coupled with slow tem-
i ent in strongly magnetized plasmas. All orderings
s poral evolution permits the common approach of
y of strongly-magnetized plasma equations show that extended-MHD modeling about a Grad-Shafranov
h the magnetic pressure and line-bending contribu-
equilibrium (e.g. Ref. [5]), as opposed to model-
p tions from J×B and the force-density from ∇p are
ingthesymmetricfieldswiththefullextended-MHD
[ the largest terms in the center-of-mass momentum
equations(e.g. Ref.[6]). Theeffectsencompassedby
equation. Assuming symmetry and nested flux sur-
1 extended-MHD are variable: extended-MHD refers
v faces allows one to derive the Grad-Shafranov equa-
to models beyond resistive MHD that include some
6 tion [1] that describes the lowest-order steady-state
combination of anisotropic thermal conduction and
7 fields. This solution is conventionally described as
stresses [7], two-fluid evolution [8], finite-Larmor-
0 the plasma equilibrium. Perturbations that drive
0 radiusclosures[9,10],and/oradvanceddrift-kinetic-
the plasma away from equilibrium launch Alfvén
0 equation closures [11].
wavesthatquicklyacttorestoretheplasmatoequi-
.
2 librium [2]. Transport occurs on slower time scales,
One challenge to simulating small fluctuations
0
while symmetry-breaking macroscopic instabilities
7 aboutanequilibriumstateisthaterrorsintheequi-
occuronintermediatetimescalesbetweenthetrans-
1 libriumcanbeontheorderoftheperturbationmag-
port and the stiff Alfvén time scale.
: nitude. ThisissueisrecognizedbyGrimmetal.[12]
v
This paradigm well-describes the experimen- with the first ‘mapping code’, a code that employs
i
X tal phenomenology, and solutions to the Grad- numerical maps to transfer fields from the spatial
r Shafranov equation are routinely calculated hun- discretization of an equilibrium code onto the spa-
a
dreds of times per plasma discharge to provide a tialdiscretizationofanothercode, wherethesecond
measureofthelocationofmagnetic-fluxsurfaces. As code typically assesses stability or the evolution of
such,thesesolutionsareindispensabletocontrolling nonlinear perturbations. Despite considerable effort
the plasma and interpreting diagnostic data. Thus to increase the accuracy of mapping codes, map-
Grad-Shafranov theory may be considered as one of ping errors are difficult, if not impossible, to com-
the most successful applications of tokamak-plasma pletelyeliminate. Inpractice,extended-MHDcodes,
theory. Measurementofboththemacro-andmicro- such as nimrod [7, 8] and m3d-c1 [13, 14], recom-
scopic perturbations confirms the small-fluctuation pute the Grad-Shafranov equilibrium with their na-
assumptionembeddedinthetheory: tearingmodes, tive spatial discretizations to circumvent these er-
for example, have perturbed magnetic energies four rors. Burke et al. [15] discusses an example of the
2
Figure 1: [color online] Toroidal current on the up-
per half of the outboard midplance from equilibria that
are mapped (left) and that use a recomputed Grad- Figure 2: [color online] Diagram of topological regions
Shafranov solution (right). From a reconstruction of of a diverted, lower-single-null shot within the DIII-D
DIII-D shot 145098 at 1800 ms. tokamak.
impactofmappingerrors. Inthatwork,thenimrod riumwithexclusivelyclosedandnested-fluxsurfaces
extended-MHDcodeisbenchmarkedwiththelinear- without diverted magnetic topology. A diagram of
MHD codes gato [16] and elite [17] on peeling- thetopologyassociatedwithcasesconsideredinthis
ballooning modes in tokamak equilibria. gato, like work, a diverted lower-single-null shot within the
other global linear-MHD codes, uses mapped equi- DIII-Dtokamak,isshowninFig.2alongwithlabels
libria while elite uses Miller equilibria [18] that of the terminology used for the topological regions
effectively act to provide the same accuracy as re- andcontours. Operationwithadivertorimpliesthe
solving the Grad-Shafranov equation. In nimrod presence of a separatrix field line that separates the
calculations with mapped equilibria, better agree- open- and closed-flux regions. The surface imme-
ment with gato is obtained, whereas when the diately inside the separatrix is the last-closed-flux
nimeq code [19] is used to recompute the Grad- surface(LCFS).TheseparatrixandLCFSareindis-
Shafranovequationforthenimrodinitialcondition, tinguishable from a practical standpoint, aside from
better agreement with elite is obtained. the separatrix field lines that extend to the divertor
While this example demonstrates the importance strike point. The open field lines between the LCFS
of high-accuracy Grad-Shafranov solutions that are and the first wall define the halo region [20, 21],
recomputedwithinacode’snativespatialdiscretiza- and in the modeling of Burke et al. this region was
tion, a further example is provided with Fig. 1. The approximated as a closed-field-line region. Tradi-
figure shows a comparison of the toroidal current tionally in linear-MHD modeling, this halo region is
density from both a mapped equilibrium along with treated as a vacuum; i.e., it has neither current nor
a case where the Grad-Shafranov solution is recom- plasma density. In extended-MHD modeling, the
puted with nimeq. This high-resolution computa- term halo region is used instead to denote that it is
tion is performed with a 72 × 512 finite-element a cold-plasma region, capable of containing current,
grid with bi-quartic elements. The bumps in the surrounding the hot, closed-field-line region.
mapped solution roughly correspond to the resolu- The goal of this work is to relax the vacuum con-
tionofthe129×129gridusedduringthereconstruc- straint of the halo region while remaining consis-
tion. Computationsofedge-modegrowthrateswith tent with the measurements that are used to pro-
the mapped equilibria give markedly different re- duce the reconstruction. For model fidelity to ex-
sults relative to using the solution recomputed with periments, it is important to carefully consider the
nimeq. vacuumassumptioninthehaloregionrelativetothe
The work of Burke et al. considered the sim- dynamics of study. Edge modeling is likely sensitive
ple case of an artificial peeling-ballooning equilib- to current and flows present in the scrape-off-layer
3
(SOL) region [22], a subregion of the halo that di- used only magnetic diagnostics external to the first
rects the energy and particle exhaust from the hot wall as a constraint [23, 24]. Later, the motional
plasma onto the divertor. Most published simula- Stark effect diagnostic [25, 26] greatly improved the
tions to date are initialized from equilibrium that accuracy of the reconstructions by providing inter-
do not have current or flow in the halo region. This nalmeasurementsofthemagneticandelectricfields.
is largely because this current is typically not in- Morerecently,measurementsofthedensityandtem-
cluded in reconstructed equilibria from the highly perature profiles via Thompson scattering [27, 28]
successful, workhorse efit code [23, 24]. Without and charge exchange recombination (CER) [29, 30]
SOL current, reconstructions must have one of two spectroscopyprovidefurtherconstraintsonthepres-
undesirableproperties: 1)eitherthereisanartificial sure profiles [31].
constraintonthecurrentwhichmustsmoothlyvan- To quantify a reconstruction’s accuracy, results
ishattheseparatrixor2)thereisacurrentdisconti- from efit Grad-Shafranov solves are applied to a
nuity at the separatrix. The former constraint leads χ-squared test against the experimental measure-
to the incorrect edge profiles, whereas the latter im- ments. In other words, the goal is to find a Grad-
pacts convergence – particularly for high-order spa- Shafranov solution that minimizes
tial discretizations such as those employed by nim-
rod and m3d-c1. Importantly, cases with either χ2 =X(cid:18)Mi−Ci(cid:19)2 (1)
constraint fail to include the measured profiles out- σi
i
side the LCFS.
whereM isthemeasuredsignal,C isthecomputed
In this paper, we discuss our method for adding i i
signal, and σ is the measurement uncertainty. In
SOL current to equilibrium reconstructions gener- i
ated by efit. One notable case that uses this this paper, we perform this χ-squared calculation
to quantify the change that arises from the numer-
methodistheDIII-DQH-modemodelinginRef.[5].
ical errors associated with mapping, our recompu-
Thedifficultieswiththecurrentandflowdiscontinu-
tation of the Grad-Shafranov solution, and the ad-
ities and a desire for more accurate modeling of the
dition of the SOL pressure profiles and associated
QH-mode instabilities motivated the development
current during the recomputation.
of our methods that include SOL profile gradients.
This paper begins by reviewing details of the efit Typically, efit reconstructions do not include
SOL current and treat the region between the sepa-
reconstructions, and discussing important metrics
ratrix and wall, i.e. the halo region, as current free.
fordeterminingthequalityofthereconstructionsin
Thisimpliesthatiftheplasmadischargehasafinite
Sec.II.Themethodsforrecomputingequilibriawith
currentontheseparatrix,whichisoftenthecasefor
separatrices along with the details on how we add
H-modedischargesthatcontainlargebootstrapand
SOLcurrenttothesenewequilibriaaredescribedin
Pfirsch-Schlüter currents, there will be a disconti-
Sec. III. Example cases that use these methods are
nuity at the LCFS. Equivalently within the context
introducedinSec.IV.Wethencomparethenewand
of the Grad-Shafranov equation, the efit profiles
original equilibria using synthetic diagnostics that
are constrained such that the pressure and toroidal
aresimilartothoseusedtoconstraintheoriginalre-
magneticfieldareconstantinthehaloregion. With
constructedequilibriuminSec.V.Finally,weexam-
finitegradientsattheLCFS,thisleadstodiscontinu-
inetheminormodificationtothelineargrowthrates
ities in the first derivatives of these profiles. These
ofpeeling-ballooningmodes(PBMs)bytheaddition
inconsistencies lead to subtle but significant issues
of SOL current in Sec. VI before making concluding
when evolving tokamak-edge unstable cases initial-
remarks in Sec. VII.
ized from efit reconstructions. For example, in
codes that retain the magneto-sonic wave physics
both the SOL current and profile gradients must be
II. OVERVIEW OF EFIT EQUILIBRIUM includedsuchthattheequilibriumisconsistentwith
RECONSTRUCTIONS forcebalance. Furthermoreduringnonlinearcompu-
tations, any inconsistencies to the Grad-Shafranov
Determination of the experimental configuration equation, including discretization errors from dis-
of tokamak plasmas has become essential for under- continuousfields,canlaunchspuriousmagneto-sonic
standing and optimizing stability and confinement waves.
in fusion research devices. Reconstruction of the It must be noted that efit has the ability to in-
experimental equilibrium from a combined set of cludeforce-free,poloidalcurrentintheSOLthrough
magnetic,temperatureanddensitymeasurementsis finitegradientsinthetoroidalmagneticflux[32,33].
computedbyminimizingtheerrorbetweenmodeled However,thiscapabilityisrarelyexercisedandthere
and observed signals. This technique, now routine, is no published work on the inclusion of finite pres-
was pioneered with the efit code which originally sure gradients in the SOL.
4
III. ALGORITHMS: BOUNDING contour. The first method uses bisection of the do-
CONTOURS AND SOL FITS main to determine where the field lines transition
from closed to open within a specified tolerance.
Initializing a nimrod computation from a recon- For the parallel implementation, it becomes an N-
section method where each core is assigned a seed
struction is a two-step process that involves map-
point between the known closed- and open-field-line
ping from the reconstructed equilibrium, and then
locations. The second method is to use the oculus
recomputation of the equilibrium. The mapping
code fluxgrid creates both a partially flux-aligned code to find the saddle point in ψ that is associated
withtheX-point,andafield-lineseededwithavan-
finite-element mesh and maps the reconstructed-
ishingly small offset towards the O-point is used as
equilibrium fields onto this mesh. This mapped
the LCFS in order to avoid stagnation of the field-
solution is refined through solution of the Grad-
Shafranov equation with the nimeq code [19]. The line tracing near the X-point. The latter method
has the advantage of being somewhat more robust
solution can then be iteratively passed between
fluxgrid and nimeq for further grid refinement forhigh-resolutioncases,butthedisadvantageofnot
beingamenabletoastraight-forwardparallelimple-
and Grad-Shafranov solves, as needed. The Grad-
mentation.
Shafranov solve is formulated as a boundary-value
problem where the boundary condition is specified Asdiscussedearlier,efitsolutionshavezeropro-
by the value of ψ from the mapped reconstruc- filegradientsoutsidetheLCFS.Acubic-splinefitof
tion. This boundary condition constrains both the this data is used to evaluate these fields within the
fields from the external coils and those generated LCFS. Equilibrium generated using these spline fits
by the internal plasma that are also determined by withconstantprofilevaluesoutsidetheLCFSclosely
the pressure and toroidal-flux profiles. With this matchthesolutiongivenby efitwhilelargelyelim-
method, the recomputed fields closely resemble the inating mapping errors. One goal of this work is
reconstructed versions; however, mapping errors are tocontrastthisrecomputedsolutionwithasolution
largelyeliminatedandtheGrad-Shafranovequation that contains SOL-profile gradients.
is satisfied up to an input tolerance. SOL-profile gradients and associated currents are
Relative to the methods described in Ref. [19], includedbydefiningboundingcontoursandnormal-
we employ extensions that identify the open- and izing the flux as an extension to the methods used
closed-flux regions of the domain in order to ap- to determine the open- and closed-flux regions. The
ply the appropriate fitted form of the pressure and SOL region is defined as the region between the
toroidal flux profiles. Within the closed-flux re- LCFS contour and contours with ψn = ψsol and
gions, these profiles are specified by efit recon- ψn = ψpf where ψsol (> 1) defines a contour(s)
structions as a function of normalized flux (ψn = at the edge of the SOL region and ψpf (< 1) de-
(ψ − ψ )/(ψ − ψ ) which is zero at the O-point fines a contour(s) in the private-flux region(s) (see
o x o
where ψ = ψ and unity on the separatrix where Fig. 2). The algorithm that determines these con-
o
ψ =ψ ). Withtheexceptionofprivate-fluxregions, toursisabletohandleanarbitrarynumberofinter-
x
regionswithψ <1containclosedfieldlinesandre- sectionsofthisregionandthecomputationalbound-
n
gionswithψ >1containopen-fieldlines. Applying ary of the domain and thus diverted topology such
n
these profiles to the domain requires identification as double null configurations is tractable. The al-
of the closed-flux region. In practice, we find the gorithmworksasfollows: TheinitialSOLbounding
bounding LCFS contour and use a simple algorithm regionisassumedtobethecomputationalboundary.
that counts the number of crossings of a line that The algorithm checks the value of ψn at each finite-
extends from a finite-element node to the boundary element node location outside the LCFS but con-
to determine if each node is enclosed by the con- tained within this ‘working’ SOL region. If the con-
tour. The separatrix contour and location of the dition ψpf <ψn <ψsol is not satisfied, the location
extrema of ψ in the core are allowed to change and betweenthewallandtheO-pointwhereψn =ψsolor
mustberecomputedaftereachiterationoftheGrad- ψn =ψpf is identified, and a new contour is traced.
Shafranov solve. As the X-point on the separatrix Absent integration error, the new contour is termi-
contour associated with diverted magnetic topology nated at the boundary. The section of the domain
consists of a stagnation point for field-line tracing, that does not encircle the O-point is removed from
weinsteadelecttofindacontourvanishinglycloseto the ‘working’ SOL region until only nodes with the
the separatrix which numerically approximates the property ψpf < ψn < ψsol remain. This new SOL
LCFS. For ourpurposes, we onlyneed tobound the contour,inadditiontotheLCFScontour,boundthe
finite-element nodes that are within the separatrix. SOL region.
We employ two methods that use field-line tracing The appropriate pressure and toroidal magnetic
of the polodial field to find this approximate LCFS flux profiles are defined within the SOL region via
5
two methods: either by fits from the experimental
data (if available) or through modified bump func-
tionfits. Twofitsareperformedforeachfield: onefit
for the SOL region immediately outside the separa-
trix with ψ >ψ , and one for the private-flux SOL
n x
region with ψ < ψ . The modified bump function
n x
uses the form
" #
−∆2
f(ψ )=f Exp +f . (2)
n 0 ψ2 −ψ2 c
n,sol n
This function has vanishing derivatives of all orders
at its endpoint and thus ensures that the current
goes smoothly to zero at the transition contour be-
tween the SOL region and the current-free region.
Figure 3: [color online] Fitted profiles (lines) from the
With this form, five free parameters are available
Thompson(electrondensityandtemperature)andCER
(ψ ,ψ ,f ,f and∆). Thevaluesofψ andψ
sol pf 0 c sol pf (ion temperature) data from shot 160414 at 3025 ms.
are inputs that influence the width of the SOL and
ThesolidlinesarefitstorawdatainsidetheLCFSwhere
may be inferred from experimental measurements.
total pressure is computed from the measured species’
The other three parameters may be determined ei- dataandincludescontributionsfromenergeticparticles.
ther by requiring C2 continuity at the LCFS or by ThedashedlinesarethefitsthatincludetheSOLregion
settingthefunctionalvalueatψ andenforcingC1 anddashediontemperatureiscomputedafterconstraint
sol
continuity at the LCFS. The former constraint has by a two temperature model and quasineutrality.
theadvantageofproducingaC1smoothcurrentpro-
file, whereas the latter method allows specification
of the density and/or temperature in the current-
freeregions. Ifthefitrequiresthatthefunctionfirst
n = 8.6×1018 m−3 and in the current-free region
reverse the sign of its derivative, a truncated Gaus- e
T = 20 eV, T = 50 eV and n = 2.1×1018 m−3
sianisfittohalfofthedomainfollowedbythebump e i e
wherethelattervaluesonlyapplytoequilibriawith
function as is used in Ref. [34].
SOL-profile gradients.
Thesecondcasestudiedisan efitreconstruction
IV. EXAMPLE CASES ofDIII-Dshot145098at1800msasshowninFig.4.
Again, the profiles shown are as included from the
efit reconstruction and for fits with SOL-profile
In order to demonstrate and quantify the impact
gradients(nimeq-SOL)alongwiththeexperimental
of our methods that recompute the Grad-Shafranov
ThomsonandCERmeasurements. Thisshotisfrom
solution and add SOL-profile gradients, we choose
aDIII-DQH-modeexperimentwithITER-likeshap-
two specific cases to study in detail: a low SOL-
ing during a period with edge harmonic oscillations
current case and a high SOL-current case. The first
(low-n perturbations). This shot contains a rela-
of these cases is an efit reconstruction of DIII-D φ
tivelylargepressuregradientfromthefittedprofiles
shot 160414 at 3025 ms with profiles as shown in
in the SOL region. Thus relative to the reconstruc-
Fig. 3. Profiles are specified as function of normal-
tion from shot 160414, we expect greater modifica-
ized flux. The profiles shown are as included from
tions to the equilibria resulting from the inclusion
theefitreconstructionandforfitswithSOL-profile
of the SOL-profile gradients in the Grad-Shafranov
gradients (nimeq-SOL) along with the experimen-
solve. In particular, at the LCFS T = 230 eV,
tal Thomson and CER measurements [42]. This e
T = 1020 eV and n = 4.8 × 1018 m−3 and in
shot is from an experiment of lithium pellet injec- i e
the current-free region T = 50 eV, T = 50 eV and
tion for edge-localized-mode pacing and the specific e i
n = 2.7×1018 m−3 where the latter values only
time represents the last 20% of the inter-ELM pe- e
apply to equilibria with SOL-profile gradients.
riod. The reconstruction contains a relatively cold
plasma at the LCFS, and thus a small pressure gra- As is clear from Figs. 3 and 4, the nimeq-
dient in the fitted profiles within the SOL region. SOLion-temperatureprofilesdonotmatchtheCER
This implies modest SOL current and modification measured data. This is a consequence of a two-
totheresultingequilibriawhentheSOLprofilesare temperature model which over-constrains the pro-
included in the Grad-Shafranov solve. In particu- files. Specifically, with a two-temperature model as-
lar, at the LCFS T = 72 eV, T = 470 eV and suming quasi-neutrality (n = Z n ) only the elec-
e i e i i
6
tributions from the other species:
imp hot
T2T =T +XnjZiT +XnhZiT . (5)
i i n j n h
e e
j h
As expected from this relation and as shown in the
plots, the nimeq-SOL ion temperature (equivalent
to T2T in Eqn. (5)) over estimates the measured
i
ion temperature. For these cases, Z is chosen such
i
thattheedgeion-temperatureprofileapproximately
matches the measured data within Figs. 3 and 4
(Z = 1.25) [43]. Thus computations in this work
i
usethenimeq-SOLfitsforψ <1andonlythecom-
n
putationswithSOL-profilegradientsusethenimeq-
SOL fits where ψ >1. Modeling that includes sep-
Figure 4: [color online] Fitted profiles (lines) from the n
arate species for hot particles and/or impurities is
Thompson(electrondensityandtemperature)andCER
required to eliminate the discrepancy with the mea-
(ion temperature) data from shot 145098 at 1800 ms.
ThesolidlinesarefitstorawdatainsidetheLCFSwhere surementsintheion-temperatureprofile,andisthus
total pressure is computed from the measured species’ planned for future study.
dataandincludescontributionsfromenergeticparticles. To further examine the reduction of mapping
ThedashedlinesarethefitsthatincludetheSOLregion errors through the recomputation of the Grad-
anddashediontemperatureiscomputedafterconstraint Shafranov solution, Figs. 5 and 6 show the resulting
by a two temperature model and quasineutrality. current-density distributions from the mapped and
recomputed (GS) cases without SOL-profile gradi-
ents, along with two cases with SOL-profile gradi-
ents that are discussed later in this section, based
tron and ion fluids contribute to the pressure:
on reconstructions from shots 160414 and 145098,
respectively. The DIII-D first wall, LCFS and SOL
n
p2T =pe+pi =neTe+ ZeTi2T . (3) contours are superimposed into the plotted com-
i putational domains. Shot 145098 uses a reversed
plasma current relative to 160414, which uses the
Thusonlythreeofthefourprofilesshowninthefig-
standardcurrentorientationforDIII-D.Thesecases
ures(n ,p,T ,andT )canbematchedexactly. Ini-
e e i
use a 72 × 64 high-order finite-element mesh with
tializing computations with the measured pressure
bi-quartic elements. For both reconstructions, the
profile is essential as both the Grad-Shafranov solu-
mapped current is clearly distorted and contains
tion and ideal stability are critically dependent on
numerical oscillations. This is particularly evident
this profile. Secondary to this, the electron temper-
for the edge current shown in the zoomed insets of
ature profile determines the resistivity profile and
the figures and outside the LCFS as shown in the
the density profile sets Alfvén speed (among other
inset figures that magnify the details near the X-
collisionality parameters).
point. The numerical bumps in the current pro-
Within the context of an extended-MHD simula-
files inside the LCFS are a result of the mapping
tion, the ion-temperature profile typically becomes
and are associated with the resolution of the efit
significant only when a two-fluid model is evolved
grid. They are not eliminated but rather only re-
and thus is often allowed to vary with respect to
solved with enhanced nimeq resolution (see Fig. 1.
experimental measurements. In experiment, the
Whileitispossibletopartiallycircumventthisissue
pressurehascontributionsfrombothimpuritiesand
withtheclosed-fluxmappedcurrentbyusingahigh-
non-Maxwellian,or‘hot’,ionparticlesfromneutral-
resolution efit (see, for example, Ref. [35, 36]), in
beam injection:
practice most efit reconstructions are generated at
relatively low resolution relative to what is required
p=p +p +p +p =n T +n T
e i imp hot e e i i for extended-MHD computations. The spatial re-
imp hot quirements to solve for Grad-Shafranov equilibria
X X
+ n T + n T . (4) are less stringent than those to solve for 3D MHD
j j h h
j h perturbations where, for example, the edge compu-
tationspresentedinSec.VIuseafiniteelementgrid
Comparing Eqns. (3) and (4), the ion temperature with 72×512 with high-order bi-quintic elements.
used to initialize our simulations includes the con- The current outside the LCFS in the mapped case
7
Figure5: [coloronline]Toroidalcurrentdensityfromshot160414at3025msplottedwithLCFS,SOL(ifapplicable)
and DIII-D limiter contours for four different cases: a mapped solution (using a finite-element computation for B
and J), resolving the Grad-Shafranov equation (GS), and resolving the Grad-Shafranov equation with two different
treatments of the SOL (labeled GS-SOL and GS-SOLpf) as described in the text. The zoomed plots of the divertor
region use a 10× smaller contour color scale to show current features.
results from the representation of the discontinuous respectively. This results in SOL widths that are
current profile on nimeq’s C0 finite-element spatial roughly consistent with the measured half width
discretization. ThemappedplotsinFigs.5and6are of the heat-flux during the later half of the inter-
generatedwithfinite-elementcalculationsthatcom- ELM period of DIII-D ELMy H-mode discharges in
pute the poloidal magnetic field and current from Ref. [37].
themappedψandRBΦ fields. Alternatively,splines Currently, we do not have diagnostic information
may be used to map the poloidal magnetic field about appropriate profiles for the private-flux re-
and current density. While this method produces gion. Two methods employing the bump function
smooth, but not consistent, fields, the derivatives of extrapolations are compared with profiles shown in
thesefieldswithnimeq’sC0 finite-elementrepresen- Fig7. Inthefirstmethod, C2 continuityisenforced
tationareusedinextended-MHDcalculations. Thus at ψ = 1 resulting in a high-pressure, high-density
n
mappedmagneticfieldsandcurrentdensitieswitha private-flux region. With the second method, C1
spline spatial representation in effect hide, but do continuity is enforced at ψ = 1 and the values at
n
not eliminate, the mapping errors. ψ are set to the same as those at ψ . In the
pf sol
For the presented cases with SOL current, we use figures and tables, these cases are referred to as GS-
ψ =1.1 and ψ =0.96 in the fits to the elec- SOL and GS-SOLpf, respectively. The profiles gen-
n,sol n,pf
tron density and temperature profiles from Thom- eratedwiththesecondmethod(GS-SOLpf)roughly
sonscatteringmeasurementsandextrapolatetheion match the measured heat-flux profile [34, 37].
temperaturetoanassumed50eV. Theresultingpro- In addition to the mapped and recomputed cases
files are shown as the dashed purple line in Figs. 3 without SOL current, Figs. 5 and 6 plot cases with
and 4. The half width of the electron pressure pro- SOL currents. With low pressure in the private-
filesareroughly3.3mmand2.7mmattheoutboard flux region the toroidal current density reverses lo-
mid-plane and 7.3 cm and 4.7 cm at the divertor cally near the divertor strike points after the Grad-
plate for the cases from shots 160414 and 145098, Shafranov solve as shown in the inset figure for the
8
Figure6: [coloronline]Toroidalcurrentdensityfromshot145098at1800msplottedwithLCFS,SOL(ifapplicable)
and DIII-D limiter contours for four different cases: a mapped solution (using a finite-element computation for B
and J), resolving the Grad-Shafranov equation (GS), and resolving the Grad-Shafranov equation with two different
treatments of the SOL (labeled GS-SOL and GS-SOLpf) as described in the text. The zoomed plots of the divertor
region use a 10× smaller contour color scale to show current features.
GS-SOLpf case. Currents near and outside the lim- fitthatenforcesC2 continuityatψ =1determines
n
iter are likely an artifact of our method. In this the toroidal-magnetic-flux profile in the SOL. The
cold-plasma region, additional terms in the momen- resultingpoloidalcurrentshaveamaximumvalueon
tum equation become large (e.g. interactions with thedivertorplateof4000and500A/m2forthecases
neutralsandthefirstwall)thatareoutsidethescope from shots 160414 and 145098, respectively. This
of the Grad-Shafranov equation. As such, the ap- current must be less than the ion saturation cur-
proximation that quatities are functions of the flux rent (J = n ec ). With a Deuteron ion species,
max e s
surfacesbreaksdown. However,atpresentourfocus a conservative calculation of the ion saturation cur-
is on the effects of the SOL-profile gradients near rent (using the vacuum temperatures and densities
the LCFS where there is the potential for interac- fromeachcase)isover2×104 A/m2 forbothcases.
tion with perturbations that originate from inside
the separatrix. With SOL-profile gradients, there is
a modest modification (less than 1%) to the plasma
V. SYNTHETIC DIAGNOSTICS
current. For the cases shown here, we re-normalize
the total current to match the value from the efit
AnalysisbetweeneachnonlinearGrad-Shafranov-
reconstruction using the method of Ref. [38]. Cases
solve iteration provides a confirmation that the
with and without current re-normalization produce
macroscopic quantities of the equilibrium (e.g. to-
similar results, where the χ2 values discussed in
tal current, toroidal flux and internal energy) are
the Sec. V are slightly smaller for cases with re-
invariant between the mapped equilibrium and the
normalization.
new solution. In the interest of showing that our
The poloidal current in the SOL that flows into methodswhichrecomputetheGrad-Shafranovequi-
and out of the divertor plate are solely an effect librium and add SOL profiles and current only min-
of the profile gradient in the toroidal magnetic flux imally impact the relative agreement with measure-
within the SOL. For these cases, a bump-function ments, we use a more advanced comparison that
9
Table I: Values of χ2/N, where N is the number of
measurements, for each diagnostic for the reconstruc-
tion,changeinplasmascurrentrelativetotheefitvalue
(-1178772 Amps), and change in the X-point and max-
imum Z value of the LCFS for shot 160414 at 3025ms.
χ2/N mapped GS GS-SOL GS-SOLpf
Thom. T 22.3 23.4 4.80 4.15
e
Thom. n 19.4 20.5 4.07 3.33
e
CER T 6.98 6.96 6.74 6.84
i
MSE 1.49 1.49 1.46 1.47
Mag. Coils 0.61 0.63 0.82 0.70
∆s mapped GS GS+SOL GS+SOLpf
∆I/I 6.95×10−5 7.97×10−4 3.22×10−7 3.22×10−7
0
∆r (cm) N/A ref. 0.72 1.07
xpt
∆r (cm) N/A ref. 0.35 0.19
zmax
Table II: Values of χ2/N, where N is the number of
measurements, for each diagnostic for the reconstruc-
tion,changeinplasmascurrentrelativetotheefitvalue
Figure 7: [color online] Fitted pressure profiles in the
(1063788 Amps), and change in the X-point and maxi-
SOL and private-flux region with two different fits, one
mum Z value of the LCFS for shot 145098 at 1800ms.
with bump function fits in the private-flux region (SOL
case) and the second with a fit that specifies a low
χ2/N mapped GS GS-SOL GS-SOLpf
pressure at the edge of the private-flux region (SOLpf
case), for two different shots, 160414 at 3025ms (top) Thom. Te 60.9 61.7 7.77 6.99
and 145098 at 1800 ms (bottom). Thom. n 2.87 5.22 11.4 9.93
e
CER T 10.2 10.3 19.7 19.6
i
MSE 1.14 1.13 1.13 1.13
Mag. Coils 1.65 1.60 4.57 3.27
computes a χ2 value. The Python code nimnos-
tics is used to calculate a χ2 from the shots and ∆s mapped GS GS+SOL GS+SOLpf
sub-cases previously discussed. nimnostics models ∆I/I0 9.86×10−5 5.46×10−3 -3.57×10−7 -3.57×10−7
the magnetic coils, MSE and Thomson scattering ∆rxpt (cm) N/A ref. 0.86 0.55
through local evaluations of the fields where linear ∆r (cm) N/A ref. 2.82 2.12
zmax
interpolation is used between finite-element nodes.
Thus the boundary of the domain is chosen as the
approximate vacuum vessel instead of the limiter in perature)andothersincreasinginχ2(Thomsonelec-
order to encompass the magnetic coil locations for tron density, MSE and coils) when comparing rela-
comparison. tive to the mapped and GS cases. The deviation of
Summaries of the χ2 value by diagnostic and theLCFScontourisalsoasmuchas2cmontheup-
case for the reconstructions from shots 160414 and per side of the contour. This deviation is apparent
145098 are shown in Tabs. I and II, respectively. in the inset figures of Fig. 6.
In both shots, the mapped and recomputed equi- In order to examine the source of the changes in
libria without SOL-profile gradients (GS cases) are χ2 in detail, Figs. 8 and 9 show the local values of
roughtly equivalent indicating that the equilibrium χ2 from Thomson electron density, MSE and mag-
resulting from the recomputation is substantially netic coils measurements for our four different cases
similar. For the cases with modest SOL-profile gra- onshots160414and145098,respectively. Withshot
dients and currents (shot 160414), the χ2 values for 160414theaggregateχ2valuesareimprovedorcom-
agivenmeasurementeitherdecreasesandthedevia- parable for every diagnostic when comparing the
tionsoftheLCFScontouraremodestorareroughly caseswithSOLgradientstothosewithout(asseenin
the same as the mapped and GS cases. However, Tab. I). We note that χ2 values are particularly im-
casesfromtheshotwithlargeSOLcurrentandpro- proved with SOL-profile gradients for the Thomson
file gradients exhibit mixed results with some mea- measurements, and Fig. 8 shows that this is aresult
surements decreasing in χ2 (Thomson electron tem- of improved agreement in the SOL region. The re-
10
Figure 8: [color online] Values of χ2 from different local measurements (crosses are from Thomson measurements
of electron density, diamonds are from MSE measurements, and squares are from coil measurements of the poloidal
magneticfield)plottedwithacolorplotoftheψ solutionandLCFS,SOL(ifapplicable)andDIII-Dlimitercontours
from shot 160414 at 3025 ms for four different cases: a mapped solution (using a finite-element computation for B
and J), resolving the Grad-Shafranov equation (GS), and resolving the Grad-Shafranov equation with two different
treatments of the SOL (GS-SOL and GS-SOLpf) as described in the text.
Figure 9: [color online] Values of χ2 from different local measurements (crosses are from Thomson measurements
of electron density, diamonds are from MSE measurements, and squares are from coil measurements of the poloidal
magneticfield)plottedwithacolorplotoftheψ solutionandLCFS,SOL(ifapplicable)andDIII-Dlimitercontours
from shot 145098 at 1800 ms for four different cases: a mapped solution (using a finite-element computation for B
and J), resolving the Grad-Shafranov equation (GS), and resolving the Grad-Shafranov equation with two different
treatments of the SOL (GS-SOL and GS-SOLpf) as described in the text.