Comparing linear ion-temperature-gradient-driven mode stability of the National Compact Stellarator Experiment and a shaped tokamak
aa r X i v : . [ phy s i c s . p l a s m - ph ] M a r Comparing linear ion-temperature-gradient-driven mode stability of the National Compact StellaratorExperiment and a shaped tokamak
J. A. Baumgaertel, G. W. Hammett, and D. R. Mikkelsen Los Alamos National Laboratory, Los Alamos, New Mexico 87544 Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543 (Dated: 6 April 2019)
One metric for comparing confinement properties of different magnetic fusion energy configurations is thelinear critical gradient of drift wave modes. The critical gradient scale length determines the ratio of the coreto pedestal temperature when a plasma is limited to marginal stability in the plasma core. The gyrokineticturbulence code GS2 was used to calculate critical temperature gradients for the linear, collisionless ion tem-perature gradient (ITG) mode in the National Compact Stellarator Experiment (NCSX) and a prototypicalshaped tokamak, based on the profiles of a JET H-mode shot and thencsx stronger shaping of ARIES-AT.While a concern was that the narrow cross section of NCSX at some toroidal locations would result in steepgradients that drive instabilities more easily, it is found that other stabilizing effects of the stellarator con-figuration offset this so that the normalized critical gradients for NCSX are competitive with or even betterthan for the tokamak. For the adiabatic ITG mode, NCSX and the tokamak had similar adiabatic ITG modecritical gradients, though beyond marginal stability, NCSX had larger growth rates. However, for the kineticITG mode, NCSX had a higher critical gradient and lower growth rates until a/L T ≈ . a/L T,crit , when itsurpassed the tokamak’s. A discussion of the results presented with respect to a/L T vs R/L T is included. I. INTRODUCTION
Two of the main magnetic fusion energy designs are theaxisymmetric tokamak and the non-axisymmetric stel-larator. Tokamaks have seen significant heat loss dueto turbulence, while stellarator losses have tradition-ally been dominated by their larger neoclassical trans-port. Studying turbulent transport in stellarators is in-creasingly important, however, as modern stellarator de-signs (such as Wendelstein 7-AS (W7-AS), Wendelstein7-X (W7-X), the National Compact Stellarator Ex-periment (NCSX), the Large Helical Device (LHD) ,and the Helically Symmetric Experiment (HSX) ) haveshown or are designed to have improved neoclassicalconfinement and stability properties. Therefore, turbu-lence could be increasingly relevant in stellarator exper-iments. Several gyrokinetic studies of drift-wave-driventurbulence in stellarator geometry have been done with a variety of gyrokinetic codes, such as GS2, GENE,
GKV-X, and FULL. These codes haveall been linearly benchmarked against each other for non-axisymmetric geometries.
Progress has even beenmade in optimizing stellarator designs to have reducedturbulent transport. Besides comparing good stellarator configurations (aswas done in Refs. 19, 21, and 29, among others), onewould like to compare stellarator confinement with thatof tokamaks. The relative benefits of each device areimportant to consider when designing the next genera-tion of experiments. A few previous comparison studieshave been done, such as those in Refs. 20 and 21. Here,the gyrokinetic turbulence code GS2 is used to comparemicroinstability of the electrostatic adiabatic ion temper-ature gradient (ITG) and the electrostatic collisionlesskinetic ITG modes in the quasi-axisymmetric NationalCompact Stellarator Experiment (NCSX) design to that of a highly-elongated tokamak. Because this tokamakand NCSX geometry differs so significantly, it is hardto pinpoint what parameter has the greatest effect, butoverall effects will be examined.ITG mode-driven turbulence has been connected ex-perimentally to measured heat losses in both tokamaks(e.g. Ref. 30) and the LHD. There is much variabil-ity in stellarator designs, and it is unclear without morestudy which modes will dominate in each. Preliminarily,Ref. 31 suggests that ITG transport in NCSX may belarger than that of ETG (see Figs. 8-9 of that paper).While only the ITG mode thresholds are compared inthis paper, further study could show that other modesdominate in this case and in other devices.In Section II, a simple comparison metric is defined foruse in this paper. Sections II A-II B describe the tokamakand stellarator configurations: a Miller equilibrium forthe highly-elongated tokamak, and a numerical equilib-rium for NCSX. Next, growth rates and critical tempera-ture gradients are compared for the ITG mode in SectionsII C-II D. Finally, the study is concluded in Section III.
II. NCSX
VS.
A SHAPED TOKAMAK
To understand the trade-offs between stellarator andaxisymmetric geometry and their confinement capabil-ities, designs can be compared computationally. Onemetric of confinement quality is the ratio of the coretemperature to the pedestal temperature, T /T ped , asfusion reactors need very high core temperatures, andhigh core temperature implies good confinement. Thisratio is related to the critical temperature gradients. If − ∂T /∂r ≈ T /L
T,crit , temperature-gradient-driven insta-bilities are marginally stable–a reasonable assumption ina reactor plasma, as temperatures inside the pedestal will1e so high that profile stiffness will ensure that gradientsare close to marginal stability. This is demonstrated byFig. 3 of Ref. 32, which shows that fusion power (andthus the temperature profile) depends primarily on thepedestal temperature and not the beam power, for thecase of balanced beams.At marginal stability (assuming 1 /L T,crit is indepen-dent of minor radius), T ( r ) = T e − r/L T,crit (1)The minimum temperature is at the edge where r is max-imum, r max = a , where a is the minor radius. In this sim-plified story, another approximation will be made, that T ped occurs at r = a . So, T ( a ) = T ped = T e − a/L T,crit .Therefore, the core temperature’s dependence on the crit-ical temperature gradient for typical tokamak values of a/R ≈ / . R/L
T,crit ≈ T /T ped = e a/L T,crit = e ( a/R )( R/L
T,crit ) ≈ e (1 / . ≈ . , (2)One wants to maximize the core temperature, T . T ped is set by non-transport mechanisms and cannot be arbi-trarily high, leaving a/L T,crit as the important parameterin equation 2. If an alternative fusion device design couldincrease a/L
T,crit by just 30%, this would increase thecentral temperature by 50%, and more than double thefusion power. (A caveat–this is a simple estimate, anddoes not take into account the fact that MHD stabilitychanges with higher pressure peakedness.) In this pa-per, the critical ion temperature gradients are comparedfor NCSX and a strongly-shaped tokamak design. Thestated stellarator minor and major radii are the averagevalues.
A. Miller equilibrium for this tokamak
NCSX runs were compared to a potential high-elongation tokamak based on a composite of ARIES-AT and JET H-mode shot (in large part because this leadsto more plasma current at fixed q ), so when designingfuture tokamaks, one would like to use the highest possi-ble values of elongation and triangularity, though elonga-tion is limited by vertical stability control if it becomestoo large. Some initial studies of shaping effects withGS2 were carried out in Ref. 35, using a range of shapesscaled from the particular JET shot ). This JET shot, describedin more detail in Refs. 41 and 42, was chosen as a repre-sentative H-mode plasma that has been studied in detailby gyrokinetics codes before. Parameters for this paper were chosen from shot q profile, while the ARIES-AT design study assumed thata reversed shear scenario can be stably maintained insteady state. The composite tokamak of this paper has aconventional q profile. The Miller equilibrium for thisprototypical or generic strongly-shaped tokamak was setup in the following way.The shaping study in Ref. 35 chose to focus on theradius r/a = 0 . r/a = 0 . κ = 1 .
73 andtriangularity of δ = 0 .
46, while the core elongationis κ core ≈ .
3. At the radius of interest, r/a = 0 . R/a = 3 . κ . = 1 . κ ′ . = 0 . δ . = 0 . δ ′ . = 0 .
60, and the Shafranov shift is ∂R /∂r = − . q . = 2 .
03 and magnetic shearˆ s = 1 . q , and ˆ s , a modifiedARIES-AT case was created using its κ = 2 .
08 and δ = 0 .
76. Assuming, from the tokamak shaping studies,that κ ′ ∝ ( κ − κ core ) and δ, δ ′ ∝ δ : κ tok . = κ JETcore +( κ . ,JET − κ JETcore ) ( κ ,tok − κ JETcore )( κ ,JET − κ JETcore ) = 1 . κ tok ′ . = κ JET ′ . ( κ ,tok − κ JETcore )( κ ,JET − κ JETcore ) = 1 .
03 (4) δ tok . = δ JET . δ tok δ JET = 0 .
31 (5) δ tok ′ . = δ JET ′ . δ tok δ JET = 0 .
99 (6)Representative flux surfaces for this prototypestrongly-shaped tokamak are shown in Figure 1.
B. NCSX geometry
Refs. 18 and 44 describe how non-axisymmetric ge-ometry input is created for GS2. The coordinate system2 s /a Z s / a FIG. 1. Illustration of flux surface shapes for a prototypestrongly-shaped tokamak at r/a = 0.8 (blue solid line), 0.9(green dashed line), and 0.98 (red dash-dot line). (color on-line) Parameter Value r/a . s ≈ ( h r/a i ) . α = ζ − qθ θ q s . s . h β i . R ≈ . a N ≈ . ma N ≈ . mB a = h B i . T TABLE I. Geometry values for the NCSX equilibrium. of the flux-tube code GS2 includes the radial coordinate, ρ = √ s ( s ≈ ( r/a ) is the normalized toroidal flux), thecoordinate aligned to the field line, θ , and the angle thatselects a flux tube, α = ζ − q ( θ − θ ) (where ζ and θ areBoozer toroidal and Boozer poloidal coordinates and θ is the ballooning parameter).The GS2 documentation defines geometrical quanti-ties in terms of a parameter d Ψ N /dρ , where ρ is the radialcoordinate and Ψ N is the normalized poloidal flux. Geo-metrical quantities in this paper follow GS2 notation andinclude d Ψ N /dρ . For more information, see Refs. 18 and44.The following figures show the magnitude of the mag-netic field (Figs. 2-3), curvature drift (Figs. 4-5), and( | k ⊥ | /k θ ) (Figs. 6-7) for both the tokamak and NCSXfield lines, for the entire domain and a close-up around θ = 0. See Table I for a complete list of geometricalquantities and their values.Notice that the bad (positive) curvature regions of −20 −10 0 10 200.80.911.11.21.3 θ (rad) B no r m FIG. 2. The NCSX (blue solid line) and tokamak (greendashed line) equilibria: normalized | B | vs. θ . (color online) −2 0 20.80.911.11.21.3 θ (rad) B no r m FIG. 3. The NCSX (blue solid line) and tokamak (greendashed line) equilibria: normalized | B | vs. θ , showing a close-up around θ = 0. (color online) NCSX are much more localized than the tokamak case.Coupled with the much stronger local magnetic shear (re-sponsible for the sharp peaks in k ⊥ ∝ ˆ s , Fig. 6), this ex-plains why NCSX’s electrostatic potential eigenfunctionsare also more localized than the tokamak’s. An exampleis shown in Figure 8. These traits could predict bettertransport properties for NCSX. C. ITG mode with adiabatic electrons
For the initial study, the ITG mode with adiabaticelectrons growth rates and their dependence on temper-ature gradient were compared. Figure 9-10 show typicalgrowth rate spectra for NCSX and this tokamak. In Fig-ure 11, the growth rate at each a/L T ( a/L T e = a/L T i )3
20 −10 0 10 20−15−10−5051015 θ (rad) ω c v , no r m FIG. 4. The NCSX (blue solid line) and tokamak (greendashed line) the curvature drift frequency ( ω cv,norm =(2 a /B N )( d Ψ N /dρ )( k ⊥ /n ) · b × [ b ·∇ b ]) along θ . (color online) −2 0 2−1−0.500.51 θ (rad) ω c v , no r m FIG. 5. The NCSX (blue solid line) and tokamak(green dashed line) equilibria: the curvature drift frequency( ω cv,norm = (2 a /B N )( d Ψ N /dρ )( k ⊥ /n ) · b × [ b · ∇ b ]) along θ , showing a close-up around θ = 0. (color online) was the highest in the range k y ρ i ∈ [0 . , .
4] for NCSXand k y ρ i ∈ [0 . , .
0] for the tokamak. These ranges werewide enough to capture the peak of the growth rate spec-trum. Growth rates shown are normalized such that( γ, ω ) = ( γ physical , ω physical )( a/v thi ). The NCSX thresh-old is a/L T,crit ≈ .
26 and the tokamak’s is a/L
T,crit ≈ .
22. This difference is not very significant. However,soon after the threshold, the NCSX growth rates surpassthose of the tokamak, indicating that for a given a/L T ,the adiabatic ITG mode is more unstable in NCSX thanin the tokamak. This implies that the transport due tothe adiabatic ITG mode would be stiffer, but the temper-ature gradients would still be expected to be very similarsince they would be set by a/L T,crit . −20 −10 0 10 2005001000150020002500 θ (rad) ( k ⊥ / k θ ) FIG. 6. The NCSX (blue solid line) and tokamak (greendashed line) equilibria: (cid:16) k ⊥ k θ (cid:17) vs. θ . (color online) −2 0 2020406080100 θ (rad) ( k ⊥ / k θ ) FIG. 7. The NCSX (blue solid line) and tokamak (greendashed line) equilibria: (cid:16) k ⊥ k θ (cid:17) vs. θ , showing a close-uparound θ = 0. (color online) D. ITG mode with kinetic electrons
The threshold of the ITG mode with kinetic electrons(with a/L n = 0) for the tokamak was somewhat lowerthan that of NCSX, but the slope of the growth-ratecurve is almost the same for both (Fig. 12). Similarto Section II C, growth rates shown were the highest ona spectrum of k y ρ i ∈ [0 . , .
4] for NCSX and k y ρ i ∈ [0 . , .
0] for the tokamak (see Figure 13-14 show typicalgrowth rate spectra). With kinetic electrons, the growthrates for the ITG mode in NCSX increased over the adi-abatic electron case (Fig. 11), while the critical gradi-ent lowered to a/L
T,crit ≈ .
21. The tokamak thresholddecreased somewhat further, to a/L
T,crit ≈ .
11. Theslope of the NCSX line is somewhat steeper, and for4 θ φ / φ FIG. 8. Comparing electrostatic eigenfunctions for NCSX( Re ( φ ): red triangles and Im ( φ ): light blue solid line) andtokamak ( Re ( φ ): blue circles and Im ( φ ): green dashed line),for an adiabatic ITG mode with a/L T = 3 , a/L n = 0. ForARIES, k y ρ i = 0 .
55, and for NCSX, k y ρ i = 1 .
0. (color online) y ρ i ω /4 γ FIG. 9. NCSX growth rate spectrum for the adiabatic ITGmode with a/L T = 3 , a/L n = 0. (color online) a/L T ≈ .
82, the NCSX growth rates are larger thanthe tokamak growth rates.The growth rate vs. a/L T plot in Figure 12 shows im-provement in the a/L T threshold of NCSX over this toka-mak by about 10%. Based on the marginal stability logicin the beginning of Section II, this corresponds to about22% more fusion power for a NCSX-based design relativeto the tokamak (with the same edge temperature anddensity assumed for the two designs, and approximatingthe fusion power as scaling as T ). Effects that mightchange this result include finite beta modifications to theequilibrium and the nonlinear Dimits shift, whichcould increase each critical gradient, but the requirednonlinear simulations are beyond the scope of this work. y ρ i ω /4 γ FIG. 10. Tokamak growth rate spectrum for the adiabaticITG mode with a/L T = 3 , a/L n = 0. (color online) a/L T γ FIG. 11. NCSX (blue crosses) and ARIES-AT-like tokamak(red circles) adiabatic ITG mode growth rate dependence ontemperature gradient. Fits obtained through piecewise lin-ear interpolation on the lowest half of the growth rate curve.(color online)
A Dimits shift has been reported for a stellarator, andhas been found in tokamak simulations. This is much better than one might have initiallyguessed based on just the local value of
R/L T in NCSX vs. a tokamak. While from equation 2 it is clear that a/L T is the relevant parameter for determining the coretemperature, in the axisymmetric community, the thresh-old for the ITG instability is usually expressed in termsof R/L T , which is often the key parameter numerically.An instability threshold R/L
T,crit can be derived fromthe dispersion relation for a local ITG mode in the badcurvature region, ignoring the parallel dynamics. Inthis limit, the critical instability parameter is the ra-tio of the temperature-gradient diamagnetic drift fre-5 a/L T γ FIG. 12. Growth rates for an ITG mode with kinetic electronsas a function of temperature gradient for NCSX (blue crosses)and an ARIES-AT-like tokamak configuration (red circles).Fits obtained through piecewise linear interpolation on thelowest half of the growth rate curve. (color online) y ρ i ω /4 γ FIG. 13. NCSX growth rate spectrum for the kinetic ITGmode with a/L T = 3 , a/L n = 0. (color online) quency ( ω ∗ T ∝ /L T i ) to the curvature drift frequency( ω d ∝ /R ). (Particles with different energies have differ-ent curvature drift velocities, which can result in Landaudamping. This criterion essentially says that the drivefrom the temperature gradient must be strong enough toovercome this damping in order to drive instabilities.)A concern could be that if an NCSX design is limitedto the same local R loc /L T,loc as in a tokamak, it wouldhave a much lower a/L T (because at some toroidal lo-cations, such as the left panel of Figure 15, the crosssection of NCSX is very narrow, with a local plasma half-width a loc ≈ a/ . /L T,loc = |∇ T | /T = ( a/L T ) /a loc is much y ρ i ω /4 γ FIG. 14. Tokamak growth rate spectrum for the kinetic ITGmode with a/L T = 3 , a/L n = 0. (color online)FIG. 15. Poloidal cross sections of NCSX for two toroidalangles. The dashed line is the location of the vacuum vesseland the solid lines are last closed flux surfaces for various ι profiles. Figure 2 of Ref. 48; reprinted with permission. Moreinformation can be found in Ref. 49. larger than the average 1 /L T . This is enhanced by thelarger average aspect ratio ( R/a ) NCSX = 4 . R/a ) tok = 3 .
42, and is partially compen-sated by the fact that the local radius of curvature ofthe magnetic field, R loc = | ˆ b · ∇ ˆ b | − = 0 .
92m (evaluatedat the outer midplane of the r/a = 0 . R = 1 .
51m in NCSX.Considering these modifications, the local R loc /L T,loc =( R loc /R )( a/a loc )( R/a )( a/L T ) = 7 . a/L T in NCSX,while R/L T = 3 . a/L T for a tokamak.Restating this concern, one may have thought that ifNCSX and a tokamak had the same normalized temper-ature gradient, a/L T , the ITG modes would be muchworse in NCSX due to a much higher R loc /L T,loc thanthe tokamak. In fact, Figure 12 showed that NCSX hasa somewhat higher critical gradient in terms of a/L T ,so the hypothesis that NCSX and a tokamak are similarwhen expressed in terms of R loc /L T,loc must be incorrect.Indeed, this is strikingly illustrated in Figure 16 (same6
10 2000.10.20.3 (R/L T ) loc γ FIG. 16. Similar to Fig. 12, except the x-axis is normalized bythe local magnetic field radius of curvature R loc , instead of a .This demonstrates that NCSX (blue crosses) performs muchbetter than would be expected if the instability was the sameat the same ( R/L T ) loc , presumably indicating that additionalstabilizing effects in the parallel dynamics are important inNCSX. Tokamak growth rates: red circles. (color online) data as Fig. 12, renormalized), which shows that NCSXhas in fact much lower growth rates than a tokamak forthe same R loc /L T,loc . This is probably because the par-allel dynamics are in fact not negligible in NCSX. Theeigenfunctions, as seen in Figure 8, are more localizedalong a field line in NCSX than in a tokamak, possi-bly through some combination of the stabilizing effectsof a narrower bad-curvature region (i.e., a shorter con-nection length between good and bad curvature regions)and stronger local magnetic shear. These effects shouldbe investigated more thoroughly in the future.
III. CONCLUSION
Cross-configuration comparisons of plasma confine-ment are important to consider for the design of futurefusion energy devices. As a simple case, the linear sta-bility of the adiabatic and kinetic ITG modes was com-pared for NCSX and a tokamak equilibrium. This par-ticular tokamak equilibrium is a composite of JET H-mode shot a/L
T,crit to the toka-mak case for ITG modes with adiabatic electrons, thoughits growth rates were higher than the tokamak’s beyondmarginal stability. However, for ITG modes with ki-netic electrons, NCSX’s critical gradient a/L T is approx-imately 9% higher than the tokamak’s, which would cor-respond to an approximately 22% increase in the fusionpower for NCSX relative to the tokamak. The growthrates in NCSX remained less than for the tokamak until a/L T & . a/L T,crit . The parameter a/L
T,crit is an important figure of meritbecause it characterizes the core to edge temperature ra-tio (if the plasma is near marginal stability as expectedin typical hot reactor regimes). While the parameter
R/L
T,crit is often a useful stability parameter in toka-mak cases, it was found that stabilizing effects in theparallel dynamics in stellarators can make it a less rele-vant measure for stellarators. Upon rescaling the kineticITG mode data as a function of
R/L T , it was found thatNCSX appears even more stable.Future work that should be done includes using GS2’snonlinear capabilities to compare heat fluxes for variousfusion energy devices. Including more physical effects,such as non-zero density gradients and collisionalities,would create a clearer picture of their relative confine-ment properties. A future study could compare stel-larators with tokamaks in various operating regimes thatmay potentially improve performance further, includingreversed magnetic shear and hybrid low-shear scenarios. IV. ACKNOWLEDGMENTS
The authors wish to thank Neil Pomphrey for creat-ing the NCSX equilibrium, Pavlos Xanthopoulos for theuse of and assistance with GIST, and W. Dorland, M.A. Barnes, and W. Guttenfelder for their help with GS2.This work was supported by the U.S. Department of En-ergy through the SciDAC Center for the Study of PlasmaMicroturbulence, the Princeton Plasma Physics Labora-tory under DOE Contract No. DE-AC02-09CH11466,and Los Alamos National Security, LLC under DOE Con-tract No. DE-AC52-06NA25396.
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