NIMROD Modeling of Quiescent H-mode: Reconstruction Considerations and Saturation Mechanism J. R. King,1 K. H. Burrell,2 A. M. Garofalo,2 R. J. Groebner,2 S. E. Kruger,1 A. Y. Pankin,1 and P. B. Snyder2 1Tech-X Corporation, 5621 Arapahoe Ave. Boulder, CO 80303, USA 2General Atomics, PO Box 85608, San Diego, CA 92186–5608, USA (Dated: draft February 2, 2017) 7 The extended-MHD NIMROD code [C.R. Sovinec and J.R. King, J. Comput. Phys. 229, 5803 1 (2010)] models broadband-MHD activity from a reconstruction of a quiescent H-mode shot on the 0 DIII-D tokamak [J. L. Luxon, Nucl. Fusion 42, 614 (2002)]. Computations with the reconstructed 2 toroidal and poloidal ion flows exhibit low-nφ perturbations (nφ ’ 1−5) that grow and saturate into a turbulent-like MHD state. The workflow used to project the reconstructed state onto the n NIMRODbasisfunctionsre-solvestheGrad-Shafranovequationandextrapolatesprofilestoinclude a scrape-off-layer currents. Evaluation of the transport from the turbulent-like MHD state leads to J a relaxation of the density and temperature profiles. Published version: Nucl. Fusion 57 022002 1 (2017) [http://dx.doi.org/10.1088/0029-5515/57/2/022002] 3 PACSnumbers: 52.30.Ex52.35.Py,52.55.Fa,52.55.Tn,52.65.Kj ] h Keywords: broadband-MHD,extended-MHDmodeling,quiescentH-mode,tokamakpedestal p - m I. INTRODUCTION into a turbulent-like state. s This paper is organized as follows. As an initial- a ItisdesirabletohaveanITER[1]H-moderegime value computation, our simulations require accu- l p that is quiescent to edge-localized modes (ELMs) rate and smooth initial conditions to avoid spuri- s. [2, 3]. ELMs deposit large, localized and impul- ous instabilities. Section II describes how the re- c sive heat loads that can damage the divertor. A constructed fields are imported into the NIMROD si quiescent regime with edge harmonic oscillations spatial discretization. One of the novel methods in y (EHO) or broadband MHD activity is observed our approach to this edge modeling is to extrapo- h in some DIII-D [4–11], JT-60U [12, 13], JET [14] late the pressure and density profiles through the p and ASDEX-U [15], discharge scenarios. These scrape-off layer (SOL) to maintain first-order con- [ ELM-free discharges have the pedestal-plasma con- tinuity and thus a continuous current profile when 1 finement necessary for burning-plasma operation in solving the Grad-Shafranov equation. The section v ITER[11]. The mode activity associated with the concludes by discussing our assumptions where the 0 EHO or broadband MHD on DIII-D is character- reconstructedfieldsareinsteadystate,andhowour 5 ized by small toroidal-mode numbers (n ’ 1−5) modeling simulates the dynamics of perturbations 0 φ andisthussuitableforsimulationwithglobalMHD aboutthissteadystate. InSec.III,themodelequa- 0 0 codes. Measurements from beam-emission spec- tions and the dynamics of our nonlinear MHD sim- . troscopy, electron-cyclotron emission, and magnetic ulations that saturate into a turbulent-like state are 2 probediagnosticsshowhighlycoherentdensity,tem- considered. The magnetic stochasticity and trans- 0 7 perature and magnetic oscillations associated with port induced from these perturbations are analyzed 1 EHO. The particle and impurity [16] transport is in Sec. IV. Finally, we conclude with a discussion of : enhanced during QH-mode, leading to essentially theimplicationsandlimitationsofourpresentmod- v steady-state profiles in the pedestal region. eling and mention future directions for this work. i X Relative to QH-mode operation with EHO, op- r eration with broadband MHD tends to occur at a higherdensitiesandlowerrotationandthusmaybe II. EXTENDED EQUILIBRIUM morerelevanttopotentialITERdischargescenarios. RECONSTRUCTION FOR NIMROD While there are computational investigations of the dischargeswithEHO[17,18], thereislesscomputa- The cross-power spectrum from magnetic fluctu- tional analysis of discharges with broadband MHD. ation probe measurements on DIII-D during shot In this paper, we investigate the broadband-MHD 145098 is shown in Fig. 1. This shot is part of state with nonlinear NIMROD [19, 20] simulations a campaign to create a QH-mode discharge with initialized from a reconstruction of a DIII-D QH- ITER-relevant parameters and shape, the ultimate mode discharge with broadband MHD. These simu- results of which are summarized in Ref. [11]. As lations include the reconstructed flow and saturate the torque is ramped down the mode activity tran- 2 Figure 3: [Color online] Density, pressure and safety- factor profiles as a function of normalized flux. SOL profilesfordensityandpressurearealsoshownforψ > n 1. Figure 1: [Color online] Cross-power spectrum plot of themagneticfluctuationprobemeasurementsfromDIII- sumptionoutsidetheLCFSandincludetemperature D shot 145098. An initial phase contains coherent EHO and density profiles with non-zero gradients which fluctuations followed by a phase with broadband-MHD generate associated small currents in the scrape-off activity. The NIMROD simulation is initialized from a layer (SOL) that cause the overall current profile reconstruction during the latter phase at 4250 ms. to be continuous. Modified-bump-function fits are used to smoothly extrapolate the pressure, electron temperatureandparticledensityintheSOLregion. Figure 2: [Color online] Derivativesofallordersvanishforthisfunctionalfit Sketch that shows a dis- at the edge of the SOL region and thus the result- continuous first derivative ingcurrentprofilesmoothlydecaystozero. Forthis in pressure causes a dis- continuous current profile case, the pressure drops from 922 to 581 Pa, the when solving the Grad- electron temperature drops from 186 to 30 eV and Shafranov equation. the density drops from 4.6×1018 to 2.5×1018 m−3 in the SOL region. The half width of the electron pressure profile is roughly 2.5 mm at the outboard sitions from EHO to broadband MHD. An EFIT mid-plane and 2 cm at the divertor plate. This re- [21, 22] reconstruction, constrained by magnetic- sults in a SOL width that is slightly smaller than probe, motional-stark-effect, Thomson-scattering, the measured half width of the heat-flux during the and charge-exchange-recombination (CER) mea- later half of the inter-ELM period of DIII-D ELMy surements, is used to specify the initial condition H-mode discharges in Ref. [24]. These profiles fits, in the NIMROD code. High quality equilibria are made with a focus on the resulting current and re- essential for extended-MHD modeling with initial- sistivityprofiles,resultinanion-temperatureprofile value codes such as NIMROD. Typically the spatial that remains above 1 keV throughout the domain. resolution requirements for extended-MHD model- This inconsistency will be removed in future model- ing, which must resolve singular-layer physics and ing. The new solution is an equilibrium that closely highlyanisotropicdiffusion,aremorestringentthan resemblestheoriginalreconstructionwiththeexcep- the resolution of equilibrium reconstructions from tion of the open-flux currents and additional quan- experimentaldischarges. Tocircumventmappinger- tification of these methods will be described in a fu- rors, we re-solve the Grad-Shafranov equation with ture manuscript [25]. This regenerated equilibrium open-flux regions using the NIMEQ [23] solver to isconsistentwiththecoreprofilesthataremeasured generate a new equilibrium while using the mapped by the high quality diagnostics on DIII-D. results for both an initial guess and to specify the The density, pressure and safety-factor profiles as boundary condition. a function of normalized flux are shown in Fig. 3, Additionally, reconstructions commonly assume where the SOL profiles for density and pressure are that the region outside the last closed flux surface included where ψ >1. The current-profile that re- n (LCFS) is current free. The pressure, temperature sults from our re-solve of the Grad-Shafranov equa- and density profiles are specified only up to the tion is plotted in Fig. 4. The SOL region con- LCFS and are assumed to be constant outside the tains small, but non-zero, currents that terminate LCFS as illustrated in Fig. 2. For discharges with poloidally on the divertor. largepedestalcurrent, asiscommonlyfound during Importantly,ourinitialconditionsincludethefull QH-mode, this can lead to a large discontinuity in reconstructedtoroidalandpoloidalflowsasshownin the current density at the LCFS that is problematic Fig.5. Experimentally,theseflowsarecriticaltothe forMHDmodeling. Duringourre-solveoftheGrad- observation of QH-mode where, in particular, large Shafranov equation, we relax the current-free as- E×B flow shear is highly correlated with quiescent 3 operation [7]. Like the thermodynamic profiles, the flow profiles are specified up to and are non-zero at theLCFS.Thusweextrapolatetheseprofilestozero within the SOL. Typically only MHD-force balance (a Grad- Shafranovsolution)isstrictlyenforcedforthesteady state. In practice, perturbations about a time- independent equilibrium are evolved, and that the time-independent equilibrium need not be a time- independent solution of the source-free resistive MHD equations [19, 26]. This effectively assumes the presence of implicit (in the sense that they are calculable but not calculated) sources, fluxes and sinks. With these assumptions, if the code is run on a MHD-stable case with n > 0 perturbations, φ themodificationtothe n =0fieldsisinsignificant. φ Alternatively, when the case is MHD-unstable, the initial n =0 fields are self-consistently modified by φ the presence of the unstable modes. The NIMROD code has the capability to com- pute the extended-MHD evolution of the recon- Figure 4: [Color online] Initial current density (contour structedfields. However,itiswell-knownthatphys- colorsfortoroidalcurrentandarrowvectorsforpoloidal ical mechanisms outside the scope of our modeling current) from the reconstructed state with an extrapo- equations mediate tokamak transport such as neo- lated SOL region. The SOL region contains small, but classical bootstrap current, toroidal viscosity, and non-zerocurrentsthatterminatepoloidallyonthediver- poloidalflowdamping,neutralbeamandRFdrives, tor. kinetic turbulence, and coupling to the scrape-off layer (SOL), neutrals, impurities and the material boundary. Including these effects requires explicit calculation of the sources, fluxes and sinks. These transport-type calculations are possible and are be- coming practical (e.g. [27–29]), but this sort of in- tegrated modeling remains in the future. Thus in this work we assume that the initial reconstructed fields are steady state and our goal is to model the evolutionofthe3Dperturbationsaroundthisstate. III. NONLINEAR EVOLUTION The nonlinear simulation uses a single-fluid, single-temperature (assuming fast equilibration in theperturbations)MHDmodelwithatemperature- dependent resistivity profile where the Lundquist number,S,inthecoreis1.1×106. HereS =τ /τ , R A where τ is the Alfvén time (τ = R /v ), v is A √ A o A A the Alfvén velocity (B/ m n µ ), τ is the resis- i i 0 R tive diffusion time (τ = R2µ /η), R = 1.748m R o 0 o is the radius of the magnetic axis, η is the electri- cal resistivity, µ is the permeability of free space, Figure5: [Coloronline]Thereconstructedtoroidal(con- 0 m is the ion mass, and n is the ion density. This tourcolors)andpoloidal(arrowvector)flowwithextrap- i i olated flows in the SOL region. The poloidal flow flips choice of resistivity is enhanced by a factor of 100 sign just inside the LCFS with flows closer to the core relativetotheSpitzervalueforcomputationalprac- proceedingcounter-clockwiseandflowsnearandoutside ticality. Themodelincludeslargeparallelandsmall the LCFS proceeding clockwise. perpendiculardiffusivitiesinthemomentumanden- ergy equations. The parallel-momentum-stress con- 4 particle and energy source in the core to maintain theprofilesattheinnerboundaryinthepresenceof fluctuation-induced transport. With respect to the outer boundary, a sheath boundary condition is not applied at the divertor and consideration of an im- proveddivertorboundaryconditionisadirectionfor future research. The energy evolution from a nonlinear NIMROD simulation, decomposed by toroidal mode number, of DIII-D QH-mode shot 145098 at 4250 ms with broadband MHD activity is shown in Fig 6. The simulationsareinitiallydominatedbyan =5per- φ turbationthatsaturatesataround30µs. Afterthis time a saturated turbulent-like state develops and the n = 1 and 2 modes become dominant through φ aninversecascade. Eachtoroidalmodeintherange of n = 1−5 is dominant at a different time and φ continued interplay between modes is observed as thesimulationprogresses,particularlyinthekinetic Figure 6: [Color online] Simulated kinetic and magnetic energy spectrum. Figure 7 plots the lower half of a energy decomposed by toroidal mode number (n ). φ poloidalcutofthe3Dpressurecontoursateightdif- ferent time slices. The first time slice (30µs) shows tribution is a coherent structure associated with the dominant n = 5 mode. By 40 µs, this structure becomes (cid:18) 1 (cid:19)(cid:16) (cid:17) φ Π =m n ν bˆbˆ− I 3bˆ·∇v ·bˆ−∇·v , sheared apart leading to a turbulent-like state at ki i i ki 3 i i later times. Higher-time resolution plots show the (1) perturbations are advected in the counter-clockwise where v is the ion velocity, bˆ = B/|B|, I is the direction consistent with the direction of the ion i identity tensor, and ν = 105 m2/s. The parallel- poloidal flow inside the LCFS with a smoke-like off- ki heat-flux contribution is gassing behavior. q =−n χ bˆbˆ·∇T , (2) k i k where T is the temperature and χ = 108 m2/s. IV. TRANSPORT INDUCED BY THE MHD k PERTURBATIONS The small perpendicular diffusivites are modeled as isotropic particle, momentum and thermal diffusivi- ties with a magnitude of 1m2/s. In the presence of these electro-magnetic MHD The 3D nonlinear simulation is performed with a perturbations, the pedestal region becomes stochas- 60×128 high-order (biquartic) finite element mesh tic as shown in Fig. 8 at t=40 µs by a magnetic packed around the pedestal region to resolve the field-line Poincaré (or puncture) plot. Field-lines poloidal plane and 24 Fourier modes in the toroidal in this figure are followed for 104 m or until they direction. The simulation is initialized from a lin- hit the wall and then are color coded by their total ear computation of modes with a restricted toroidal length. Theregionfromnearthetopofthepedestal modenumberrange(n =1−8). Themodeenergies out to the LCFS becomes stochastic. As the inset φ at t = 0s are small, the largest energy is contained figure shows, a homoclinic tangle structure [30–32] within the n = 4 mode which has a spectral ki- develops near the divertor x-point. Given the large φ netic energy content of 4.2×10−5 J and a spectral parallel thermal conductivity and stochastic mag- magnetic energy content of 4.4×10−6 J. netic fields, one expects significant energy transport The boundary conditions, on both the inner an- within the pedestal region to result. However, as nulus and outer wall, are no-slip for the velocity, shown and discussed next, this is not the case and Dirichletforthedensityandtemperatureandaper- the energy transport is relatively small when com- fectly conducting wall boundary condition for the pared with the particle transport. magnetic field. Linear computations show that the Figure 9 shows the toroidal average of the den- modegrowthratesareunaffectedbypresenceofthe sity, temperature, toroidal- and poloidal-rotation, innerboundary,howeverthereisanimportanteffect and current-density profiles on the outboard mid- in nonlinear computations. The Dirichlet condition plane for the initial conditions and average values ondensityandtemperatureprovidesanunrestricted duringthreetimewindows. Thelargestprofilemod- 5 Figure7: [Coloronline]Asliceofthe3Ddomainthatshowstheevolutionofthepressurecontoursateightdifferent times. During the initial stages (t=30 and 40µs), eddies of hot, high density plasma are ejected from the pedestal and are advected poloidally in the counter-clockwise direction. Later in time these eddies are sheared apart and the dynamicsbecomeamoreturbulent-likestatewithsmoke-likeoffgassingwhichprecessespoloidallycounter-clockwise. ifications occur with the density and current pro- by effects from confined particles with large banana files,whereastheflowprofilesarelargelyunchanged. orbits from the higher-density pedestal region. Consistent with experimental observations during QH-mode[11],thesimulatedstateleadstolargepar- ticle transport relative to the thermal transport. V. DISCUSSION AND CONCLUSIONS However,thissimulationresultissomewhatpuzzling giventhestochasticmagneticfieldregionwithinthe pedestal. Wepositthreepotentialexplanationsthat With regard to the saturation mechanism of the require further investigation. The first possibility is unstable modes, there are two avenues to satura- that the simulation time is too short for the pro- tionfromaspectralenergyperspective: (1)Theun- files to reach a fully relaxed state. The second the- stable perturbations can directly modify the mean ory is that the large thermal conductivity changes fields and eliminate the source of free energy by re- the phase of the temperature perturbation relative laxingtheprofilegradients; or(2)theperturbations to the density perturbation in such a way that the can couple to stable modes that dissipate the en- flux-surface-averagedad´vectivetransport(wherethe ergy (again through modification of the mean fields particle flux is (d/dV) n˜v˜ ·∇ψ/|∇ψ|dV and the or through energy flow to the boundary of the do- thermal flux is roughly (Vd/dV)´ T˜v˜·∇ψ/|∇ψ|dV; main). Figure9showsthattheperturbationsinthis V case make non-trivial modifications to the n = 0 here V is the volume enclosed by a flux surface) is φ fields that relax the pressure gradient and current largefortheplasmadensitybutsmallfortheplasma profile leading to saturation. energy. A third hypothesis is that the stochastic One complication to this picture arises when con- transport is small because the temperature in the sidering the steady-state nature of the initial state open-fieldlineregionislarge(’1keV)andthusthe from the reconstructed fields. As mentioned in effective temperature gradient along the field-lines Sec. II, these fields are assumed to be time indepen- is small. For this computation, the pressure profile dent given the presence of sources, sinks and fluxes (and thus implicitly the temperature profile) in the that are outside the scope of our modeling. For a SOL was chosen to minimize the SOL currents. Fu- discharge state with broadband MHD activity, the ture computation will include temperatures in the contributionofthefluxfromtheMHDperturbations open-flux region that are at least an order of mag- is included in the steady-state assumption. In this nitudesmaller. Theexperimentallyrelevantvalueis sense, our modeling of the transport from the MHD somewhat difficult to determine as the ion tempera- perturbations constitutesa ‘double counting’of this ture profile is not well constrained in the open-field- flux. For studies that compare the level of this flux line region. Given the relatively low density outside to experiment there are two approaches to resolve theLCFS,theCERmeasurementscanbecorrupted this inconsistency: (1) The initial profiles could dif- 6 Figure 8: [Color online] Poincaré plot of magnetic field- linesatt=40µs. Thepedestalregionbecomesstochas- tic where the original LCFS and SOL-region bounding contoursareshownforreference. Insetshowshomoclinic tangle structure near the divertor x-point. fer from the experiment and be more unstable such that the MHD perturbations relax the profiles to a state that resembles the reconstructed profiles; or (2) the n =0 modifications from the MHD pertur- φ bationscouldbecancelled(orignored)suchthatthe final profiles match the reconstructed values. This first approach suffers from the difficulty of finding the more unstable state, a priori, that relaxes to the reconstructed state. The second approach is the traditional way turbulent flux calculations are performed and is of interest for future studies. As this approach eliminates mode saturation through modification of the mean fields, the stability of the mode spectrum becomes critical. In particular, it is likely that simulations must be performed with an extended-MHD model that includes two-fluid, first- orderfinite-Larmour-radiuseffectsthatstabilizethe Figure 9: [Color online] Toroidal average of the den- intermediate-n modes. sity, temperature, toroidal- and poloidal-rotation, and φ current-densityprofilesontheoutboardmidplaneforthe Our simulations produce an MHD turbulent-like initial condition and average values during three time state, whichisagoodcandidatetoatleastpartially windows. explain the broadband-MHD phenomena. However, additional comparisons to experimental data are re- quired to confidently claim these simulations truly model the discharge dynamics. Prior attempts to 7 compare with magnetic probe data proved unsuc- Stuart Hudson and Todd Evans for discussions re- cessful as the probe measurement temporal resolu- garding magnetic field structure and the reviewers tion (200 kHz) is approximately two orders of mag- for detailed comments on the manuscript. This ma- nitude smaller than our nonlinear simulation time terial is based on work supported by the U.S. De- period. Higher time-resolution measurements that partment of Energy Office of Science and the Sci- make local measurements of the perturbations (e.g. DAC Center for Extended MHD Modeling under beam-emission spectroscopy and Doppler reflectom- contract numbers DE-FC02-06ER54875, DE-FC02- etry) are a more promising avenue to pursue valida- 08ER54972 (Tech-X collaborators) and DE-FC02- tion and will be the subject of future studies. 04ER54698 (General Atomics Collaborators). 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