 such stability has been obtained, in the large-aspect-ratio circular limit, by means of a variational refinement of the lower bound for the energetic particle potential energy. We also investigate the effect of various profiles for the hot particle pressure on stability, and we explore the stability of off-angle modes. Moderately energetic particles, however, can destabilize the plasma through resonant interaction at their curvature drift fre- quency. We study these so-called "balloon-bone" modes, using a delta function model for their resonant response. The complete forms of the Mercier solutions in the MHD region are obtained analytically and numerically. Matched onto the inertia layer, these solutions give a dispersion relation valid for finite shear and poloidal beta values, which then is analyzed by the Nyquist technique. Results are presented for the limit in which Alfvenic effects are negligible, namely, (,)(omega)(,dh) << (omega)(,A), where (,)(omega)(,dh) is the curvature drift frequency and (omega)(,A) is the Alfven frequency, and in which the energetic particles are modeled with a slowing-down distribution in energy. Finally, even if the ideal modes and resonantly -excited modes can be simultaneously stabilized, resistive ballooning instabilities may persist

We are pursuing a cross-benchmarking and validation initiative for tokamak 3D equilibrium calculations, with 11 codes participating: the linearized tokamak equilibrium codes IPEC and MARS-F, the time-dependent extended MHD codes M3D-C1, M3D, and NIMROD, the gyrokinetic code XGC, as well as the stellarator codes VMEC, NSTAB, PIES, HINT and SPEC. Dedicated experiments for the purpose of generating data for validation have been done on the DIII-D tokamak. The data will allow us to do validation simultaneously with cross-benchmarking. Initial cross-benchmarking calculations are finding a disagreement between stellarator and tokamak 3D equilibrium codes. Work supported in part by U.S. DOE under Contracts DE-ACO2-09CH11466, DE-FC02-04E854698, DE-FG02-95E854309 and DE-AC05-000R22725.

Attaining the highest performance in poloidally diverted tokamaks requires resonant magnetic perturbation coils to avoid core instabilities (locked
such stability has been obtained, in the large - aspect - proportion round terminus ad quem, by means of a variational refinement of the lower bound for the energetic atom potential energy. We also investigate the consequence of various profiles for the blistering atom pressure on stability, and we research the stability of off - angle mode. reasonably energetic particles, however, can destabilize the plasma through resonant interaction at their curvature drift fre- quency. We analyze these so - called " balloon - bone " manner, using a delta function model for their evocative response. The complete class of the Mercier solutions in the MHD area are obtained analytically and numerically. equal onto the inertia layer, these solutions hold a dispersion relation valid for finite shear and poloidal beta values, which then is analyzed by the Nyquist technique. result are presented for the limit in which Alfvenic effects are negligible, namely, (,) (omega)(,dh) < < (omega)(,A), where (,) (omega)(,dh) is the curvature drift frequency and (omega)(,A) is the Alfven frequency, and in which the energetic particles are modeled with a slowing - down distribution in energy. Finally, even if the ideal modes and resonantly -excited modes can be simultaneously stabilize, resistive ballooning instabilities may persevere 

 We are quest for a cross - benchmarking and validation initiative for tokamak three-d equilibrium calculations, with 11 codes participating: the linearize tokamak equilibrium codes IPEC and MARS - F, the time - dependent extended MHD codes M3D - C1, M3D, and NIMROD, the gyrokinetic code XGC, as well as the stellarator codes VMEC, NSTAB, PIES, HINT and SPEC. Dedicated experiments for the purpose of generating datum for validation have been dress on the DIII - five hundred tokamak. The datum will allow us to do validation simultaneously with cross - benchmarking. Initial cross - benchmarking calculations are finding a disagreement between stellarator and tokamak three-d equilibrium codes. Work supported in share by U.S. DOE under Contracts DE - ACO2 - 09CH11466, DE - FC02 - 04E854698, DE - FG02 - 95E854309 and DE - AC05 - 000R22725. 

 Attaining the gamey performance in poloidally diverted tokamak requires resonant magnetic perturbation coil to avoid effect instabilities (locked
 sufh stability has been obuained, in the laryw-aspecv-ratio dircular limit, by means of a variatilnql redinement of the lower cound for the eneegetmc particle poteifial encxgy. Ws alsm investigate tme effect ox various profhlds for the hot particle pressure on seabilitu, wnd we explore the ftabjlity of off-angle modes. Moderately energeuic particles, howefer, can destabilize the plwsma through resonant lnteraction at eyeir curvatufe drift fgz- quency. We study these so-called "balloon-bove" mobes, using a dwltw function midel sor their resonant rasponse. The complete norms of the Mercier solutionv in the MHD regiog are obtdiued analytically and bunericdlly. Matzyed onuo vhe inertla mayer, thess solutions give a dispersion tejqtion valid fkr finytq shear and poloidal beta values, which uhen js analyzed by the Nyquust technique. Results are presqnted for the limit in which Alfvenic effects are neglmgkblt, kwowlj, (,)(omega)(,dh) << (omega)(,A), where (,)(omega)(,dh) is the curvattde dgift frequency anb (omega)(,A) is the Slvvrg frequency, avd in cgidh the energetic pwrticlef are modeled rith a slowing-down distribution in energy. Fpnaloy, even if the idecl modes and respnantky -excited modes can be simhltaneously stabilizsa, resistive balluonpng hnstabilities may persist

Wq are purwuiny a crosr-benvhmarkyng and vapidatlmn initiative for hokamck 3D aquilibriul calculations, with 11 codes partmripating: the kiteagized tokcmak eauilibrium coqes IPEC and MCRS-F, the time-aependent vxtended KHD codes M3Q-C1, M3D, and NIMSLD, the gyrokmnetic coqe XTC, aw well xr the stellarayor codes VMEC, NSTAV, PIES, HINT and SPCC. Deajcated experimeuus dor the purpose of gegegatmng dwda for validdtiov hxfe bedn done on bhe DIIO-D tokamak. The data fill allow us to do vakiqation sumultanejusly with crpss-benchmarking. Inltial cruss-bemchiarking calculations are findihg a disahrecment between steolarator and tokamak 3D equilibrium codes. Work suppocted in part by U.S. DOE ynder Contracts DE-AEO2-09GH11466, DE-FC02-04E854698, DE-FJ02-95E854309 and DE-AC05-000R22725.

Attdining the highest pwrformance in polpidally diverted tokajaks raquirfs resonant magnetic perturbation coils to avoid core instabilities (lockee
such stability has been obtained, in the limit, means of variational refinement of energetic potential energy. We investigate the effect various profiles for the hot particle on stability, and we explore the stability of off-angle modes. Moderately energetic particles, can destabilize the plasma through resonant interaction at their curvature drift fre- quency. study so-called modes, a delta function model for their resonant response. The complete forms of the Mercier solutions in MHD region are obtained analytically and numerically. Matched the inertia layer, these give a dispersion relation valid finite and poloidal values, then analyzed by the technique. Results are presented for the limit in which Alfvenic effects are negligible, namely, (,)(omega)(,dh) << (omega)(,A), (,)(omega)(,dh) is drift frequency (omega)(,A) the frequency, and in energetic particles are modeled with a energy. Finally, even if the ideal modes and -excited modes be simultaneously stabilized, resistive ballooning instabilities persist We are pursuing a cross-benchmarking and validation for tokamak 3D equilibrium calculations, with 11 codes participating: the linearized tokamak equilibrium codes IPEC the time-dependent extended MHD M3D-C1, M3D, and the code as as the codes VMEC, NSTAB, PIES, HINT and SPEC. Dedicated experiments for the of generating data for validation have been done on the The will allow us do validation simultaneously with Initial calculations are finding a stellarator tokamak Work in by U.S. DOE under DE-ACO2-09CH11466, DE-FC02-04E854698, DE-FG02-95E854309 and DE-AC05-000R22725. the highest performance in magnetic perturbation coils to avoid core instabilities (locked
 such stability has been obtaiNed, in the laRge-asPecT-raTiO cirCulaR limit, by means oF A varIational refinement of thE loweR bOUnd fOR tHe eneRgetic pARtICLe pOtEnTiaL eNErGy. We aLso InvestiGate the effEct Of Various profiLEs For the hot pArtIcle pressure On sTabiliTy, And WE explOre The stAbilitY Of off-aNgle modes. moDEratelY EnergetIC PaRticLes, however, can destABiLIze the plasma thRough rEsONaNT IntEraCtion at theIr CurvaTUre drifT FrE- QUEncY. we study these sO-called "ballOOn-bOne" modEs, UsiNG a deltA funcTiON moDel for their ResoNant respoNse. The COmplete FOrms of tHe MercIer SolUtioNS iN tHe MhD REgiON aRe oBTaiNed analyTiCaLly anD numERICAlly. matChed Onto tHe inertia layeR, thEse sOLutIons gIve a dIspeRsIon reLation Valid FoR finite shear and PoloIdal beta vAluEs, WhiCh Then iS AnalyzEd bY thE NyquisT techniQUe. REsULTS aRe presented for the lImIT In Which AlfVenic eFFeCtS Are negliGiBle, NameLY, (,)(Omega)(,Dh) << (omEGa)(,a), where (,)(omEga)(,dh) iS ThE cUrvaturE dRift frEqUenCy aNd (omeGA)(,A) is The AlfVen frequEncy, aND in which the eneRGetic particleS ArE MOdELed wIth A slowing-dowN disTRibuTion IN eNerGY. FinaLly, evEn IF tHE ideal modes and resonAnTly -excIted mOdes can be simuLtaneously STABilized, rEsisTIvE Ballooning instAbiliTies may perSIst

We are PursuIng a crosS-benchmarKINg and valIdaTioN inItiATIvE for tokamak 3D eQUIlibRiUm calcuLatIons, witH 11 coDes ParTicIpAting: the lInearizeD tOkAmAk EquIlibrIUm codes IpEc anD MaRS-f, the tIMe-depeNdent ExteNdEd mhD cOdes M3D-C1, m3d, aND nIMRoD, ThE gyrOkiNeTic coDe XGc, As wEll as thE stellaraTor COdes vMeC, nSTAB, PIeS, HINT and SPEC. deDicated expErImeNts for THE purpose Of generating data for valiDAtion haVe bEen doNe on The DIII-D tOkaMak. The DatA Will alLow us tO do vaLiDatION simuLTAnEouSlY with cross-BENchMarkiNg. initIal crosS-benchmarking calcuLAtiOns are finding A diSagrEEMeNt bETwEEn sTeLLarATOr and tokamak 3D eqUilibrium cOdES. WOrk supportED in PaRt by U.S. DoE under contrACts DE-ACo2-09CH11466, DE-FC02-04E854698, De-FG02-95E854309 and DE-aC05-000r22725.

AttAINinG the highesT performAnce in polOIdallY DiVerteD toKamaks ReQuiRes reSonant MAgnEtic pErturbAtIon coiLs to aVoId core inStabilities (locked
 such stability has been o btained, i n the  la rge -a spec t-ra tio circular l i mit,  by means of a variati onalre f inem e nt  of t he lowe r  b o u ndfo rthe  e n er getic  pa rticlepotentialene rg y. We also i n ve stigate th e e ffect of var iou s prof il esf or th e h ot pa rticle pressu re on sta bi l ity, a n d we ex p l or e th e stability of of f -a n gle modes. Mod eratel ye ne r g eti c p articles,ho wever ,  can de s ta b i l ize the plasma th rough reson a ntintera ct ion at the ir cu rv a tur e drift fre - qu ency. Westudyt hese so - called"ballo on- bon e" m o de s,  us in g  ad el taf unc tion mod el  f or th eirr e s o nant  re spon se. T he complete f orm s of the  Merc ier s olut io ns in  the M HD re gi on are obtained  ana lytically  an dnum er icall y . Matc hed  on to theinertia lay er , t he se solutions givead i sp ersion r elatio n  v al i d for fi ni teshea r and p oloi d al  beta va lues,w hi ch  then i sanalyz ed  by  th e Nyq u isttechni que. Res ultsa re presented f o r the limit i n  w h i ch Alfv eni c effects a re n e glig ible ,  n ame l y, (, )(ome ga ) (, d h) << (omega)(,A),wh ere (, )(ome ga)(,dh) is t he curvatu r e drift fr eque n cy and (omega)(,A ) isthe Alfven frequenc y, an d in whi ch the en e r getic pa rti cle s a rem o de led with a sl o w ing- do wn dist rib ution i n e ner gy.  Fi na lly, even  if theid ea lmo des  andr esonantl y-ex ci ted  mode s  can b e sim ulta ne ou s lystabili z ed , resi st iv e ba llo on ing i nsta b ili ties ma y persist 

W e  are  p ur suing a  cross-benchm ar king and v al ida tion i n i tiativefor tokamak 3D equilibr i um calc ula tions , wi th 11 cod espartic ipa t ing: t he lin eariz ed  to k a mak e q u il ibr iu m codes IP E C  an d MAR S- F, t he time -dependent extende d  MH D codes M3D-C 1,M3D, a nd  NI M RO D , t he gyr o k inetic code XGC , as wellas th e stellara t orco des VME C, NSTA B, PI E S, HINT  and SPEC . Dedicat ed  exp e r ime nts for th e purpos e of gene r ating da ta fo r v alidat io n h ave b een do n e o n the  DIII- Dtokama k. Th edata wil l allow us to do valida tion s imult ane ously wit h c r oss -benchmar king . Initialcro ss- bench mar k ing c alcu l at ion s  arefind i ng a disa g re eme n t  b etween stel l a r ato r and  to k amak 3 D eq uilibrium codes.W ork supportedin p a r t b y U . S. D OE  under Contrac tsDE - A CO2-09CH 11 466, DE-FC0 2-04E854 69 8 , DE- FG02-9 5E8543 09 andD E -A C 05-000 R227 25. 

Attaini ngth e  highes tpe r forman ce i npoloid ally d i vert e d  tokamaks requir es re s o nantm agn eticpe rturbat i on c oils to av oid core in stabil itie s (lo cked
 such_stability has_been obtained, in the_large-aspect-ratio circular_limit,_by means_of_a variational refinement_of the lower_bound for the energetic_particle potential energy._We_also investigate the effect of various profiles for the hot particle pressure on stability,_and_we explore_the_stability_of off-angle modes. Moderately energetic_particles, however, can destabilize the_plasma through_resonant interaction at their curvature drift fre- quency._We_study these so-called_"balloon-bone" modes, using a delta function model for their_resonant response. The complete forms of_the Mercier solutions_in_the_MHD region are obtained_analytically and numerically. Matched onto the_inertia layer, these solutions give a_dispersion relation valid for finite shear and_poloidal beta values, which then is_analyzed by the Nyquist technique._Results are_presented for the limit in_which Alfvenic effects_are negligible,_namely, (,)(omega)(,dh) <<_(omega)(,A), where (,)(omega)(,dh) is the curvature_drift frequency and_(omega)(,A) is the Alfven frequency, and_in_which the energetic_particles_are_modeled with_a slowing-down distribution_in_energy. Finally,_even_if the ideal modes and resonantly_-excited_modes can be simultaneously stabilized, resistive ballooning_instabilities may persist

We are_pursuing_a cross-benchmarking and validation_initiative for tokamak 3D equilibrium_calculations, with 11 codes participating: the_linearized tokamak_equilibrium codes_IPEC and MARS-F, the time-dependent extended MHD codes M3D-C1, M3D, and_NIMROD, the gyrokinetic code XGC, as_well as the stellarator_codes VMEC,_NSTAB,_PIES, HINT and_SPEC._Dedicated experiments_for the purpose of generating data for_validation have_been done on the DIII-D tokamak._The data will allow_us_to do validation simultaneously with cross-benchmarking._Initial cross-benchmarking calculations are finding a_disagreement between stellarator and tokamak_3D_equilibrium_codes. Work supported in part_by U.S. DOE under Contracts DE-ACO2-09CH11466,_DE-FC02-04E854698, DE-FG02-95E854309 and_DE-AC05-000R22725.

Attaining the highest performance in poloidally diverted_tokamaks_requires resonant magnetic perturbation coils to_avoid_core instabilities (locked