Microturbulence in optimised stellarators

Transcription

1 Q Josefine H. E. Proll, Benjamin J. Faber, Per Helander, Samuel A. Lazerson, Harry Mynick, and Pavlos Xanthopoulos Many thanks to: T. M. Bird, J. W. Connor, T. Go rler, W. Guttenfelder, G.W. Hammett, F. Jenko, A. Ban o n Navarro, G. G. Plunk, M.J. Pu schel, D. Told PP Greifswald Stellarator Theory Berlin, June 2014 PP Greifswald 1/21

2 Q Particle orbits in optimised stellarators Quasi-symmetry e.g. NCSX ( Quasar ) Quasi-isodynamicity e.g. Wendelstein 7-X (W7-X) PP Greifswald 2/21

3 Q The problem with microinstabilities... thanks to stellarator optimisation, stellarators might show reduced neoclassical transport, comparable to tokamaks? next issue: anomalous transport investigate microinstabilities which trigger small-scale turbulence How does the optimisation influence microinstabilities? PP Greifswald 3/21

4 Q The problem with microinstabilities... thanks to stellarator optimisation, stellarators might show reduced neoclassical transport, comparable to tokamaks? next issue: anomalous transport investigate microinstabilities which trigger small-scale turbulence How does the optimisation influence microinstabilities? Electrostatic microinstabilities on-temperature-gradient mode (TG) - limits T Trapped-electron mode (TEM) - limits n PP Greifswald 3/21

5 Q Outline Quasi-isodynamicity Analytical stability analysis HSX vs. W7-X - collaboration with B. Faber (Madison) TEM optimisation - collaboration with H. Mynick and S. Lazerson (PPPL) PP Greifswald 4/21

6 Q Outline Quasi-isodynamicity Analytical stability analysis HSX vs. W7-X - collaboration with B. Faber (Madison) TEM optimisation - collaboration with H. Mynick and S. Lazerson (PPPL) PP Greifswald 5/21

7 Q Properties of quasi-isodynamic stellarators Contours of constant B = ψ α poloidally closed ψ = toroidal flux, radial coordinate α = field line label, binormal coordinate bounce averaged radial drift vanishes vd ψ = 0 Subbotin et al. Nucl. Fusion , courtesy of Y. Turkin PP Greifswald 6/21

8 Q Properties of quasi-isodynamic stellarators Contours of constant B = ψ α poloidally closed ψ = toroidal flux, radial coordinate α = field line label, binormal coordinate bounce averaged radial drift vanishes vd ψ = 0 Action integral of the bounce motion adiabatic invariant R J(ψ) = mvk dl Subbotin et al. Nucl. Fusion , courtesy of Y. Turkin in maximum-j-configurations with J/ ψ < 0: PP Greifswald 6/21

9 Q Properties of quasi-isodynamic stellarators Contours of constant B = ψ α poloidally closed ψ = toroidal flux, radial coordinate α = field line label, binormal coordinate bounce averaged radial drift vanishes vd ψ = 0 Action integral of the bounce motion adiabatic invariant R J(ψ) = mvk dl in maximum-j-configurations with J/ ψ < 0: Subbotin et al. Nucl. Fusion , courtesy of Y. Turkin direction of the precessional drift ω a ω da < 0 d ln na - diamagnetic frequency dψ ω da kα J - precessional drift frequency ψ ω a kα PP Greifswald 6/21

10 Q Properties of quasi-isodynamic stellarators Contours of constant B = ψ α poloidally closed ψ = toroidal flux, radial coordinate α = field line label, binormal coordinate bounce averaged radial drift vanishes vd ψ = 0 Action integral of the bounce motion adiabatic invariant R J(ψ) = mvk dl in maximum-j-configurations with J/ ψ < 0: favourable bounce-averaged curvature for all orbits Subbotin et al. Nucl. Fusion , courtesy of Y. Turkin direction of the precessional drift ω a ω da < 0 d ln na - diamagnetic frequency dψ ω da kα J - precessional drift frequency ψ ω a kα PP Greifswald 6/21

11 Q Outline Quasi-isodynamicity Analytical stability analysis HSX vs. W7-X - collaboration with B. Faber (Madison) TEM optimisation - collaboration with H. Mynick and S. Lazerson (PPPL) PP Greifswald 7/21

12 Q The set of gyrokinetic equations Electrostatic collisionless gyro-kinetic equation (in ballooning space) for a perturbation with frequency ω = ωr + iγ ea φ T ivk k ga + (ω ωda )ga = J0 (k v /Ωa ) ω ω a fa0 Ta with ga the non-adiabatic part of the distribution function of species a diamagnetic frequency 3 E T ω a = ω a 1 + ηa Ta 2 Ta kα d ln na with ω a = ea dψ d ln Ta d ln na and ηa = dψ dψ magnetic drift frequency ωda = k vd,a with vda 2 v ln B + vk2 κ 2 and κ = b b b = Ωa PP Greifswald 8/21

13 Q The set of gyrokinetic equations Electrostatic collisionless gyro-kinetic equation (in ballooning space) for a perturbation with frequency ω = ωr + iγ ea φ T ivk k ga + (ω ωda )ga = J0 (k v /Ωa ) ω ω a fa0 Ta with ga the non-adiabatic part of the distribution function of species a diamagnetic frequency 3 E T ω a = ω a 1 + ηa Ta 2 Ta kα d ln na with ω a = ea dψ d ln Ta d ln na and ηa = dψ dψ magnetic drift frequency ωda = k vd,a with vda b = Ωa 2 v ln B + vk2 κ 2 and κ = b b Close the system with the quasi-neutrality equation X X na ea2 X Z δna ea = 0 φ= ea J 0 ga d3 v T a a a a PP Greifswald 8/21

14 Q Energy transfer We define Z Z dl d3 v ea vk b + vd φ J0 fa1 =j E B as rate of the gyrokinetic energy transfer from the field to particles of species a Pa = Re [Proll, Helander, Connor and Plunk, PRL 2012] und [Helander, Proll and Plunk, PoP 2013] PP Greifswald 9/21

15 Q Energy transfer We define Z Z dl d3 v (ivk k ga ωda ga )φ J0 B as rate of the gyrokinetic energy transfer from the field to particles of species a Pa = ea m [Proll, Helander, Connor and Plunk, PRL 2012] und [Helander, Proll and Plunk, PoP 2013] PP Greifswald 9/21

16 Q Energy transfer We define Z Z dl d3 v (ivk k ga ωda ga )φ J0 B as rate of the gyrokinetic energy transfer from the field to particles of species a P Pa < 0 for a destabilising species a, also a Pa γ Pa = ea m [Proll, Helander, Connor and Plunk, PRL 2012] und [Helander, Proll and Plunk, PoP 2013] PP Greifswald 9/21

17 Q Energy transfer We define Z Z dl d3 v (ivk k ga ωda ga )φ J0 B as rate of the gyrokinetic energy transfer from the field to particles of species a P Pa < 0 for a destabilising species a, also a Pa γ Pa = ea m if 0 < ηa < 2/3 and ω a ωda < 0 [Proll, Helander, Connor and Plunk, PRL 2012] und [Helander, Proll and Plunk, PoP 2013] PP Greifswald 9/21

18 Q Energy transfer We define Z Z dl d3 v (ivk k ga ωda ga )φ J0 B as rate of the gyrokinetic energy transfer from the field to particles of species a P Pa < 0 for a destabilising species a, also a Pa γ Pa = ea m if 0 < ηa < 2/3 and ω a ωda < 0 only possibbilty: ωω i > 0, and then Pe > 0 Pi < 0 [Proll, Helander, Connor and Plunk, PRL 2012] und [Helander, Proll and Plunk, PoP 2013] PP Greifswald 9/21

19 Q Energy transfer We define Z Z dl d3 v (ivk k ga ωda ga )φ J0 B as rate of the gyrokinetic energy transfer from the field to particles of species a P Pa < 0 for a destabilising species a, also a Pa γ Pa = ea m if 0 < ηa < 2/3 and ω a ωda < 0 only possibbilty: ωω i > 0, and then Pe > 0 Pi < 0 only ion driven mode, no classical TEM [Proll, Helander, Connor and Plunk, PRL 2012] und [Helander, Proll and Plunk, PoP 2013] PP Greifswald 9/21

20 Q Energy transfer We define Z Z dl d3 v (ivk k ga ωda ga )φ J0 B as rate of the gyrokinetic energy transfer from the field to particles of species a P Pa < 0 for a destabilising species a, also a Pa γ Pa = ea m if 0 < ηa < 2/3 and ω a ωda < 0 only possibbilty: ωω i > 0, and then Pe > 0 Pi < 0 only ion driven mode, no classical TEM No TEMs and C-TPM in quasi-isodynamic configurations if 0 < ηa < 2/3! [Proll, Helander, Connor and Plunk, PRL 2012] und [Helander, Proll and Plunk, PoP 2013] PP Greifswald 9/21

21 Q Outline Quasi-isodynamicity Analytical stability analysis HSX vs. W7-X - collaboration with B. Faber (Madison) TEM optimisation - collaboration with H. Mynick and S. Lazerson (PPPL) PP Greifswald 10/21

22 Q Gyrokinetic Electromagnetic Numerical Experiment (GENE) F. Jenko et al. (Garching) solves nonlinear 5D-Vlasov-Maxwell system on fixed grid contains arbitrary number of gyro-kinetic particle species realistic magnetic geometry flux tube as a good approximation for kk k entire flux surface of a stellarator possible as well Example for a flux tube M. Barnes, PhD thesis 2008 PP Greifswald 11/21

23 Q Simulated geometries Helically Symmetric Experiment quasi-helically symmetric stellarator aspect ratio: A = 8 Magnetic field strength B, red = Bmax, blue = Bmin. The visible cut shows the so-called bean plane. PP Greifswald 12/21

24 Q Simulated geometries Helically Symmetric Experiment quasi-helically symmetric stellarator aspect ratio: A = 8 if ω e < 0 here, see plot: ω e ωde > 0 in the trapping wells ω de on the flux surface at half radius r = 0.71rmax, as a function of bounce point location. PP Greifswald 12/21

25 Q Simulated geometries Helically Symmetric Experiment quasi-helically symmetric stellarator aspect ratio: A = 8 if ω e < 0 here, see plot: ω e ωde > 0 in the trapping wells bad curvature and magnetic overlap perfectly 0.2 B κ κ[a.u.] 1.35 B[a.u.] z Magnetic field strength B and curvature κ along a magnetic field line. z = 0 in the outboard midplane of the bean plane. PP Greifswald 12/21

26 Q Simulated geometries Wendelstein 7-X approaching quasi-isodynamicity aspect ratio: A = 10 trapped particles in the almost straight sections Magnetic field strength B, red = Bmax, blue = Bmin. The visible cut shows the so-called bean plane. PP Greifswald 13/21

27 Q Simulated geometries Wendelstein 7-X approaching quasi-isodynamicity aspect ratio: A = 10 trapped particles in the almost straight sections if ω e > 0 here, see plot: ω e ωde > 0 also in the straight sections, but less bad ω de on the flux surface at half radius r = 0.71rmax, as a function of bounce point location. PP Greifswald 13/21

28 Q Simulated geometries Wendelstein 7-X aspect ratio: A = 10 trapped particles in the almost straight sections if ω e > 0 here, see plot: ω e ωde > 0 also in the straight sections, but less bad bad curvature and magnetic well locally separated B κ κ[a.u.] approaching quasi-isodynamicity B[a.u.] z Magnetic field strength B and curvature κ along a magnetic field line. z = 0 in the outboard midplane of the bean plane. PP Greifswald 13/21

29 Q on temperature gradient modes a/ln = 1.0, a/lti = 3.0 both configurations: bean flux tube more unstable TGs with adiabatic electrons PP Greifswald 14/21

30 Q on temperature gradient modes a/ln = 1.0, a/lti = 3.0 both configurations: bean flux tube more unstable kinetic electrons: HSX vs. W7-X : destabilisation vs. stabilisation TGs with kinetic electrons PP Greifswald 14/21

31 Q on temperature gradient modes a/ln = 1.0, a/lti = 3.0 both configurations: bean flux tube more unstable kinetic electrons: HSX vs. W7-X : destabilisation vs. stabilisation TGs with adiabatic electrons PP Greifswald 14/21

32 Q on temperature gradient modes a/ln = 1.0, a/lti = 3.0 both configurations: bean flux tube more unstable kinetic electrons: HSX vs. W7-X : destabilisation vs. stabilisation TGs with kinetic electrons PP Greifswald 14/21

33 Q Trapped-electron modes a/ln = 2.0, a/lte = 1.0 Trapped-electron modes PP Greifswald 15/21

34 Q Trapped-electron modes a/ln = 2.0, a/lte = 1.0 HSX vs. W7-X: ordinary TEMs with ωω e > 0 vs. ion-driven mode with ωω i > 0 no ordinary TEMs in W7-X Trapped-electron modes PP Greifswald 15/21

35 Q TG-turbulence a/ln = 1.0, a/lti = W7-X - bean W7-X - triangle 7 Qes /Qgb Qes /Qgb 8 HSX - bean HSX - triangle t (a/ci) TG heat flux in HSX t (a/ci) TG heat flux in W7-X PP Greifswald 16/21

36 Q TG-turbulence a/ln = 1.0, a/lti = W7-X - bean - QNL W7-X - triangle - QNL W7-X - bean - QQL*0.1 W7-X - triangle - QQL* Qes /Qgb Qes /Qgb 0.4 HSX - bean -QNL HSX - triangle - QNL HSX - bean - QQL*0.1 HSX - triangle - QQL* kyρi TG heat flux spectrum in HSX with QQL γ/ky kyρi TG heat flux spectrum in W7-X with QQL γ/ky2 PP Greifswald 16/21

37 Q TG-turbulence a/ln = 1.0, a/lti = 3.0 Contour plot of potential in HSX zonal flows? Contour plot of potential in W7-X zonal flows? PP Greifswald 16/21

38 Q TG-turbulence a/ln = 1.0, a/lti = W7-X - bean - with ZF W7-X - bean - w/o ZF 14 Qes /Qgb Contour plot of potential in HSX zonal flows? t (a/cs) TG heat flux in W7-X - with and without zonal flows PP Greifswald 16/21

39 Q Outline Quasi-isodynamicity Analytical stability analysis HSX vs. W7-X - collaboration with B. Faber (Madison) TEM optimisation - collaboration with H. Mynick and S. Lazerson (PPPL) PP Greifswald 17/21

40 Q The TEM proxy function STELLOPT: optimisation of 3D equilibria via proxy functions H. Mynick, N. Pomphrey, S. Ethier PoP (2002) PP Greifswald 18/21

41 Q The TEM proxy function STELLOPT: optimisation of 3D equilibria via proxy functions H. Mynick, N. Pomphrey, S. Ethier PoP (2002) dea: reduce energy transfer rate maximise average good curvature PP Greifswald 18/21

42 Q The TEM proxy function STELLOPT: optimisation of 3D equilibria via proxy functions H. Mynick, N. Pomphrey, S. Ethier PoP (2002) dea: reduce energy transfer rate maximise average good curvature proxy function similar to energy transfer rate near marginal stability Z Z πea2 d` T Pa = ) J0 φ 2 fa0 d3 v δ(ω ω da )ω da (ω da ω a Ta B PP Greifswald 18/21

43 Q The TEM proxy function STELLOPT: optimisation of 3D equilibria via proxy functions H. Mynick, N. Pomphrey, S. Ethier PoP (2002) dea: reduce energy transfer rate maximise average good curvature proxy function similar to energy transfer rate near marginal stability Z Z πea2 d` T Pa = ) J0 φ 2 fa0 d3 v δ(ω ω da )ω da (ω da ω a Ta B choose Q as proxy and minimise this for each flux tube Z 1/Bmin Q= ω d (λ)dλ 1/Bmax Z +`0 ω d (λ) = H `0 1 d` B(`) ωd (λ, `) p λ 1 λb(`) PP Greifswald 18/21

44 Q The TEM proxy function STELLOPT: optimisation of 3D equilibria via proxy functions H. Mynick, N. Pomphrey, S. Ethier PoP (2002) dea: reduce energy transfer rate maximise average good curvature proxy function similar to energy transfer rate near marginal stability Z Z πea2 d` T Pa = ) J0 φ 2 fa0 d3 v δ(ω ω da )ω da (ω da ω a Ta B choose Q as proxy and minimise this for each flux tube Z 1/Bmin Q= ω d (λ)dλ 1/Bmax Z +`0 ω d (λ) = `0 H 1 d` B(`) ωd (λ, `) p λ 1 λb(`) minimise Q for different flux tubes on different flux surfaces PP Greifswald 18/21

45 Q Optimising Quasar Magnetic field strength of Quasar (initial) Magnetic field strength of Quasar (TEM-optimised) PP Greifswald 19/21

46 Q Outline Quasi-isodynamicity Analytical stability analysis HSX vs. W7-X - collaboration with B. Faber (Madison) TEM optimisation - collaboration with H. Mynick and S. Lazerson (PPPL) PP Greifswald 20/21

47 Q Conclusions: comparison HSX and W7-X: enhanced TEM stability in W7-X compared with HSX TG turbulence level slightly higher in HSX as well PP Greifswald 21/21

48 Q Conclusions: comparison HSX and W7-X: enhanced TEM stability in W7-X compared with HSX TG turbulence level slightly higher in HSX as well TEM optimisation: developed proxy function first linear tests of optimised Quasar are currently being conducted PP Greifswald 21/21

49 Q Conclusions: comparison HSX and W7-X: enhanced TEM stability in W7-X compared with HSX TG turbulence level slightly higher in HSX as well TEM optimisation: developed proxy function first linear tests of optimised Quasar are currently being conducted Outlook simulate entire plasma volume instead of flux tubes TG turbulence (with adiabatic electrons) on the s = 0.5 flux surface of W7-X (courtesy of P. Xanthopoulos) PP Greifswald 21/21

50 Q Conclusions: comparison HSX and W7-X: enhanced TEM stability in W7-X compared with HSX TG turbulence level slightly higher in HSX as well TEM optimisation: developed proxy function first linear tests of optimised Quasar are currently being conducted Outlook simulate entire plasma volume instead of flux tubes include kinetic electrons in nonlinear simulations TG turbulence (with adiabatic electrons) on the s = 0.5 flux surface of W7-X (courtesy of P. Xanthopoulos) PP Greifswald 21/21

51 Q Conclusions: comparison HSX and W7-X: enhanced TEM stability in W7-X compared with HSX TG turbulence level slightly higher in HSX as well TEM optimisation: developed proxy function first linear tests of optimised Quasar are currently being conducted Outlook simulate entire plasma volume instead of flux tubes include kinetic electrons in nonlinear simulations theory and simulations including collisions electromagnetic effects TG turbulence (with adiabatic electrons) on the s = 0.5 flux surface of W7-X (courtesy of P. Xanthopoulos) PP Greifswald 21/21