Gyrokinetic theory for particle transport in fusion plasmas

Transcription

1 Gyrokinetic theory for particle transport in fusion plasmas Matteo Valerio Falessi 1,2, Fulvio Zonca 3 1 INFN - Sezione di Roma Tre, Via della Vasca Navale, 84 (00146) Roma (Roma), Italy 2 Dipartimento di Matematica e Fisica, Roma Tre University, Via della Vasca Navale, 84 (00146) Roma (Roma), Italy 3 ENEA, Fusion and Nuclear Safety Department, C. R. Frascati, Via E. Fermi, 45 (00044) Frascati (Roma), Italy M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

2 Introduction: purpose statement & motivations Obtain a closed set of equations describing the dynamics of a thermonuclear plasma on the energy confinement time of a burning plasma experiment; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

3 Introduction: purpose statement & motivations Obtain a closed set of equations describing the dynamics of a thermonuclear plasma on the energy confinement time of a burning plasma experiment; By means of the equations that we are going to derive it is possible to: study self-consistently collisional and turbulent transport; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

4 Introduction: purpose statement & motivations Obtain a closed set of equations describing the dynamics of a thermonuclear plasma on the energy confinement time of a burning plasma experiment; By means of the equations that we are going to derive it is possible to: study self-consistently collisional and turbulent transport; investigate structure formation on meso-scales, i.e. scale separation between reference state and fluctuations is not postulated; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

5 Introduction: purpose statement & motivations Obtain a closed set of equations describing the dynamics of a thermonuclear plasma on the energy confinement time of a burning plasma experiment; By means of the equations that we are going to derive it is possible to: study self-consistently collisional and turbulent transport; investigate structure formation on meso-scales, i.e. scale separation between reference state and fluctuations is not postulated; describe transport processes in realistic geometry. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

6 Introduction: self-consistency and spatio-temporal scales Parameter Minor radius Ionic Larmor radius energy conf. time Fluctuation frequency Collision frequency ee ITER cm cm 3.5 sec Hz Hz Table : Courtesy of Abel et al M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

7 Introduction: self-consistency and spatio-temporal scales Parameter Minor radius Ionic Larmor radius energy conf. time Fluctuation frequency Collision frequency ee ITER cm cm 3.5 sec Hz Hz Table : Courtesy of Abel et al predictive simulations require collisions and fluctuations to be treated on the same footing; on a sufficiently long time, collisions can produce a modification of the reference state of the same order of the variation produced by fluctuations/turbulence; self-consistency is crucial also due to mutual interactions, e.g. increased collisional transport due to the fluctuations... M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

8 Introduction: what are meso-scales? Radial transport requires the study of flux surface averaged equations: t n ψ = (nv ) ψ ; finite contributions only from non-linear terms with vanishing toroidal and poloidal mode-numbers; in principle they have arbitrary length-scale; postulating a scale-separation between reference state and fluctuations is not necessarily correct. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

9 Introduction: outline transport processes on the reference state length-scale by means of the moments method; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

10 Introduction: outline transport processes on the reference state length-scale by means of the moments method; transport processes on an arbitrary length-scale by means of kinetic theory. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

11 Gyrokinetic transport: notation Following Hazeltine and Meiss 2003, we adopt this notation: nv = dv vf, P = dv vvf, R = dv 1 2 mv2 vvf,... F = dv mv C, G = dv 1 2 mv2 v C, W = dv 1 2 mv2 C,.... M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

12 Gyrokinetic transport: notation Following Hazeltine and Meiss 2003, we adopt this notation: nv = dv vf, P = dv vvf, R = dv 1 2 mv2 vvf,... F = dv mv C, G = dv 1 2 mv2 v C, W = dv 1 2 mv2 C,.... Moments of the distribution function have the following evolution equations: t n + (nv ) = 0, t (nmv ) + P en (E + V B/c) = F, ( 3 t 2 p + m ) 2 nv 2 + Q ene V = W + F V,... M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

13 Gyrokinetic transport: notation Following Hazeltine and Meiss 2003, we adopt this notation: nv = dv vf, P = dv vvf, R = dv 1 2 mv2 vvf,... F = dv mv C, G = dv 1 2 mv2 v C, W = dv 1 2 mv2 C,.... Moments of the distribution function have the following evolution equations: t n + (nv ) = 0, t (nmv ) + P en (E + V B/c) = F, ( 3 t 2 p + m ) 2 nv 2 + Q ene V = W + F V,... in principle, studying the hierarchy of moments equations is equivalent to study the kinetic equation; long-wavelength structures are conveniently studied by means of moments equation. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

14 Gyrokinetic transport: ordering assumptions In the core the plasma is magnetized, thus: f = F 0 + δf ρ l ln F 0 ρ l /L O(δ) 1; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

15 Gyrokinetic transport: ordering assumptions In the core the plasma is magnetized, thus: f = F 0 + δf ρ l ln F 0 ρ l /L O(δ) 1; perturbations have arbitrary length-scale; no scale-separation assumption between reference state and perturbations; main difference with previous works; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

16 Gyrokinetic transport: ordering assumptions In the core the plasma is magnetized, thus: f = F 0 + δf ρ l ln F 0 ρ l /L O(δ) 1; perturbations have arbitrary length-scale; no scale-separation assumption between reference state and perturbations; main difference with previous works; Following Hinton and Hazeltine 1976, Frieman and Chen 1982, we describe slowly varying reference states: and similarly for n 0,... ω 1 t ln p 0 = ω 1 p 1 p 0 0 t O(δ2 ) M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

17 Gyrokinetic transport: ordering assumptions We introduce the following ordering for fluctuating quantities (see Frieman and Chen 1982): ce / t δb Bv th Ω B k O(δ). k M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

18 Gyrokinetic transport: ordering assumptions We introduce the following ordering for fluctuating quantities (see Frieman and Chen 1982): ce / t δb Bv th Ω B k O(δ). k The ordering assumptions lead to the following relations: f = f M + O(δ), {nv, F, Q, [P Ip], [R I(5/2)p(T/m)]} O(δ). M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

19 Gyrokinetic transport: flux coordinates realistic geometry is crucial; flux coordinates, i.e. (ψ, θ, φ), are the natural coordinates; Figure : Courtesy of J.Ball et al. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

20 Gyrokinetic transport: flux coordinates realistic geometry is crucial; flux coordinates, i.e. (ψ, θ, φ), are the natural coordinates; coordinate basis is ( ψ, θ, φ); B 0 = F φ + φ ψ; equilibrium quantities do not depend on the toroidal angle φ. Figure : Courtesy of J.Ball et al. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

21 Gyrokinetic transport: the moment approach Taking the cross product of the force balance equation with b, see Hinton and Hazeltine 1976, we obtain: nv = 1 mω b [ t (nmv ) + P ene F ]. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

22 Gyrokinetic transport: the moment approach Taking the cross product of the force balance equation with b, see Hinton and Hazeltine 1976, we obtain: nv = 1 mω b [ t (nmv ) + P ene F ]. Applying the ordering: (nv ) 1 = 1 b ( p + en φ) mω only information about the zeroth-order distribution function is required to calculate the lowest order perpendicular flux; we will use this methodology to calculate second order fluxes across magnetic surfaces; distribution function must be known up to the first order to describe transport processes on the energy confinement time. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

23 Gyrokinetic transport: density transport In order to apply this method to descibe the transport of particles, following Hinton and Hazeltine 1976, we act with the projector R 2 φ on the momentum equation and we take the flux surface average: R 2 φ t (nmv ) ψ + R 2 φ P ψ = R 2 φ (ene + F ) ψ + R 2 φ (en/cv B) ψ. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

24 Gyrokinetic transport: density transport In order to apply this method to descibe the transport of particles, following Hinton and Hazeltine 1976, we act with the projector R 2 φ on the momentum equation and we take the flux surface average: R 2 φ t (nmv ) ψ + R 2 φ P ψ = R 2 φ (ene + F ) ψ + R 2 φ (en/cv B) ψ. Applying the ordering, after some calculations (see Falessi 2017), we obtain: (en/c)v ψ ψ = (ene + F ) R 2 φ ψ + + (en/c)v (B 0 B) R 2 φ ψ M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

25 Gyrokinetic transport: density transport In order to apply this method to descibe the transport of particles, following Hinton and Hazeltine 1976, we act with the projector R 2 φ on the momentum equation and we take the flux surface average: R 2 φ t (nmv ) ψ + R 2 φ P ψ = R 2 φ (ene + F ) ψ + R 2 φ (en/cv B) ψ. Applying the ordering, after some calculations (see Falessi 2017), we obtain: (en/c)v ψ ψ = (ene + F ) R 2 φ ψ + + (en/c)v (B 0 B) R 2 φ ψ In general the term R 2 φ t (nmv ) is non negligible; ψ only for long-wavelength structures (N.B.: different from long-wavelength turbulence), i.e. k L 1, this term can be neglected and thus the moment method is convenient. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

26 Gyrokinetic transport: density transport Collisional fluxes are given by the following expressions (see Hinton and Hazeltine 1976): (en/c)v ψ ψc = F R 2 φ ψ (en/c)v ψ ψnc = (ene 0 + F ) R 2 φ ψ. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

27 Gyrokinetic transport: density transport Collisional fluxes are given by the following expressions (see Hinton and Hazeltine 1976): (en/c)v ψ ψc = F R 2 φ ψ (en/c)v ψ ψnc = (ene 0 + F ) R 2 φ ψ. The fluctuation induced flux is given by (see Falessi 2017): (en/c)v ψ ψgk = (ene ene 0 ) R 2 φ ψ + + (en/c)v (B 0 B) R 2 φ ψ. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

28 Gyrokinetic transport: density transport Collisional fluxes are given by the following expressions (see Hinton and Hazeltine 1976): (en/c)v ψ ψc = F R 2 φ ψ (en/c)v ψ ψnc = (ene 0 + F ) R 2 φ ψ. The fluctuation induced flux is given by (see Falessi 2017): (en/c)v ψ ψgk = (ene ene 0 ) R 2 φ ψ + + (en/c)v (B 0 B) R 2 φ ψ. These relations generalizes the derivation by Hinton and Hazeltine 1976, to a non-quiescent plasma; neoclassical transport theory is obtained as long-wavelength limit. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

29 Gyrokinetic transport: density transport t n ψ = 1 ] [V V nv ψ ψ ψc + V nv ψ ψnc + V nv ψ ψgk M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

30 Gyrokinetic transport: density transport t n ψ = 1 ] [V V nv ψ ψ ψc + V nv ψ ψnc + V nv ψ ψgk additive form does not imply the mutual independence of collisional and turbulent fluxes; in general fluctuations must be taken into account in the calculation of collisional fluxes; fluctuations may enhance the deviation of system from local thermodynamic equilibrium increasing collisional transport; collisions damp fluctuation induced long lived structures which, in turn, regulate turbulent transport; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

31 Gyrokinetic transport: density transport t n ψ = 1 ] [V V nv ψ ψ ψc + V nv ψ ψnc + V nv ψ ψgk additive form does not imply the mutual independence of collisional and turbulent fluxes; in general fluctuations must be taken into account in the calculation of collisional fluxes; fluctuations may enhance the deviation of system from local thermodynamic equilibrium increasing collisional transport; collisions damp fluctuation induced long lived structures which, in turn, regulate turbulent transport; at this order of the expansion, if the gyrokinetic ordering hold, it can be shown, see Sugama and Horton 1996, that the corrections to collisional fluxes are negligible. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

32 Gyrokinetic transport: fluctuation induced flux the calculation of the fluxes in terms of the distribution function requires kinetic theory; in the next slides we will calculate fluctuation-induced fluxes; in order to describe transport on the energy confinement time we need to calculate the distribution function up to O(δ); low frequency, i.e. ω Ω, fluctuations are treated by means of gyrokinetics. M. V. Falessi Figure : Courtesy of Y. Xiao et al., PoP Gyrokinetic transport theory 13/06/ / 32

33 Gyrokinetic transport: pull-back we need to express particle distribution function, i.e. f, in terms of the gyrocenter distribution function, i.e. F ; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

34 Gyrokinetic transport: pull-back we need to express particle distribution function, i.e. f, in terms of the gyrocenter distribution function, i.e. F ; the gyrocenter distribution function F must satisfy F ( z) = F (z) = f(ẑ) and the following relation (see Brizard and Hahm 2007, Frieman and Chen 1982): f = e ρ F =e ρ F e [ ] [ e F e ( + δφ + δφ v m E m c δa ( F e ρ δψ gc m E + 1 B 0 F µ ) + ) 1 B 0 F µ + δa b B 0 F where δψ gc = δφ gc v c δa gc = e ρ ( δφ v c δa) e ρ δψ and... is the average over the gyrophase. ] M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

35 Gyrokinetic transport: moments representation The gyrophase average involves the introduction of Bessel functions as integral operators. At the leading order: e ρ (...) = Î0(...) e ρ v(...) = Î0v b(...) + mc e µî1b (...). where Îng(r) dke ik r Î n (λ)ĝ(k) and În(x) (2/x) n J n (x); using these relations we calculate the moments of the distribution function n = f v, nv = v f v,... M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

36 Gyrokinetic transport: moments representation... obtaining the following expressions: f v = + e m [ Î 0 F e ( F m F δφ + e E v m + δa b B 0 F v E + 1 F B 0 µ 1 F B 0 µ ) ] δψ gc v + ( δφ v c δa ) v + v f v = mc [ e b µî1 F e ( F m E + 1 F ) ] δψ gc B 0 µ v which we substitute in the fluctuation induced flux expression... M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

37 Gyrokinetic transport: fluctuation induced flux... obtaining, at the leading order, the following expressions for the flux induced by electric field fluctuations: (ene ene 0 ) R 2 φ R ψ = e 2 φ δφ Î0 (λ)δḡ where Ḡ = F e F m E δψ gc, and by magnetic field fluctuations: (en/c)v (B0 B) R 2 φ ψ = ( e R 2 φ v ) ( m ) c δa Î0 (λ)δḡ + e µδb Î1 (λ)δḡ v ψ v ψ M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

38 Gyrokinetic transport: fluctuation induced flux... obtaining, at the leading order, the following expressions for the flux induced by electric field fluctuations: (ene ene 0 ) R 2 φ R ψ = e 2 φ δφ Î0 (λ)δḡ where Ḡ = F e F m E δψ gc, and by magnetic field fluctuations: (en/c)v (B0 B) R 2 φ ψ = ( e R 2 φ v ) ( m ) c δa Î0 (λ)δḡ + e µδb Î1 (λ)δḡ v ψ only symmetry breaking perturbations generate transport; v ψ M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

39 Gyrokinetic transport: fluctuation induced flux... obtaining, at the leading order, the following expressions for the flux induced by electric field fluctuations: (ene ene 0 ) R 2 φ R ψ = e 2 φ δφ Î0 (λ)δḡ where Ḡ = F e F m E δψ gc, and by magnetic field fluctuations: (en/c)v (B0 B) R 2 φ ψ = ( e R 2 φ v ) ( m ) c δa Î0 (λ)δḡ + e µδb Î1 (λ)δḡ v ψ only symmetry breaking perturbations generate transport; these expressions can be calculated explicitly knowing the distribution function up to the first order. v ψ M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

40 Gyrokinetic transport: energy transport Following the same methodology we obtain an equation for the energy transport: t p + p /2 + 1 [V ( )] ψ V Q eff ψ ψ ψ = W ψ where W ψ is a collisional heating source and: Q eff ψ ψ Q ψ ψ + cp E R 2 φ ψ. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

41 Gyrokinetic transport: energy transport Following the same methodology we obtain an equation for the energy transport: t p + p /2 + 1 [V ( )] ψ V Q eff ψ ψ ψ = W ψ where W ψ is a collisional heating source and: Q eff ψ ψ Q ψ ψ + cp E R 2 φ ψ. The following relations hold: Q eff ψ ψc = (mc/e) G R 2 φ ψ Q eff ψ ψnc = (ce 0 (p + p /2) + (mc/e)g ) R 2 φ ψ Q eff ψ ψgk = c(e E 0 )(p + p /2) R 2 φ ψ Q (B B 0 ) R 2 φ ψ M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

42 Gyrokinetic transport: energy transport Following the same methodology we obtain an equation for the energy transport: t p + p /2 + 1 [V ( )] ψ V Q eff ψ ψ ψ = W ψ where W ψ is a collisional heating source and: Q eff ψ ψ Q ψ ψ + cp E R 2 φ ψ. The following relations hold: Q eff ψ ψc = (mc/e) G R 2 φ ψ Q eff ψ ψnc = (ce 0 (p + p /2) + (mc/e)g ) R 2 φ ψ Q eff ψ ψgk = c(e E 0 )(p + p /2) R 2 φ ψ Q (B B 0 ) R 2 φ ψ The expressions are formally the same! We can substitute the moments representation... The reason will be described in the next slides using a kinetic approach. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

43 Gyrokinetic transport: comparison with previous works this derivation generalize the work by Hinton and Hazeltine 1976 by including fluctuation-induced transport; a limited amount of works have studied in a self-consistent way collisional transport and fluctuation induced transport. In particular Sugama et al and, recently, Plunk 2009 and Abel et al. 2013; the derivation technique that we have used is significantly more compact and intuitive; results are consistent with previous works and they are derived in the framework of gyrokinetic field theory (see Sugama 2000). Therefore they can be calculated directly from gyrokinetic simulations; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

44 Gyrokinetic transport: comparison with previous works this derivation generalize the work by Hinton and Hazeltine 1976 by including fluctuation-induced transport; a limited amount of works have studied in a self-consistent way collisional transport and fluctuation induced transport. In particular Sugama et al and, recently, Plunk 2009 and Abel et al. 2013; the derivation technique that we have used is significantly more compact and intuitive; results are consistent with previous works and they are derived in the framework of gyrokinetic field theory (see Sugama 2000). Therefore they can be calculated directly from gyrokinetic simulations; differently from these works we did not introduce spatio-temporal averages and, consequently, scale separation assumptions; thus we have shown that postulating a separation of scales between the characteristic spatio-temporal scales of the reference state and of the fluctuations is not required; in the second part of the talk we will show that this is correct only for fluctuations such that kl < δ 1/2 but not in the general case. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

45 Gyrokinetic transport: longer time-scales in a modern magnetic fusion device, i.e. ITER, the time scale of validity of the transport equations that we derived is of the order of the seconds; short compared the expected duration of a pulse, i.e. > 3000s... predictive simulations require an accuracy up to O(δ 3 ) on the fluxes and up to O(δ 2 ) on the distribution function; it would be useful to extend this derivation up to the next order... M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

46 Gyrokinetic transport: longer time-scales in a modern magnetic fusion device, i.e. ITER, the time scale of validity of the transport equations that we derived is of the order of the seconds; short compared the expected duration of a pulse, i.e. > 3000s... predictive simulations require an accuracy up to O(δ 3 ) on the fluxes and up to O(δ 2 ) on the distribution function; it would be useful to extend this derivation up to the next order... anyway the equations already derived could be used to build actuators based on reduced models for the real time control of plasma dynamic evolution. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

47 Gyrokinetic transport: longer time-scales Extending this derivation to longer timescales requires to modify the: Fluctuation induced fluxes conceptually clear, see Brizard and Hahm 2007; an expression for the pull-back operator correct up to O(δ 2 ) is required; in principle derived by means of gyrokinetic field theory (see Sugama 2000); not yet derived for electromagnetic fluctuations. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

48 Gyrokinetic transport: longer time-scales Extending this derivation to longer timescales requires to modify the: Fluctuation induced fluxes conceptually clear, see Brizard and Hahm 2007; an expression for the pull-back operator correct up to O(δ 2 ) is required; in principle derived by means of gyrokinetic field theory (see Sugama 2000); not yet derived for electromagnetic fluctuations. Collisional fluxes classical and neoclassical transport theories provide a framework to derive linear relations connecting the thermodynamic forces and the fluxes satisfying Onsager symmetry; higher order terms, in principle, may lead to non-linear closure relations; non-linear closure relations can be studied by means of Thermodynamic Field Theory, see Sonnino M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

49 Gyrokinetic transport: recap We have derived a set of transport equations describing the evolution of the reference state on the energy confinement time using the moment approach; no scale separation assumptions have been postulated; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

50 Gyrokinetic transport: recap We have derived a set of transport equations describing the evolution of the reference state on the energy confinement time using the moment approach; no scale separation assumptions have been postulated; the derivation scheme allows to treat collisions and fluctuations on the same footing in arbitrary geometry; we have shown the interplay between collisional and fluctuation induced transport; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

51 Gyrokinetic transport: recap We have derived a set of transport equations describing the evolution of the reference state on the energy confinement time using the moment approach; no scale separation assumptions have been postulated; the derivation scheme allows to treat collisions and fluctuations on the same footing in arbitrary geometry; we have shown the interplay between collisional and fluctuation induced transport; in the next slides we will derive a set of transport equation describing microscales and mesoscales applying the theory of phase space zonal structures, see Chen and Zonca 2016, neglecting the effect of collisions. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

52 Phase space zonal structures: definition Zonal structures Mode mode coupling between fluctuating fields can generate toroidal symmetric structures in the density and temperature profiles unaffected by rapid collision-less dissipation, i.e. Landau damping, called zonal structures; zonal structures must satisfy k 0 everywhere, e.g. δe r = ψ ψ δφ(ψ). M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

53 Phase space zonal structures: definition Zonal structures Mode mode coupling between fluctuating fields can generate toroidal symmetric structures in the density and temperature profiles unaffected by rapid collision-less dissipation, i.e. Landau damping, called zonal structures; zonal structures must satisfy k 0 everywhere, e.g. δe r = ψ ψ δφ(ψ). Phase space zonal structures Their counterpart in the phase space are called phase space zonal structures and they importantly regulate turbulence saturation level by scattering instability turbulence to shorter radial wavelength stable domain; they are coherent micro/meso-scale radial corrugations of the reference state, see Chen and Zonca They are associated to deviations from the local thermodynamic equilibrium and thus they enhance collisional transport; therefore they must be properly accounted for a self-consistent description of gyrokinetic transport and they cannot be neglected by means of a scale separation assumption. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

54 Phase space zonal structures: evolution equation Considering µ F0 = 0, and the low β tokamak ordering, we define the zonal structure response: δf z = e ρ δḡz + e m δφ F 0 0,0 E M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

55 Phase space zonal structures: evolution equation Considering µ F0 = 0, and the low β tokamak ordering, we define the zonal structure response: δf z = e ρ δḡz + e m δφ F 0 0,0 E δḡz evolves according to the Frieman-Chen nonlinear gyrokinetic equation: ( t + v + v d ) δḡz = e m F 0 E t δψ gc z c b δψ gc δḡ B 0 ( where δψ gc z = Î0 δφ0,0 v c δa ) 0,0 + m e µî1δb 0,0. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

56 Phase space zonal structures: evolution equation Considering µ F0 = 0, and the low β tokamak ordering, we define the zonal structure response: δf z = e ρ δḡz + e m δφ F 0 0,0 E δḡz evolves according to the Frieman-Chen nonlinear gyrokinetic equation: ( t + v + v d ) δḡz = e m F 0 E t δψ gc z c b δψ gc δḡ B 0 ( where δψ gc z = Î0 δφ0,0 v c δa ) 0,0 + m e µî1δb 0,0. starting from this expression we will derive an equation describing the transport of particles on the energy confinement time. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

57 Phase space zonal structures: transport equation Particle drift velocity due to the equilibrium fields can be written in the following form: v d = v Ω (bv ) thus, exploiting the toroidal symmetry of δḡz, we can write: v d = v [ ( ) F v JΩ θ ψ ( F v ψ B 0 B 0 ) θ Up to the leading order we can re-write the free streaming operator: ( ) F v t + v + v Ω ψ. ]. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

58 Phase space zonal structures: transport equation Particle drift velocity due to the equilibrium fields can be written in the following form: v d = v Ω (bv ) thus, exploiting the toroidal symmetry of δḡz, we can write: v d = v [ ( ) F v JΩ θ ψ ( F v ψ B 0 B 0 ) θ Up to the leading order we can re-write the free streaming operator: ( ) F v t + v + v Ω ψ. ]. Introducing the (drift/banana center) decomposition δḡz = e iqz δḡ z and imposing: [ v Q z = F (ψ) Ω ( v Ω ) ] k z dψ/dr where k z ( i r ), [...] τ 1 dl b v [...] is the average along the unperturbed orbits, we obtain... M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

59 Phase space zonal structures: transport equation... the following equation for the drift/banana center zonal structure response δḡ z : ( t δḡ z = [e iqz e F 0 m E t δψ gc z c )] b δψ gc δḡ. B 0 M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

60 Phase space zonal structures: transport equation... the following equation for the drift/banana center zonal structure response δḡ z : ( t δḡ z = [e iqz e F 0 m E t δψ gc z c )] b δψ gc δḡ. B 0 Using the pull-back expression and the following relation connecting the flux surface average with the bounce average: f v ψ = 4π2 V τ b f n=0 dµde. v / v =± we obtain (see Falessi 2017) the equation for the transport of particles: t δf z v ψ = e m F0 tδφ 0,0 + 4π2 E V τ b dµde ( e iqzî ) [ 0 e m where V = dv/dψ. v v / v =± F ( 0 e E iqzî ) ( c 0 t δφ 0,0 e iqz b δψ gc δḡ B 0 ) ], M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

61 Phase space zonal structures: transport equation... the following equation for the drift/banana center zonal structure response δḡ z : ( t δḡ z = [e iqz e F 0 m E t δψ gc z c )] b δψ gc δḡ. B 0 Using the pull-back expression and the following relation connecting the flux surface average with the bounce average: f v ψ = 4π2 V τ b f n=0 dµde. v / v =± we obtain (see Falessi 2017) the equation for the transport of particles: t δf z v ψ = e m F0 tδφ 0,0 + 4π2 E V τ b dµde ( e iqzî ) [ 0 e m where V = dv/dψ. v v / v =± F ( 0 e E iqzî ) ( c 0 t δφ 0,0 e iqz b δψ gc δḡ B 0 The derivation for the energy transport is analogue. All the transport equations are obtained by taking the moments of this expression. ) ], M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

62 Phase space zonal structures: long-wavelength limit At the leading order: c B b δψ gc δḡ c ( R 2 φ δψ gc ψ δḡ). Therefore, after some calculations (see Falessi 2017), the transport equation can be re-written in a form which remind the results obtained with the moments method: t δf z v ψ = e m 1 V ( V e iqzî 0 ψ [ ( 1 e iqzî ) ( 0 e iqzî ) ] F 0 0 E tδφ 0,0 ) [ce iq zr2 φ δψ gc v δḡ] + v ψ M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

63 Phase space zonal structures: long-wavelength limit At the leading order: c B b δψ gc δḡ c ( R 2 φ δψ gc ψ δḡ). Therefore, after some calculations (see Falessi 2017), the transport equation can be re-written in a form which remind the results obtained with the moments method: t δf z v ψ = e m 1 V ( V e iqzî 0 ψ [ ( 1 e iqzî ) ( 0 e iqzî ) ] F 0 0 E tδφ 0,0 ) [ce iq zr2 φ δψ gc v δḡ] this equation describes the radial oscillations on any length-scale of the density profile in the absence of collisions; + v ψ M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

64 Phase space zonal structures: long-wavelength limit At the leading order: c B b δψ gc δḡ c ( R 2 φ δψ gc ψ δḡ). Therefore, after some calculations (see Falessi 2017), the transport equation can be re-written in a form which remind the results obtained with the moments method: t δf z v ψ = e m 1 V ( V e iqzî 0 ψ [ ( 1 e iqzî ) ( 0 e iqzî ) ] F 0 0 E tδφ 0,0 ) [ce iq zr2 φ δψ gc v δḡ] this equation describes the radial oscillations on any length-scale of the density profile in the absence of collisions; mesoscales are spontaneously created by turbulence; + v ψ M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

65 Phase space zonal structures: long-wavelength limit At the leading order: c B b δψ gc δḡ c ( R 2 φ δψ gc ψ δḡ). Therefore, after some calculations (see Falessi 2017), the transport equation can be re-written in a form which remind the results obtained with the moments method: t δf z v ψ = e m 1 V ( V e iqzî 0 ψ [ ( 1 e iqzî ) ( 0 e iqzî ) ] F 0 0 E tδφ 0,0 ) [ce iq zr2 φ δψ gc v δḡ] this equation describes the radial oscillations on any length-scale of the density profile in the absence of collisions; mesoscales are spontaneously created ( by turbulence; taking the long wavelength limit, i.e. e iqzî ) 0 1 (N.B. this is different from studying long-wavelength turbulence), the RHS reduces to the flux calculated with the moment approach; in particular k z L < δ 1/2 is required. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32 + v ψ

66 Summary we have derived a set of equations describing the evolution of the reference state on the energy confinement time taking into account collisions and fluctuations; no scale-separation assumption is required; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

67 Summary we have derived a set of equations describing the evolution of the reference state on the energy confinement time taking into account collisions and fluctuations; no scale-separation assumption is required; we have shown the mutual interactions between fluctuations and collisions; nv ψ ψc + nv ψ ψnc +... M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

68 Summary we have derived a set of equations describing the evolution of the reference state on the energy confinement time taking into account collisions and fluctuations; no scale-separation assumption is required; we have shown the mutual interactions between fluctuations and collisions; the notion of spatio-temporal scale of variation of the reference state is naturally introduced; nv ψ ψc + nv ψ ψnc +... M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

69 Summary we have derived a set of equations describing the evolution of the reference state on the energy confinement time taking into account collisions and fluctuations; no scale-separation assumption is required; we have shown the mutual interactions between fluctuations and collisions; the notion of spatio-temporal scale of variation of the reference state is naturally introduced; we have shown how the derivation technique could be applied to derive equations valid on loger time-scales. nv ψ ψc + nv ψ ψnc +... M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

70 Summary we have described fluctuations induced transport on an arbitrary length-scale by means of phase space zonal structures; meso-scales are spontaneously generated by the dynamics eventually invalidating a scale separation assumption; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

71 Summary we have described fluctuations induced transport on an arbitrary length-scale by means of phase space zonal structures; meso-scales are spontaneously generated by the dynamics eventually invalidating a scale separation assumption; finally we have shown how the long-wave limit of the zonal structure evolution is consistent with the moment approach; ( e iqzî ) 0 1 M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

72 Future perspectives study the evolution equation for phase space zonal structures in the collisional case; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

73 Future perspectives study the evolution equation for phase space zonal structures in the collisional case; discuss the choice of the collision integral; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

74 Future perspectives study the evolution equation for phase space zonal structures in the collisional case; discuss the choice of the collision integral; obtain the expression for the collisional fluxes by taking the long wave-length limit of this equation thus providing a unified theory of collisional and fluctuation induced transport; M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

75 Future perspectives study the evolution equation for phase space zonal structures in the collisional case; discuss the choice of the collision integral; obtain the expression for the collisional fluxes by taking the long wave-length limit of this equation thus providing a unified theory of collisional and fluctuation induced transport; development of a reduced model for numerical implementation. Evolution equations for the reference state and for the mesoscales of the system, i.e. PSZS; comparison with existing multiscale gyrokinetic transport codes, i.e. Trinity, and global gyrokinetic codes. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32

76 Thank you for your attention. M. V. Falessi Gyrokinetic transport theory 13/06/ / 32