Nonlinear entropy transfer via zonal flows in gyrokinetic plasma turbulent transport

Transcription

1 Nonlinear entropy tranfer via zonal flow in gyrokinetic plama turbulent tranport Motoki Nakata 1, Tomo-Hiko Watanabe, Hideo Sugama, and Yauhiro Idomura 1 1 Rokkaho-Site of Japan Atomic Energy Agency National Intitute for Fuion Science 1t APTWG International Conference, June 147th, 11

2 Summary Nonlinear entropy tranfer procee in the toroidal ETG/ITG turbulence have been invetigated by mean of gyrokinetic Vlaov imulation. Then, highlighted finding in the preent tudy are (1) Elongated pectrum (higher-kx) of the ITG-driven heat flux while the confined pectrum (kx~) of the ETG-driven heat flux, () Succeive entropy tranfer to higher-kx mode via the trong triadinteraction with high-amplitude zonal flow in toroidal ITG turbulence, (3) Entropy tranfer within low-wavenumber mode due to the three-modeinteraction among non-zonal mode in toroidal ETG turbulence. The nonlinear entropy tranfer analyi provide one with a ytematic method for examining both the nonlinear ZF-DW interaction and their effect on the turbulent tranport level. ( The reult of thi work i ubmitted to PoP )

3 Introduction - Anomalou (or turbulent) heat tranport driven by micro-intabilitie - main caue of ion heat tranport --> Ion temperature gradient (ITG) driven mode - main caue of electron heat tranport --> Trapped electron mode (TEM) --> Electron temperature gradient (ETG) driven mode The zonal-flow dynamic in the turbulent tranport procee i a key iue. Zonal flow v. Drift-wave turbulence - For quantifying the effect of zonal flow on the tranport regulation, ytematic analye of nonlinear interaction among zonal and non-zonal mode are neceary. --> Nonlinear entropy tranfer analyi i applied to ITG and ETG turbulence k a ρti ρte MHD ITG TEM ETG e.g., Bipectrum analyi Y. Nagahima et al., PRL(5)

4 Introduction cont. -- Why entropy and it tranfer in turbulent plama? - The entropy variable δs δ f (g) k reflect the phae-pace tructure of the ditribution function, and i directly connected with the tranport flux and the colliional diipation. - Nonlinear interaction among vortice/flow in kinetic turbulent tranport procee are decribed naturally by the entropy tranfer function, which i regarded a a kinetic extenion of the Reynold tre. (cf. Slide 3 & 4) Recent work on entropy tranfer procee - Numerical obervation of the hell-to-hell local tranfer of the entropy variable in ub- ρ cale of toroidal ITG turbulence. A. B. Navarro et al., PRL(11) --> Analye with iotropic hell in D k -pace obcure the zonal-flow dynamic. In thi tudy, the critical role of zonal flow on the entropy tranfer procee i revealed. cf., for example, J. A. Kromme et al., PoP(1994), H. Sugama et al., PoP(9) T l, l l T l, l, l k ρ1 Shell-to-hell local entropy tranfer A. B. Navarro et al., PRL(11)

5 GK eq. Gyrokinetic entropy balance equation -3- Entropy balance eq. t + b + iω D µ m b B t Navier-Stoke eq. Energy balance eq. Energy tranfer func. δ f (g) k c M p,q B δψ p δ f q (g) (g) C δ f k = F M iω T iω D b e δψ k δs k + W k = η Q k + T k + D k Entropy variable heat flux Entropy tranfer function J. A. Kromme et al., PoP(1994) H. Sugama et al. PoP(9) A. A. Schekochihin et al., PPCF(8) Analogy with iotropic fluid-turbulence ytem T t u k + M k u p u q = F k νk u k t E Vicou diipation k = T k + D k T k = δ k+p+q= Re M k u p u q u k p q Colliional diipation k-pace tructure of the energy tranfer function S. Kida, JFM(1997)

6 The triad entropy tranfer function t J[k p, q ] Symmetrie: Detailed Balance Relation: δs k + W k = η Q k + T k + D k T k = p c B b (p q ) q δ k +p +q = J[k p, q ] J[k p, q ] = J[k q, p ] = J[ k q, p ] J[k p, q ] + J[p q, k ] + J[q k, p ] = kzf : zonal flow p : turbulence q : turbulence ky tranport-driving mode q p kzf -4- entropy tranfer function d 1 Re[δψ p h q h k δψ q h p h k ] F M triad entropy tranfer function kx

7 Outline of the entropy tranfer procee intbilityaturation phae -5- J[p q, k zf ] toroidal ITG (Tranport-driving mode) J[k zf p, q ] J[p q, k ] toroidal ETG (Tranport-driving mode) J[q k, p ] (High-kx mode) Zonal mode (with high-amplitude and high-kzf ) (Low-kx mode) (Low-kx mode) teady tate J[p q, k zf ] (Tranport-driving mode) J[k zf p, q ] J[p q, k ] (Tranport-driving mode) J[q k, p ] (High-kx mode) Zonal mode (work a a mediator on the entropy tranfer) (Low-kx mode) (Low-kx mode)

8 mode k x δφk x, ky -6- [(c) and (d)] for the ITG (upper row) and ETG (lower row) d!s dw T D Term in Eq. (1) Comparion of entropy balance in toroidal dbs"trb# D"trb# "trb# ITG andetg dw turbulence. /dt d Q "zf# T (g) (trb) 1 d (trb) (trb) δs + W = η Q T + D(trb), δs dt 1 η Q toroidal ITG turbulence with =.4 Turbulence(non-zonal)-part dbs"trb# D"trb# Term in Eq. (1) Term in Eq. () (1) 1 1 dw"trb# T"zf# d Q T = Term in Eq. (1) Term in Eq. () Entropy balance relation: Turbulence-part (trb) D > k(trb) d δ fk /FM toroidal ETG turbulence with =.4 Turbulence(non-zonal)-part Time t [Ln/vt] ime evolution of each term in the entropy balance relation of the zonal-flow part, Eq Time t [Ln!/vt]

9 finite k x(lower modeflow kx δφkx, ky [(c) and (d)] for the ITG (upper row) and ETG (lo δφkx, ky [(c) and (d)] for (upperinteraction row)of and ETG row) mechanim turbulence uppreion duethe to ITG nonlinear with zonal x mode kof x Entropy balance relation: Zonal flow-part cued in Sec. 5.4, baed on the pectral analyi of the entropy tranfer function. -7- Term in Eq. () Term in Eq. () d! S /dt d! S Comparion of vortex tructure and zonal flow in toroidal ITG and Comparion of entropydwbalance in toroidal ITG and ETG turbulence. dw T (g) T ETG turbulence d 1 δs + W =D T + D, δs d δ f /F 1 MD k number pectra of turbulent dt vortice and zonal flow in the ITG and ETG turbulence are k intenity are in thi ection. The time evolution of ky -pectra of the fluctuation hown T = D > [ and ] and the intenity toroidal ITG turbulence with =.4 Zonal flow-part d!s dw T D 11. () Time t [Ln/vt] Term in Eq. () Term Termin ineq. Eq.() () 5.4 (d), where the treamer intenity δφkx =, ky kx δφkx, ky [(c) and (d)] for the ITG (upper row) and ETG (lower row) x mode toroidal ETG turbulence with =.4 Zonal flow-part Time t [Ln/vt] Time evolution of each term in the entropy balance relation of the zonal-flow Eq. of each term in the entropy balance relation of the zonal-flow FIG. 5.3: Timepart, evolution

10 The role of ZF in the intability-aturation -8- d dt δs (trb) + W (trb) = η Q T Heat flux Tranfer to ZF Col.-diipation + D (trb) d dt δs + W = T ZF-generation + D The ratio T /( D (trb) ) i an important factor indicating the effect of ZFgeneration on the aturation of turbulent tranport level. T i T e /( D (trb) i ) = /( D (trb) e ) = for ITG for ETG In the ITG cae, the contribution of ZFgeneration to the tranport aturation i much larger than that in the ETG cae. + W / δφ kzf k zf ZF-inertia defined by M δs indicate a difficult-togenerate factor of the zonal flow for a given ource of T. M i /M e = ITG-driven ZF i much tronger than ETG-driven one.

11 -9- Comparion of vortex and flow tructure Vortex and flow tructure in the toroidal ITG and ETG turbulence Contour plot of the potential fluctuation in the teady tate toroidal ITG turbulence toroidal ETG turbulence Streamer dominated tructure (High tranport level) ZF-dominated tructure (Low tranport level)

12 e..4.6 ηe-. Qek /η Q(max) wavenumber kx [! (c) ETG ηi Qik /ηi Q(max) wavenumber kx [! For the ITG cae, the broad-kx.4 pectrum i formed ( through the trong. interaction with ZF. ) ky [! wavenumber ky [! wavenumber ky [! ITG wavenumber kx [! wavenumber kx[! (d) : ForWavenumber the ETG cae, the confined FIG. pectra of the turbulent hea.4 pectrum i formed in the lowerfluctuation δφk [(c) and (d)] in the teady tate wavenumber region.. 1 ky [! wavenumber ky [!.6 w Wavenumber pectra of turbulent heat flux 5.4 Nonlinear entropy tranfer via zonal mode

13 Entropy tranfer to ZF in aturation phae 1- T /! Q aturation phae toroidal ITG toroidal ETG Time t [L n /v t ] - In the aturation phae, the entropy tranfer from non-zonal to zonal mode i ignificant in ITG, and i much larger than that in ETG, i.e.,. J i [k zf p, q ]/η i Q i J e [k zf p, q ]/η e Q e Entropy tranfer from non-zonal to zonal mode: J[kzf p,q]/ηq.6 1x1-4 wavenumber q y [! wavenumber q y [! wavenumber q x [!.6 k zf = k zf =.141 ITG ETG wavenumber q x [! te] gain 5x1-5 x1-5x1-5 lo x1-4 1x1-4 gain 5x1-5 x1-5x1-5 lo x1-4

14 Entropy tranfer to ZF in teady phae - T /! Q toroidal ITG toroidal ETG teady phae Time t [L n /v t ] - In the teady phae, the entropy tranfer from non-zonal to zonal mode become quite weak in ITG, i.e. J i [k, zf p, q ]/η i Q i 1. - Low-k mode interaction are dominant in ETG. Entropy tranfer from non-zonal to zonal mode: J[kzf p,q]/ηq.6 1x1-4 wavenumber q y [! wavenumber q y [! wavenumber q x [!.6 k zf = k zf =.141 ITG ETG wavenumber q x [! te] gain 5x1-5 x1-5x1-5 lo x1-4 1x1-4 gain 5x1-5 x1-5x1-5 lo x1-4

15 x1 3- wavenum wavenum x1 Entropy tranfer to NZ-mode in teady phae of ITG -5x x Spectra of Ji [p q, kzf ]/ηi Qi (entropy tranfer to non-zonal(nz) mode) x1 x1 er 5 cf. Nonlinear entropy tranfer via zonal flow in toroidal plama turbulence J [p q, k ] + J [q k, p ] = J [k q, q ] the detailed balance relation: i. zf zf i. zf i -.6 px=. py= x1-5. x x px=.141 py=.5 wavenumber qx [! (c).6-4 1x1 wavenumber qy [!.4 wavenumber qy [! wavenumber qy [!.6 wavenumber qx [!.6-4 1x x1. px=.8 py=.5 1x1-4 5x1-5 x1 x x x x1-4 x1 x wavenumber q [! wavenumber q [! ] wavenumber qx [! ] x x ti ti (c) x1 1x1 p =.8 p =.141 number qy [! number qy [! -.4 In the teady tate of ITG.4 turbulence, the entropy of the primary mode 5x1 5x1-5 (tranport-driving mode) i tranferred to the ihigher-k x! t! 3.. ucceively. toroidal ITG turbulence, where the time-average taken over mode (with le contribution x1the triad-interaction x1 to heat tranport) via with a a mediator. ) zonal mode. ( ZF work x py=.5 x Wavenumber pectra of the triad tranfer function normalized by the FIG. 5.11: py=.5 J i [ p q, kzf ]/ηi Qi, for three -5 different p with py =.5ρ 1 ti (fixed) in the

16 x1 x1 4- wavenum wavenum Entropy tranfer to NZ-mode in teady phae of ETG -5x x Spectra of Je [ p q, kzf ]/ηe Qe (entropy tranfer to non-zonal mode) x1-4 x1 er 5 cf. Nonlinear tranfer via zonal flow in relation: toroidal plamaj turbulence, kzf.4 ] +.6 Je [q -.6 kzf, -.4 p ]-. = J q,.6 q ] theentropy detailed balance e [p q. e [k.zf x1. x x x py=.5 wavenumber qx [! (c) (c).6-4 1x1 wavenumber qy [! px=. py=.5 wavenumber qy [!.4.4 5x1-5. px=.141 py=.5 1x1-4 5x1-5 x1 x x x x x1-4 x wavenumber qx [! wavenumber qx [! wavenumber qx [! (c) x1.6 1x1 p =.75 p = x py=.5 number qy [! number qy [! wavenumber qy [!.6 wavenumber qx [! (c).6-4 1x1 p =.75 x Wavenumber pectra of J e [p q, kzf ]/ηe Qe, for three different FIG. 5.13: py= ρ 1 the teady tate of toroidal ETG turbulence, where the of ETG the inucceive entropy tranfer totimete (fixed) 5x1 turbulence, 5x1-5 over! t! 3.and the nonlinear interaction i no.longer oberved, - In the teady tate. the higher-kx mode x1 among low-wavenumber non-zonal mode are x1 dominant

17 -5x1 wav Comparion between ITG and ETG -.6 x kzf =.141 py = qy =. 1x x1 x1-5x x wavenumber px [l From.6 1x1-4 To wavenumber px [l kzf =.75 py = qy =. 1x1-4 5x1-5 x1-5x x wavenumber px [l From - Succeive entropy tranfer to the higher-kx mode kzf =.75.4 and the entropy tranfer within in ITG, low-kx region in -5 5x1 ETG, are reponible for the different pectral hape. of turbulent heat flux. er qx [l -4 Je-.6 [p -.4 q -., kzf ]/η (p x, q x ) eq.e.4ge.6 wavenumber qx [l wavenumber qx [l To Ji [p q, kzf ]/ηi Qi Gi (p x, q x )