Intermittent Oscillations Generated Intermittent Oscillations Generated by ITG-driven Turbulenceby ITG-driven Turbulence

US-Japan JIFT WorkshopDecember 15th-17th, 2003

Kyoto University

Kazuo Takeda, Sadruddin Benkadda*, Satoshi Hamaguchi and Masahiro Wakatani

Graduate School of Energy Science, Kyoto University*CNRS URA 773, Université de Provence, France

Table of ContentsTable of Contents

1. Motivation Background and earlier studies

2. Model equations Extension of the earlier model

3. Highlight data Our low degree-of-freedom model

4. Summary

MotivationMotivation

1. Backgroundi. Ion-temperature-gradient (ITG)-driven turbulence and self-

generated sheared plasma flowsii. Low degree-of-freedom models are useful to understand nonlinear

physics.

2. Earlier studiesi. 11 ODE model given by Hu and Horton G. Hu and W. Horton, phys. Plasmas 4, 3262 (1997)

3. Extensioni. Increasing the degree of freedom from 11 to 18 in

order to include 3rd harmonics

EquationsEquations

where

Toroidal ITG mode is described by the following equations, W. Horton, D-I. Choi and W. Tang, Phys. Fluids 24, 1077 (1981)

()()[]()222222,1,,,1,lnlniiiiieinTiiipgKgtyyppKptyKTTLLdTdnφφφφφμφφφκηη ⊥⊥⊥⊥⊥⊥≡∂∂∂⎡⎤ −+ =−+ −+∇ ∇ ∇ ∇∇⎣⎦∂∂∂∂∂+=−+∇∂∂≡+=

iKφμ: electrostatic potential (fluctuation): ion temperature gradient: viscosity

pgκ: ion pressure (fluctuation): effective gravity: thermal conductivity

Normalization

The standard drift wave units used for normalization.()()()()()000000000,,,,,,.ssnsnesniiesnesnesxxyyttLceLBTppLTpTeLBTeLBT≡ρ≡ρ≡φ≡φρ≡ρμ≡μρκ≡κρ

And are assumed for numerical calculation.

.,.,.g===005004001μκ

Linear Stability AnalysisLinear Stability Analysis

The most unstable wave number is estimated as

In this study, the following wave number is assumed,

From the dispersion relation, the linear growth rate is as follows.

()2221xyikkkgK⊥=−=:

()(){} ()()(){} ()2224222224221121,1141yikkyiiykgKkikkkDkDkgKkikkkkgKkωκμκμ⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥⎡⎤=−−−++±+⎣⎦⎡⎤=−−++−−+⎣⎦

()()()()215,215xyxiyikkkgKkgK= =−=−⇒

Linear Growth RateLinear Growth Rate

Low-degree-of-freedomLow-degree-of-freedom ModelModel

If are neglected, these agree with the 11 ODE model.

G. Hu and W. Horton, Phys. Plasmas 4, 3262 (1997)

Low-order modes to describe nonlinear behaviour of a toroidal ITG mode in a simple slab model are as follows,

{} 0,,0,,,33cscspφ

()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()φ=φ+φ++φ+φ+φ+φ+φ+φ+φ=++0001212133303200121,,sinsin2sincossin2cossinsinsin2sin,,sinsinsin3sin3cossi2sn3sin,snin3icccscssxxxyxyxxxyxyxyxyxxxtkxtkxkytxyttkxtkxtkxkytkxkytkxkytkxkykxkyptkxpxytptkxptkxptkx()()()()()()()()()()()()()()()()+++++21323cossin2cossisin3cossin3nsinsinsin.2sincscssxyyxyxyxxyykypptkxkypttkxkyptkxkyptkxxkkkyy

Numerical Procedures

011xVxuipvdpNpvdVdxK⎛⎞κ=+⎜⎟κ⎝⎠≡+∫

(),xyyxyRyyRSvvdyLvvvxStx≡=∂∂=∂∂∫

For studying the anomalous transport induced by the nonlinear ITG mode,Nusselt number have been calculated.

The vorticity equation contains the Reynolds stress which generates sheared flows.

Bifurcation Process I-IBifurcation Process I-I

Time evolution of(a) kinetic energy K and (b) Nusselt number Nu in the case of Ki=0.3, whereThe system converges to a steady state.

()1.=+eiiiKTTη

Bifurcation Process I-IIBifurcation Process I-II

Phase space ‘K0 (m=0 mode)-Nu’ and power spectrum of K1 (m=1 mode) for Ki=0.3, where

()1.=+eiiiKTTη

Bifurcation Process II-IBifurcation Process II-I

Time evolution of(a) kinetic energy K and (b) Nusselt number Nu in the case of Ki=0.4, where The system converges to a periodic oscillation.

()1.=+eiiiKTTη

Bifurcation Process II-IIBifurcation Process II-II

Phase space ‘K0 (m=0 mode)-Nu’ and power spectrum of K1 (m=1 mode) for Ki=0.4, where

()1.=+eiiiKTTη

Bifurcation Process III-IBifurcation Process III-I

Time evolution of(a) kinetic energy K and (b) Nusselt number Nu in the case of Ki=0.5, where More modes are excited.

()1.=+eiiiKTTη

Bifurcation Process III-IIBifurcation Process III-II

Phase space ‘K0 (m=0 mode)-Nu’ and power spectrum of K1 (m=1 mode) for Ki=0.5, where

()1.=+eiiiKTTη

Bifurcation Process IV-IBifurcation Process IV-I

Time evolution of(a) kinetic energy K and (b) Nusselt number Nu in the case of Ki=0.6, where Chaotic oscillations appear.

()1.=+eiiiKTTη

Bifurcation Process IV-IIBifurcation Process IV-II

Phase space ‘K0 (m=0 mode)-Nu’ and power spectrum of K1 (m=1 mode) for Ki=0.6, where

()1.=+eiiiKTTη

Intermittent Behaviour IIntermittent Behaviour I

Time evolution ofkinetic energy K for Ki=4, where

Intermittent bursts (so called avalanche) are observed.

()1.=+eiiiKTTη

Intermittent Behaviour IIIntermittent Behaviour II

Time evolution of (a) Nusselt number Nu and(b) Reynolds stress SR in the case of Ki=4. Nu and SR burst at the time when the ITG mode grows rapidly.

Intermittent Behaviour IIIIntermittent Behaviour III

Real space contours of at (a) bursting, (b) laminar and (c) tilting phase for Ki=4, where

()1.=+eiiiKTTηφ

Intermittent Behaviour IV

This intermittency is caused by the competition of the following 3 factors;

1. Generation of sheared flows due to nonlinear coupling between higher harmonics, and suppression of the ITG turbulence by the sheared flows.

2. Gradual reduction of sheared flows due to viscosity.3. Rapid re-growth of the ITG modes due to the reduction

of the stabilizing effect by the sheared flows.

Scaling LawScaling Law

The scaling law

log Nu 3 log ∝ Ki

is obtained.It is suggested that the improved confinement is related to the intermittency of the system.

SummarySummary

1. Sheared flows are generated by the nonlinear mode coupling, and a bifurcation corresponding to an L-H transition has been obtained.

2. In the strongly turbulent regime, an intermittent behaviour appears. This intermittency is caused by the competition of the 3 factors.

3. A scaling law, log Nu 3 log ∝ Ki, has been obtained between Nu and Ki. Improved confinements are related to the intermittency of the system.

4. Essential nonlinear behaviour of the system can be at least qualitatively accounted for by nonlinear interaction of several low order harmonics.

New physical phenomena can be obtained by using 18 ODE model.

Intermittent Behaviour VIntermittent Behaviour V

Phase space ‘K0-Nu’ at (a) bursting, (b) laminar and (c) tilting phase in the case of Ki=4.

18 ODEs I18 ODEs I

()()()()()()()()()(),,,φφφφφφφφφμφφφφφφμφφφφφφμφφφφφφ⎡⎤+=−+−−⎣⎦+=−−+=−−+=−−−−+20xy22cssccssc401xx12122323x10222cssc402xxyx1313x20222sccs403xxyx1212x3c22220s220s11xy2x121x21kd1kk35kdt2d14k8kk16kdtd919kkk81kdt2d1kkkk2k4kdt()()()()()()()()()(),,φφφφφμφφφφφφφφφφφμφφ⎡⎤−−−−−−+−⎣⎦⎡⎤+=−−−+−−−+−−−−⎣⎦+=−−220s220s2ss4c3x232x32i1y1y111s22220c220c220c220c2cc4s11xy2x121x213x232x32i1y1y111c22222xy12k4k3k9k1gKkkgkpkd1kkkk2k4k2k4k3k9k1gKkkgkpkdtd1kkkkdt()()()()()()()()()()(),,φφφφφφφμφφφφφφφφφμφφφφ⎡⎤+−−−−−−+−⎣⎦⎡⎤+=−+−−−+−−−−⎣⎦+=−−+20s220s220s2ss4cx113x131x31i2y2y222s22220c220c220c2cc4s22xy1x113x131x31i2y2y222c22220s33xy2x12kk3k9k1gKkkgkpkd1kkkkkk3k9k1gKkkgkpkdtd1kkkk2kdt()()()()()(),,φφφμφφφφφφφμφ⎡⎤−−−−+−⎣⎦⎡⎤+=−+−+−−−−⎣⎦220s2ss4c1x21i3y3y333s22220c220c2cc4s33xy2x121x21i3y3y3334k1gKkkgkpkd1kkkk2k4k1gKkkgkpkdt

18 ODEs II18 ODEs II

where,,,.=+=+=+=2222222222123xyxyxyxyxyκκκκ4κκκ9κκκκκ2

()()(),,,φφφφφφφφκφφφφφφκφφφφκ=−+−+−+−−=−+−+−−=−+−−=20xysccssccssccssccs2011212212132322323x102cssccssccssc202xy111113133131x2 20xycssccssc20312122121x3c21xykdpppppppppkpdt2dpkpppppp4kpdt3kdppppp9kpdt2dpkpdt()()()()()()()()(),,φφφφφφφφφκφφφφφφφφφκφφφ⎡⎤−−−+−−−−−⎣⎦⎡⎤=−−−−+−−−+−⎣⎦=−+0ss00ss00ss00ss0s2c1221211223323223iy111s20cc00cc00cc00cc0c2s1xy1221211223323223iy111c20ss00s2xy111213p2pp2pp3ppKkkpdpkpp2pp2pp3ppKkkpdtdpkpppdt()()()()()()(),,,φφφφκφφφφφφφκφφφφφκφ⎡⎤−−−−−⎣⎦⎡⎤=−−+−−−+−⎣⎦⎡⎤=−+−−−⎣⎦=−−s00ss0s2c313113iy222s20cc00cc00cc0c2s2xy111113313113iy222 c20ss00ss0s2c3xy12212112iy333s20cc3xy122p3ppKkkpdpkpppp3ppKkkpdtdpkpp2ppKkkpdtdpkppdt()(),φφφφκ⎡⎤+−+−⎣⎦00cc0c2s12112iy3332ppKkkp