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Analytical and computational paradigms for plasma turbulence-I A Thyagaraja UKAEA/EURATOM Fusion...
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Transcript of Analytical and computational paradigms for plasma turbulence-I A Thyagaraja UKAEA/EURATOM Fusion...
Analytical and computational paradigms for plasma turbulence-I
A Thyagaraja
UKAEA/EURATOM Fusion Association
Culham Science Centre, Abingdon, OX14 3DB, UK
Trieste Plasma School, October, 2003
Acknowledgements
• Professor Swadesh Mahajan for inviting me• Peter Knight,Terry Martin, Jack Connor, Chris Lashmore-Davies
(Culham) • Marco de Baar, Erik Min, Hugo de Blank, Dick Hogeweij, Niek Lopes
Cardozo (FOM)• Xavier Garbet, Paola Mantica, Luca Garzotti (EFDA/JET)• Nuno Loureiro (Imperial College)• Michele Romanelli (Frascati)• Dan McCarthy (USEL)• EPSRC (UK)/EURATOM
Synopsis of Part I
• What is plasma turbulence?
• What are the key problems to be addressed?
• Simple example of the advection-diffusion equation and “phase-mixing effects of flows”
• Analytical paradigm for zonal flow generation
• Summary
What is plasma turbulence?
• In principle, a plasma can be maintained (driven) by sources against collisional (dissipative) losses.
• Resulting current/pressure profiles are strongly unstable.
• Instability spontaneously breaks symmetry in space & time.
• Growing modes nonlinearly saturate, leading to turbulent fluxes, spectral cascades and anomalous transport.
• Equilibrium and turbulence cross-talk on a range of scales, especially in the mesoscales.
Characteristics of tokamak turbulence
• “Universal”, electromagnetic (dn/n and dj/j comparable!), between system size and ion gyro radius; between confinement (s) and Alfvén (ns) times:
• Plasma is “self-organising”, like planetary atmospheres (Rossby waves=Drift waves).
• Transport barriers connected with sheared flows, rational q’s, inverse cascades/modulational instabilities (Hasegawa).
• Analogous to El Nino, circumpolar vortex, “shear sheltering” (J.C.R Hunt et al).
))(/( kLv snth
Why is turbulence important?
• Usually, though not invariably, turbulent losses are more severe than neoclassical.
• Magnetic shear (q’) and E x B flow shear seem to play key roles in formation and dynamics of high gradient regions called Transport Barriers (ETB’s or ITB’s) identified in experiments.
• Understanding and control crucial to power plant issues: economics, divertor loading, ash removal etc.
• Difficult unsolved problem. Much recent progress through complementary approaches, close theory/expt interaction.
Key Concepts: q and zonal flow
• “Mode rational surface” when m=nq; long wave length MHD modes may occur. “Magnetic shear” dq/dr, an important stability parameter;dynamo effects.
• Plasma knows “number theory”, resonances analogous to Saturn’s rings occur -KAM theory
• Radial electric field associated with sheared zonal flow (from ExB drifts); influences stability: Taylor flow analogy!
• Inverse and direct cascades determine turbulent saturation and transport.
Challenges for Theory
• Explain observations, scalings, thresholds.
• Predict phenomena (ITB’s, transitions, sawteeth, ELM’s, impurity behaviour, pinches..)
• Calculate with adequate accuracy, faster than experiment, consistent with both qualitative and quantitative facts.
• Suggest new diagnostics, improved performance, better engineering design.
Challenges for Experiments
• Comprehensive, time-space resolved diagnostics of T, n, q, E, Z needed.
• Measurements of turbulent spectra (high & low k).
• Transients: pellets, modulated heating.
• Adequate inter and intra machine comparisons.
• Only starting to be met in JET, ASDEX, TORE-Supra, DIIID, MAST, NSTX, JT-60U, TEXTOR, FTU..
What are zonal flows?
• Poloidal E x B flows, driven by turbulent Reynolds stresses: “Benjamin-Feir” type of modulational instability, “inverse cascade” recently explained in Generalized Charney Hasegawa Mima Equation.
• Highly sheared transverse flows “phase mix” and lead to a “direct
cascade” in the turbulent fluctuations.
• Enhances diffusive damping and stabilizes turbulence linearly and nonlinearly.
• Confines turbulence to low shear zones.
)()(x
fD
xy
fx
t
fvy
The Advection-Diffusion Equation
Sheared velocity in combination with diffusion changes spectrum
“Reynolds number” measures shear/diffusion: DxR Lv xy/))(
2'(
Damping rate is proportional to Dv y
3/13/2
'
Spectrum discrete, “direct cascade due to phase mixing”
“Jets” in velocity lead to “ghetto-isation/confinement” to low shear regions
Eigenvalue spectrum for Vy(x)=10x; D=0.001. For D=0, it will lie on the real axis!
)()(x
fD
xy
fx
t
fvy
Dispersion curves of growth rate versus zonal flow wave number are plotted for
different value of alpha=a/Ln; note maximum growth at intermediate wave numbers
The pump wave is depleted by the zonal flow and side-bands; the four-wave system has two invariants which stay constant for several growth times. Full nonlinear evolution involves numerical solution of GCHME
Plots of side-band and zonal flow amplitudes and the Enstrophy invariant for 12 growth times. Note slow growth of the “four-wave invariant” towards the end.
Surprising phenomenon of “beat generation” of long-wavelength modes by high-k ones at later times. Note mq=1 is not the fastest growing mode initially. At later times it “catches up”!
Discussion• Even the simplest, linear advection-diffusion equation reveals important
structural features of the effects of zonal flows.
• Shear-induced damping, “ghetto” effects of jets.
• Generalized Charney Hasegawa Mima equation is the simplest nonlinear, conservative 2-d model for drift wave generation of zonal flows.
• Modulational instability of a pump demonstrated.
• Conservative model but displays many generic features!
• Clear example of “inverse cascade” and beat generation.• • Very similar mechanism thought to lead to turbulent dynamo effects in
induction equation “zonal current models”
Conclusions
• We have looked at basic concepts of zonal flows and their effect using analytically tractable models which give insight.
• There is a lot more to zonal flow generation mechanisms and effects!
• In the next lecture we will consider a “first-principles based” approach to simulations of electromagnetic turbulence in current fusion devices called tokamaks.
• The issues which arise in such simulations will be discussed in the next Lecture in the light of the simpler models which will be contrasted with the “real thing”.