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Analytical and computational paradigms for plasma turbulence-I

This paper discusses the analytical and computational approaches to understanding and modeling plasma turbulence, with a focus on tokamak turbulence and zonal flows.

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Analytical and computational paradigms for plasma turbulence-I

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  1. 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

  2. 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

  3. 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

  4. 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.

  5. 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).

  6. 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 calledTransport 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.

  7. 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.

  8. 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.

  9. 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..

  10. 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.

  11. The Advection-Diffusion Equation Sheared velocity in combination with diffusion changes spectrum “Reynolds number” measures shear/diffusion: Damping rate is proportional to Spectrum discrete, “direct cascade due to phase mixing” “Jets” in velocity lead to “ghetto-isation/confinement” to low shear regions

  12. Eigenvalue spectrum for Vy(x)=10x; D=0.001. For D=0, it will lie on the real axis!

  13. Dotted lines initial disturbance, solid line rescaled solution for large t

  14. 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

  15. 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

  16. 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.

  17. 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”!

  18. 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”

  19. 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”.

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