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

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

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”.