Hiérarchies des modèles fluides

M. Ottaviani

(Ass. EURATOM-CEA, DRFC, Cadarache)

Plan de l’exposé

• Rappel: échelles spatiales et temporelles

• Développement de basse fréquence 1: des équations de Braginskii aux modèles fluides de basse fréquence

• Développement de basse fréquence 2: de l’équation girocinétique aux modèles girofluides

• Exemples (issus de mes travaux, anciens ou en cours)

• Conclusions

Space and time scales in magnetic confinement

Low frequency dynamics: scales of interest

epea i c/

ITG

ETG

TEM

short- wave length ITG

plasma profile

zonal flows

zonal flows

E

a/vTe

a/vTi

Time

Space (r)

three scales

global

blobs

MHD

macro micromeso

From kinetic to fluid: classical approach

• Kinetic equation (collision dominated) ==>

• Chapman-Enskog expansion ==>

• Braginskii equations (1960) ==>

• low frequency expansions (drift ordering) ==>

• models, eventually ad hoc closures

From kinetic to fluid: gyrofluid approach (since 1990)

• Kinetic equation, non collisional (Vlasov) ==>

• low frequency expansion ==>

• gyrokinetic equation ==>

• moments, closures ==>

• hierarchy of gyrofluid models

Braginskii equations

Drift expansion

Generalized Ohm’s law

Hasegawa Mima model (1978)

Hasegawa-Wakatani model (1982)

Four field model, Hazeltine et al (1985)

Problems

Gyrofluid approach

Moments of the GKE (m+n models), Dorland (1992)

The issue of GF closures

Further developments

The 3+1 model from Beer (1994)

Hasegawa-Mima, revisited

Example 1. Collisionless reconnection: sawtooth crash

Result (Ottaviani & Porcelli, 1993)Current sheet formation on a fast timescale, controlled by el. inertia.

Example 2. Tearing-modes at low collisionality

• Analysis of tearing modes (reconnecting modes) in regimes of low collisionality. The drift frequency exceeds the collision frequency. Electron inertia comes into play

• Employs the four-field model

• Study linear stability criteria.

• Search for bistability (coexistence of states).

• Coupling to drift waves

Electromagnetic solution, islandlocalized electric potential

Electrostatic solution, small islandde-localized electric potential, drift-wave

Example 3. Ion temperature gradient turbulence

• General goal: determine the dependence of turbulent thermal transport as a function of dimensionless parameters (Manfredi & Ottaviani, 1997, and foll.)

• ITG model:

Conclusions (personal)

• Conventional low frequency fluid models still useful, at least for a first approach to a given problem

• Gyrofluid models more flexibles, take naturally into account the anisotropy

• Good practical closures for the parallel dynamics exist, if sufficiently high order momenta are kept, especially when magnetic fluctiations are present

• FLR closures still involved, complicated. Problematic at very short wavelengths (below the ion Larmor radius)

• More work ?