Frank Jenko

with contributions from F. Merz, T. Görler,M.J. Püschel, T. Hauff, and D. Told

Max-Planck-Institut für Plasmaphysik, GarchingUniversität Ulm

Porquerolles, France, 20 April 2009

Turbulence inMagnetized Fusion PlasmasNew Insights and Future Challenges

Plasma turbulence – an ubiquitous phenomenon> 99% of the visible universe is inthe plasma state, mostly turbulent

ITER is one of the most challenging scientific projects

It is currently being built in Cadarache

Plasma turbulencedetermines its energy confinement time

ITER and plasma turbulence

www.iter.org

α heating must compensate energy losses:• Electromagnetic radiation• Turbulent transport

Key requirements:

• Large central pressure(limited by onset oflarge-scale instabilities)

• Large energy confinementtime (limited by small-scaleturbulence):

τ = E / PE plasma loss

1 10 100 1000 T [million degrees]

TokamaksStellaratoren

MCF: Towards ignition

www.ipp.mpg.de/~fsj/gene

Plasma turbulence – a Grand Challenge

Hydrodynamic turbulence

According to a famous statement by Richard Feynman…

…and a survey by the British “Instituteof Physics” among many of the leadingphysicists world-wide…

TURBULENCE:A challenging topic for both basic and applied research

Turbulence – one of the most important unsolved problems in modern physics

“Millennium Issue”(December 1999)

What is turbulence?

Turbulence…

• is an intrinsically nonlinear phenomenon

• occurs (only) in open systems

• involves many degrees of freedom

• is highly irregular (chaotic) in space and time

• often leads to a (statistically) quasi-stationarystate far from thermodynamic equilibrium

These properties make it a very complicated problem –neither Dynamical Systems Theory nor Statistics applies!

The Navier-Stokes equation

The NSE in its ‘classical’ form:

Expressed in terms of vorticity :

Reynolds number as single dimensionless parameter:

Turbulence as a local cascade in wave number space…

„Big whorls have little whorls, little whorls have smaller whorlsthat feed on their velocity, and so on to viscosity“

Much turbulence research addresses the cascade problem.

The Richardson cascade

kE

ηk k

energy flux

fk

driverange

dissipationrangeinertial

rangeComputationaleffort

E = v2 d3x = E(k) dk⌠⌡

∞12V

⌠⌡0

Kolmogorov’s theory from 1941

K41 is based merely on intuition and dimensional analysis –it is not derived rigorously from the Navier-Stokes equation

Key assumptions:

• Scale invariance – like, e.g., in critical phenomena• Central quantity: energy flux ε

E(k) = C ε2/3 k-5/3

This is the most famous turbulence result: the “-5/3” law.However, K41 is fundamentally wrong: scale invariance is broken!

Direct numerical simulations

Wilczek

et al. 2008

Structure formation and broken scale invariance

Key open issues: Drive range• Often, one is interested mainly in the large scales. Here,

one encounters an interesting interplay between linear(drive) and nonlinear (damping) physics. – Is it possibleto remove the small scales?

• Yes: LES, Dynamical Systems Approach etc.

Ore

llano

& W

engl

e, J

T 2

001

Gyrokinetic turbulence

Vlasov-Maxwell equations

Removing the fast gyromotion[Frieman, Chen, Lee, Hahm, Brizard et al., 1980s]

Charged rings as quasiparticles;gyrocenter coordinates

Dilute and/or hot plasmas are almost collisionless.

Therefore, (3D) fluid theory is not applicable,and one has to use a (reduced) kinetic description!

Reduced kinetic description

Nonlinear integro-differential equations in 5 dimensions...

Appropriate field equations

X = gyrocenter positionV� = parallel velocityµ = magnetic moment

Advection/Conservation equation

The nonlinear gyrokinetic equations

In the year 2009…

� GK has emerged as the standard approach to (core) turbulence

� a variety of nonlinear GK codes is being used and (further) developed

� code development has become a team effort –help by computer experts

GYRO GTCGEM GS2

etc.

GENEORB5GYSELAELMFIRE

etc.

GKVG5Detc.

US

Europe Japan

These codes differ by their numerical schemes and physics…

Code benchmarking

� Are we solving the equations right?� Yes, according to a recent comparison between 4 codes

(GENE, GYRO, GS2, PG3EQ)� Such efforts are (sometimes) a bit painful but necessary

• GENE is a physically comprehensive and well benchmarked code

• GENE is publicly available (www.ipp.mpg.de/~fsj/gene)

• various applications in fusion research (tokamaks and stellarators) and astrophysics; can be run as a (radially) local or global code

• the differential operators are discretized via a combination of spectral, finite difference, and finite volume methods; the time stepping is done via a (non-standard) 4th-order explicit Runge-Kutta method

• in addition, GENE can also be run as a linear eigenvalue solver

• two main goals: deeper understanding of fundamental physics issues and direct comparisons with experiments (interfaces to MHD codes)

The simulation code GENE

• Parallelization due to high-dimensional domain decomposition

(either pure MPI or mixed MPI/OpenMP paradigm)

• GENE runs very efficiently on a large number of parallel platforms

(including IBM BlueGene, IBM Power6, Cray XT4, SGI Altix etc.)

• GENE is part of the European DEISA benchmark suite and the

EU-Japanese IFERC benchmark suite

GENE is a massively parallel code

0

2

4

6

8

10

12

14

16

0 5000 10000 15000 20000

number of processors

spee

du

pGENE on BlueGene/L (strong scaling)

(problem size: ~300-500 GB;measurements in co-processor mode at IBM Watson Research Center)

Speedup: 14.2 / 16

Turbulent fluctuations are quasi-2DReason: Strong background magnetic field

Possible simulation volume: flux tube, annulus, full (or fractional) torus

ExB drift velocity

(random walk/mixinglength estimates)

potential contourspotential contours==

streamlines of streamlines of ExBExB velocityvelocity

Typical heat and particle diffusivities are of the order of 1 m2/s.

Turbulent mixing in a tokamak

GradientsGradients→→ fluctuationsfluctuations

→→ transporttransport

Plasma microturbulence:Linear drive

Perpendicular dynamics: de-/stabilization in out-/inboard regions

stable

unstable Rayleigh-Taylor instabilityAnalogy in a plasma:

Parallel dynamics: localization in outboard regions

Gradient-driven microinstabilities

G. Hammett

Trapped Electron Modes(TEMs)

Electron Temperature Gradient (ETG) Modes

Ion Temperature Gradient (ITG) Modes

Different kinds of microinstabilities drive different kinds of plasma turbulence

Drive (= transport)range dynamics isnot universal.

Plasma turbulenceis actually a multi-scale problem.

Basic properties of ITG modes

In the context of fusion energy,gaining a better understandingof plasma turbulence is crucial

Existence of critical temperature gradients

Temperature profiles tend to be ‘stiff’ (cp. solar convection zone).

Typical space scales: several ion gyroradii (not system size)

Plasma microturbulence:Nonlinear saturation?

Structure formation

Emergence of zonal ExB flows (due to symmetry breaking!)

They are linearly neutrallystable but excited nonlinearly

Zonal flows in geo-/astrophysics

Effect on turbulent transport

Zonal flows may reduce or evensuppress the turbulent transportby means of vortex shearing

Saturation of ITG modes: zonal flows

Secondary instabilities & ZF generation

Strintzi & Jenko 2007Xanthopoulos et al. PRL 2007

• Large-amplitude streamers are Kelvin-Helmholtz unstable[Cowley at al. 1991; Dorland & Jenko PRL 2000]

• This secondary instability contains a zonal-flow component• Near-equivalence to 4-mode and wave-kinetic approaches

Simpletokamak

W7-Xstellarator

Features of TEM turbulence

Saturated phase of TEM turbulence simulations:

- In the drive range, nonlinear and linear frequencies are identical

- In the drive range, there is no significant shift of cross phases w.r.t. linear ones

– No dependence of transport level on zonal flows [Dannert & Jenko 2005]

Zonal flows suppressed

linear

nonlinear

� Both weak and strong turbulence theories suggest that the ExB nonlinearity can be represented by a coherent part and a random noise part

� and are fluctuating quantities; minimizing the model error , we obtain

Theory-motivated statistical analysis

Low-ky drive range: large transport contributions, but small random noise; here, one finds:

Saturation of TEMs: “eddy damping”

~ky2

Merz & Jenko, PRL 2008

This is in line with various theories, including Resonance Broadening Theory (Dupree), MSR formalism (Krommes), Dressed Test Mode Approach (Itoh).

Plasma microturbulence:Sub-ion-gyroradius scales?

Multiscale plasma turbulence

trapped electron modes

ETG modes

ITG modes

Not shown here:

- drift waves

- ballooning modes

low k

pure TEM turbulence(GENE simulation)

Surprising finding: ETG transportexceeds mixing length expectations [Jenko et al., PoP/PRL 2000]

Coexistence of ITG and ETG modes

ITG/TEM/ETG turbulence: Large fraction of electron heat trans-port is carried by electron scales (cmp. recent experiments).

[Görler & Jenko, PRL 2008]

Reduced mass ratio (400),but still > 100,000 CPU-h.

box size: ~64 ρi resolution: ~2ρe

ASDEX Upgrade #20431 (ρpol = 0.98)

separatrix

ρpol = 0.98

Edge transport barrier region:

• kyρs < 0.1 → ITG mode

• kyρs ~ 0.15 → microtearing mode

• kyρs > 0.2 → ETG mode

F.J., APS Invited Talk 2008

H-mode edge: Electron heat diffusivity

ETG turbulence is able to explain the residualelectron heat transport in H-mode edge plasmas.

F.J., APS Invited Talk 2008

Transport barrier:Long-wavelength

turbulence issuppressed viavortex shearing

Plasma microturbulence:Magnetic field fluctuations?

Gyrokinetic turbulence at high beta

Nonlinear drop clearly exceeds (quasi-)linear expectations;this is likely due to destructive ITG/TEM interference.

Finite-beta CycloneBase Case simulations

[Pueschel et al., PoP 2008]

Radial diffusion of magnetic field lines

For ITG turbulence, the field lines exhibit normal diffusion;remark: microtearing modes exhibit more complex behavior.

Poloidal cross section MSD versus number of toroidal turns

Note: Field line dynamics can also be studied by Hamiltonian maps.

Transport along fluctuating field lines

Magnetic transport is well described by Rechester-Rosenbluthtype model – confirmed by inspection of cross phases.

Rechester-Rosenbluth type model M.J. Püschel, PhD Thesis (2009)

Note: Interesting applications in astrophysics.

Plasma microturbulence:Transport of fast particles?

Diffusivities and correlationsTest particle approach

Turbulent structures and energetic particle trajectories

Turbulent structures and energetic particle trajectories (cont’d)

Validity of orbit averaging

Scaling laws: Diffusivities vs energy

Analytical theoryis confirmed by

GENE simulations

Magnetic transport of beam ions is independent of energy!

Final remarks

Some achievements in gyrokineticturbulence theory/simulation

� Throughout the 1990s, ITG turbulence has served as a reference point for turbulence simulations; it is now fairly well understood (critical gradients, zonal flow generation, nonlinear upshift shift, large stiffness)

� Over the last 10 years, other drive mechanisms and transport channels have been investigated and are at least partially understood (but not yet their interaction)

� Preliminary sim/exp comparisons look promising

NL GK(full space-time scales)

NL GK(reduced space-time scales)

Quasilinear models based on L/NL GK/GF

Computationaleffort

Numberof runs

Sim/exp comparisons are and will be based on a hierarchy of models

Space-time scale reduction and/or quasilinear modelling is necessary!

Some outstanding open issues� Prediction of steady state density and temperature profiles

(coupling to transport codes; see talk by M. Barnes)

� Nonlocal and finite-size effects

� Core and edge transport barriers

� Reduced descriptions as well as multi-scale, multi-physics models (turbulence, MHD instabilities, neoclassics etc.)

� Ultimate goal: virtual fusion plasma

Predictive capability requires solid understanding of the

basic physics and close interaction between theory/simulation,

applied math/computer science, and experiment.