Fusion, space, and solar plasmas as

complex systems

Richard Dendy

Euratom/UKAEA Fusion Association

Culham Science Centre

Abingdon, Oxfordshire OX14 3DB, U.K.

Centre for Fusion, Space and Astrophysics

Department of Physics, Warwick University

Coventry CV4 7AL, U.K.

Work supported in part by UK Engineering and Physical Sciences Research Council

Fusion, space, and solar plasmas as complex systems

Richard Dendy

22nd Canberra International Physics Summer School, December 2008

Part I

A brief introduction to fusion, space and solar plasmas

-How do they arise?

-What do they look like?

-What range of lengthscales and timescales is involved?

-How is the plasma state described mathematically?

Fusion, space, and solar plasmas as complex systems

Richard Dendy

22nd Canberra International Physics Summer School, December 2008

Part IA

Fusion, space and solar plasmas

-How do they arise?

-What do they look like?

A rapid introduction to fusion plasma physics…

The Joint European Torus at Culham: a very complex system:

Principles of toroidal magnetic confinement of plasmaBasic flow: toroidal current Basic magnetic field: appliedBasic flow: toroidal current Basic magnetic field: applied toroidaltoroidal

JET from the inside

Fusion power is not perpetually n decades away

ITER: to zeroth order, a scaled-up JET

A rapid introduction to space plasma physics

Basic flow: the solar wind Basic magnetic field: Earth’s dipole

The Aurora: terrestrial consequence of magnetospheric plasma activity

Aurora borealis from below Aurora australis from space

A rapid introduction to solar plasma physicsBasic flow: corona and solar wind Basic magnetic field: from solar dynamo

X-ray image of plasma in magnetic loops rising Ultraviolet image of solar disc

into the solar corona from the photosphere at time of major flare

Fusion, space, and solar plasmas as complex systems

Richard Dendy

22nd Canberra International Physics Summer School, December 2008

Part 1B

A crash course in the fundamentals of plasma

physics

-What types of model description are appropriate?

-What is their mathematical implementation?

-What a priori conclusions can we draw in relation to complex

systems approaches?

Fundamental features of the plasma state

• Fully ionised matter; intermingled particles of opposite electric charge; overall charge neutrality

• Local electrical non-neutrality is possible, associated with local fluctuations in the numbers of electrons and ions

• Self consistency: electrons and ions dynamically respond to, and give rise to, localised electric fields E

• Cyclotron motion response to magnetic fields B: helical paths spiralling along magnetic field lines

• High electrical conductivity →internal currents j→internal magnetic fields through Ampère’s law

• Self consistency again: plasma particle motion responds to and gives rise to magnetic fields

• High temperature e.g. 10keV ≈ 100 million OC →kinetic effects →keep track of velocity distribution

• Levels of description:

1. Individual particle dynamics 2. Kinetic 3. Fluid and current flows

• Plasma is the fourth state of matter: the physical and mathematical

models adopted retain features used for all three other states.

Self consistency in plasmas: example – the Debye length

• Insert a positive charge into plasma that has overall charge neutrality, with equal number density n0 of electrons and ions (protons). The inserted charge attracts electrons e and repels ions i→ local dynamics create a screen of negative charge, along with local electrostatic potential φ

• Assume Boltzmann distribution of particle energies E: n(E) ~ exp(-E/kBT). Since energy E = charge times local electrostatic potential φ,

→for electrons ne(E) ~ n0exp(eφ/kBT); →for ions ni(E) ~ n0exp(-eφ/kBT)

• Local charge density ρ = e(ni - ne) = n0eexp(-eφ/kBT) - exp(eφ/kBT) ~ -n0e2φ/kBT

• Self consistency requires φ and ρ must also satisfy Poisson’s equation

• For spherically symmetric φ, defining λD = (ε0T/n0e2)1/2 Debye length, this becomes

• Solutions are where A is constant: λD governs screening

( )T

ennn

eei

0

20

00

2

εφ

εερφ =−−=−=∇

20

202

2

1

DT

en

dr

dr

dr

d

r λφ

εφφ

≡=

r

eA

Dr λ

φ/−

=

Single particle level of description: 1 of 2Motion of nonrelativistic particle (mass m, charge q) is described by Lorentz force law

Consider prescribed electric field E and magnetic field B. Simplest case:

zero electric field E; uniform and constant B

Let direction of B define the z-direction, and differentiate with respect to t:

→simple harmonic oscillation of perpendicular components of velocity at the

cyclotron frequency (or gyrofrequency) Ω = qB/m

Convenient to use complex representation for vx etc., hence write solutions of (2) as

v = constant; phase δxmay be set equal to zero, i.e.

Lorentz equation gives

[ ],),(),( ttqdt

dm rBvrEv

×+= (1)vr=

dt

d

Bvv

×= qdt

dm

xx vm

qBv

2..

−=

yy vm

qBv

2..

−=

( )yxyx itivv ,, exp δ+Ω= ⊥

(2)

( ).

exp xtivvx =Ω= ⊥

( )...

exp1

ytiivvvqB

mv xxy =Ω±=

Ω±== ⊥

Single particle level of description: 2 of 2Integrating components of the Lorentz force equation with respect to t

Taking real parts:

rL = v/Ω = m v/(qB) is the Larmor radius of the particle’scircular orbit around its guiding centre at (x0, y0).

If B is directed into slide, electrons appear

to rotate clockwise, ions anti-clockwise.

There is also a velocity vz along B which is

unaffected by B: hence full particle trajectory

is a helix.

Cyclotron resonant energy transfer from electromagnetic waves having frequency ω can occur, provided:

- frequency resonance between wave and gyro motion, ω = Ω - kz vz, and

- wave has right circularly polarised component (to heat electrons), or

- wave has left circularly polarised component (to heat ions)

tievixx Ω⊥

Ω−=− 0

tiev

yy Ω⊥

Ω±=− 0

trxx L Ω=− sin0 tryy L Ω±=− cos0

From F F Chen, Introduction to Plasma Physics

& Controlled Fusion, Plenum

Trapped particle dynamics in tokamak magnetic field

Trapped particle dynamics: an example underpinning complexity

• Multiple lengthscales and timescales associated with three adiabatic invariants of the underlying charged particle dynamics:

1. Cyclotron frequency 10 MHz; Larmor radius 1 cm for ions

2. Bounce frequency 100 kHz; banana width 5 cm for ions

3. Toroidal precession 10 kHz; system scale metres

• Collisions may convert trapped particles (non-current-carrying)

into toroidally circulating (hence current-carrying) particles

• Canonical toroidal angular momentum pφ = mvφ + Aφ(r)

where A(r) is local magnetic vector potential.

Hence collisional changes in vφ → changes in position r

• Levels of description: if we sum over many particles, trapped and passing fluids (electron and ion) exert frictional forces on each other

Kinetic description of plasma: Vlasov equationDefine the distribution function f(x,y,z,vx,vy,vz,t) such that:

fdvxdvydvz is the number of particles per unit volume, at position (x,y,z), at time t,

with velocity components in the narrow range vx to vx+dvx, vy to vy+dvy, vz to vz+dvz.

Thus f is a function of seven variables (t,x,y,z,vx,vy,vz); particles move in six dimensional

phase space (x,v). Example of a distribution function f: suppress x and t, and recall Maxwellian

In this multidimensional phase space

For charged particles

in E, B fields

Hence where

All particles with given initial (x,v) have identical phase space trajectories; the value of f is a

fixed initial condition which cannot change, hence Vlasov equation

[ ]TvvvmT

mnvvvf zyxzyx 2/)(exp

2),,( 222

2/3

++−

=

π

z

z

y

y

x

xzyx

v

f

dt

dv

v

f

dt

dv

v

f

dt

dv

z

fv

y

fv

x

fv

t

f

dt

df

∂∂

+∂∂

+∂∂

+∂∂

+∂∂

+∂∂

+∂∂

=

[ ]BvEv

×+= qdt

dm

zyx vvv ∂∂

+∂∂

+∂∂

=∂∂ ∧∧∧

zyxv

( )v

BvEv∂∂

⋅×++∇⋅+∂∂

=f

m

qf

t

f

dt

df

( ) 0=∂∂

⋅×++∇⋅+∂∂

vBvEv

f

m

qf

t

f

Self consistent kinetic description of plasmaSelf consistent (and nonlinear) description requires closure via Maxwell’s equations

Charge density ρ and current density j are moments of the distribution function f:

where f1 is the perturbed part of f = f0 + f1 + …

The full nonlinear self-consistent kinetic model is thus the Vlasov-Maxwell system.

The Vlasov equation contains the basis for a fluid description. Take its zeroth velocity moment

where n(x,t) is the particle number density

where the fluid velocity

is the particle velocity averaged over the particle distribution. Since the moment of the

remaining term in Vlasov is zero, we have the fluid continuity equation

0ερ

=⋅∇ Et

c∂∂

+=×∇Ej

B0

2

ε

vdfe 310 ∫∫∫−=⋅∇ Eε vdvfej 3

1∫∫∫−=

t

ndf

td

t

f

∂∂

=∂∂

=∂∂

∫∫ vv 33

( )uvvvv ndffd ⋅∇≡⋅∇=∇⋅ ∫∫ 33 vvvu33 / dfdf ∫∫=

0)( =⋅∇+∂∂

unt

n

Magnetohydrodynamic force equationTo obtain the fluid equation of motion, take the next moment of the Vlasov equation by

multiplying through by mv and then integrating over all particle velocities:

First term

Second term

This contains pressure tensor and nonlinear fluid advection terms

Third term

Note the implicit signposts to complexity:

-Nonlinear fluid advection underlies turbulence

-Pressure term embodies physics on lengthscales and timescales outside the fluid model

-System not closed since equation for mth velocity moment draws in the (m + 1)th

-Need an Ohm’s law to relate E, B and u, and an equation of state for P

( ) 0)( 333 =∂∂

⋅×++∇⋅+∂∂

∫∫∫ vv

BvEvvvvvv df

qfdmdt

fm

)(33uvvvv n

tmdf

tmd

t

fm

∂∂

=∂∂

=∂∂

∫∫

( ) ( ) ( ) vvvvvvvvv333 dfdffd ∫∫∫ ⋅∇=⋅∇=∇⋅

( ) ( ) ( ) P⋅∇+∇⋅+⋅∇=∇⋅∫ uuuuvvv mnnmfdm 3

( ) ( ) )(33BuEvBvEv

vBvEv ×+−=×+−=

∂∂⋅×+ ∫∫ ndfdf

High temperature plasma physics overview: complex systems inferences

• Mathematics: systems of coupled nonlinear equations at all levels of description

• Different levels of description needed to capture physics within a specific range of lengthscales and timescales

• Physics: very broad range of lengthscales and timescales, with coupling across lengthscales and timescales, e.g. particle dynamics can resonate with bulk MHD waves

• Observed phenomenology includes, as we shall discuss:

-Self organisation: few-degree-of-freedom behaviour emerging from very high dimensional plasma systems. E.g. coherent nonlinear structures and resilient global properties such as temperature profiles

-Intermittency: various forms of bursty and avalanching phenomena associated with the spatio-temporal concentration of energy release and with non-local transport properties.

• Thus on a priori and observational grounds, plasmas invite interpretation from acomplex systems perspective (but arrived late to this particular party...).

Fusion, space, and solar plasmas as complex systems

Richard Dendy

22nd Canberra International Physics Summer School, December 2008

Part 2

Fusion plasmas as complex systems:

phenomenology and modelling

-What types of observed plasma behaviour are particularly interesting

from a complex systems modelling perspective?

-What is a sandpile and why is it relevant?

Fusion, space, and solar plasmas as complex systems

Richard Dendy

22nd Canberra International Physics Summer School, December 2008

Part 2A

Plasmas as complex systems

-Questions and motivation

Complex systems science: two quotations

• “The dream arising from the breathtaking progress of physics during the past

two centuries combined with the advances of modern high-speed computers –

that everything can be understood from first principles – has been thoroughly

shattered”

- Per Bak, How Nature Works: O.U.P 1997, p.6

• “Complexity science: a new and fast-growing area of interdisciplinary science

that seeks to understand those aspects of natural systems that are dominated

by their collective interactions rather than their individual parts”

- Global Science in the Antarctic Context: British Antarctic Survey Core

Programme 2005-2010, p.20

Fusion, space and solar plasmas are complex systems

• Behaviour is governed by multiple nonlinear physical processes, that interact with each other, and operate across an exceptionally wide range of

-length scales: e.g. millimetre to metre in laboratory

-time scales: e.g. tens of Gigahertz to 100’s of seconds in laboratory

• First principles mathematical models for plasmas are both complicated and necessarily reduced (i.e. truncated), implying uncertainty as to whether all the key physics has been captured. What is the minimal key physics?

• What constitutes quantitative (or indeed qualitative) agreement between,

for example, a nonlinear mathematical model and experimental observations

of plasma turbulence? Upon what basis is one model better than another?

• Require quantitative methods that extract model-independent information

from observed nonlinear signals from real plasmas: in particular, statistics

Complex systems science and plasma physics: two main lines of enquiry

• Are there fundamental similarities between

-plasma systems

-other, non-plasma, complex systems (geosciences, life sciences,... )?

In particular, as regards overall global behaviour?

• Plasma behaviour is often

- physically nonlinear

- multi-lengthscale, multi-timescale

- statistically non-Gaussian

Fresh quantitative methods from complex systems science are needed to quantify, compare, and ultimately understand this.

The first challenge: Universal models from complex systems science for plasma physics

• Identify simple universal models that capture the key physics of:

- extended macroscopic systems, governed by

- multiple coupled nonlinear processes, that operate across

- a wide range of spatial and temporal scales

• In such systems it is often the case that:

- energy release occurs intermittently in bursty events

- phenomenology can exhibit scaling, i.e. self similarity

• Guided by knowing the dominant plasma physics processes:

- construct minimalist models that yield relevant global behaviour

- address questions that are inaccessible to analytical treatment, and are too demanding on computational resources for numerical treatment

Off-axis electron cyclotron heating in DIII-D tokamak: temperature profile

C C Petty and T Luce, Nuclear Fusion 34, 121 (1994)

• Strongly centrally peaked

when ECH applied far off-axis

• “The electron temperature

does not respond to the

localised heating as expected

for a diffusive system”

JET temperature profiles as turbulence evolves

JET pulse 46767 has sequential Ohmic, L-mode preheat, delayed ELMy H-mode,

ITB, and post-ITB phases which correlate with measured turbulence; from Conway et

al, EPS Maastricht 1999.

Profiles between 5.0s and 6.5s

15 MW NBI + ICRH from 4.7s

Turbulence drops at 5.7s

ITB at 6.2s, terminates at 6.5s

Enhanced confinement and ELMs in JET

Pulse 39638: 12MW NBI, gas puffing until 18s, 2.5MA, 2.3T; from Fishpool,

Nucl Fusion 38, 1373 (1998)

Gas rate (electron/sec)

Confinement

enhancement H89

Deuterium Balmer-alpha

emission

The second challenge: Strongly nonlinear signals from fusion, space, solar and astrophysical plasmas

• These signals reflect strongly nonlinear plasma behaviour that is

– turbulent, or pulsed, or intermittent, or bursting:

– coherent phenomena play a minor role

• Time series of order 10,000 measurements of signal intensity, gathered over

– tens of milliseconds (e.g. MAST tokamak edge turbulence)

– several years (e.g. astrophysical X-ray sources)

• Datasets challenge theory and interpretation because they

arise from

– multiple interacting nonlinear plasma processes,

– operating on a wide range of lengthscales and timescales

Strongly nonlinear signals from solar and astrophysical plasmas: key examples

Full disk solar EUV/XUV

emission

X-ray binary Cygnus X-1

Microquasar GRS 1915

Strongly nonlinear signals from fusion plasmas: key examples

Edge localised modes in

the Mega Amp Spherical

Tokamak (MAST)

Edge localised modes in

the Joint European Torus

(JET)

0 2 40

1

2

x 1015

Time (s)

ELM

em

issi

on

Nonlinear signals from fusion, space, and astrophysical plasmas:a fundamental practical challenge for complex systems science

• Requires quantitative data analysis techniques that capture nonlinearity

• In this strongly nonlinear context:

– What are the criteria for agreement between a given model and

a given dataset?

– Upon what fair basis is one model better than another?

• Progress requires quantitative methods that are not conditioned by

prior model assumptions: hence model-independent

• Practical motivation: essential for constructing a rigorous interpretive and predictive capability