Fusion, space, and solar plasmas as complex systems

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 Uniersity Coentry 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 22 nd 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 inoled? -How is the plasma state described mathematically?

Fusion, space, and solar plasmas as complex systems Richard Dendy 22 nd 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 ery complex system:

Principles of toroidal magnetic confinement of plasma Basic flow: toroidal current Basic magnetic field: applied toroidal

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 actiity Aurora borealis from below Aurora australis from space

A rapid introduction to solar plasma physics Basic flow: corona and solar wind Basic magnetic field: from solar dynamo X-ray image of plasma in magnetic loops rising Ultraiolet 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 22 nd 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; oerall 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 gie rise to, localised electric fields E Cyclotron motion response to magnetic fields B: helical paths spiralling along magnetic field lines High electrical conductiity internal currents j internal magnetic fields through Ampère s law Self consistency again: plasma particle motion responds to and gies rise to magnetic fields High temperature e.g. 10keV 100 million O C kinetic effects keep track of elocity distribution Leels of description: 1. Indiidual 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 positie charge into plasma that has oerall charge neutrality, with equal number density n 0 of electrons and ions (protons). The inserted charge attracts electrons e and repels ions i local dynamics create a screen of negatie charge, along with local electrostatic potential φ Assume Boltzmann distribution of particle energies E: n(e) ~ exp(-e/k B T). Since energy E = charge times local electrostatic potential φ, for electrons n e (E) ~ n 0 exp(eφ/k B T); for ions n i (E) ~ n 0 exp(-eφ/k B T) Local charge density ρ = e(n i - n e ) = n 0 e{exp(-eφ/k B T) - exp(eφ/k B T)} ~ -n 0 e 2 φ/k B T Self consistency requires φ and ρ must also satisfy Poisson s equation 2 2 ρ e ne 0 φ φ = = ( ni ne) = ε ε εt 0 0 0 For spherically symmetric φ, defining λ D = (ε 0 T/n 0 e 2 ) 1/2 Debye length, this becomes 1 r 2 d dr r 2 d φ dr = 2 n 0e φ ε T 0 φ λ 2 D e φ= A r/ λ D Solutions are where A is constant: λ D goerns screening r

Single particle leel of description: 1 of 2 Motion of nonrelatiistic particle (mass m, charge q) is described by Lorentz force law d dr m = q[ E( r, t) + B( r, t) ], = (1) dt dt Consider prescribed electric field E and magnetic field B. Simplest case: zero electric field E; uniform and constant B d m = q B dt Let direction of B define the z-direction, and differentiate with respect to t:.. qb = m 2 x x.. qb = m y y simple harmonic oscillation of perpendicular components of elocity at the cyclotron frequency (or gyrofrequency) Ω = q B/m 2 Conenient to use complex representation for x etc., hence write solutions of (2) as = iωt + i (2) = constant; phase δ x may be set equal to zero, i.e. Lorentz equation gies ( ) x, y exp δ x, y m qb 1 Ω ( iωt) x exp = x =.. y = x =± x =± i exp( iωt) =. y.

Single particle leel of description: 2 of 2 Integrating components of the Lorentz force equation with respect to t x x 0 Taking real parts: iωt = i e y y Ω x x = rl sinωt 0 r L = /Ω = m /( q B) is the Larmor radius of the particle s circular orbit around its guiding centre at (x 0, y 0 ). iωt =± e Ω y y = ± rl cosω 0 0 t If B is directed into slide, electrons appear to rotate clockwise, ions anti-clockwise. There is also a elocity z along B which is unaffected by B: hence full particle trajectory is a helix. From F F Chen, Introduction to Plasma Physics & Controlled Fusion, Plenum Cyclotron resonant energy transfer from electromagnetic waes haing frequency ω can occur, proided: - frequency resonance between wae and gyro motion, ω = Ω - k z z, and - wae has right circularly polarised component (to heat electrons), or - wae has left circularly polarised component (to heat ions)

Trapped particle dynamics in tokamak magnetic field

Trapped particle dynamics: an example underpinning complexity Multiple lengthscales and timescales associated with three adiabatic inariants 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 conert trapped particles (non-current-carrying) into toroidally circulating (hence current-carrying) particles Canonical toroidal angular momentum pφ = mφ + Aφ(r) where A(r) is local magnetic ector potential. Hence collisional changes in φ changes in position r Leels of description: if we sum oer many particles, trapped and passing fluids (electron and ion) exert frictional forces on each other

Kinetic description of plasma: Vlaso equation Define the distribution function f(x,y,z, x, y, z,t) such that: fd x d y d z is the number of particles per unit olume, at position (x,y,z), at time t, with elocity components in the narrow range x to x +d x, y to y +d y, z to z +d z. Thus f is a function of seen ariables (t,x,y,z, x, y, z ); particles moe in six dimensional phase space (x,). 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 gien initial (x,) hae identical phase space trajectories; the alue of f is a fixed initial condition which cannot change, hence Vlaso equation [ ] T m T m n f z y x z y x 2 ) / ( exp 2 ),, ( 2 2 2 3/ 2 + + = π z z y y x x z y x f dt d f dt d f dt d z f y f x f t f dt df + + + + + + = [ ] B E + = q dt d m z y x + + = z y x ( ) B E + + + = f m q f t f dt df ( ) 0 = + + + B E f m q f t f

Self consistent kinetic description of plasma Self consistent (and nonlinear) description requires closure ia Maxwell s equations ρ 2 E= j E c B = + ε 0 ε 0 t Charge density ρ and current density j are moments of the distribution function f: ε 0 E = e f where f 1 is the perturbed part of f = f 0 + f 1 + 1 d 3 j= e The full nonlinear self-consistent kinetic model is thus the Vlaso-Maxwell system. f 1 d 3 The Vlaso equation contains the basis for a fluid description. Take its zeroth elocity moment f t d 3 fd = t = f d fd 3 = n t ( nu) where n(x,t) is the particle number density 3 3 3 3 u = fd / fd where the fluid elocity is the particle elocity aeraged oer the particle distribution. Since the moment of the remaining term in Vlaso is zero, we hae the fluid continuity equation n + ( nu ) = 0 t

Magnetohydrodynamic force equation To obtain the fluid equation of motion, take the next moment of the Vlaso equation by multiplying through by m and then integrating oer all particle elocities: m First term f t Second term This contains pressure tensor and nonlinear fluid adection terms m Third term m d 3 + m f t d ( ) fd Note the implicit signposts to complexity: 3 = m t -Nonlinear fluid adection underlies turbulence 3 + q fd 3 ( E+ B) d = 0 -Pressure term embodies physics on lengthscales and timescales outside the fluid model -System not closed since equation for m th elocity 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 3 = m ( nu) t f 3 3 3 ( ) fd = ( f) d = ( f) d 3 ( ) fd = mu ( nu) + mn( u ) u+ P f + 3 3 ( E B) d = f( E+ B) d = n( E+ u B)

High temperature plasma physics oeriew: complex systems inferences Mathematics: systems of coupled nonlinear equations at all leels of description Different leels of description needed to capture physics within a specific range of lengthscales and timescales Physics: ery broad range of lengthscales and timescales, with coupling across lengthscales and timescales, e.g. particle dynamics can resonate with bulk MHD waes Obsered phenomenology includes, as we shall discuss: -Self organisation: few-degree-of-freedom behaiour emerging from ery high dimensional plasma systems. E.g. coherent nonlinear structures and resilient global properties such as temperature profiles -Intermittency: arious forms of bursty and aalanching phenomena associated with the spatio-temporal concentration of energy release and with non-local transport properties. Thus on a priori and obserational grounds, plasmas inite interpretation from a complex systems perspectie (but arried late to this particular party...).

Fusion, space, and solar plasmas as complex systems Richard Dendy 22 nd Canberra International Physics Summer School, December 2008 Part 2 Fusion plasmas as complex systems: phenomenology and modelling -What types of obsered plasma behaiour are particularly interesting from a complex systems modelling perspectie? -What is a sandpile and why is it releant?

Fusion, space, and solar plasmas as complex systems Richard Dendy 22 nd Canberra International Physics Summer School, December 2008 Part 2A Plasmas as complex systems -Questions and motiation

Complex systems science: two quotations The dream arising from the breathtaking progress of physics during the past two centuries combined with the adances of modern high-speed computers that eerything 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 collectie interactions rather than their indiidual parts - Global Science in the Antarctic Context: British Antarctic Surey Core Programme 2005-2010, p.20

Fusion, space and solar plasmas are complex systems Behaiour is goerned 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 quantitatie (or indeed qualitatie) agreement between, for example, a nonlinear mathematical model and experimental obserations of plasma turbulence? Upon what basis is one model better than another? Require quantitatie methods that extract model-independent information from obsered 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 oerall global behaiour? Plasma behaiour is often - physically nonlinear - multi-lengthscale, multi-timescale - statistically non-gaussian Fresh quantitatie methods from complex systems science are needed to quantify, compare, and ultimately understand this.

The first challenge: Uniersal models from complex systems science for plasma physics Identify simple uniersal models that capture the key physics of: - extended macroscopic systems, goerned 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 eents - phenomenology can exhibit scaling, i.e. self similarity Guided by knowing the dominant plasma physics processes: - construct minimalist models that yield releant global behaiour - 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 diffusie system

JET temperature profiles as turbulence eoles 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 behaiour 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 oer tens of milliseconds (e.g. MAST tokamak edge turbulence) seeral 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) ELM emission 2 1 x 10 15 0 0 2 4 Time (s)

Nonlinear signals from fusion, space, and astrophysical plasmas: a fundamental practical challenge for complex systems science Requires quantitatie data analysis techniques that capture nonlinearity In this strongly nonlinear context: What are the criteria for agreement between a gien model and a gien dataset? Upon what fair basis is one model better than another? Progress requires quantitatie methods that are not conditioned by prior model assumptions: hence model-independent Practical motiation: essential for constructing a rigorous interpretie and predictie capability