Magnetohydrodynamics of Laboratory and Astrophysical Plasmas

Home , Astrophysical plasma, Magnetohydrodynamics

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Hans Goedbloed FOM-Institute for Plasma Physics ‘Rijnhuizen’ & Astronomical Institute, Utrecht University

Lectures at Centro Brasileiro de Pesquisas F´ısicas, Rio de Janeiro March – June 2006

Notes by J.P. Goedbloed and R. Keppens based on PRINCIPLES OF MAGNETOHYDRODYNAMICS by J.P. Goedbloed & S. Poedts (Cambridge University Press, 2004) 0-2 Contents

1. Introduction [book: Chap. 1] plasma: deﬁnitions, occurrence, conditions 2. Elements of Plasma Physics [book: Chap. 2] charged particles, collective interactions, ﬂuid description 3. MHD model [book: Chap. 4] laboratory and astrophysical plasmas from one point of view 4. Spectral Theory [book: Chaps. 5–7] waves and instabilities in inhomogeneous plasmas 5. Magnetic Structures [book: Chap. 8] tokamak, sun, planetary magnetospheres, stellar winds, pulsars 6. Flowing Plasmas [ future Volume 2 ] waves and instabilities of stationary plasmas, shocks 7. Toroidal Plasmas [ future Volume 2 ] equilibrium and stability of tokamaks and accretion disks 0-3 Literature

Introductory plasma physics:

¯ F.C. Chen, Introduction to Plasma Physics and Controlled Fusion (1984).

¯ J.A. Bittencourt, Fundamentals of Plasma Physics (1986).

¯ R.J. Goldston and P.H. Rutherford, Introduction to Plasma Physics (1995). Magnetohydrodynamics:

¯ J.P. Freidberg, Ideal Magnetohydrodynamics (1987).

¯ D. Biskamp, Nonlinear Magnetohydrodynamics (1993).

¯ J.P. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics (2004). http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521626072 www.rijnh.nl/users/goedbloed (ErrataPrMHD.pdf) Plasma astrophysics:

¯ E.R. Priest, Solar Magnetohydrodynamics (1984).

¯ A.R. Choudhuri, The Physics of Fluids and Plasmas, intro for Astrophysicists (1998).

¯ R.M. Kulsrud, Plasma Physics for Astrophysics (2004). 0-4 Plasma physics on www

¯ Fusion energy www.fusie-energie.nl (nuclear fusion and ITER, in Dutch)

¯ Solar physics dot.astro.uu.nl (Dutch Open Telescope) www.spaceweathercenter.org (space weather)

¯ Plasmas general www.plasmas.org (basics, applications of plasmas)

¯ These notes mesonpi.cat.cbpf.br/cbpfindex (downloadable pdf files) or www.rijnh.nl/users/goedbloed (downloadable pdf files) Introduction: Overview 1-1 Chapter 1: Introduction

Overview

¯ Motivation: plasma occurs nearly everywhere, magnetized plasma unifying theme for laboratory and astrophysical plasma physics; [ book: Sec. 1.1 ]

¯ Thermonuclear fusion: fusion reactions, conditions for fusion, magnetic conﬁne- ment in tokamaks; [ book: Sec. 1.2 ]

¯ Astrophysical plasmas: the standard view of nature, why it fails, examples of astrophysical plasmas; [ book: Sec. 1.3 ]

¯ Definition of plasma: usual microscopic definition (collective interactions), macroscopic definition (the magnetic field enters). [ book: Sec. 1.4 ] Introduction: Motivation 1-2

Plasma

¯ Most common (9¼± ) state of matter in the universe.

¯ On earth exceptional, but obtained in laboratory thermonuclear fusion experiments

at high temperatures (Ì ½¼ Ã ).

¯ Crude deﬁnition: Plasma is a completely ionised gas, consisting of freely moving positively charged nuclei and negatively charged electrons.

Applications

¯ Magnetic plasma conﬁnement for (future) energy production by Controlled Ther- monuclear Reactions.

¯ Dynamics of astrophysical plasmas (solar corona, planetary magnetospheres, pulsars, accretion disks, jets, etc.).

¯ Common ground: Plasma interacting with a magnetic ﬁeld. Introduction: Nuclear fusion (1) 1-3

Reactions of hydrogen isotopes

n + n + n + n + n n + +

D T He n 3.5 MeV 14.1 MeV

Two products

¯ Charged « particles:

capture in plasma magnetic ﬁeld µ « particle heating

¯ Neutrons:

6 ¿

capture in Li blanket µ fusion energy · T breeding Introduction: Nuclear fusion (2) 1-4

Why plasma?

¯ To overcome electrostatic repulsion

of nuclei need k ½¼ eÎ 2 8 + n D µ Ì ½¼ Ã (ionisation at ½4 eÎ ).

µ Plasma completely ionised gas consisting of freely moving positively charged nuclei and negatively charged electrons.

How to conﬁne?

¯ Magnetic ﬁelds: 1. charged particles gyrate around ﬁeld lines;

2. ﬂuid and magnetic ﬁeld move together (“B frozen into the plasma”);

? 3. thermal conductivity: k .

µ Need: Closed magnetic geometry. Introduction: Nuclear fusion (3) 1-5

Power balance

Power contributions (Ì in units of k eÎ ):

¾ ¾

¯ È = h Ú i Ò E Ò f ´ Ì µ ; E ¾¾ : 4 ÅeÎ ;

thermonuclear output Ì Ì Ì

¾ ½ = ¾ ¾9 ½ = ¾ ¿ ½

¯ È = Ò « Ì ; « ¿ : 8 ¢ ½¼ Â Ñ × ;

Bremsstrahlung losses B

¯ È = ¿ = Ò Ì :

heat transport losses Ä E

(a) Original idea (Lawson): three power contributions externally available for conversion

into electricity and back again into plasma heating with efﬁciency ¼ : ¿¿ ,

È · È = ´ È · È · È µ

B Ä Ì B Ä (1)

µ ignition condition:

¿ Ì

Ò = :

E (2)

½ = ¾

e e

= ´ ´½ µµ f ´ Ì µ « Ì (b) Present approach (more restrictive): ignition when power losses are balanced by

« È

-particle heating « ,

¾ ¾

È · È = È = Ú h i Ò E Ò f ´ Ì µ ; E ¿ : 5 ÅeÎ

B Ä « « « (3)

µ formally condition (2) still applies, but now with new f and ¼ : ½¿5 : Introduction: Nuclear fusion (4) 1-6

Power balance (cont’d)

¯ Fusion power ¸ radiation · transport losses: (a) Lawson criterion: lower curve, (b) Modern approach: upper curve.

¯ Upper curve at minimum ( Ì k ¾¼ ! eÎ ):

¾¼ ¿

Ò ¿ ¢ ½¼ Ñ ; ×

E typically:

¾¼ ¿

Ò ½¼ Ñ ! ! × ¿

µ Magnetic ﬁelds provide the only way to conﬁne matter of such high temperatures during such long times. Introduction: Nuclear fusion (5) 1-7

Interaction of currents and magnetic ﬁelds

j B j

¯ Schematic history of fusion experiments: B z - pinch: θ - pinch: very unstable end-losses (remains so in a torus) (in torus: no equilibrium)

Tokamak: delicate balance between equilibrium & stability Introduction: Nuclear fusion (6) 1-8

Tokamak

¯ Magnetic conﬁnement:

poloidal coils producing toroidal magnetic field iron transformer core

transformer winding (primary circuit)

p p plasma current (secondary circuit) B pol : poloidal magnetic field

B tor : toroidal magnetic field resultant plasma contained by magnetic field helical field Introduction: Nuclear fusion (7) 1-9

Tokamak (cont’d)

¯ Goal is electricity producing power plants: Introduction: Nuclear fusion (8) 1-10

Progress in fusion research

¯ Progress made in controlled fusion over the years shows the same impressive advance as other ﬁelds recognized as world leaders. (from: CRPP Annual Report 2000) Introduction: Astrophysical plasmas (1) 1-11

The Standard View of Nature

Nuclear forces

· quarks / leptons

½5

nuclei (· ) / electrons ( ) ½¼ Ñ

Electrostatic forces

atoms / molecules ½¼ Ñ (ordinary matter: electrically neutral) . . . . .

Gravity

9 ½¿

stars / solar system ½¼ = ½¼ Ñ

¾¼ ¾¿

galaxies / clusters ½¼ = ½¼ Ñ

¾6

universe ½¼ Ñ However, ... Introduction: Astrophysical plasmas (2) 1-12

The universe does not consist of ordinary matter

¯ > 9¼± is plasma: electrically neutral, where the nuclei and electrons are not tied in atoms but freely move as ﬂuids.

¯ The large scale result is Magnetic ﬁelds (example: interaction solar wind – magnetosphere).

Geometry

¯ Spherical symmetry of atomic physics and gravity (central forces) not present on the plasma scale:

Ö ¡ B = ¼ is not compatible with spherical symmetry (example: solar ﬂares). Introduction: Astrophysical plasmas (3) 1-13

Example: The Sun

a magnetized plasma! (sunatallwavelengths.mpeg) Introduction: Astrophysical plasmas (4) 1-14

Example: Coronal loops

[ from Priest, Solar Magnetohydrodynamics (1982)] Introduction: Astrophysical plasmas (5) 1-15

Example: Coronal loops (cont’d)

[ from recent observations with TRACE spacecraft ] Introduction: Astrophysical plasmas (6) 1-16

Example: Stellar wind outﬂow (simulation)

¯ Axisymmetric magnetized wind with a ‘wind’ and a ‘dead’ zone [ Keppens & Goedbloed, Ap. J. 530, 1036 (2000)] Introduction: Astrophysical plasmas (7) 1-17

Example: Magnetosphere

Æ Introduction: Astrophysical plasmas (8) 1-18

Example: Polar lights

Beauty of the polar lights (a1smallweb.mov)

Solar wind powering auroral displays (fuvmovie.mpeg) Introduction: Astrophysical plasmas (9) 1-19

Example: Accretion disk and jets (YSO)

Young stellar object

Å ½ Å

( £ ¬ ): accretion disk ‘seen’ edge-on as dark strip, jets colored red. Introduction: Astrophysical plasmas (10) 1-20

Example: Accretion disk and jets (AGN)

Å ½¼ Å

Active galactic nucleus ( £ ¬ ): optical emission (blue) centered on disk, radio emission (red) shows the jets. Introduction: Astrophysical plasmas (11) 1-21

Example: Accretion disk and jets (simulation)

Stationary end state from the simulation of a Magnetized Accretion Ejection Structure: disk density surfaces (brown), jet magnetic surface (grey), helical ﬁeld lines (yellow), accretion-ejection particle trajectory (red). [ Casse & Keppens, Ap. J. 601, 90 (2004)] Introduction: Deﬁnitions of plasma (1) 1-22

Crude deﬁnition:

Plasma is an ionized gas.

¿ = ¾

Ò ¾ Ñ k

Í =k Ì

i e

Rate of ionization: = e (Saha equation)

Ò Ò

Ò i

½¾¾

¾5 ¿

Ì Ã = ¿¼¼ Ò = ¿ ¢ ½¼ Ñ Í = ½4 eÎ : 5 µ Ò =Ò ¾ ¢ ½¼

– air: , Ò , i i Ò (!)

½¿

8 ¾¼ ¿

Ì = ½¼ Ã Ò = ½¼ Ñ Í = ½¿ : 6 eÎ µ Ò =Ò ¾ : 4 ¢ ½¼

– tokamak: , i , i i Ò

Microscopic deﬁnition:

Æ Plasma is a quasi-neutralgas of charged and neutral particles which exhibits collective behaviour (Chen).

(a) Long-range collective interactions dominate over binary collisions with neutrals

Ò Z Ò

(b) Length scales large enough that quasi-neutrality ( e i ) holds (c) Sufﬁciently many particles in a Debye sphere (statistics) Introduction: Deﬁnitions of plasma (2) 1-23

Collective behavior

Conditions: n

N D = 1

½ 1032

core sun

(a) Ò

Ò Ú - 8 Ò Øh λD = 10 m

tokamak: ¾ : 4 ¢ ½¼ × 8 N D = 10 τn = 1 s 24

¾¼ 10

corona: ¾ ¢ ½¼ ; ×

λD = 10- 4 m

Ö tokamaktokamak

¯ Ì k

(b) D ¾ 16 τ 17 e Ò 10 n = 6 x 10 s

5 λ

= 7 ¢ ½¼ Ñ D = 1 m

tokamak: D corona sun

; = Ñ ¼ : ¼7

corona: D 16 N D = 10 108

¿ 4

Æ Ò ½

Æ = ½ : 4 ¢ ½¼

tokamak: D 1 T

9 2 4 6 8 10 10 10 10 Æ = ½ : 4 ¢ ½¼ : 10 10

corona: D air Introduction: Deﬁnitions of plasma (3) 1-24

So far, only the electric ﬁeld appeared. (LOCAL)

Macroscopic deﬁnition:

Æ For a valid macroscopic model of magnetized plasma dynamical configurations, size, duration, density, and magnetic field strength should be large enough to establish fluid behavior and to average out the microscopic phenomena (i.e. collective plasma oscillations and cyclotron motions of electrons and ions).

Now,themagneticﬁeldenters: (GLOBAL!)

½ ½

(a) ª B (time scale longer than inverse cyclotron frequency);

Ê B

(b) i (length scale larger than cyclotron radius).

µ MHD magnetohydrodynamics