0-1 Magnetohydrodynamics of Laboratory and Astrophysical Plasmas 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 « ¨ ªOverview © ¯ 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), macro- scopic definition (the magnetic field enters). [ book: Sec. 1.4 ] Introduction: Motivation 1-2 « ¨ ªPlasma © ¯ Most common (9¼±) state of matter in the universe. ¯ On earth exceptional, but obtained in laboratory thermonuclear fusion experiments 8 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 Æ ­ ¯ Magnetic plasma confinement for (future) energy production by Controlled Ther- monuclear Reactions. ¯ Dynamics of astrophysical plasmas (solar corona, planetary magnetospheres, pul- sars, accretion disks, jets, etc.). ¯ Common ground: Plasma interacting with a magnetic field. Introduction: Nuclear fusion (1) 1-3 ¯ ¬ Reactions of hydrogen isotopes Æ ­ n + n + n + n + n n + + D T He n 3.5 MeV 14.1 MeV ¯ ¬ Two products Æ ­ ¯ 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? Æ ­ ¯ To overcome electrostatic repulsion of nuclei need ½¼ keÎ 2 8 + n D µ Ì ½¼ à (ionisation at ½4 eÎ ). µ Plasma completely ionised gas consisting of freely moving positively charged nuclei and negatively charged electrons. « ¨ ªHow 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 « ¨ ªPower balance © e Power contributions (Ì in units of keÎ): ½ ¾ ¾ e ¯ È = h Ú iÒ E Ò f ´Ì µ ; E ¾¾:4 ÅeÎ ; thermonuclear output Ì Ì Ì 4 ¾ ½=¾ ¾9 ½=¾ ¿ ½ e ¯ È = «Ò Ì ; « ¿:8 ¢ ½¼ Â Ñ × ; Bremsstrahlung losses B e ¯ È = ¿ÒÌ = : 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: e ¿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 ½ e È · È = È = h Ú iÒ E Ò f ´Ì µ ; E ¿:5 ÅeÎ B Ä « « « (3) 4 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) Æ ­ ¯ Fusion power ¸ radiation · transport losses: (a) Lawson criterion: lower curve, (b) Modern approach: upper curve. e ¯ Upper curve at minimum ( Ì ¾¼ keÎ !): ¾¼ ¿ Ò ¿ ¢ ½¼ Ñ × ; E typically: ¾¼ ¿ Ò ½¼ Ñ ! ¿ × ! E 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 Æ ­ B 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 « ¨ ªTokamak © ¯ 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) Æ ­ ¯ Goal is electricity producing power plants: Introduction: Nuclear fusion (8) 1-10 ¯ ¬ Progress in fusion research Æ ­ ¯ Progress made in con- trolled 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 « ¨ ªThe Standard View of Nature © Nuclear forces · quarks / leptons ½5 nuclei (·) / electrons ( ) ½¼ Ñ Electrostatic forces · 9 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 Æ ­ ¯ > 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 Æ ­ ¯ 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 Æ ­ a magnetized plasma! (sunatallwavelengths.mpeg) Introduction: Astrophysical plasmas (4) 1-14 ¯ ¬ Example: Coronal loops Æ ­ [ from Priest, Solar Magnetohydrodynamics (1982)] Introduction: Astrophysical plasmas (5) 1-15 ¯ ¬ Example: Coronal loops (cont’d) Æ ­ [ from recent observations with TRACE spacecraft ] Introduction: Astrophysical plasmas (6) 1-16 ¯ ¬ Example: Stellar wind outflow (simulation) Æ ­ ¯ 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 Æ ­ 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) Æ ­ Young stellar object Å ½Å ( £ ¬ ): accretion disk ‘seen’ edge-on as dark strip, jets colored red. Introduction: Astrophysical plasmas (10) 1-20 ¯ ¬ Example: Accretion disk and jets (AGN) Æ ­ 8 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) Æ ­ 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 « ¨ ªCrude definition: © Plasma is an ionized gas. ¿=¾ ¿=¾ Ò ¾ Ñ k Ì Í =k Ì i e i Rate of ionization: = e (Saha equation) ¾ Ò Ò Ò i h ½¾¾ ¾5 ¿ Ì = ¿¼¼ Ã Ò = ¿ ¢ ½¼ Ñ Í = ½4:5 eÎ 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)