PLASMA RELAXATION The Super-Comic Edition

1. PLASMA RELAXATIONThe Super-Comic Edition Loren Steinhauer University of Washington Plasma Physics Summer School Los Alamos, 10 August 2006

2. Sleepy Lumpy Baby With permission from… Acknowledgments

3. Roadmap The players Behavioral psychology “Good” behavior The secret of good behavior Some essential math What happy plasmas look like Why aren’t all plasmas happy? Relaxed states (and more)

4. Our friendly plasma Part 1: The Players Plasma($) Definition: an “exotic” fluid because… there is a lot of extra weird stuff going on

5. KERPLUNK time How weird is weird? Reference point: a simple fluid: only one player: the fluid itself Stuff that happens: Waves: only sound waves Friction: viscosity($) = a fluid dragging against itself

6. ghost Plasmas($) are weird Three players:  ion fluid (+)  electron fluid ()  electromagnetic (EM) field All three occupy the same space at the same time

7. Weird… Ionand electron fluids: (a) motion energy (kinetic): a high-grade energy form (b) thermal energy (heat): low-grade energy low-grade = highly disorganized; high entropy($) Re second law of thermodynamics All three players carry their own kinds of energy EM field: (a) electric field energy: high-grade energy (b) magnetic field energy: high-grade energy Fusion: the ions are an isotope of hydrogen

8. Blah blah Blah blah Blah blah How do the three actors get along? • ion fluidtoelectron fluid: drag against each other = electrical resistance($) Ions talk to electrons; both talk to the EM field  ion fluidtoEM field: ions push against the EM field, and vice versa = Lorentz force ($) electron fluidtoEM field: ditto

9. Waves in a plasma Simple fluids have sound waves, but a plasma has many other kinds Fluids talking to EM field gives rise to… sound waves, Alfven waves($) and many many more

10. Oof! Biff! Sock! Ugh! Sibling rivalry Get along: they must stay close together, otherwise gigantic electric fields jerk them back together: quasi-neutrality($) Ions and electrons get along, sort of… Fight: they drag against each other: = electrical resistance ($)

11. Energy crisis Energy implications: Electrical resistance burns upEM fieldenergy (high-grade energy) and dumps it intoionandelectron thermal($) energy (low-grade energy) Electrical resistance is an entropy($)-builder

12. Fusion and space plasmas Very hot! experimental plasmas ~ 100 eV 1 eV($) = 11,600 oK fusion plasmas ~ 10,000 eV = 10 keV In hot plasma the drag between ions and electrons is very weak, i.e. low electrical resistance ($)

13. Hmmph! Pah! Snobs In high-temperature plasmas the ions and electrons hardly even speak to each other

14. “Blah blah blah!” “Blah blah blah!” Tell him “Blah blah blah!” Tell her “Blah blah blah!” Swiss Embassy EM field: intermediary between ions and electrons.

15. Peace! Peace! My toys My toys break Important exception Ions and electrons can have a modest amount of direct communication, if the EM field is tough This can happen with plasma relaxation($)

16. Part II: Behavioral Psychology

17. Two fundamental classes of behavior Waves and friction (1) Waves: very complicated in a plasma because there are three players, not just one. sound waves($): energy of ions and electrons Alfven waves($): energy of magnetic field Energy: waves move energy around from one place to another. How fast?

18. How fast do things happen with waves? Timescale($) for wave dynamics: wave ~ L/V L = size scale of plasma (~ 20 cm) V = (say) speed of sound: Our friendly plasma  100 eV hydrogen plasma: V ~ 1800 km/s Timescale for wave dynamics is wave ~ 1 s

19. He He! YOW! ZAP! magic pencil Classes of behavior… (2) Friction: electrons drag against ions electrical resistance($) Energy: electrical resistance burns up EM field energy and turns it into low-grade “heat” energy

20. Dissipation Dissipation moves energy around too It always takes things downhill Electrical resistance burns up EM field energy (high grade) and converts it into plasma heat energy (low grade) Thermal conduction moves heat energy from one place to another; this also is an entropy builder too, making the temperature uniform (less organized) Viscous friction burns up the flow energy (high-grade form of plasma energy) and turns it into heat (low-grade)

21. How fast do things happen with dissipation? Timescale for burning up EM field energy: resistivity ~ L2/D L = length scale D = resistive “diffusivity($)” In our friendly plasma D ~ 1.2 m2/s; Timescale for burning up EM field energy resistivity ~ 30 ms Much slower than the timescale for waves

22.  Lu = 30,000 A fancy name: Lundquist number($) Lundquist number: Lu resistivity/wave Waves move energy around 30,000 times faster than resistance burns up EM field energy.

23. Push me again Papa The swing Wave: oscillating motion of the swing Dissipation: friction in the air and rope If Lu = 30,000, Baby would swing back and forth ~30,000 times before the swing more-or-less stops.

24. break ? Key question If Lu >> 1 (waves much faster than dissipation) can we ignore dissipation? Answer: partly… but not entirely: In Plasma Relaxation, Lu is large but dissipation still plays a key role More later…

25. Part III “Good” Behavior

26. Four types of behavior (1) Totally good behavior: the “ideal case” Our friendly plasma as a compliant little angel This hardly ever happens

27. Keep it down. Poke Poke Type-2 Behavior Low-level scrapping that the baby-sitter tolerates “A dull roar”

28. Oof! Ugh! Type-3 Behavior 3) Violent scrap The baby-sitter is a casualty; the kids end up wasted.

29. Oof! I survived! I’m happy I’m happy Ugh! Type-4 Behavior 4) Violent scrap but the result is different… Babysitter manages to make some adjustments, after which things are more or less peaceful

30. Things like this can happen in a plasma #1) Totally good behavior. Say Lu = 30,000. Waves equalize everything out in about 1 s and then hardly anything happens for the next 30,000 microseconds. Good, but it hardly ever happens!

31. Type-2 Behavior 2) Low-level scrapping No major violence, but continuous, small-scale scrapping between ions and electrons = plasma turbulence($). This leads to a plasma that leaks energy faster than you would like.

32. Type-3 Behavior Plasma starts off more or less quiet: quiescent($) Suddenly, a violent instability (BIG WAVE) appears Plasma goes crazy; A lot of dissipation($) takes place. The EM field intervenes, but gets knocked out: Death ensues = disruption($)

33. Type-4 Behavior Plasma starts off more or less quiet: quiescent($) Suddenly, a violent instability (BIG WAVE) appears, but the result is different Some dissipation($) takes place. The EM field intervenes, and calms things down: The plasma survives in a relaxed state($)

34. I need a vacation Energy Type-4 behavior, but at a price It wears down the EM field The cost of relaxation($) Getting toward …the topic of this lecture

35. Nasty little gremlin L But but but… How can all this violence happen at high–Lu? Timescale formula dissipation ~ L2/D ~ 30 ms for L = 25 cm (plasma size). What if a wave process has a much smaller scale, say L ~ 1/2 cm? Then dissipation ~ 20 s …almost as fast as the wave time. Example: reconnection($)

36. BRRRR! break Small is big (or beautiful?) Can small stuff affect the whole? (The mouse that roared) Exampleinsulating your house Install R100 insulation keep single-pane windows Even though Lwindow << Lhouse most of the heat loss is through the windows. The house remains cold and drafty

37. Part IV The secret of “Good” Behavior

38. Om The mystery of the relaxed state What’s the difference between type-3 behavior (passed out) and type-4 (relaxed state) The secret - EM field: a hidden personality trait of the babysitter The esoteric mathematics of topology.

39. two twists no twist one twist Simple illustration #1: Möbius strip “squashed donut” Consider a particle moving along an edge. In making one complete loop (long way), how many times does a particle circle around the short way? Magnetic fields can get twisted around themselves like this.

40. Gmmp! mini-break (2) Knottedness Magnetic fields can get tied up in knots too. Magnetic helicity($100)Km measures the knottedness. Mobius: untwisted = zero one twist = some two twists = more How to express this mathematically?

41. Part VI: The math of relaxed states

42. You’re welcome. Thanks B! B A Magnetic helicity: the magic personality trait Km = ABdV B = magnetic field($) (vector) A = “vector potential($)”: B = A A exists because B = 0 One of Maxwell’s equations($)

43. domain boundaryS domain volume V What is a “domain” “DOMAIN”: volume occupied by the plasma inside some well-specified boundary Example of an idealized domain boundary: a perfectly-electrically conducting metal wall.

44. magnetic field (magnitude of B) Natural constant: permeability of free space Magnetic energy Another important volume integral: total magnetic energy in the domain Wm is a high-grade form of energy

45. I cannot tell a lie Properties of the two integrals In the perfectly ideal case (only waves, no dissipation) Wm = constKm = const i.e. they are constants of motion($) or integrals of motion($) Caveat: not quite true for Wm, (see later “postscript”)

46. I’m not Brian Taylor; he’s not nearly as handsome. Ta da! Taylor theory Named after Brian Taylor Truth telling: Taylor’s Conjecture

47. house of straw house of bricks Same big bad wolf Key elements of Taylor theory All animals are equal, but some are more equal than others Principle #1: SELECTIVE DECAY($100) (a) It isn’t a perfect world, but some things play in our favor i.e. you must take dissipation into account even if Lu is large Go elephants! (b) Wm is less rugged than Km i.e., when dissipation goes to work, Wm burns down a lot faster than Km

48. Wow, they never taught that in preschool Taylor: principle #2: MINIMUM ENERGY($100) How far down does Wm burn? As low as possible while keeping fixed Km Minimize Wm subject to fixed Km  relaxed state($) (Taylor state) Math: constrained minimization problem($) variational calculus($)method of Lagrange multipliers($)

49. You may not get all this today. Finding Taylor states($): do the math  (WmKm) = 0 variation of (…) Lagrange multiplier (constant) Substitute

50. The math (cont.) Remember B = A Apply the variations