1. Unstable Relationships (Part 1) Outcomes i.Give examples of common instabilities(esp. plasmas) ii. Recognise generic susceptibility to instability iii. Describe the nature of population instabilities iv. Describe the nature of modal instabilities v. Distinguish between absolute and convective instability N St J Braithwaite QuAMP ’06 5. Unstable modes 1. Ideas about instability 2. Unstable populations I: (nonlinear) differential equations 3. What is plasma ? 4. Unstable populations II: species in a plasma 6. Normal modes and instability

2. Common examples of instability Balloon buoyancy/burst Falling off a log Rayleigh Taylor – heavier fluid over lighter fluid Population explosion Audio ‘feedback’ Squeaking balloon ‘Wheel wobble’ Kelvin Helmholtz – differential fluid flow(s) Ideas about instability

3. convective instability absolute instability stable

4. Independent population growth (rabbits) number of rabbits Unstable populations

5. Coupled population growth (foxes and rabbits) number of foxes Unstable populations

6. Coupled population growth (foxes and rabbits) fox as predator tends to reduce rabbits population rabbit as food tends to increase fox population foxes compete for food Unstable populations

7. Equilibrium population (foxes and rabbits) equilibrium points equilibrium conditions Unstable populations

8. Oscillatory population (foxes and rabbits) Time evolution Phase plane Unstable populations

9. Sinusoidal oscillations (electric circuits) C L i v Unstable populations

10. Sinusoidal oscillations (electric circuits) C L i v Unstable populations

11. Sinusoidal oscillations Unstable populations

12. C L i v fox rabbit Instability: weakly-damped, undamped or growing interchanges of energy or information between two (or more) reservoirs back Unstable populations

13. The Tarantula Nebula (Hubble Space Telescope) The Aurora Borealis (Jan Curtis, 6/9/96) Lightning over Oxford (A A Goruppa, 1994) What’s a plasma ? Fusion research plasmas (JET)…hot physics with hot engineering A Plasma Ball… High pressure 1-10 kPa Ar 100 V 13.56 MHz

14. Plasma in a nut shell an ionised gas a condition of matter beyond gaseous (amounting for >99% of the matter of the visible universe) exist from astronomical to microscopic scales behave as quasineutral mixture of charged fluids and neutral gas components are hot enough to radiate electromagnetic energy (glow) particularly interesting when not in equilibrium (like solids, liquids and gases) What’s a plasma ?

15. fully ionised partially ionised 10 Pa = 75 mtorr 0.005% ionised gas density 2  10 20 m -3 plasma density 10 16 m -3 What’s a plasma ? E F = q(E + v  B) q natural time scale cyclotron frequency B Lorentz force

16. an electron passing a single -/+ charge at the origin at 110 km h –1 (30 m s –1 ) Binary collisions What’s a plasma ? elastic inelastic

17. What’s a plasma ? You are about here, but neutral

18. surface volume Making charges A B B What’s a plasma ? Losing charges neutralisation Volume loss rate depends on concentrations Surface loss rate depends on fluxes

19. Steady state but not thermodynamic equilibrium...characterising plasmas - sustaining the steady state... What’s a plasma ?

20. unstable production–loss

21. Stable plasmas Electrode voltage envelope for stable oxygen plasma. Stable Oxygen plasma (500 mT, 25 W) Nigham & Wiegand, 1974 “Changes in the electron density lead to a change in the electron temperature due to the quasi- steady nature of the electron energy” Volume production = wall loss Ionization rate: K i (T e ) – strong function n e … T e and then n e (negative feedback). unstable production–loss

22. Unstable Plasma An output voltage envelope for unstable oxygen plasma. [Attachment about 100 x faster than electron impact detachment and ion- ion recombination] Unstable Oxygen plasma (500 mT, 150 W) unstable production–loss

23. The electron density peaks where light output peaks. 500 mT, 150 W 600 mT, 50 W 500 mT, 50 W Oxygen plasma Filtered Photo diode signal 500 mT, 150 W unstable production–loss Photodiode signal shows 3-5 kHz instabilities in oxygen plasma. Oxygen plasma (500 mT, 150 W) (a) Unfiltered signal(b) Filtered signal

24. A Descoeudres, L Sansonnens and Ch Hollenstein Plasma Sources Sci. Technol. 12 (2003) 152–157 RF on unstable production–loss instability seen growing to saturation within a few cycles of switch on

25. Unstable Relationships (Part 2) Outcomes i.Give examples of common instabilities(esp. plasmas) ii. Recognise generic susceptibility to instability iii. Describe the nature of population instabilities iv. Describe the nature of modal instabilities v. Distinguish between absolute and convective instability N St J Braithwaite QuAMP ’06 5. Unstable modes 1. Ideas about instability 2. Unstable populations I: (nonlinear) differential equations 3. What is plasma ? 4. Unstable populations II: species in a plasma 6. Normal modes and instability

26. unstable modes C S Corr, P G Steen and W G Graham Plasma Sources Sci. Technol. 12 (2003) 265–272 P Chabert, A J Lichtenberg, M A Lieberman and A M Marakhtanov Plasma Sources Sci. Technol. 10 (2001) 478–489

27. unstable production–loss–energy-input C S Corr, P G Steen and W G Graham Plasma Sources Sci. Technol. 12 (2003) 265–272

28. C S Corr, P G Steen and W G Graham Plasma Sources Sci. Technol. 12 (2003) 265–272 unstable production–loss–energy-input

29. P Chabert, A J Lichtenberg, M A Lieberman and A M Marakhtanov Plasma Sources Sci. Technol. 10 (2001) 478–489 unstable modes

30. P Chabert, A J Lichtenberg, M A Lieberman and A M Marakhtanov Plasma Sources Sci. Technol. 10 (2001) 478–489 back

31. Electromagnetic modes in plasmas

32. Maxwell =1 = ? Electromagnetism in plasmas

33. Electromagnetic modes in plasmas E, H ~ exp –i  t But at HF, the conductivity term can be neglected Dielectric function permittivity for a plasma

34. Dielectric model: Motion of a bound electron in an E-M field Electromagnetic modes in plasmas losses resonances Dielectric function permittivity for a plasma

35. Electromagnetic modes in plasmas Polarization: Dielectric function: Dielectric function permittivity for a plasma

36. Dielectric model: free electron in an e-m field Electromagnetic modes in plasmas = 0 for free electron = 0 for low collisionless Dielectric function permittivity for a plasma

37. Electromagnetic waves in unmagnetized, collisionless plasma Electromagnetic modes in plasmas

38. Polarization in magnetized plasma Dielectric properties of plasma now dependent on polarization of E-M radiation

39. Electromagnetic modes in plasmas Electromagnetic waves in magnetized, collisionless plasma weakly magnetized strongly magnetized

40. Electromagnetic modes in plasmas What makes the electromagnetic modes unstable? An available source of energy: ‘free energy’ flows of heat flows of particles – beams large amplitude waves back