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Unstable Relationships (Part 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

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Unstable Relationships (Part 1)

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  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 1. Ideas about instability 2. Unstable populations I: (nonlinear) differential equations 3. What is plasma ? 4. Unstable populations II: species in a plasma 5. Unstable modes 6. Normal modes and instability N St J Braithwaite QuAMP ’06

  2. Common examples of instability Ideas about 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)

  3. Ideas about instability stable convectiveinstability absoluteinstability

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

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

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

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

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

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

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

  11. Sinusoidal oscillations Unstable populations

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

  13. The Tarantula Nebula (Hubble Space Telescope) What’s a plasma ? The Aurora Borealis (Jan Curtis, 6/9/96) A Plasma Ball… Lightning over Oxford (A A Goruppa, 1994) High pressure1-10 kPa Ar 100 V 13.56 MHz Fusion research plasmas (JET)…hot physics with hot engineering

  14. Plasma in a nut shell What’s a plasma ? • 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)

  15. fully ionised partially ionised What’s a plasma ? 10 Pa = 75 mtorr 0.005% ionised gas density 2  1020 m-3 E plasma density 1016 m-3 F= q(E + vB) q Lorentz force natural time scale cyclotron frequency B

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

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

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

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

  20. unstable production–loss

  21. Stable plasmas ne … Te and then ne Stable Oxygen plasma (500 mT, 25W) unstable production–loss (negative feedback). • 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: Ki(Te) – strong function Electrode voltage envelope for stable oxygen plasma.

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

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

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

  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 1. Ideas about instability 2. Unstable populations I: (nonlinear) differential equations 3. What is plasma ? 4. Unstable populations II: species in a plasma 5. Unstable modes 6. Normal modes and instability N St J Braithwaite QuAMP ’06

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

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

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

  31. Electromagnetic modes in plasmas

  32. Electromagnetism in plasmas Maxwell = ? Electromagnetic modes in plasmas =1

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

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

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

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

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

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

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

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

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