Unstable Relationships(Part 1)

Outcomes

i. Give examples of common instabilities(esp. plasmas)ii. Recognise generic susceptibility to instabilityiii. Describe the nature of population instabilitiesiv. 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 instability2. Unstable populations I: (nonlinear) differential equations

3. What is plasma ?

4. Unstable populations II: species in a plasma

6. Normal modes and instability

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)

Idea

s ab

out

inst

abili

ty

Idea

s ab

out

inst

abili

ty

convectiveinstability

absoluteinstability

stable

Independent population growth (rabbits)

number of rabbits

Uns

tabl

e po

pula

tions

Coupled population growth (foxes and rabbits)

number of foxes

Uns

tabl

e po

pula

tions

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 foodU

nsta

ble

popu

latio

ns

Equilibrium population (foxes and rabbits)

equilibrium points

equilibrium conditionsUns

tabl

e po

pula

tions

Oscillatory population (foxes and rabbits)

Time evolution Phase plane

Uns

tabl

e po

pula

tions

Sinusoidal oscillations (electric circuits)

dv

dt=

iC, v=−L

didt

C L

i

v

&x =k1y, &y=−k2x

&&x=k1&y=−k1k2x&&x=−kx

x=Asinωt+ Bcosωt

Uns

tabl

e po

pula

tions

Sinusoidal oscillations (electric circuits)

dv

dt=

iC, v=−L

didt

C L

i

v

&x =k1y, &y=−k2x

&&x=k1&y=−k1k2x&&x=−kx

x=Asinωt+ Bcosωt

Uns

tabl

e po

pula

tions

Sinusoidal oscillations

&x =k1y, &y=−k2x

&&x=−k1&y=−k1k2x&&x=−kx

x=Asinωt+ Bcosωt

Uns

tabl

e po

pula

tions

C L

i

v

foxrabbit

Instability: weakly-damped, undamped or growing interchanges of energy or information between two (or more) reservoirs

back

Uns

tabl

e po

pula

tions

The Tarantula Nebula

(Hubble Space Telescope)

The Aurora Borealis

(Jan Curtis, 6/9/96)

Lightning over Oxford

(A A Goruppa, 1994)

Wha

t’s a

pla

sma

?

Fusion research plasmas (JET)…hot physics with hot engineering

A Plasma Ball…

High pressure1-10 kPa Ar 100 V 13.56

MHz

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)

Wh

at’s

a

pla

sma

?

fully ionisedpartially ionised

10 Pa = 75 mtorr 0.005% ionised

gas density 2 1020 m-3

plasma density 1016 m-3

Wh

at’s

a

pla

sma

?

E

F = q(E + v B)q

natural time scale

mdv

dt=q v×B ~mv/ τ

τ ce =meB

ωce =eBm

cyclotron frequency

B

Lorentz force

an electron passing a single -/+ charge at the origin at

110 km h–1 (30 m s–1)

-2

-1

0

1

2

-2 -1 0 1 2

-2

-1

0

1

2

-2 -1 0 1 2

Binary collisions

Wh

at’s

a

pla

sma

?

elastic

inelastic

Wha

t’s a

pla

sma

?

You are about here, but neutral

surfacevolume

Making charges

A +B±→ A+ +B±+ e-

A +hν → A+ +e-

A +B* → A+ +B+e-

efast- +A → A+ +eslow

- +eslow-

A

B

B

Wh

at’s

a

pla

sma

?

Losing charges

A + e- → A -+hνAB* + e- → A -+B

e-+A+ + e- → A+efaster-

e-+A+B→ AB-

AB-+C+ → ABC

neutralisation

Volume loss rate depends on concentrations

Surface loss rate depends on fluxes

Steady state but not thermodynamic equilibrium

...characterising plasmas - sustaining the steady state...

Wh

at’s

a

pla

sma

?

&ne =dne

dt=K iznenO2

+ Ko-detnOn– + Ke-detnen– + Km-dretnmn–

−Katt2nenO2−Krec2nen+

unst

able

pro

duct

ion–

loss

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: Ki(Te) – strong function

ne … Te and then ne

(negative feedback).

unst

able

pro

duct

ion–

loss

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)

unst

able

pro

duct

ion–

loss

The electron density peaks where light output peaks.

500 mT, 150 W600 mT, 50 W500 mT, 50 W

Oxygen plasma

Filtered Photo diode signal

500 mT, 150 W

unst

able

pro

duct

ion–

loss

Photodiode signal shows 3-5 kHz instabilities in oxygen plasma.

Oxygen plasma (500 mT, 150 W)(a) Unfiltered signal (b) Filtered signal

A Descoeudres, L Sansonnens and Ch Hollenstein Plasma Sources Sci. Technol. 12 (2003) 152–157

RF on

unst

able

pro

duct

ion–

loss

instability seen growing to saturation within a few cycles of switch on

Unstable Relationships(Part 2)

Outcomes

i. Give examples of common instabilities(esp. plasmas)ii. Recognise generic susceptibility to instabilityiii. Describe the nature of population instabilitiesiv. 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 instability2. Unstable populations I: (nonlinear) differential equations

3. What is plasma ?

4. Unstable populations II: species in a plasma

6. Normal modes and instability

unst

able

mod

es

C S Corr, P G Steen and W G GrahamPlasma Sources Sci. Technol. 12 (2003) 265–272

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

unst

able

pro

duct

ion–

loss

–ene

rgy-

inpu

t

C S Corr, P G Steen and W G GrahamPlasma Sources Sci. Technol. 12 (2003) 265–272

C S Corr, P G Steen and W G GrahamPlasma Sources Sci. Technol. 12 (2003) 265–272

unst

able

pro

duct

ion–

loss

–ene

rgy-

inpu

t

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

unst

able

mod

es

unst

able

mod

es

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

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Ele

ctro

mag

netic

mod

es in

pla

smas

Ele

ctro

mag

netic

mod

es in

pla

smas

Maxwell

div D = ρdiv B = 0

curl E = −∂B∂t

curl H = J f+∂D∂t

D =εε0E

B =μμ0H

=1

= ?

Electromagnetism in plasmas

Ele

ctro

mag

netic

mod

es in

pla

smas

Ele

ctro

mag

netic

mod

es in

pla

smas

E, H ~ exp –iωt

But at HF, the conductivity term can be neglected

Dielectric function permittivity for a plasma

Dielectric model: Motion of a bound electron in an E-M field

Ele

ctro

mag

netic

mod

es in

pla

smas

lossesresonances

Dielectric function permittivity for a plasma

Ele

ctro

mag

netic

mod

es in

pla

smas

P =(ε −1)E

Polarization:

Dielectric function:

Dielectric function permittivity for a plasma

Dielectric model: free electron in an e-m field

Ele

ctro

mag

netic

mod

es in

pla

smas

= 0 for free electron

= 0 for low collisionless

Dielectric function permittivity for a plasma

Electromagnetic waves in unmagnetized, collisionless plasma

Ele

ctro

mag

netic

mod

es in

pla

smas

Ele

ctro

mag

netic

mod

es in

pla

smas

Polarization in magnetized plasma

Dielectric properties of plasma now dependent on polarization of E-M radiation

Ele

ctro

mag

netic

mod

es in

pla

smas

Electromagnetic waves in magnetized, collisionless plasma

weakly magnetized strongly magnetized

Ele

ctro

mag

netic

mod

es in

pla

smas

What makes the electromagnetic modes unstable?

An available source of energy: ‘free energy’• flows of heat• flows of particles – beams• large amplitude waves

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