Lecture 5

Instability

• Why can some solar features stay long time like prominences and coronal loops, while some produce quick events such as flares? This is the problem of instability.

• Basic Methods– Linearization method

Consider one dimension potential energy waveforce :

for small displacement

is first order approximation of F.

dxdWFxm

dxdWF −==−=

..

)(10

2

2..xF

dxWdxxm

x

=

−=

=

)(1 xF

StableNeutrally0Unstable00

Stablen,Oscillatio00 if

1

solution) mode (normal Assume

2

2

02

2

02

22

0

=>

=

=

ω

ω

ω

ω

dxWd

dxWd

m

exx ti

• Energy Method

unstable - point onefor 0stable -nt displaceme small allfor 0

)(21

2)0()( 1

02

22

<>

−=

=−=

WW

xxFdxWdxWxWW

δδ

δ

• Linearized Equations• This is very similar to the study of waves

( )

( )

0,0,

0

=

=⋅∇

×∇=

=∇+∂∂

+×+−∇=

××∇=∂∂

rP

DtDBBj

vt

gBjPdtDv

BvtB

ρµ

ρρ

ρρ

( )

( )

( )

∇⋅−

∂∂∂

=∂∂

=∇×∇

=

=⋅∇=∂∂

+×+×+−∇=∂∂

××∇=∂∂

+=+=+==+=

=⋅∇×∇

=

+×+∇=

γρρ

ρ

µ

ρρ

ρρ

ρρρ

µ

ρ

0

01

0

01

11

1

101

1100111

0

011

101010110

00

000

log10

0,

0

:

,,,,

by mequilibriu thePerturb

0

0

mequilibriuAt

PvPt

PtP

BBj

vt

gBjBjPtv

BvtBThen

jjjBBBPPPvv

BBj

gBjP

e

( ) ( )

( )( )

( )( )

( ) ( )( )( )( ) ( )( )

( ) ( ) tiertr

BB

BBgP

BjBjgPtrF

trFt

BBt

v

zz

yy

xx

ωξξµ

ξµ

ξξρ

ρξ

ξξρ

ξρρξ

ξ

00

00

0001

101110

02

2

0

01001

1

0

, solution

,

,

becomesmotion ofEquation

Define

ˆˆˆorder lowest To

vv

vvv

vvvvv

vvvvvv

vv

vvvv

vv

=

××∇××∇+

×××∇×∇+∇+−∇=

×+×++−∇=

=∂∂

⋅−∇=××∇=∂∂

=

∂∂

+∂∂

+∂∂

=∇

( ) ( )( )

( ) ( ) ( )( )( )

unstable0stable - oscillates system0

constants. are cose special a As

2

200

200

20

000

0002

<

>

×××∇×∇+∇−⋅∇∇=

=−

ω

ω

ξρξρξρ

ρξξρω

BBvgcF

BPrFr

As

vvvvvv

v

ξv

0rv

rv

O

mequilibriu

• Rayleigh-Taylor Instability• See Fig 7.3 interface between two medium

( ) ( )

( ) ( ) stable

stable0if

00

00

un

B−+

−+

>

( )

[ ] [ ][ ] [ ]

[ ]

)(0

)(1

)(1)(

02

2

011

02

2

12

01

1111

)(1

)(11

101

101

12

0

0

0

So

,0 7.10from

0interfaceover integrate

+++

+

−+

+

−=

+

−=

−=

−==+

←−=

+−=

+−=−

ρρω

ρρω

ωρ

ρω

ρω

ρωωρ

zz

zz

z

yyz

z

zz

gvdzdv

k

gvdzdv

k

dzdv

kiP

ikPviikvdzdv

jumpPPP

vgPi

vdzdg

dzdPv

( )( )

( )

unstable h wavelengtsmall --- stable

form general more

like grow

00

0,0,00

uniform are,, valuemequilibriu initial if

2)(0

)(0

)(0

2)(0

22

)(1

2

112

21

2

21

11

000

critical

critical

criticalx

ykxkti

tgk

kz

kz

zzz

kkkk

BgkBkgk

ezve

egk

zeze

vvkdzvd

vvtt

BPB

yx

<>

=+−=

−=

<>

=+=

=∇=×∇=∂∂

=∂∂

−

+

+

−

+−

−

µρµρ

ω

ω

ρρ

ω

vv

Case 2: Uniform field (Fig. 7.3)It can be shown that

( )( )

c

c

c

x

kk

kkB

gk

kBgk

21 mode growingfastest

0 unstable is interface2

2

20

)(0

)(0

)(0

)(0

220

)(0

)(0

)(0

)(02

=

• Energy Method Applications

( ) ( )

( )

( )[ ]

( ) ( )( ) ( ) ( ) dVgPBjBdVBjBjgP

dVF

dVgzUPBW

ertrBBPP

WstableW

ti

⋅∇⋅+∇⋅∇+×−=

×+×⋅−⋅−∇⋅=

⋅−=

++=

=+++

<>

∫

∫

∫

+

ξρξξξρρ

γξµ

ξρξξ

ξ

ρµ

δ

ξξρρ

δδ

ω

vvvvvvv

vvvv

000

010

21

10011

000

20

00101010

2121

212

,,,,above ason perturbati

unstable point,oneleastatin00

• Kink instability• Consider a cylindrically symmetric magnetic flux tube. If it is

kinked, point A has stronger field than point B. ---- perturbation can go further.In polar coordinate system

( ) ( )

( )

( )( ) ( )( )01

02

11

202

020

00

00

0

000

& ignore2

2

0 condition free force

ˆ1ˆ

ˆˆ

BB

PgdVBBBWRB

BBdRd

Bj

zRBdRd

RdRdBj

zRBRBB

z

zz

z

××∇=

×∇×⋅−=

−=+

=×

+−=

+=

∫−ξ

ξµδ

φ

φ

φφ

φ

vvv

vv

v

v

( ) ( )

( ) ( )( )( ) ( ) ( )( )[ ] ( )

( )

+++

−+

++−

++−

−+

−=

++++

=

−++

+−

= ∫

∞

222

20

22202

02

222

20

2

222

200

200

222

20

32200

0

20

2222

2

1

11

1

1W

steps dcomplicate several

RhkBRhB

dRdRBh

RhkRBh

RhkRkRBB

RRkBB

RG

RhkBRhkRBBR

RF

dRHBRRhkRG

dRdF

zz

z

R

φ

φφ

φ

ξξξδ

( ) ( ) ( )

( )

7.8 & 7.7 Figsin shown assuch results numerical gettingby done becan This

0

solvecan we,conditionsboundary with

2

1,,

0

22

0000222

=+

−

=

∝=

+−−⋅

++=

∫∞

RR

RR

kz

z

R

z

RRR

GdRdF

dRd

dRGdRdFW

eL

h

BkRBR

BBkRdRd

RhkR

dRdRH

ξξ

ξξδ

ξπ

ξξξξ φ

• Summary of Instabilities1. Interchange instability

An interface between two plasma with different P&B. If two neighboring bundle can be interchanged, if the wrinkles have a wavenumber k, radius of curvature Rc interface, then

unstable :Short wavestable : waveLong

2L. distance

a with ends in twodown anchored are fields magnetic if

2

2

22

2

Lv

RPk

RPk

A

c

c

+−=

−=

ρω

ρω

2. Reyleigh-Taylor instability (Fig 7.3)

( )

( )

( )20

)(0

)(0

)(0

)(0

220

)(0

)(0

)(0

)(02

000

)(0

)(0

)(0

)(02

)(0

)(0

)(0

)(02

2

condition critical

2yinstabilit changecan field horizontal uniform

0

yinstabilit changenot dobut rategrowth modify fields, verticaluniform

rateat growon Perturbati

Bgk

kBgk

Bg

ik

gkk

gk

e

c

x

ti

µρρ

ρρµρρρρω

ρρµ

ω

ρρρρω

ρρρρω

ω

−+

−+−+

−+

−+

−+

−+

−+

−+

−=

++

+−

−=

−=∞=

+−

−==

+−

−=

( )

stable are wavesLong

restore tories tension tmagnetic ,00 unstablemost

: thenbelow, fields magneticby supported is plasma if

2

2

0

20

22

gLvk

kgkik

PBkgk

A

x

x

x

<

≠

==

+−= +−

ω

ρµω

3. Pinched dischargevertical current produces azimuthal fields plasma is confined –pinched

inwards is balanced by pressure gradient outwardsIf there is no magnetic fields inside plasma the growth rate of sausage instability is

for disturbance

The pinch can be stabilized by large enoughaxial field

modified dispersion relation

for stability, if requires

Bjvv

×

radius :221

0 aakPi

=

ρω

a

P

iak

21

0

0

12

,

=≈ −ρ

ω

µµφ

22

220

0

BBPB zz =+

20

20

0

02 2a

BaP z

µρρω +−=

220 2

1φBB z >

4. Flow InstabilityLaminar viscous flow between rigid boundaries such as walls of Lab channels becomes unstable and develop into a turbulent flowit occurs when critical Reynolds number*ee RR >

( )

( )( ) ( )( )2

*e

21

00

*e

22

if unstable

yinstabilit Helmhotz -Kelvin - speeds flow 2 baring flow inviscid uniform have weIf

Ha500RflowB//

numberHartman :LBHa

Ha000,50Rflow

−++−

+−

−

−>

=

=

=

⊥

UUgk

v

B

ρρρρ

ρσ

5. Resistive Instability

( )

( )

( )

( )21

0

0

12

31

44

22

20

1

2

A

,,,

rateGrowth

on.straficatidensity produce sheet tocurrent totransverseˆn gravitatio - moden gravitatio

stable ifˆ force restoring

force drivingydiffusivit magnetic

scale timeaon plasma

throughdiffused width ofsheet current ain field

−−

−

−====

=

>−=×=

=

=

dxdgl

vl

kli

xgFF

xBvBjF

F

llB

GdA

A

Gd

A

dL

xc

d

ρρ

τµσηη

ττ

τττω

ρ

εσ

µση

ητ

vv

( )

( )

( )[ ]beating. coronal on,reconnecti leadcan rategrowth fastest hasvelength longest wa

1,

,requirenot does

1 when occursonly :mode tearing

ˆ

y diffusivit magneticin sheet theacross variationspatial a is there:mode rippling

41

51

223

0

51

32

24

0

0

200

0

0

6. Convective Instability

yinstabilit convective0if

frequency Vaisala-Brunt

37.4

only Consider 4.chapter in thisdiscussed have We

200

00

0

2

→>

>

=

−=

ω

ω

ad

ad

dzdT

dzdTN

dzdT

dzdT

TgN

g

7. Radiatively driven thermal instabilityIf thermal conduction were in effective, thermal instabilities would occur in the corona and upper atmosphere, due to the radiative loss term in energy equation.

Suppose plasma is initially in equilibrium with balance between mechanical heating (per unit volume) and opticallythin radiation leads to

00 ,ρTρh

αχρT αχρ 00Th =

yinstabilit thermal- continues,on perturbati

,0tT makes)T(T T,in decrease small a 1, if

1

pressureconstant at on perturbati afor

0

10

1

00

0

<∂∂

( )

( )

instable

length critical

stabalized

timeconduction

fields. magnetic along conductionheat by prevented isy instabilit Thermal7.1 Table see10Tfor 1 Typically,

scale Time

max

max

21

20

2/700

2/500

2

2/50////

1-

5

100

LL

LTkL

TkcL

kkTk

K

Tc

radc

Pc

Prad

>

=

=

<=

=∇∇

><

=

−

−

χρ

ττρτ

ρ

α

γρτ

α

α

Homework

• Try to use normal mode method to analyze the kink instability.