equilibrium j→✗B→=…p +

b-

/

¹⁰

Stability

^assessment

/

^{Ideal MHD}

formulation equilibrium j→✗B→=Ép

^{+ (}

pñ

^{. -55't . - )}

don't know yet if ^{it's stable or not}

* ~

linearly

^{unstable & Kirk, ballooning}

)

linearly

^stable _fearing

big

perturbation

VE

RET

non

linearly

^{unstable non}

linearly

^stable

e neoclassical tearing ^mode

)

^{( niicro turbulence)}

* Formulation

of

^linear

stability

- Adds ^a small perturbation ^{and see what happens} A→eñ,-4 ⁼ A→ocx→^{. + I}

?ex→

^,

-4

^{A. →A}_,

i Assume exponential growth ^or

damping

A-ioir-y-A.ci

, e-^iwte

• Im ^{(w ) > 0}

Instability

^lInhomogeneous ^{media )}

{

^{waves Chimo}_generous ^media

)

* Ideal ^{MHD waves}

of

,e+

^i.

tip t.pe?i7--o

up^, =p

.ci?u?7WPi--dpocE?up

,

Got ^used ]

zp ^el)

jet

ÑÉP -14pct

^'-

Ñ)=o

2 B→

get

^-

Ñ×É=

^{+ (ÑxB→}₎ ^WB

?

^{= - Éxlñi+BT ) ez,}

No

_j=ÉtB→ wµoj→

^{, = -iv.+ (}

É+É+Éo

^{)) ez,}

Pp¥eñ .EE/---jxB'--Epwpoiei--il-ii-B?7-

^{ÉP, e4)}

differential of ^{→ Algebraic} eq^.

Assume

homogeneous

^{media ,}

B→o=BoÉ

→

To ^{= 0} _, U→o=0 ^,

ÑPo=0

_,

Épo=0

Let

A.ee?t)=A,eick?x-'-w-C

⁾

note _,

Ige

^{→ -Tw É → TÉ for perturbed quantities.}

also ^note _, vi.ÉA^' =

u-i.si#+u?.-FA?=oUo--o

homogeneity

en en ,,, y^,,,,, , ^{, ,,} gy,

WB

?

^{= -}

UT

^Cri?

Bj's

⁺

BICE

^?u?

)

wteoj ?

^{= -}

itfxu-ijcri.BJJ-ifk-i-B%cE.ci

^{, )}

Wao ^✗

BY

^{= - T}

[

^ñ

? cri.is?)-k-Cu-?.Bo5fCk-?Bo

⁾

+ T

[ BICE .BY

^{) -}

EBI ]eE?ñi )

¥ uiporeoñi

⁼

[ ñui .is?y-ricui?B-oycri.Bi7

-

[ Bick ?Ñ

^{) -}

EBI ]cÑ .ie?j+MotPor?lE?u-?g

Let

vi=µ÷p

^.

ci=%÷

A /fue ^{n sound}

(4)

/ Bi

^→

µ

^-

Kiva )u?

=

k-Ieri.ie?3vI-rickuUiii)vI+k-ck?u-?)g2

Let

{

^ñ

k→

_, ⁼⁼

unitary KIF

^{+ K,}

,É 5+42-2

^{w/o loss}

of generality

(

^w

'

-

Kivi

_)uix=o

eat

^-

kicf-rivijuiy-cki-ki.ci

³⁴²⁼⁰

- ( Kiki,

Cst

)

Uiy

⁺

cut

^-

KEI

^)U,z=o

at

^{= K}

? .VE

^{shear Alfuen}

Ñ= IKTvi-cs.be/--ei- F)

^£

) magnetos

^on:c

where _,

ñ=4%÷%¥÷;y

- o < 5<1 ^{→ w'} > ^o

→ Imcw)=o ^{→ no instable.}

growth

-cy

^or

damping

→

expected

^{since no free}

energy

^source

* Magneto ^{sonic wave}

for

^low

p

p ^-

¥

^{, " '}

&

fast^. Compressional

Al-fve.nu

^- ~~ (

KI

^-1 _{KI )Va} let ~~ KFC

}

^{Slow . Sound}

CI ^{) shear} Alfven ^wave

WE Kiva

Independent

of

^Kt

even

if

^{k+⇒ Ka}

4×1=0 ,

Uig

^=4z=o

É

.ie?=oE-u,=oPi=Pc=onotewBT=-u-iLr?-B-o

^{) ✗}

ÉÉ ]

Bench ^U,

Big

^{__ Biz -0}

vii. BY

⁺

É

^{transverse wave}

A-

Ife

>

z # _field

-

perpendicular

^kinetic

energy

←> magnetic

bending energy

- most dangerous

Ñ=(BotBÑ=Bft2Ñ-

_-

BF

→

easy

^{to excite}

E.

ii.

^{→ o}

µii~¥÷i%

^,

Pil

^→

ñi To

ltt ) compressional

^{Alfven wave}

afar

(

Kitko ? )vI for pal

Uta ^IO , but

Uiyto

^Uczto

Bix ^to _Peko ^, f. ^to

- but MOP^'

/

^BoB

,z<<

^\

mostly

^{due to} compression

of

^magnetic

field ^{, not} plasma

- note

typically

^{K+ → Ki, C: 2412072A )}

T ^T

major

^radiusradius^minor

-

Wig

^{→ Uit}

nearly

^{transverse wave}

y

_-?

B→

:

2-

÷

n wave propagation

1T¥

)

sound

^weave

WE KEI

_,

www.cky-ro.is?--o

longitudinal

^wave

Y ✗ pressure contour

>

> 2- >

*

Kivi

^→

Kivi

^{→ •}

kici

compressional

^{shear sound}

Altuer

Alfuen

Ideal

^MHD

equilibrium / stability

is _as

fast

^{as at least}

Soundwave

mostly

^{Acfuen timescale}

5/12 Ideal ^MH17 stability

/ energy

^principle

1. For

general geometry

^{, Ideal Mitt with}

u→o=o

If

^{we express it,}

=2§=

^{, , where Éeñ,-4 plasma}

displacement

• not

Ige

^*

iii.

^I'

ftp.e-i?u-i7=oei---eii--ipo-pofi-E7euaed)zPi

get vii.

^E'

pottP.tv?ui7--op.=-q'.-Epo--tp.l-E.e5)ei72BTge---v-

^✗

CUT

^✗

BT

^{) is}

?

^{= I>+}

lÉ+B→o

^{? ez)}

Moji

^{= -5+5}

?

µ^.

_É=Ñ×É×eei×ñi

^{) 137}

Po0¥e

^:

-5×13%+5%-5 ?

^-

Ep

^, ey

el)~e3 ) into ^C.

4)

p.

Jeg

jeg ⁼

¥[ÉxÑ✗lÑ✗B%]✗B?t%¢É+B%×Ñ+oÉ×Éo7]+ Efei.Epo-tpoe-i.es

^'s

)

I

ÉeÉ

₎ ^{: ideal MHD force operator}

note ^{one can no} longer §

xe.tk?*-w-c

^{) (✗} _,

but

s+:ueane

^{g-' = eiex, e-iwe}

cripes

^{' = -}

ÉeÉ

^{) : normal mode equation}

→

numerically

^{more amenable than I. v. p}

→ still too much

information

→ works

only

^{for the system unstable}

→

typically

^{, we just want know}

if ^the equilibrium ^{is stable or not} ,

2. Energy

principle

el) ^note

ÉeÉ

^{) is}

self-adjoint

^.

meaning Sri

^-

Éeéisdx

^'

=/ eiiiiñidñ

^'

# arbitrary

^{trial fens .}

- expected for ^conserved system

- can be

directly

^{proved with}

ÉÑ

⁾

12)

self-adjoint

^{ness means}

real eaj ^w

'

is ^{real , w} is

purely (

imaginary

-: w

'

/ %ÑFdñ=

^-

)É&Éc53dñ

(a)^*

fpolei

^{Pdx" =}

-g§.Éc§*]dx→

→ (E) ⁼_@2)^*

W= 0 at marginal point

( not Recw^{)-1-0 .} Imco)=o

)

lb) Discrete normal modes _are

orthogonal

^.

2 (weight

fen

^. Po ^]

Let Wmm ^=/

WE

↳

^-

f- win

.es?n----iceiin))aijes-m.(-wip9-i---Fos?3)ex-'cwm--Wn7Jpoo-s' m.es?dx-'--o

→

gqg-i.es?dx-'--o

13) Variational

formulation

^.

note a-=

-

£s¥ÉoE)dx→

* eleiid.is =%%÷÷g→}-

Then §ex→) makes ^maximum

( _minimum

)

^{0£ .}

is ^an

eigenfen of

^{normal mode problem}

•: At extremism,

5^{' →} £+05^{' then at→ W'+}

rat

uterus =

oweoi%53-orwcrei%53-owCH.org?g-clCdJKlB*ihi

) ⁺ k(dÉ*iÉ^)-1K

(05909-7+002)

rw(Ñ5§)+dwtE9Ñ^)-

ut(k05¥É)+k(É*iÉ

⁾

)

0 ⁼ out ~

kC5*cÉ

)

Sai {051-1-+755+85} -1054-+755+075*19=0

hold

if ñpÉ=

^-

ÉcÉ

]

Now _, you ^can

just

^minimize rwc§*iE ⁾

w '

=

¥ É

to find ^{the most unstable mode}

(

^worknot ^whenwhen

really

^{stable unstabledue continuum mode} (Toed ^bloed & Poedts ⁽²⁰⁰⁴)

* _{since one can} always ^K

without changing ^DW through §^,,

(4)

Energy

^principle

(

^{Bernstein '58}

)

Equilibrium ^{is stable}

if -5

^.

dwtÉ*§

^{) > o}

for ^{all allowable trial}

ferns §

( ^even

if

^{Ñ is not an}

eogenfen )

If É making

^{JWLO exists , it's unstable}

• :

If ^{the modes are discrete}

any q→= snares ?

owcei.es#-)--tz-znlariwntJwL0

_means at least ^{one Wn} Lo

- : complete

proof by

^{Laval ' 65,}

(Freidberos ^ch 8)

Now ^one can

Just

^{minimize JW}

3 ^{. Standard}

form of

^sweeties)

m→=§*

dw-r-tfri.fg.EE/-5?y-B-i-teoe-Fxi51xB?--Jfei- ÉP

^to

-5.57µF

① ③ ^②

Let

É=

^hi,

5+91

^is

?

^-.

-5×(9,9×15)

④

fri.CI?-BT7xB?--f(-JxB?3-CB-xoI)--f-v-?(B?xcExnIy-fB

^?.

-5×(5×75)

=/ (57-05×513)

^{.de -}

JB

^?

.is?cnIs---fn-i.ds-'(ri.B?eeEs)-fB-ieEs.B?cnI

⁾

②

fnI.DE/np-E-9-)-fopc-i?53c-i.ri7

③ E.

[ I

^✗B

?

^{+ I'}

eei.JP?)=oV

①e②x③ ^{leads to} _,

nI=q→I

Jw^: Jury^{. + Iws}

Swf

^:

1) F. Bite riskiest

^-

ii. I'

⁺⁵

?

⁺

§

.es?i-)eesI.-Ep)dx-'JWs=Egfei*.d5' ftp.E.B?-tp-i.ei-eii;-ip)dS

4. _Intuitive

form

^{. Let}

ÉEB ?

É= ÉÉ

^{+ Qais} _,

I

^-

-5

, ⁺

jab

orw-i-F.ffa-it-EI-i.es?--izesIr?T+ieonpl-i-eiF

^{① ② ③}

-

aieoeéiitip )o5¥É

^{] - rig}

.EE/-5).aIGdx--

④ ⑤

① ^{: shear Aetuen}

/#→

② ^:

compressional

^Allfven

③ ^{: Sound wave}

④ ^{: pressure - drroen}

instability

+

^{: current - driven}

instability

* minimization ^:O

-1

^{ow →}

(-5-5)=0 by choosing

^Bo,

£

. Self-adjoint ^form

read Freid

berg

^{ch - 8}