gdhszmuia-ciifcaptfdv.mu?vIscap

4/12/2021

^{, current}

/

^charge

(

^drift

Fent

^{n - E}

)

^{-- so}

→ ←s -7

mandate

^{: en}

LE 't iixis 's

^{- v. p + R,}

+5

^{, - moi's.}

dqfzpj-5zpio.ie

^{) e i. E'i'+}

E' jar

^.-ii.

pies

^{. -ii.}

5. etgmuso

* Two ^- fluids ^{relations s =} (^x.

p

⁾

Cl) momentum conservation

Ei fodvamiincap III

I

,

=

I' apt EI

^{. =}

fair maiiacapempvpcpa

^{-- o}

L2)

energy

conservation

Rs

Jd 've fmri cap

=

gdhszmuia-ciifcaptfdv.mu?vIscap

→

I

Qapt Fap

E.

^{Rs --}

Qapto-patu%FFptiej.FI 's

^so

e3 ) ^current &

Resistivity

- f-^{rook collision}

approximation

IIe

^{-- -}

msnarapcui

^-

rip

⁾

I _c.elision

frequency

- fur ^{electron d-- e} ,

DIT

^ions

p

^{-- T}

Fei

^{- - mene}

Vertue

^-

city

- momentum conservation

EI

_, ⁺

I'

ie ^-- o → menever ^-- minivie

if

near Vei ^⇒ Vie

-

-

proportional

^{to currents}

j

^{'s -}

enlai

^-

iii

⁾

Feit me I

^'

•

parallel

^{momentum EG .}

for

^electron

medc=

^{renee. + is It}

Fei

^{. "}

tf

small ^inertia _p ^- homogeneous

F- 9

' Ea ⁼

etaeEei.ci

^{' a}

'

.

" " "

yay

^{- E→ ( Oke~✓}

.io#..m 159

^Dm

I

^{' .}

I

^'

÷ :)

^{n... -}

worm

6,2261 ^i.e _Me, ^I 2ha

' In

general

^,

I

^{' can't be determined →}

p

alone

by

^I

need

E'

_p

147 ^Ohmic

heating

energy

conservation

Rei ^e are ^{= -}

ii. Fei

^{. -}

iii. Iie

= ^-

Eirene

^{- i'}is

= _-

meveeij

^{' .}

ten )

^--

nj

e5) ^charge

density

Typically

^{. we don't use E. E- E}

'

= ecni ^-ne )

to determine E' ^{= -} _of .

Rather ,

parallel

^{momentum balance}

gives ^a useful

info

^for

ol steady

^{state . cnet.it Vape = o}

const^.

Te

^{. ene Jud = J. . Pe = Tea, ne}

ne ^a

etpleolte

^?

at ^a ete

le

₎ ^{en Ene}

?

Very

^good approximation ^.

167 Draft of fluid

In

steady

^{state , w/o source}

(

^exchange

B'

^{+ ( en}

tenures

^-

Ep

³⁼⁰

→

iii.

^r. ₊

5'Ii

^'

vi. evi

^{. ④ is}

→ t

-099 for

E-⁺ B'^' drift diamagnetic £ ₉

d.rife Up

Q ) where ^{are B} , curvature drifts _?

consistent ^with particle ^motion ?

vigor

^.

VI.

^{t + Kuru}→

* As sump

g Itis

^{' ro emagneticlow -p assumptionfield}

by

^externalcoils⁾ Maxwell^ion

EA) charge accumulation

A

simplified

^{example .}

^

if

^{, per) I}

°

^{- Z consider Ton drifts}

<##

^- particle picture

←

F

¥# E

Igor

^,

25253×-813=21

^"

measure _q

Yaffa

^t

BT

^e.

Brit

charge

^{q t →}

accumulation

_N^{q opto f F}

dfs.ge

^→

Tee

^--

envgc

f-

^{← ← -}

we^:*, ⁼

:L

^{. - I}

Tp ^soonstrfo _robo

like

pedestal

- flood picture ^v

;

⁼

B→xo

en Bt

Ep

⁼

jpr

⁼

envpr

^{" = en BE x e-}

HpbE

*

=

-4¥

^:

= ^{- r} kept ^a

- r

to Bo

dear

^{. -}

I

^'

reader :# )

^--

2÷÷

''

-

-

fee :÷

⁼

:÷

^{-- do}

:#

A bit ^more

generally

^,

- wi

's

⁼

E. vj ?

^e.

try

^it

for

⁺

E'

^so

)

CB

) drift

^{( or current}

) itself

^?

→

④ B Goldston ^{'s book} _.

①

1¥

,

chapter 7.4.

→

cancelled ^since Us

remains same

(

Laboratory

^frame

) lmoviny frame

⁾

A bit ^more

generally

^,

ji

^.

I'

^{got +}

It

^.

Ei

^{= -}

547=-573

en

E.ie?..I.vgo

^{wi --}

via

^Exit

en

Ext

^{. . - E'+}

(

^E'

:3

^.

)

⁼

Baton

^-

2Pi5×E!3

iii.

⁼

then :tE#

⁺

ten .it:7

^-

seeing

4/14/21

^{Single fluid}

Magnetohydrodynamics

^{CMH b)}

Ton & ^electron

. plasma

density

^{n = he = hi}

- Mass

density

^{pm =}

Is

^{msns = Mini}

- charge

density

^{6 =}

jeans

^{= o}

- Mass

velocity

^is

g msn.is/pm=U7tme-m..iie

- current

density j

^{' =}

Is

^esnsie

's

⁼

eneui

^{-iie 7}

§

→ ^→ →

wire ^at

mermaid

{ Eerie

^-

Een

- y.ee pressure ^tensor

95

⁼

Is

_Msn, Liv_'s^-E'_vis^- it's>

- Total ^heat flat

af

^,

ggtzmsns (

^(Js-

Tiflis

^-ie

)

( Ignore source terms )

Cl )

continuity

Is

^{Ms +}

( continuity

^{for each 7}

→ '

feet E

^'.

cpmii

^{) - o}

Es

^{es x (}

continuity

^{for each}

)

→

agree

^E'-

j

^{' so}

123

Equation of

^motion

Is

^{l momentum} conservation

for

^each

)

→

pond :{

^{= 6}

E'

⁺

j 't E

^{' -}

E. Y

^'

TERI

e.37 Equation

of energy

Is

^{e energy} conservation

for

^each

)

→

Feel 'zp

^{) +}

5zpc-E.ci

⁾

EET

^:

use of

^'

=

ej

^{- hi' ) - CE'eEtB' ? t}

-3k€

147

Generalized

^{Ohm's law}

Is Engine

^momentum conservation

for

^each

)

or electron equation

of

^{motion l two species)}

mee=

^{- ene}

E

^{' - ene}_Ts

B

^-

E. ietenenj

^'

electron inertia if ^- I

ene

E' tu 'xB→=nj

^{' ,}

I

ene ^← Hall term

t Matu^ell

equations

E'

ni

^--

Mj 't I. IIe

+

E

^{' = - 25'}

to :* . :÷ :

Approximation

for 4¥

^low

frequency : :c .is?::...y

^phenomena

a ^: scale

of interest

^{6 -}

EoE/a

w ^._-

Ier Vania

^,

ii

^-

E' YI

,

-

E

. ^.

-

j

^-

II

_, ^{- e}

;ua

CA) ^small gyro ^radius

1a

^KI

EB) ^high collisional

icy ÷

^{, n}

tha

^{k /}

cc) low

resistivity

^?

→

Goldston

Friedberg

lo) ^Maxwell

Y÷¥÷

^-

EEE

^-

uve÷ ⇐

^{, a}

"^"

→

E' xei-rojtt.IE/

Ll)

continuity

n

:÷÷¥ne¥e÷÷

^{Veni .}

*

eyed

^{. -}

term

^-

II al

→

¥e I

^so

many

^{insta. eieies}

* .

w→=o

^←

Incompressible

^.

( defence

^{. so ←}

Effect

^-

Pacino

EZ) ^momentum

:÷÷÷:w÷n÷i÷ ⇐⇒

^{. .}

Pm ^ai

'

be

-z→

ⁿ

pmuvth.la

^{~ a-}

Pm ^Ven? la ^Vthi

Pm ii. E' ^E

TE ~Yn÷z { cgchzj

^{Ol') MHDdrift-}

ordering

^-MHD

ordering

"

Hazeltine

u :÷n←÷ii÷x¥÷

^{p a "}

pen

^{momentum ~}

plz

^,,

viscosity

pm

DII

^{= +}

Exit

^-

Ep

^-

E3)

Energy

^Millen

?

" ÷w"n#¥÷

Puth^-. la

/ ⇐ It÷÷t

^.

kn ^-

Plmevei

,

Ki ^-

( Iie )£k

^,

fake )tIapcE

^.

tEEtE

=

of

^-

E' taxis )

=

ng

²

( 47 ^{Ohm 's law}

→

u¥ .sn

.

:::÷÷ii÷i7 ⇒ ¥÷It

since

jxis

^{' -}

E'

^per

E'

^p,

:¥;÷

^-

minister

( OCI )

^{MHD -}

ordering

OCD

^{drift MHD}

un O

( Ia )

^{- Ken,-}

E' tuxes -mjeIxEe

→ resistive MH17

for

parallel

^{motion .}

J

^{, ~}_Mod

Be

→

n^-

u¥e

^-

Foa

^-

Tn

Magnetic ^Reynolds

( Lnundqaist )

number

I

^{' + it}

B'

^to _{→ ideal MH17}

* MHD

equations

E'

⁺

E' =µj

→

by eliminating

^{E' -}

j

^'

E'

^c-

E'

^{= -a'}

I

see dem

Fet

^{pm CE}

'

-i' 7=0

defeat pm

^-

E)

^{" O} _de

PmdII=jx5

^'-

Ep

^→

Eet Esp

'i' 7=0

dIetEp Ezo

"

diet 4tB%

^{.> sis.}

Bye

^.

E'

^{+ E'}

xpgen.jo III

- E^'

utBCE.u7-z.IE

( 4 equations ₈ equations

for ⁽ E^{'. -}

E.

^ii.

j.pm.PT

^{for c. ii.}

5.

^{pm . P)}

simple

geometry (

^{ie→ -}

15×-547%5

^Bt

B=bz5t5xEy

l 4. &.bz , -47 Pa ^{, P}

four field equations^.