Winter Quarter 2009Vassilis Angelopoulos

Robert Strangeway

Date Topic [Instructor]

1/5 Organization and Introductionto Space Physics I [A&S]

1/7 Introduction to Space Physics II [A]

1/12 Introduction to Space Physics III [S]

1/14 The Sun I [A]

1/21 The Sun II [S]

1/23 (Fri) The Solar Wind I [A]

1/26 The Solar Wind II [S]

1/28 --- First Exam [A&S]

2/2 Bow Shock and Magnetosheath [A]

2/4 The Magnetosphere I [A]

ESS 200C - Space Plasma Physics

Date Topic2/9 The Magnetosphere II [S]

2/11 The Magnetosphere III [A]

2/18 Planetary Magnetospheres [S]

2/20 (Fri) The Earth’s Ionosphere [S]

2/23 Substorms [A]

2/25 Aurorae [S]

3/2 Planetary Ionospheres [S]

3/4 Pulsations and waves [A]

3/9 Storms and Review [A&S]

3/11 --- Second Exam [A&S]

Schedule of Classes

ESS 200C – Space Plasma Physics

• There will be two examinations and homework assignments.• The grade will be based on

– 35% Exam 1– 35% Exam 2– 30% Homework

• References– Kivelson M. G. and C. T. Russell, Introduction to Space Physics,

Cambridge University Press, 1995.– Chen, F. F., Introduction of Plasma Physics and Controlled Fusion, Plenum

Press, 1984– Gombosi, T. I., Physics of the Space Environment, Cambridge University

Press, 1998– Kellenrode, M-B, Space Physics, An Introduction to Plasmas and Particles in

the Heliosphere and Magnetospheres, Springer, 2000.– Walker, A. D. M., Magnetohydrodynamic Waves in Space, Institute of

Physics Publishing, 2005.

Space Plasma Physics

• Space physics is concerned with the interaction of charged particles with electric and magnetic fields in space.

• Space physics involves the interaction between the Sun, the solar wind, the magnetosphere and the ionosphere.

• Space physics started with observations of the aurorae.– Old Testament references to auroras.– Greek literature speaks of “moving accumulations

of burning clouds”– Chinese literature has references to auroras prior

to 2000BC

• Aurora over Los Angeles (courtesy V. Peroomian)Cro-Magnon “macaronis” may be earliestdepiction of aurora (30,000 B.C.)

– Galileo theorized that aurora is caused by air rising out of the Earth’s shadow to be illuminated by sunlight. (He also coined the name aurora borealis meaning “northern dawn”.)

– Descartes thought aurorae are reflections from ice crystals.– Halley suggested that auroral phenomena are ordered by

the Earth’s magnetic field. – In 1731 the French philosopher de Mairan suggested they

are connected to the solar atmosphere.

• By the 11th century the Chinese had learned that a magnetic needle points north-south.

• By the 12th century the European records mention the compass.

• That there was a difference between magnetic north and the direction of the compass needle (declination) was known by the 16th century.

• William Gilbert (1600) realized that the field was dipolar.

• In 1698 Edmund Halley organized the first scientific expedition to map the field in the Atlantic Ocean.

By the beginning of the space age auroral eruptions had been placed in the context of

the Sun-Earth Connection

• 1716 Sir Edmund Halley Aurora is aligned with Earth’s field…

• 1741 Anders Celsius and has magnetic disturbances.

• 1790 Henry Cavendish Its light is produced at 100-130km

• 1806 Alexander Humboldt but is related to geomagnetic storms

• 1859 Richard Carrington and to Solar eruptions.

• 1866 Anders Angström Auroral eruptions are self-luminous and…

• 1896 Kristian Birkeland … due to currents from space:

• 1907 Carl Störmer in fact to field-aligned electrons…

• 1932 Chapman & Ferraro accelerated in the magnetosphere by…

• 1950 Hannes Alfvén the Solar-Wind–Magnetosphere dynamo.

• 1968 Single Satellites Polar storms related to magnetospheric activity

• 1976 Iijima and Potemra … communicated via Birkeland currents

• 1977 Akasofu Magnetospheric substorm cycle is defined

• 1997 Geotail Resolves magnetic reconnection ion dynamics

• 1995-2002 ISTP era Solar wind energy tracked from cradle to grave

• 2002- Cluster era Space currents measured, space-time resolved

• 2008 THEMIS Substorm onset is due to reconnection

It All Starts from the Sun…Solar Wind Properties:• Comprised of protons (96%),

He2+ ions (4%), and electrons.• Flows out in an Archimedean

spiral.

• Average Values:– Speed (nearly Radial): 400 - 450 km/s– Proton Density: 5 - 7 cm-3

– Proton Temperature 1-10eV (105-106 K)

Shaping Earth’s Magnetosphere • Earth’s magnetic field is an

obstacle in the supersonic magnetized solar wind flow.

• Solar wind confines Earth’s magnetic field to a cavity called the “Magnetosphere”

Auroral Displays: Direct Manifestation ofSpace Plasma Dynamics

Societal Consequences of Magnetic Storms

• Damage to spacecraft.

• Loss of spacecraft.

• Increased Radiation Hazard.

• Power Outages and radio blackouts.

• Damage to spacecraft.

• Loss of spacecraft.

• Increased Radiation Hazard.

• Power Outages and radio blackouts.

• GPS Errors• GPS Errors

Stellar wind coupling to planetary objects is ubiquitous in astrophysical systems

MERCURY: 10 min EARTH: 3.75 hrs JUPITER: 3 weeks

ASTROSPHERE GALACTIC CONFINEMENT

SUBSTORM RECURRENCE:

Magnetized wind coupling to stellar and galactic systems is common thoughout the Universe

Mira (a mass shedding red giant)and its 13 light-year long tail

In this class we study the physics that enable and control such planetary and stellar interactions

Introduction to Space Physics I-III

• Reading material– Kivelson and Russell Ch. 1, 2, 10.5.1-10.5.4– Chen, Ch. 2

The Plasma State

• A plasma is an electrically neutral ionized gas.– The Sun is a plasma– The space between the Sun and the Earth

is “filled” with a plasma.– The Earth is surrounded by a plasma.– A stroke of lightning forms a plasma– Over 99% of the Universe is a plasma.

• Although neutral a plasma is composed of charged particles- electric and magnetic forces are critical for understanding plasmas.

The Motion of Charged Particles

• Equation of motion

• SI Units– mass (m) - kg– length (l) - m– time (t) - s– electric field (E) - V/m– magnetic field (B) - T– velocity (v) - m/s– Fg stands for non-electromagnetic forces (e.g. gravity) -

usually ignorable.

gFBvqEqdtvdm

rrrrr

+×+=

• B acts to change the motion of a charged particle onlyin directions perpendicular to the motion.– Set E = 0, assume B along z-direction.

ycy

y

xcxy

x

xy

yx

vm

Bvqv

vm

Bvqm

Bvqv

Bqvvm

Bqvvm

22

22

22

22

Ω−=−=

Ω−=−==

−=

=

&&

&&&

&

&

mBq

c||

=Ω

• Solution is circular motion dependent on initial conditions. Assuming at t=0:

and

– Equations of circular motion with angular frequency Ωc (cyclotron frequency or gyro frequency). Above signs are for positive charge, below signs are for negative charge.

• If q is positive particle gyrates in left handed sense• If q is negative particle gyrates in a right handed sense

⊥== vvv yx ;0

)cos()sin(tvvtvv

cy

cx

Ω±=Ω±=

⊥

⊥

)sin(

)cos(

0

0

tvyy

tvxx

cc

cc

ΩΩ

±=−

ΩΩ

=−

⊥

⊥m

• Radius of circle ( rc ) - cyclotron radius or Larmorradius or gyro radius.

– The gyro radius is a function of energy.– Energy of charged particles is usually given in electron volts

(eV)– Energy that a particle with the charge of an electron gets in

falling through a potential drop of 1 Volt- 1 eV = 1.6X10-19

Joules (J).• Energies in space plasmas go from electron Volts to

kiloelectron Volts (1 keV = 103 eV) to millions of electron Volts (1 meV = 106 eV)

• Cosmic ray energies go to gigaelectron Volts ( 1 GeV = 109 eV).

• The circular motion does no work on a particle

qBmv

v

c

cc

⊥

⊥

=

Ω=

ρ

ρ

0)()( 221

=×⋅==⋅=⋅ Bvvqdtmvdv

dtvdmvF

rrrrr

rr

Only the electric field can energize particles!Particle energy remains constant in absence of E !

• The electric field can modify the particles motion.– Assume but still uniform and Fg=0.– Frequently in space physics it is ok to set

• Only can accelerate particles along• Positive particles go along and negative particles go

along • Eventually charge separation wipes out

– has a major effect on motion. • As a particle gyrates it moves along and gains energy • Later in the circle it losses energy.• This causes different parts of the “circle” to have different radii -

it doesn’t close on itself.

• Drift velocity is perpendicular to and• No charge dependence, therefore no currents

0≠Er

0=⋅ BErr

Er

Br

EE−

E

⊥EEr

2BBEuE

rrr ×

=

Er

Br

Br

Y X

Z

• Assuming E is along x-direction, B along z-direction:

• Solution is:

• In general, averaging over a gyration:

)(2

2

yx

cy

xcx

xcy

ycxx

vBEv

vv

vv

vqEv

+Ω−=

Ω−=

Ω=

Ω±=

&&

&&

m&

&

BEtvv

tvv

xcy

cx

−Ω±=

Ω±=

⊥

⊥

)cos(

)sin(

BEVB

BEvBvqEqdtvdm ×≡

×=⇒=×+= 20

rrrrrrr

• Note that VExB is:– Independent of particle charge, mass, energy– VExB is frame dependent, as an observer moving

with same velocity will observe no drift.– The electric field has to be zero in that moving frame.

• Consistent with transformation of electric field in moving frame: E’=γ(E+VxB). Ignoring relativistic effects, electric field in the frame moving with V= VExB is E’=0.

• Mnemonics:– E is 1mV/m for 1000km/s in a 1nT field

• V[1000km/s] = E[mV/m] / B [nT]– Thermal velocities (kT=1/2 m vth

2):• 1keV proton = 440km/s• 1eV electron = 600km/s

– Gyration:• Proton gyro-period: 66s*(1nT/B)• Electron: 28Hz*(B/1nT)

– Gyroradii:• 1keV proton in 1nT field: 4600km*(m/mp)1/2*(W/keV)*(1nT/B)• 1eV electron in 1nT field: 3.4km *(W/eV)*(1nT/B)

• Any force capable of accelerating and decelerating charged particles can cause an average (over gyromotion) drift:

– e.g., gravity

– If the force is charge independent the drift motion will depend on the sign of the charge and can form perpendicular currents.

– Forces resembling the above gravitational force can be generated by centrifugal acceleration of orbits moving along curved fields. This is the origin of the term “gravitational” instabilities which develop due to the drift of ions in a curved magnetic field (not really gravity).

– In general 1st order drifts develop when the 0th order gyration motion occurs in a spatially or temporally varying external field. To evaluate 1st order drifts we have to integrate over 0th order motion, assuming small perturbations relative to Ωc, rL

2qBBgmug

rrr ×

=

FVqB

BFvqFqBvqdtvdm

rrv

rvrrr

≡×

=⇒=+×= 20)/(

• The Concept of the Guiding Center– Separates the motion ( ) of a particle into motion

perpendicular ( ) and parallel ( ) to the magnetic field.– To a good approximation the perpendicular motion can

consist of a drift ( ) and the gyro-motion ( )

– Over long times the gyro-motion is averaged out and the particle motion can be described by the guiding center motion consisting of the parallel motion and drift. This is veryuseful for distances l such that and time scales τ such that

vr

⊥v v

Dvc

vΩ

( )cc

vvvvvvvv gcD ΩΩ⊥ +=++=+=rrrrrrrr

1<<lcρ( ) 11 <<Ω −τc

• Inhomogenious magnetic fields cause GradB drift.– If changes over a gyro-orbit rL will change.– gets smaller when particle goes into stronger B.

– Assume ∇B is along Y– Average force over gyration:

– u∇B depends on charge, yields perpendicular currents.

Br

qBmvrLc

⊥==ρ

32

21

32

21

qBBBmv

qBBBmvu B

∇×=

×∇= ⊥⊥

−∇

vrrvr

x

Y

ZB=Bzz

-

+

BB

mvF

yB

Bmv

yBtvqF

yBtvBtqvyBqvF

Fy

ByBzyBzB

zzc

cy

zc

czczxy

x

zzz

∇−=

∂∂

−=∂

∂>Ω<

Ω−>=<

>=∂

∂Ω

Ω±Ω±<−>=<−>=<

>=<∂

∂+==

⊥

⊥⊥

⊥⊥

v

r

2

02

022

00

00

21

.21)(sin

))sin()(sin()(

0

)(ˆ)(ˆ

• The change in the direction of the magnetic field along a field line can also cause drift (Curvature drift).– The curvature of the magnetic field line introduces a drift

motion.• As particles move along the field they undergo centrifugal

acceleration.

• Rc is the radius of curvature of a field line ( ) where

, is perpendicular to and points away from the center of curvature, is the component of velocity along

• Curvature drift can also cause currents.

cc

RR

mvF ˆ

2

=r

bbRn

c

ˆ)ˆ(ˆ

∇⋅−=

BBbr

=ˆ n̂ Br

v Br

2

2

2

2 ˆˆ)ˆ(

qBR

nBmv

qB

bbBmvu

cc

×−=

∇⋅×=

rrr

• At Earth’s dipole u∇B , uc are same direction and comparable:

– u∇B , uc are ~ 1RE/min=100km/s for 100keV particle at 5RE, at 100nT

– The drift around Earth is: 0.5hrs for the same particle at the same location

• At Earth’s magnetotail current sheet, u∇B , uc are opposite each other:– Curvature dominates at Equator, Gradient dominates further away from equator

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛⋅≈∇ keV

WBnT

rRskmRu E

EB 1001005/100~min/1

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=∇ W

keVnT

BRrh

ECB

1001005

5.02

,τ

• Parallel motion: Inhomogeniety along B (∇║B)– As particles move along a changing field they experience force

• Parallel force is Lorenz force due to small δB perpendicular to nominal B (δW║)• Force along particle gyration is due to dB/dt in frame of gyrocenter (δW┴)• Total particle energy is conserved because there is no electric field in rest frame

– Consider 1st order force on 0th order orbit in mirror field• B change due to presence of ∇║B must be divergence-less (∇B=0) so:

– In cylindical coordinates gyration is: – The mirror field is:– The Lorenz force is:– From gyro-orbit averaging:

– Defining: we get:

– This is the mirror force. Note that is conserved during the particle motion:

trrvv cΩ=== ⊥⊥ mm θθ ;;

BsBF II∇−=∂∂−>=< μμ )(

)();();( rzrzzrzr BvqFBvBvqFBvqF θθθ −=+−==[ ] 0210)(1;0 =∂∂⋅⋅−=⇒=∂∂+∂∂=∇= rzrzr zBrBzBrrBrBBθ

Bmv2

21 ⊥≡μ

[ ] [ ]zBBmvzBrvqF

vBFBvqF

zrzz

rrrzr

∂∂⋅⋅−=∂∂><⋅−>=<

==>==<><>=<

⊥=

=

/2121

)0(;02

0

0

θ

θθ

( ) 022

222

2

=⇒−=−⇒−=⎟⎟⎠

⎞⎜⎜⎝

⎛−⇒−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

⇒−=⎟⎟⎠

⎞⎜⎜⎝

⎛⇒−=

∂∂

∂∂

−=⇒∂∂

−=><

⊥⊥

dtd

dtdBB

dtd

dtdBmv

dtd

dtdBmvW

dtd

dtdBmv

dtd

dtdB

ts

sB

dtdvmvv

sBvF IIII

IIIIII

μμμμμ

μμμμ

Y XZ

r dr

dθdz

• Another way of viewing μ– As a particle gyrates the current will be

where

– The force on a dipole magnetic moment is

Where

I.e., same as we derived earlier by averaging over gyromotion

cTqI =

ccT

Ω=

π2

μ

ππ

π

π ===

Ω==

⊥Ω

Ω

⊥

⊥

BmvIA

vrA

c

cqv

cc

2

2

2

2

22

2

2

dzdBBF μμ −=∇⋅=

vrr

b̂μμ −=r

In a Magnetic Mirror:• The force is along and away from the direction of

increasing B.• Since and kinetic energy must be conserved

a decrease in must yield an increase in • Particles will turn around when• The loss cone at a given point is the pitch angle below

which particles will get lost:

Br

0=E

v ⊥vμ2

21 mvB =

)( 22212

21

⊥+= vvmmv

mmii RBB /1/sin2 ==α

• Time varying fields: B– As particle moves into changing field or– As imposed/background field increases (adiabatic

compression/heating)

• An electric field appears affecting particle orbit

• Along gyromotion, speed v┴ increases, but μ is conserved:

• Flux through Larmor orbit: Φ=πrL2B=(2 π m/q2) μ remains constant

tBE

∂∂

−=×∇r

r

( ).

0

2

)(2

_:,)(

22

_0_0

_0_0

_

2

constBB

BrdtdBqmv

SddtBdqSdEqldEqmv

elementlineldwheredtldEqvEqv

dtdmv

Lgyrationgyration

eforifor

surface

eforifor

surfaceoverorbitgyrogyration

=⇒=⇒=

⇒=⎟⎠⎞

⎜⎝⎛±=⎟⎟

⎠

⎞⎜⎜⎝

⎛

⇒−=×∇==⎟⎟⎠

⎞⎜⎜⎝

⎛

⇒==⋅=

⊥

><

><

−

⊥

⊥⊥⊥

∫∫∫−

+

−

+

μδμμδμδ

μδπδ

δv

vvvvvv

vv

vvvvv

• Time varying fields: E– See Chen, Ch 2

• Maxwell’s equations– Poisson’s Equation

• is the electric field• is the charge density• is the electric permittivity (8.85 X 10-12 Farad/m)

– Gauss’ Law (absence of magnetic monopoles)

• is the magnetic field– Faraday’s Law

– Ampere’s Law

• c is the speed of light.• Is the permeability of free space, H/m• is the current density

0ερ

=⋅∇ Er

Er

0=⋅∇ Br

Br

ρ0ε

tBE

∂∂

−=×∇r

r

JtE

cB

rr

r02

1 μ+∂∂

=×∇

0μ 7

0 104 −×= πμJr

• Maxwell’s equations in integral formNote: use Gauss and Stokes theorems; identities A1.33; A1.40 in Kivelson and Russell)

– is a unit normal vector to surface: outward directed for closed

surface or in direction given by the right hand rule around C for

open surface, and is magnetic flux through the surface. – is the differential element around C.

000

1ε

ρεε

ρ QSdEdVdVESdEEAV

AV

=⋅⇔=⋅∇=⋅⇔=⋅∇ ∫∫∫ ∫vrrvrr

tldE

tSd

tBSdEldE

tBE

SdBB

A

C

A

∂Φ∂

−=⋅

⇔∂Φ∂

−=⋅∂∂

−=⋅×∇=⋅⇔∂∂

−=×∇

=⋅⇔=⋅∇

∫

∫∫ ∫

∫

vv

vr

vrvvr

r

vrr

)(

00

Sdv

ldv

ISdEt

ldB

SdJSdEt

SdBldBJtE

cB

cC

cC

01

01

02

2

2)(1

μ

μμ

+⋅∂∂

=⋅

⇔⋅+⋅∂∂

=⋅×∇=⋅⇔+∂∂

=×∇

∫∫

∫∫∫∫vrvr

vvrvvvvrrr

r

Φ

• The first adiabatic invariant– says that changing drives (electromotive

force). This means that the particles change energy in

changing magnetic fields.

– Even if the energy changes there is a quantity that remains constant provided the magnetic field changes slowly enough.

– is called the magnetic moment. In a wire loop the magnetic moment is the current through the loop times the area.

– As a particle moves to a region of stronger (weaker) B it is accelerated (decelerated).

EtB rr

×−∇=∂∂

Er

.2

21

constBmv

== ⊥μ

μ

Br

• For a coordinate in which the motion is periodic the action integral

is conserved. Here pi is the canonical momentum: where A is the vector potential).

• First term: • Second term:

• For a gyrating particle:

Note: All action integrals are conserved when the properties of the system change slowly compared to the period of the coordinate.

constant == ∫ iii dqpJ

μπqmsdpJ 2

1 =⋅= ∫ ⊥rr

Aqvmprrr

+=

qmmvdsmv μππρ 42 == ⊥⊥∫qm

qBvmSdBqSdAqsdAq μππ 22 22

−=−=⋅=⋅×∇=⋅ ∫∫ ∫∫∫ ⊥rrrrrr

• The second adiabatic invariant– The integral of the parallel momentum over one complete

bounce between mirrors is constant (as long as B doesn’t change much in a bounce).

• Bm is the magnetic field at the mirror point

– Note the integral depends on the field line, not the particle– If the length of the field line decreases, u|| will increase

• Fermi acceleration

.22

222

2

1

2

1

2

1||||

consdsBBmJ

dsBWmdsmvdspJm

m m

m

m

m

m

∫

∫∫∫=−=

⇒−===

μ

μ

• The total particle drift in static E and B fields is:

• For equatorial particle in electrostatic potential ϕ (E=-∇ϕ):

– Particle conserves total (potential plus kinetic) energy

• Bounce-averaged motion for particles with finite J

– Particle’s equatorial trace conserves total energy

2||22

3||32

22||3

221

2

ˆ2

)(2

ˆ

BqRBrW

qBBB

BBE

qBBbBW

qBBBW

BBE

BqRBrmv

qBBBmv

BBEuuuu

c

c

c

ccBBED

vvrrv

vv)vvrrv

vvrrvrrrr

×+

∇×+

×

=∇×

+∇×

+×

=×

+∇×

+×

=++=

⊥

⊥∇×

μ

222

)()()(qB

WqBqB

BBqB

qBuD+∇×

=∇×

+∇×

=ϕμϕ

vrvrvrr

2

)),,()((qB

rJWrqBvD

vvvrv μϕ +∇×

=

• Drift paths for equatorially mirroring (J=0) particles, or• … for bounce-averaged particles’ equatorial traces

in a realistic magnetosphere.

– As particles bounce they also drift because of gradient and curvature drift motion and in general gain/lose kinetic energy in the presence of electric fields.

– If the field is a dipole and no electric field is present, then their trajectories will take them around the planet and close on themselves.

• The third adiabatic invariant– As long as the magnetic field

doesn’t change much in the time required to drift around a planet the magnetic flux inside the orbit must be constant.

– Note it is the total flux that is conserved including the flux within the planet.

∫ ⋅=Φ SdBvr

• Limitations on the invariants– is constant when there is little change in the field’s

strength over a cyclotron path.

– All invariants require that the magnetic field not change much in the time required for one cycle of motion

where is the cycle period.

τ11

<<∂∂

tB

B

hrsms

s

J

1010~min11~1010~ 36

−−

−

Φ

−−

ττ

τ μ

τ

μ

cBB

ρ1

<<∇

• A plasma as a collection of particles– The properties of a collection of particles can be described

by specifying how many there are in a 6 dimensional volume called phase space.

• There are 3 dimensions in “real” or configuration space and 3 dimensions in velocity space.

• The volume in phase space is• The number of particles in a phase space volume iswhere f is called the distribution function.

– The density of particles of species “s” (number per unit volume)

– The average velocity (bulk flow velocity)

dxdydzdvdvdvdvdr zyx=dvdrtvrf ),,( rr

dvtvrftrn ss ),,(),( rrr∫=

dvtvrfdvtvrfvtru sss ∫∫= ),,(/),,(),( rrrrrrr

The Properties of a Plasma

– Average random energy

– The partial pressure of s is given by

where N is the number of independent velocity components (usually 3).

– In equilibrium the phase space distribution is a Maxwelliandistribution

where

( ) ( )⎥⎦

⎤⎢⎣

⎡ −=

s

ssss kT

uvmAvrf2

21

exp,rr

rr

∫∫ −=− dvtvrfdvtvrfuvmuvm ssssss ),,(/),,()()( 2212

21 rrrrrrrr

))(2( 221

sss

s uvmNn

p rr−=

( )23

2 TkmnA ss π=

• For monatomic particles in equilibrium

• The ideal gas law becomes

– where k is the Boltzman constant (k=1.38x10-23 JK-1)• This is true even for magnetized particles.

• The average energy per degree of freedom is:

– for a 1keV, 3-dimensional proton distribution, we mean:kT=1keV, get Eaverage=1.5keV, and define vthemal=(2kT/mp)1/2=440km/s

– for a 1keV beam with no thermal spreadkT=0 and V=440km/s

2/212/)(

21 22 ⎟

⎠⎞

⎜⎝⎛==−= thermalsssaverage vmkTuvmE rr

2/)( 221 NkTuvm ss =−

rr

sss kTnp =

– Other frequently used distribution functions.• The bi-Maxwellian distribution

– where – It is useful when there is a difference between the distributions

perpendicular and parallel to the magnetic field • The kappa distribution

Κ characterizes the departure from Maxwellian form.– ETs is an energy.– At high energies E>>κETs it falls off more slowly than a Maxwellian

(similar to a power law)– For it becomes a Maxwellian with temperature kT=ETs

( )( ) ( )

⎥⎦

⎤⎢⎣

⎡ −−

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−=

⊥

⊥⊥

s

ss

s

ssss kT

uvmkT

uvmAvrf

221

221

' expexp,rr

rr

( ) ( ) 1221

1,−−

⎥⎦

⎤⎢⎣

⎡ −+=

κ

κ κ Ts

ssss E

uvmAvrfrr

rr

∞→κ

⎟⎟⎠

⎞⎜⎜⎝

⎛= ⊥

21

23

'sssss TTTAA

• What makes an ionized gas a plasma?– The electrostatic potential of an isolated charge q0

– The electrons in the gas will be attracted to the ion and will reduce the potential at large distances, so the distribution will differ from vacuum.

– If we assume neutrality Poisson’s equation around q0 is

– The particle distribution is Maxwellian subject to the external potential

• assuming ni=ne=n∞ far away

– At intermediate distances (not at charge, not at infinity):

– Expanding in a Taylor series for r>0 for both electrons and ions

( ) )(00

2ei nner −−=

−=∇

εερϕ r

rq

04πεϕ =

1<<kTeϕ

( )

ie

ie

TTT

kTne

kTe

kTeenr

111

110

2

0

2

+=

=⎥⎦

⎤⎢⎣

⎡+−+=∇ ∞∞ ϕ

εϕϕ

εϕ r

[ ]eieieieiei kTqnnkTqvmAvf ,,,2,, /exp/)

21(exp)( ϕϕ ∞=⇒⎥⎦

⎤⎢⎣⎡ +−=v

•The Debye length ( ) is

where n is the electron number density and now e is the electron charge. Note: colder species dominates.•The number of particles within a Debye sphere

needs to be large for shielding to occur

(ND>>1). Far from the central charge the electrostatic

force is shielded.

rqe D

r

04 επϕ

λ−

=

34 3

DD

nN λπ=

Dλ

ie

ieD TT

TTTne

kT+

=⎟⎠⎞

⎜⎝⎛= ;

21

20ελ

• The plasma frequency– Consider a slab of plasma of thickness L.– At t=0 displace the electron part of the slab by <<L and

the ion part of the slab by <<L in the opposite direction.

– Poisson’s equation gives

– The equations of motion for the electron and ion slabs are

ie δδδ −=

δε0

0enE =

δεε

δδδ

δ

δ

)(0

02

0

02

2

2

2

2

2

2

2

2

2

2

ione

ie

iion

ee

mne

mne

dtd

dtd

dtd

eEdt

dm

eEdt

dm

+−=−=

=

−=

eδiδ

– The frequency of this oscillation is the plasma frequency

– Because mion>>me

ionpi

epe

pipep

mne

mne

0

02

2

0

02

2

222

εω

εω

ωωω

=

=

+=

pep ωω ≈

• Useful formulas:

) ( 21043/)/(/

)(;9.8

)(),(;107.1

)(),(;4.7

3

32/1

2/39

321

protonsfornHzfmmsqrtff

cmnnkHzf

cmneVTnTN

cmneVTnTm

pieipepi

pe

D

D

⋅=⇒=

⋅=

×=

⎟⎠⎞

⎜⎝⎛=

−

−

−λ

• A note on conservation laws– Consider a quantity that can be moved from place to place.

– Let be the flux of this quantity – i.e. if we have an element of area then is the amount of the quantity passing the area element per unit time.

– Consider a volume V of space, bounded by a surface S.

– If σ is the density of the substance then the total amount in the volume is

– The rate at which material is lost through the surface is

– Use Gauss’ theorem

– An equation of the preceeding form means that the quantity whose

density is σ is conserved.

fr

Ar

δAfrr

δ⋅

∫V

dVσ

∫ ⋅S

Adfrr

∫ ∫ ⋅−=V S

AdfdVdtd rr

σ

0=⎭⎬⎫

⎩⎨⎧ ⋅∇+

∂∂

∫ dVftV

rσ

ft

r⋅−∇=

∂∂σ

• Magnetohydrodynamics (MHD)– The average properties are governed by the basic

conservation laws for mass, momentum and energy in a fluid.

– Continuity equation

– Ss and Ls represent sources and losses. Ss-Ls is the net rate at which particles are added or lost per unit volume.

– The number of particles changes only if there are sources and losses.

– Ss,Ls,ns, and us can be functions of time and position.– Assume Ss=0 and Ls =0, where Ms is

the total mass of s and dr is a volume element (e.g. dxdydz)

where is a surface element bounding the volume.

sssss LSunt

n−=⋅∇+

∂∂ r

sssss Mdrnm == ∫ ρρ ,

sdut

Mdrut

Mss

sss

s rrr⋅+

∂∂

=⋅∇+∂

∂∫∫ ρρ )(

sdr

– Momentum equation

where is the charge density, is the current density, and the last term is the density of non-electromagnetic forces.

– The operator is called the convective derivative and gives the total time derivative resulting from intrinsic time changes and spatial motion.

– If the fluid is not moving (us=0) the left side gives the net change in the momentum density of the fluid element.

– The right side is the density of forces• If there is a pressure gradient then the fluid moves toward lower

pressure.

• The second and third terms are the electric and magnetic forces.

sgssqsssssssss

s mFBJEpLSumuut

u ρρρ +×++−∇=−+∇⋅+∂

∂ rrrrrrr

)()(

ssqs nq=ρ ssss unqJ rr=

)( ∇⋅+∂∂

sut

r

( )( ) sgssqssssstu mFBJEpuuss

rrrrrrr

ρρρρ +×++−∇=⋅∇+∂∂

– The term means that the fluid transports momentum with it.

• Combine the species for the continuity and momentum equations– Drop the sources and losses, multiply the continuity

equations by ms, assume np=ne and add.

Continuity

– Add the momentum equations and use me<<mp

Momentum

ss uu rr∇⋅

0)( =⋅∇+∂∂ u

trρρ

mFBJpuutu

gρρ +×+−∇=∇⋅+∂∂ rrrrr

)(

• Energy equation

where is the heat flux, U is the internal energy density of the monatomic plasma and N is the number of degrees of freedom– adds three unknowns to our set of equations. It is usually

treated by making approximations so it can be handled by the other variables.

– Many treatments make the adiabatic assumption (no change in the entropy of the fluid element) instead of using the energy equation

or

where cs is the speed of sound and cp and cv are the specific heats at constant pressure and constant volume. It is called the polytropic index. In thermodynamic equilibrium

mFuEJqupuUuUut g

rrrrrrr⋅+⋅=+++⋅∇++

∂∂ ρρρ ])[()( 2

212

21

)2/( nNkTU =qr

qr

.constp =−γρ)(2 ρρ

∇⋅+∂∂

=∇⋅+∂∂ u

tcpu

tp

srr

ργ pcs =2vp cc=γ

3/5)2( =+= NNγ

• Maxwell’s equations

– doesn’t help because

– There are 14 unknowns in this set of equations -– We have 11 equations.

• Ohm’s law– Multiply the momentum equations for each individual species

by qs/ms and subtract.

where and σ is the electrical conductivity

JB

EtB

rr

rr

0μ=×∇

×−∇=∂∂

0=⋅∇ Br

0)(=×∇⋅−∇=

∂⋅∇∂ EtB rr

puJBE ,,,,, ρrrrr

)]}([11){( 2 uJtJ

nemBJ

nep

neBuEJ e

err

rrrrrrr

⋅∇+∂∂

−×−∇+×+= σ

sss

s unqJ rr∑=

– Often the last terms on the right in Ohm’s Law can be dropped

– If the plasma is collisionless, may be very large soσ)( BuEJ

rrrr×+= σ

0=×+ BuErrr

• Frozen in flux– Combining Faraday’s law ( ), and

Ampere’ law ( ) with

where is the magnetic viscosity– If the fluid is at rest this becomes a “diffusion” equation

– The magnetic field will exponentially decay (or diffuse) from a

conducting medium in a time where LB is the system size.

tBE

∂∂

−=×∇r

r

JBrr

0μ=×∇

BButB

m

rrrr

2)( ∇+××∇=∂∂ η

01 μση =m

BtB

m

rr

2∇=∂∂ η

mBD L ητ 2=

)( BuEJrrrr

×+= σ

– On time scales much shorter than

– The electric field vanishes in the frame moving with the fluid.– Consider the rate of change of magnetic flux

– The first term on the right is caused by the temporal changes in B

– The second term is caused by motion of the boundary– The term is the area swept out per unit time– Use the identity and Stoke’s theorem

– If the fluid is initially on surface s as it moves through the system the flux through the surface will remain constant even though the location and shape of the surface change.

)( ButB rrr

××∇=∂∂

∫∫ ∫ ×⋅+⋅∂∂

=⋅=Φ

CA AlduBdAn

tBdAnB

dtd

dtd )(ˆˆ

rrrr

r

ldurr

×

0ˆ)( =⋅⎟⎟⎠

⎞⎜⎜⎝

⎛××∇−

∂∂

=Φ

∫ dAnButB

dtd

A

rrr

Dτ

BACCBArrrrrr

×⋅=×⋅

• Magnetic pressure and tension

since

– A magnetic pressure analogous to the plasma

pressure ( )

– A “cold” plasma has and a “warm”

plasma has

– In equilibrium

• Pressure gradients form currents

0021 )(2)(

0μμμ BBBBBBJFB

rrrrrrr∇⋅+∇−=××∇=×=

0

2

2μBpB =

p∇−

0

2

2μβ

Bp

≡ 1<<β

1≥β

pBJ ∇=×rr

( ) ( ) ( )BAABBArrrrrr

∇⋅−⋅∇=×∇×

Frozen-in Theorem Recap – 2

Rate of change of line element:

( ) 112 UrUU ∇⋅+= d

r1 r2

r2'r1'

dr

dr'

U1dt U2dt

r1' = r1 + U1dt

r2' = r2 + U2dt

Taylor Expansion:

( ) 112 Urrrrr ∇⋅+=′−′=′∴ ddtdd

( )Urr∇⋅= d

DtDdei .,.

Frozen-in Theorem Recap – 3

Rate of change of magnetic field:

0 , =×+−=×∇ BUEBE t∂∂

( ) ( ) ( ) ( ) ( )BUUBUBBUBUB ∇⋅−⋅∇−∇⋅+⋅∇=××∇=∴ t∂∂

( ) UBBBUBB⎟⎟⎠

⎞⎜⎜⎝

⎛∇⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛+∇⋅=∴

ρρρ

ρDtDei

DtD

DtD .,. ,

Mass Conservation:

( ) ( ) ( ) DtDteit ρρρ∂∂ρρ∂∂ρ =⋅∇−=∇⋅+=⋅∇+ UUU .,. ,0

( ) ( )UBUBB⋅∇−∇⋅=∴

DtD

( )Urr∇⋅= d

DtDdfc ..

• Magnetic pressure and tension

since

– A magnetic pressure analogous to the plasma

pressure ( )

– A “cold” plasma has and a “warm” plasma

has

– In equilibrium

• Pressure gradients form currents

r F B =

r J ×

r B = 1

μ0(∇ ×

r B ) ×

r B = −∇B2 2μ0 + (

r B ⋅ ∇)

r B μ0

0

2

2μBpB =

p∇−

0

2

2μβ

Bp

≡ 1<<β

1≥β

pBJ ∇=×rr

r A × ∇ ×

r B ( )= ∇

r B ( )⋅

r A −

r A ⋅ ∇( )

r B

– cancels the parallel component

of the term. Thus only the perpendicular component of the magnetic pressure exerts a force on the plasma.

– is the magnetic tension and is

directed antiparallel to the radius of curvature (RC) of the

field line. Note that is directed outward.

0

2

0 2ˆˆˆ

μμBbbBBb ∇=∇⋅

r

0

2

2μB∇−

)ˆ

(ˆˆ)(0

2

0

2

CRBnbbB

μμ −=∇⋅

n̂

– The second term in can be written as a sum of two terms

bbBBBbBB ˆˆ)(ˆ])[(0

2

00∇⋅+∇⋅=∇⋅

μμμrrrBJrr

×

•Some elementary wave concepts

–For a plane wave propagating in the x-direction with wavelength and frequency f, the oscillating quantities can be taken to be proportional to sines and cosines. For example the pressure in a sound wave propagating along an organ pipe might vary like

–A sinusoidal wave can be described by its frequency

and wave vector . (In the organ pipe the frequency is f and . The wave number is ).

ωkr

)}(exp{),( 0 trkiBtrB ω−⋅=rvrr ( ))sin()cos(),( 0 trkitrkBtrB ωω −⋅+−⋅=

rrrrrr

λ

)sin(0 tkxpp ω−=

fπω 2= λπ2=k

• The exponent gives the phase of the wave. The phase velocity specifies how fast a feature of a monotonic wave is moving

• Information propagates at the group velocity. A wave can carry information provided it is formed from a finite range of frequencies or wave numbers. The group velocity is given by

• The phase and group velocities are calculated and waves are analyzed by determining the dispersion relation

kk

v ph

r2

ω=

kv g r

∂∂

=ω

)(kωω =

• When the dispersion relation shows asymptotic behavior toward a given frequency, , vg goes to zero, the wave no longer propagates and all the wave energy goes into stationary oscillations. This is called a resonance.

resω

• MHD waves - natural wave modes of a

magnetized fluid

– Sound waves in a fluid

• Longitudinal compressional oscillations which propagate

at

• and is comparable to the thermal speed.

21

21

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=ρ

γρ

ppcs

21

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛=

mkTcs γ

– Incompressible Alfvén waves• Assume , incompressible fluid with background

field and homogeneous

• Incompressibility

• We want plane wave solutions b=b(z,t), u=u(z,t), bz=uz=0

• Ampere’s law gives the current

• Ignore convection ( )=0

∞→σ 0Br

z

y

x

B0

J b, u

0=⋅∇ ur

izbJ ˆ

0

1

∂∂

−= μ

r

∇⋅ur

BJptu rrr

×+−∇=∂∂ρ

• Since and the x-component of momentum becomes

• Faraday’s law gives

• The y-component of the momentum equation becomes

• Differentiating Faraday’s law and substituting the y-component of momentum

0,0 ==∂∂

yJxp 0=zJ

iuBE

juu

JbBJxp

tu

zyyx

ˆ

ˆ

0)(

0

0

−=

=

=−+∂∂

−=∂

∂

r

r

ρ

jzuBj

zEj

tb x ˆˆˆ

0 ∂∂

=∂

∂−=

∂∂

zbBBJ

tuy

∂∂

=×−=∂

∂

ρμρ 0

00

1 rr

2

2

0

22

02

2

zbB

tzuB

tb

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛=

∂∂∂

=∂∂

ρμ

where is called the Alfvén velocity.

• The most general solution is . This is a

disturbance propagating along magnetic field lines at the

Alfvén velocity.

2

22

2

2

zuC

tb

A ∂∂

=∂∂

21

0

2

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ρμBCA

)( tCzbb A±=

• Compressible solutions– In general incompressibility will not always apply.– Usually this is approached by assuming that the system

starts in equilibrium and that perturbations are small.• Assume uniform B0, perfect conductivity with equilibrium

pressure p0 and mass density 0ρ

EE

JJ

uubBB

ppp

T

T

T

T

T

T

rr

rr

rr

rrr

=

=

=+=

+=+=

0

0

0 ρρρ

– Continuity

– Momentum

– Equation of state

– Differentiate the momentum equation in time, use Faraday’s law and the ideal MHD condition

where

)(0 ut

r⋅∇−=

∂∂ ρρ

))((10

00 bBp

tu rrr

×∇×−−∇=∂∂

μρ

ρρρ

∇=∇∂∂

=∇ 20)( sCpp

0)))((()(

)()(

22

2

0

=××∇×∇×+⋅∇∇−∂∂

××∇=×∇−=∂∂

AAs CuCuCtu

BuEtb

rrrrr

rrrr

21

0 )( ρμ

BAC

rr=

0BuErrr

×−=

– For a plane wave solution

– The dispersion relationship between the frequency ( ) and the propagation vector ( ) becomes

This came from replacing derivatives in time and space by

)](exp[~ trkiu ω−⋅rrr

ωkr

−ω 2r u + (Cs2 + CA

2 )(r k ⋅

r u )

r k + (

r C A ⋅

r k )[(

r C A ⋅

r k )

r u − (

r C A ⋅

r u )

r k − (

r k ⋅

r u )

r C A ] = 0

×→∇×

⋅→∇⋅

→∇

−→∂∂

ki

ki

ki

it

r

r

r

ω

– Case 1

• The fluid velocity must be along and perpendicular to

• These are magnetosonic waves

kr

0Br

kr

ur0Br

21

)()( 22Asph CCkkv +±== ωr

0Bkrr

⊥

kukCCu As

rrrr ))(( 222 ⋅+=ω

– Case 2

• A longitudinal mode with with dispersion relationship

(sound waves)

• A transverse mode with and

(Alfvén waves)

0Bkrr

(k2CA

2 −ω 2)r u + ((Cs

2 CA2 ) −1)k 2(

r C A ⋅

r u )

r C A = 0

kurr

ωk

= ±Cs

0=⋅uk rr

ωk

= ±CA

• Alfven waves propagate parallel to the magnetic field.

•The tension force acts as the restoring force.

•The fluctuating quantities are the electromagnetic field and the current density.

– Arbitrary angle between and kr

0Br

0Br

F

ISVA

Phase Velocities

VA=2CS