ESS 200C - Space Plasma Physics Winter Quarter 2009 Vassilis Angelopoulos Robert Strangeway Schedule of Classes

Date Topic [Instructor] Date Topic 1/5 Organization and Introduction 2/9 The Magnetosphere II [S] to Space Physics I [A&S] 2/11 The Magnetosphere III [A] 1/7 Introduction to Space Physics II [A] 2/18 Planetary Magnetospheres [S] 1/12 Introduction to Space Physics III [S] 2/20 (Fri) The Earth’s Ionosphere [S] 1/14 The Sun I [A] 2/23 Substorms [A] 1/21 The Sun II [S] 2/25 Aurorae [S] 1/23 (Fri) The Solar Wind I [A] 3/2 Planetary Ionospheres [S] 1/26 The Solar Wind II [S] 3/4 Pulsations and waves [A] 1/28 --- First Exam [A&S] 3/9 Storms and Review [A&S] 2/2 Bow Shock and Magnetosheath [A] 3/11 --- Second Exam [A&S] 2/4 The Magnetosphere I [A] 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 Cro-Magnon “macaronis” may be earliest • Aurora over Los Angeles (courtesy V. Peroomian) depiction 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 • In 1698 Edmund Halley Chinese had learned that organized the first a magnetic needle points scientific expedition to north-south. map the field in the • By the 12th century the Atlantic Ocean. 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. • 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 By the beginning • 1859 Richard Carrington and to Solar eruptions. of the space age • 1866 Anders Angström Auroral eruptions are self-luminous and… auroral eruptions • 1896 Kristian Birkeland … due to currents from space: had been placed • 1907 Carl Störmer in fact to field-aligned electrons… in the context of • 1932 Chapman & Ferraro accelerated in the magnetosphere by… the Sun-Earth • 1950 Hannes Alfvén the Solar-Wind–Magnetosphere dynamo. Connection • 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 of Space Plasma Dynamics Societal Consequences of Magnetic Storms

• Damage to spacecraft. • Loss of spacecraft. • Increased Radiation Hazard. • Power Outages and radio blackouts. •GPS Errors Stellar wind coupling to planetary objects is ubiquitous in astrophysical systems

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

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

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

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 dvr r r r m = qE + qvr × B + F dt g • 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. • B acts to change the motion of a charged particle only in directions perpendicular to the motion. – Set E = 0, assume B along z-direction.

mv&x = qvy B

mv&y = −qvx B 2 2 qv&y B q vx B 2 v&&x = = − = −Ωc vx m m2 2 2 q vy B 2 v&&y = − = −Ωc vy m2 | q | B Ω = c m • Solution is circular motion dependent on initial conditions. Assuming at t=0: vx = 0;vy = v⊥

vx = v⊥ sin(±Ωct) v = v cos(±Ω t) and y ⊥ c

v⊥ x − x0 = m cos(Ωct) Ωc

v⊥ y − y0 = ± sin(Ωct) Ωc – 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 • Radius of circle ( rc ) - cyclotron radius or Larmor radius or gyro radius. v⊥ = ρcΩc mv ρ = ⊥ c qB – 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 r dvr d( 1 mv2 ) r F ⋅vr = m ⋅vr = 2 = qvr ⋅(vr × B) = 0 dt dt Only the electric field can energize particles! Particle energy remains constant in absence of E ! • The electric field can modify the particles motion. r r – Assume E ≠ 0 but B still uniform and F =0. g r r – Frequently in space physics it is ok to set E ⋅ B = 0 r r • Only E can accelerate particles along B • Positive particles go along E and negative particles go along − E • Eventually charge separation wipes out E

–E⊥ has a major effect on motion. r • As a particle gyrates it moves along E 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. r r E × B ur = E B2 r r • Drift velocity is perpendicular to E and B • No charge dependence, therefore no currents Z

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

v&x = qEx ± Ωcvy

v&y = mΩcvx 2 v&&x = −Ωcvx

2 Ex v&&y = −Ωc ( + vy ) B • Solution is:

vx = v⊥ sin(±Ωct) E v = v cos(±Ω t) − x y ⊥ c B • In general, averaging over a gyration: r r r dv r r r r E × B m = qE + q v × B = 0 ⇒ v = ≡ V E×B dt B2 • 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] 2 – Thermal velocities (kT=1/2 m vth ): • 1keV proton = 440km/s • 1eV electron = 600km/s – Gyration: • Proton gyro-period: 66s*(1nT/B) • Electron: 28Hz*(B/1nT) – Gyroradii: 1/2 • 1keV proton in 1nT field: 4600km*(m/mp) *(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: v r dvr r v F × B r m = q vr × B + q(F / q) = 0 ⇒ vr = ≡ V dt qB2 F

– e.g., gravity r mgr × B ur = g qB2

– 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 th to integrate over 0 order motion, assuming small perturbations relative to Ωc, rL • The Concept of the Guiding Center – Separates the motion ( vr ) of a particle into motion perpendicular ( ) and parallel ( v ) to the magnetic field. v⊥ – To a good approximation the perpendicular motion can consist of a drift ( v ) and the gyro-motion ( v ) D Ωc vr = vr + vr = (vr + vr )+ vr = vr + vr ⊥ D Ωc gc Ωc – 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 very useful for distances l such that ρ c l << 1 and time scales τ −1 such that ()Ωcτ <<1 • Inhomogenious magnetic fields cause GradB drift. r – If B changes over a gyro-orbit rL will change. mv –ρ = r = ⊥ gets smaller when particle goes into stronger B. c L qB – Assume ∇B is along Y Y + – Average force over gyration:

r ∂B - Z x B = zˆB (y) = zˆ(B + y z0 ) z z0 ∂y B=Bzz

< Fx >= 0

v⊥ ∂Bz0 < Fy >= − < qvx Bz (y) >= − < qv⊥ sin(±Ωct)(Bz0 ± sin(Ωct) ) >= Ωc ∂y 2 2 v⊥ 2 ∂Bz0 1 mv⊥ ∂Bz0 < Fy >= −q < sin (Ωct) > = − . Ωc ∂y 2 B ∂y 2 1 mv v F = − ⊥ ∇B 2 B

–u∇B depends on charge, yields perpendicular currents. v r r v ∇B× B B×∇B ur = −1 mv2 = 1 mv2 ∇B 2 ⊥ qB3 2 ⊥ qB3 • 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. mv2 r ˆ F = Rc Rc nˆ ˆ ˆ •Rc is the radius of curvature of a field line ( = − ( b ⋅ ∇ )b ) Rc wherer B r bˆ = , n ˆ is perpendicular to B and points away from the center B r of curvature, v is the component of velocity along B r r mv2 B×(bˆ⋅∇)bˆ mv2 B× nˆ r uc = 2 = − 2 qB RcqB • Curvature drift can also cause currents. • 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

⎛ 5RE ⎞⎛100nT ⎞⎛ W ⎞ u∇B ≈1RE / min ~ 100km / s ⋅⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ r ⎠⎝ B ⎠⎝100keV ⎠ – The drift around Earth is: 0.5hrs for the same particle at the same location

2 ⎛ r ⎞ ⎛ B ⎞⎛100keV ⎞ ⎜ ⎟ τ ∇B,C = 0.5h⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ 5RE ⎠ ⎝100nT ⎠⎝ W ⎠

• At Earth’s magnetotail current sheet, u∇B , uc are opposite each other: – Curvature dominates at Equator, Gradient dominates further away from equator • 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 ∇θ B must be divergence-less (∇B=0) so: ║ θ Z – In cylindical coordinates gyration is: vθ = mv⊥ ;r = r⊥ ;θ = mΩct – The mirror field is: B = 0;∇B =1 r ∂(rB ) ∂r + ∂B ∂ z = 0 ⇒ B = −1 2⋅r ⋅[∂B ∂ z] Y X θ r z r z r=0 – The Lorenz force is: Fr = q(v Bz ); F = q(−vr Bz + vz Br ); Fz = q(−vθ Br ) θ – From gyro-orbit averaging: < Fr >= q < v > Bz =< F >= 0;(Br = vr = 0)r=0 μ θ 2 < Fz >= −1 2⋅q < vθ r > []∂Bz ∂ z r=0 = −1 2⋅mv⊥ / B ⋅[]∂Bz ∂ z 1 mv2 – Defining: μ ≡ ⊥ we get:μ < F >= −μ(∂B ∂ s) = −μ∇ B 2 μB II μ – This is the mirror force. Note that is conserved during the particle motion: ∂B dv ∂B ∂μs dB d ⎛ mv2 ⎞ μ dB II ⎜ II ⎟ < F > vII = − vII ⇒ mvII = − = − ⇒ ⎜ ⎟ = − ⇒ ∂s dt ∂s ∂t dt μdt ⎝ 2 ⎠ dt μ ⎛ 2 ⎞ ⎛ 2 ⎞ dz d mv⊥ dB d mv⊥ dB d dB dμ ⎜W − ⎟ = − ⇒ − ⎜ ⎟ = − ⇒ − ()B = − ⇒ = 0 dθ dt ⎝ 2 ⎠ dt dt ⎝ 2 ⎠ dt dt dt dt

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

q 2π I = where T = π T c Ω c π c 2 2 v⊥ A = rc = 2 Ωc 2 π 2 v⊥ q Ωc mv⊥ IA = 2 = = μ 2π Ωc 2B – The force on a dipoleμ magnetic moment is r v dB F = r ⋅∇B = −μ dz Where μ r = −μ bˆ

I.e., same as we derived earlier by averaging over gyromotion In a Magnetic Mirror: r • The force is along B and away from the direction of increasing B. • Since E = 0 and kinetic energy must be conserved 1 2 1 2 2 2 mv = 2 m(v + v⊥ )

a decrease in v must yield an increase in v⊥ 1 2 • Particles will turn around when B = 2 mv μ • The loss cone at a given point is the pitch angle below 2 which particles will get lost: sin αi = Bi / Bm =1/ Rm • 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 r r ∂B ∇× E = − ∂t

• Along gyromotion, speed v┴ increases,v but μ is conserved: d v v dl v vv m (vv ) = qE ⋅vv = qE , where : dl = line _ element ⇒ ⊥ dt ⊥ ⊥ dt δ 2 v ⎛ mv⊥ ⎞ v v v v v dB v ⎜ ⎟ = ∫ qEdl = q ∫ (∇× E)dS = −q ∫ dS ⇒ ⎝ 2 ⎠gyration gyro−orbit over _ surface surface dt <0 for _ i+ <0 for _ i+ >0 for _ e− >0 for _ e− ⎛ mv2 ⎞ ⎛ dB ⎞ δ ⎜ ⊥ ⎟ 2 ⎜ ⎟ = ±q⎜ ⎟ rL = B ⇒ ⎝ 2 ⎠gyration ⎝ dt ⎠gyration δ μ()B = B ⇒ = 0 ⇒ μδ μ = const. π δμ μδ

2 2 • Flux through Larmor orbit: Φ=πrL B=(2 π m/q ) μ remains constant • Time varying fields: E – See Chen, Ch 2 • Maxwell’s equations r ρ – Poisson’sr Equation ∇ ⋅ E = •E is the electric field ε 0 •ρ is the charge density -12 •ε 0 is the electric permittivity (8.85 X 10 Farad/m) – Gauss’ Law (absence of magnetic monopoles) r r∇ ⋅ B = 0 •B is the magnetic field – Faraday’s Law r r ∂B ∇× E = − ∂t – Ampere’s Law r r 1 ∂E r ∇× B = + μ J c2 ∂t 0 • c is the speed of light. μ •μ Is the permeability of free space, = 4 π × 10 − 7 H/m r 0 0 •J is the current density ε

• Maxwell’s equations in integralε ρ form Note: use Gauss and Stokes theorems; identities A1.33; A1.40 in Kivelson and Russell) r ρ r v r 1 r v Q ∇ ⋅ E = ⇔ E ⋅dS = ∇ ⋅ E dV = dV ⇔ E ⋅dS = ∫∫A ∫ ∫ 0 V 0 V A ε 0 r r v ∇⋅ B = 0 ⇔ B ⋅d S = 0 ∫A r r r ∂B v v r v ∂B v ∂Φ ∇× E = − ⇔ E ⋅dl = (∇× E)⋅d S = − ⋅d S = − ⇔ ∂t ∫∫C ∫ ∂t ∂t v v ∂Φ E ⋅dl = − ∫ ∂t A rμ r 1 ∂E r r v v v v 1 ∂ r v v ∇× B = + 0 J ⇔ B ⋅dl = (∇× B)⋅dS = 2 E ⋅dS + 0 J ⋅dS ⇔ c2 ∂t ∫C ∫ c ∂t ∫ ∫

r v 1 ∂ r v B ⋅dl = 2 E ⋅dS + μ 0 I ∫C c ∂t ∫ v –dS is a unit normal vector to surface: outward directed for closedμ surface or in direction given by the right hand rule around C for open surface, and Φ is magnetic flux through the surface. v –dl is the differential element around C. • The first adiabatic invariant r ∂B r r r –= −∇× E says that changing B drives E (electromotive ∂t 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. 1 mv2 μ = 2 ⊥ = const. B

–μ 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). • For a coordinate in which the motion is periodic the action integral J = p dq = constant i ∫ i i r r r is conserved. Here pi is theπρ canonical momentum: p = m v + q A where A is the vector potential). π • First term: mv ds = 2 mv = 4 mμ πq ∫ ⊥ ⊥ • Second term: r r r r r 2 m2v2 qA⋅dsr = q∇× A⋅dS = qB ⋅dS = − ⊥ = −2π mμ q ∫ ∫∫ ∫∫ qB • For a gyrating particle: 2πm J = pr ⋅dsr = μ 1 ∫ ⊥ q

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

m2 m2 J = p||ds = 2 mv||ds = 2 2m W − Bds ⇒ ∫ ∫m ∫m 1 1

m2 J = 2 2mμ Bm − Bds =cons. ∫m 1 •Bm is the magnetic field at the mirror point μ – Note the integral depends on the field line, not the particle

– If the length of the field line decreases, u|| will increase • Fermi acceleration • The total particle drift in static E and B fields is: v r r v v r r r r E × B 1 2 B×∇B 2 rˆc × B uD = uE×B + u∇B + uc = 2 + 2 mv⊥ 3 + mv|| 2 = B qB qRc B v r r v v ) v v E × B B×∇B B×(b∇)B +W + 2W = B2 ⊥ qB3 || qB3 v r r v v E × B B×∇B rˆc × B 2 + μ 2 + 2W|| 2 B qB qRc B ϕ • For equatorial particleμ in electrostatic potential ϕ (E=-∇ϕ): r v r v r v B×∇(q ) B×∇( B) B×∇(qϕ +W ) ur = + = D qB2 qB2 qB2

– Particle conserves total (potential plus kinetic) energy

• Bounce-averagedϕ motion for particles with finite J r v B×∇(q (rv) +W (μ, J,rv)) vv = D qB2

– Particle’s equatorial trace conserves total energy • 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 rplanetv the magnetic flux Φ = ∫ B ⋅ d S inside the orbit must be constant. – Note it is the total flux that is conserved including the flux within the planet.

• Limitations on the invariants –μ is constant when there is little change in the field’s strength over a cyclotron path. ∇B 1 << B ρ c

– All invariants require that the magnetic field not change much in the time required for one cycle of motion 1 ∂B 1 << B ∂t τ

where τ is the cycle period. τ −6 −3 μ ~ 10 −10 s τ J ~ 1s −1min

τ Φ ~ 10m −10hrs The Properties of a Plasma • 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 dvdr = dvxdvy dvz dxdydz • The number of particles in a phase space volume is f (rr,vr,t)dvdr where f is called the distribution function. – The density of particles of species “s” (number per unit volume) n (rr,t) = f (rr,vr,t)dv s ∫ s – The average velocity (bulk flow velocity) ur (rr,t) = vrf (rr,vr,t)dv / f (rr,vr,t)dv s ∫ s ∫ s – Average random energy 1 m (vr − ur )2 = 1m (vr − ur )2 f (rr,vr,t)dv / f (rr,vr,t)dv 2 s s ∫ 2 s s s ∫ s – The partial pressure of s is given by

ps 2 1 r r 2 = ( 2 ms (v − us ) ) ns N where N is the number of independent velocity components (usually 3).

– In equilibrium the phase space distribution is a Maxwellian distribution ⎡ 1 r r 2 ⎤ r r 2 ms (v − us ) fs ()r,v = As exp⎢ ⎥ ⎣ kTs ⎦ 3 2 where As = ns (m 2π kT ) • For monatomic particles in equilibrium 1 r r 2 2 ms (v − us ) = NkT / 2 • The ideal gas law becomes

ps =ns kTs – 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:

1 r r 2 ⎛ 1 2 ⎞ Eaverage = ms (v − us ) = kT / 2 = ⎜ msvthermal ⎟ / 2 2 ⎝ 2 ⎠ – for a 1keV, 3-dimensional proton distribution, we mean: 1/2 kT=1keV, get Eaverage=1.5keV, and define vthemal=(2kT/mp) =440km/s – for a 1keV beam with no thermal spread kT=0 and V=440km/s – Other frequently used distribution functions. • The bi-Maxwellian distribution 2 ⎡ 1 m (v − u ) ⎤ ⎡ 1 r r 2 ⎤ r r ' 2 s s 2 ms ()v⊥ − u⊥s fs ()r,v = As exp⎢− ⎥ exp⎢− ⎥ kT kT ⎣⎢ s ⎦⎥ ⎣ ⊥s ⎦ 3 ⎛ 1 ⎞ A' = A T 2 ⎜T T 2 ⎟ – where s s s ⎜ ⊥s s ⎟ ⎝ ⎠ – It is useful when there is a difference between the distributions perpendicular and parallel to the magnetic field κ • The kappa distribution −κ −1 ⎡ 1 r r 2 ⎤ r r 2 ms ()v − us fs ()r,v = A s ⎢1+ ⎥ ⎣ κETs ⎦ 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 κ → ∞ it becomes a Maxwellian with temperature kT=ETs • What makes an ionized gas a plasma? ϕ q – The electrostatic potential of an isolated charge q0 = 4πε 0r – 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 2 r −ερ e ∇ ϕ()r = = − (ni − ne ) ϕ 0 ε 0 – The particle distribution is Maxwellian subject to the external potential

v ⎡ 1 2 ⎤ f (v) = Aexp − ( mi,evi,e + q ) / kTi,e ⇒ ni,e = n∞ exp[]qϕ / kTi,e ⎣⎢ 2 ⎦⎥ • assuming n =n =n far away ϕ i e ∞ – At intermediate distances (not at charge, not at infinity): eϕ kT <<1 ε ϕ – Expanding in a Taylor series forϕ r>0 for both electrons and ions 2 2 r en∞ ⎡ e e ⎤ e n∞ ∇ ()r = ⎢1+ −1+ ⎥ = ε ϕ 0 ⎣ kTe kTi ⎦ 0kT 1 1 1 = + T Te Ti −r ϕ λ qe D = π 4 ε 0r

•The Debye length ( λ D ) is

1 λ 2 ⎛ ε 0kT ⎞ TeTi D = ⎜ 2 ⎟ ;T = ⎝ ne ⎠ Te +Ti 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 4π nλ3 N = D needs to be large for shielding to occur D 3

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

• The plasma frequency – Consider a slab of plasma of thickness L.

– At t=0 displace the electron part of the slab by δ e <

δ = δ e −δ i – Poisson’s equationδ gives E = en0 δ ε0 – The equationsδ of motion for the electron and ion slabs are

2 d e meδ 2 = −eE dt δ d 2 δ m i = eE ion dt 2 d 2 d 2 d 2 eε2n e2n e i 0 ε 0 2 = 2 − 2 = −( + )δ dt dt dt 0me 0mion – The frequencyω of this oscillation is the plasma frequency ω 2 2 2 p = ωpe + pi ω e2n 2 = 0 pe ε 0me ω 2 2 e n0 pi = ε 0mion

– Because mion>>me ω p ≈ ω pe • Useful formulas:

1 2 ⎛ T ⎞ −3 λD = 7.4m⎜ ⎟ ;T (eV ),n(cm ) ⎝ n ⎠ T 3/ 2 N =1.7×109 ;T (eV ),n(cm−3 ) D n1/ 2 −3 f pe = 8.9kHz ⋅ n;n(cm )

f pi = f pe / sqrt(mi / me ) ⇒ f pi / 43 = 210Hz ⋅ n ( for protons) • A note on conservation laws – Consider a quantity that can be moved from place to place. r r – Let f be the flux of this quantity – i.e. if we have an element of area δA r r then f ⋅ δ A 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 ∫σ dV V r r – The rate at which material is lost through the surface is ∫ f ⋅dA d r r S ∫∫σ dV = − f ⋅dA dt VS ⎧∂σ r⎫ – Use Gauss’ theorem ∫ ⎨ + ∇ ⋅ f ⎬dV = 0 V ⎩ ∂t ⎭ ∂σ r = −∇⋅ f ∂t – An equation of the preceeding form means that the quantity whose density is σ is conserved. • Magnetohydrodynamics (MHD) – The average properties are governed by the basic conservation laws for mass, momentum and energy in a fluid. – Continuity equation ∂n s + ∇ ⋅n ur = S − L ∂t s s s s

–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. ρ –Ss,Ls,ns, and us can be functions of time and position. = m n , ρ dr = M – Assume Ss=0 and Ls =0,ρ s s s ∫ s s where Ms is the total mass of s and dr is a volume element (e.g. dxdydz) ∂M ∂M s + ∇ ⋅( ur )dr = s + ρ ur ⋅dsr ∂t ∫ s s ∂t ∫ s s where d sr is a surface element bounding the volume. ρ ρ

ρ – Momentum equation ∂ρ ur r r r r s s r r ρ ( ∂t + ∇⋅()susus )= −∇ps + qs E + J s × B + ρs Fg ms ∂ur r r r ( s + ur ⋅∇ur ) + m ur (S − L ) = −∇p + E + J × B + ρ F m s ∂t s s s s s s s qs s s g s r r where ρ qs = q s n s is the charge density, J s = q snsus is the current density, and the last term is the density of non- electromagnetic forces. ∂ – The operator ( + u r ⋅ ∇ ) is called the convective derivative ∂t s 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. r r – The u s ⋅ ∇ u s 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.

∂ρ + ∇ ⋅(ρ ur) = 0 ρ Continuity ∂t

– Add the momentum equations and use me<

ρ

• Energy equation ∂ r r r (1 u2 +U)+∇⋅[(1 u2 +U)ur + pur +qr] = J ⋅E + ρur⋅F m ∂t 2 2 g where qr is the heat flux, U is the internal energy density of the monatomic plasma ( U = nNkT / 2 ) and N is the number of degrees of freedom –qr 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 ∂p r 2 ∂ρ r + u ⋅∇p = cs ( + u ⋅∇ρ ) pρ −γ = const. or ∂t ∂t 2 γ γ = c c where cs is the speed of sound c s = p ρ and p v

cp and cv are the specific heats at constant pressure and constant volume. It is called the polytropic index. In thermodynamic equilibrium γ = (N + 2) N = 5 / 3 • Maxwell’s equations r ∂B r = −∇× E ∂t r r ∇× B = μ0 J r –∇ ⋅ B = 0 doesn’t helpr because ∂(∇ ⋅ B) r = −∇ ⋅∇× E = 0 ∂t r r r – There are 14 unknowns in this set of equations - E, B, J,ur, ρ, p – We have 11 equations. •Ohm’s law – Multiply the momentum equations for each individual species by q /m and subtract. s s r r r r 1 1 r r m ∂J r J = σ {(E + ur × B) + ∇p − J × B − e [ + ∇ ⋅(Jur)]} ne e ne ne2 ∂t r r where J = ∑ q s n s u sand σ is the electrical conductivity s – Often the last terms on the right in Ohm’s Law can be dropped r r r J = σ (E + ur × B) – If the plasma is collisionless, σ may be very large so r r E + ur × B = 0 • Frozen in flux r r ∂B – Combining Faraday’s law ( ∇ × E = − ), and ∂t r r r r r r Ampere’ law (∇ × B = μ J ) with J = σ ( E + u × B) r 0 ∂B r r = ∇×(ur × B) +η ∇2 B ∂t m

where η m = 1 σ μ 0 is the magnetic viscosity – If the fluid is at rest thisr becomes a “diffusion” equation ∂B r =η ∇2 B ∂t m – The magnetic field will exponentially decay (or diffuse) from a τ 2 conducting medium in a time D = L B η m where LB is the system size. – On time scales much shorter than τ D r ∂B r = ∇×(ur × B) ∂t – The electric field vanishes in the frame moving with the fluid. – Consider the rate of change of magnetic flux r dΦ d r ∂B r r = B ⋅nˆdA = ⋅nˆdA + B ⋅(ur × dl ) dt dt ∫∫AA∂t ∫C – The first term on the right is caused by the temporal changes in B – The second term is caused by motion of the boundary r – The term ur × d l is the area swept out per unit time r r r r r r – Use the identity A ⋅ B × C = C ⋅ A × B and Stoke’s theorem r dΦ ⎛ ∂B r ⎞ = ⎜ − ∇×(ur × B)⎟⋅nˆdA = 0 ∫A ⎜ ⎟ dt ⎝ ∂t ⎠ – 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. • Magnetic pressure and tension μ r r r 1 r r 2 r r FB = J × B = μ (∇× B)× B = −∇B 2 0 + (B ⋅∇)B μ0 r 0 r r r r r since A×(∇× B)= (∇B)⋅ A − (A⋅∇)B

B2 –pB = A magnetic pressure analogous to the plasma 2μ0 pressure ( − ∇ p ) β p ≡ –B2 A “cold” plasma has β < < 1 and a “warm” 2μ0 plasma has β ≥1 r r – In equilibrium J × B = ∇p

• Pressure gradients form currents Frozen-in Theorem Recap – 2

Rate of change of line element: dr' r ' 2 r1' = r1 + U1dt r1'

U1dt r2' = r2 + U2dt U2dt dr r Taylor Expansion: 1 r2

U 2 = U1 + (dr ⋅∇)U1

′ ′ ∴dr′ =r2 −r1 = dr+dt()dr⋅∇ U1

Dd r i.e., = ()dr ⋅ ∇ U Dt Frozen-in Theorem Recap – 3

Rate of change of magnetic field:

∇ × E = − ∂B ∂t , E + U × B = 0

∴∂B ∂t = ∇ × (U × B)= U(∇ ⋅ B)+ (B ⋅∇)U − B(∇ ⋅ U )− (U ⋅∇)B DB ∴ = ()B ⋅∇ U − B ∇ ()⋅ U Dt Mass Conservation: ρ ρ ∂ρ ∂t + ∇⋅(ρU)= 0, i.e., ∂ρ ∂t + (U⋅∇)ρ = −ρ(∇⋅U)= Dρ Dt ρ D B B D ⎛ B ⎞ ⎛ B ⎞ ∴ = ()B ⋅ ∇ U + , i.e., D ⎜ ⎟ Dt = ⎜ ⋅ ∇ ⎟U Dt Dt ⎝ ⎠ ⎝ ρ ⎠

Dd r c. f . = ()dr ⋅ ∇ U Dt • Magnetic pressure and tension r r r r r r r F = J × B = 1 (∇×B ) × B =−∇B2 2μ + (B ⋅∇)B μ B μ0 0 0 r r r r r r since A ×∇×()B =∇( B )⋅ A − (A ⋅∇)B

B2 –pB = A magnetic pressure analogous to the plasma 2μ0 pressure ( − ∇ p ) β p ≡ –B2 A “cold” plasma has β << 1 and a “warm” plasma 2μ0 has β ≥1 r r – In equilibrium J × B = ∇p

• Pressure gradients form currents r r – The second term in μ J × B can be written as a sum of two terms r r r μ 2 [(B ⋅∇)B] = bˆB ⋅∇B + (B )bˆ⋅∇bˆ μ μ 0 0 0 ˆ r 2 – bB ⋅∇B = bˆbˆ∇B cancels the parallel component 0 2μ0

2 of the − ∇ B term. Thus only the perpendicular component 2μ0 of the magnetic pressure exerts a force on the plasma.

μ 2 B2 ˆ ˆ nˆB – ( )b ⋅∇b = −( ) is the magnetic tension and is 0 μ0 RC directed antiparallel to the radius of curvature (RC) of the field line. Note that n ˆ is directed outward. •Some elementary wave concepts –For a plane wave propagating in the x-direction with wavelength λ 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

p = p0 sin(kx −ωt) –A sinusoidal wave can be described by its frequencyω r and wave vector k . (In the ωorgan pipe the frequency is f and ω = 2 π f . The wave number is k = 2 π λ ). r r r r r r B(r,t) = B0 (cos(k ⋅r − t) + isin(k ⋅r −ωt)) r r v r B(r,t) = B0 exp{i(k ⋅r −ωt)} • The exponent gives the phase of the wave. The phase velocity specifies how fast a feature of a monotonic wave is moving ω r v = k ph k 2 • 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 ∂ω v g = r ∂k • The phase and group velocities are calculated and waves are analyzed by determining the dispersion relation ω = ω(k) • When the dispersion relation shows asymptotic behavior toward a given frequency,ω res , vg goes to zero, the wave no longer propagates and all the wave energy goes into stationary oscillations. This is called a resonance. • MHD waves - natural wave modes of a magnetized fluid

– Sound waves in a fluid

• Longitudinal compressional oscillations which propagate at 1 1 ⎛ ∂ρp ⎞ 2 ⎛γ p ⎞ 2 cs = ⎜ ⎟ = ⎜ ⎟ ⎝ ∂ ⎠ ⎝ ρ ⎠

1 2 •⎡ ⎛ kT ⎞⎤ and is comparable to the thermal speed. cs = ⎢γ ⎜ ⎟⎥ ⎣ ⎝ m ⎠⎦ – Incompressible Alfvén waves r • Assume σ → ∞ , incompressible fluid with B 0 background field and homogeneous z

B0 y J b, u x • Incompressibility ∇ ⋅ur = 0

• We want plane wave solutions b=b(z,t), u=u(z,t), bz=uz=0 • Ampere’s law gives the current r ∂ b J = − 1 iˆ μ 0 ∂ z • Ignore convection ( ur ⋅ ∇ )=0 ∂ur r r ρ = −∇p + J × B ∂t ∂p • Since = 0 , J = 0 and J = 0 the x-component of ∂x y z momentum becomes ∂u ∂p ρ x = − + (J B − b J ) = 0 ∂t ∂x y 0 y z ur = uˆj r ˆ E = −uB0i • Faraday’s law gives ∂b ∂E ∂u ˆj = − x ˆj = B ˆj ∂t ∂z 0 ∂z • The y-component of the momentum equation ρ ∂u y 1 r r B ∂b becomes = − J × B = 0 0 μ ∂t 0 ρ ∂z

• Differentiating Faraday’s law and substituting the y- component of momentum ∂ 2b ∂ 2u ⎛ B2 ⎞ ∂ 2b = B = ⎜ ⎟ 2 0 ⎜ μ ⎟ 2 ∂t ∂z∂t ⎝ 0 ρ ⎠ ∂z ∂ 2b ∂ 2u = C 2 ∂t 2 A ∂z 2 1 ⎛ B2 ⎞ 2 ⎜ ⎟ where C A = ⎜ μ ⎟ is called the Alfvén velocity. ⎝ 0 ρ ⎠ • The most general solution is . This is a b = b(z ± CAt)

disturbance propagating along magnetic field lines at the

Alfvén velocity. • 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 ρ pressure p0 and mass density 0

ρT = ρ0 + ρ

pT = p0 + p r r r BT = B0 + b r r uT = u r r JT = J r r ET = E ∂ρ – Continuity = −ρ (∇ ⋅ur) ρ∂t 0 ∂ur 1 r r –Momentum 0 = −∇p − (B0 ×(∇×b)) ∂t μ0 ρ ∂pρ – Equation of state ∇p = ( ) ∇ = C 2∇ρ ∂ 0 s

– Differentiate the momentum equation in time, use Faraday’s r r r law and the ideal MHD condition E = −u × B0 r ∂b r r = −(∇× E) = ∇×(ur × B ) ∂t 0 ∂ 2ur r r − C 2∇(∇ ⋅ur) + C ×(∇×(∇×(ur ×C ))) = 0 ∂t 2 s A A r r B where C A = 1 μ 2 ( 0 ρ ) r r r – For a plane wave solution u ~ exp[i(k ⋅r −ω t)] – The dispersion relationshipr between the frequency ( ω ) and the propagation vector ( k ) becomes r r r r r r r r r r −ω 2ur + (C 2 + C 2 )(k ⋅ ur )k + (C ⋅ k )[(C ⋅ k )ur − (C ⋅ ur )k − (k ⋅ ur )C ] = 0 s A A A A A ‰ This came from replacing derivatives in time and space by ∂ → −iω ∂t r ∇ → ik r ∇⋅ → ik ⋅ r ∇× → ik × r r – Case 1 k⊥B0 2 r 2 2 r r r ω u = (Cs + CA )(k ⋅u)k r r • The fluid velocity must be along k and perpendicular to B0

r r B k 0 ur

r 2 2 1 • These are magnetosonic waves v = k (ω ) = ±(C +C ) 2 ph k s A r r – Case 2 k B0

r r (k 2C 2 −ω 2 )ur + ((C 2 C 2 ) −1) k 2(C ⋅ ur )C = 0 A s A A A r • A longitudinal mode with ur k with dispersion relationship ω =±C k s (sound waves) r • A transverse mode with k ⋅ ur = 0 and ω =±C k A (Alfvén waves) • 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. r r – Arbitrary angle between k and B0 r B0 VA=2CS V A F S I

Phase Velocities