Basic Plasma Physics Principles

Summary

This lecture document outlines the foundational principles of plasma physics with applications to high-energy solar physics. It covers single particle orbits and drifts, adiabatic invariants and magnetic mirroring, magnetohydrodynamics (MHD) and force-free fields, resistive diffusion in solar coronal loops, kinetic theory including the Boltzmann and Vlasov equations, wave dispersion relations, and plasma instabilities such as the two-stream instability.

Page 1: Introduction, Particle Orbits, and Drifts

Basic Plasma Physics Principles Gordon Emslie Oklahoma State University

Outline Single particle orbits; drifts Magnetic mirroring MHD Equations Force-free fields Resistive Diffusion The Vlasov equation; plasma waves

Single particle orbits E and B fields are prescribed; particles are “test particles”

Single particle orbits F = q (E + v × B) Set E = 0 (for now) F = q v × B Since F ⊥ v, no energy gain (F.v = 0) Particles orbit field line mv²/r = qvB r = mv/qB (gyroradius) ω = v/r = qB/m (gyrofrequency)

Motion in a Uniform Magnetic Field Is this an electron or an ion?

Drifts F = q (E + v × B) Now let E ≠ 0. Relativistic transformation of E and B fields: E’ = γ(E + (v/c) × B) B’ = γ(B –(v/c) × E) E’² – B’² = E² – B² If E < B (so that E² – B² < 0), transform to frame in which E = 0: v = c (E × B)/B² In this frame, we get simple gyromotion So, in ‘lab’ frame, we get gyro motion, plus a drift, at speed vD = c (E × B)/B²

E × B drift

Drifts Exercise: What if E > B?

Drifts: vD = c (E × B)/B² E is “equivalent electric field” Examples: (1) E is actual electric field: vD = c (E × B)/ B² (independent of sign of q) (2) Pressure gradient: qE = -∇p: vD = -c∇p × B/qB² (dependent on sign of q) (3) Gravitational field: mg = qE: vD = (mc/q) g × B/B² (dependent on m and sign of q) Puzzle: in absence of magnetic field, particles subject to g accelerate at the same rate and in the same direction; particles subject to E accelerate in opposite directions at a rate which depends on their mass. Why is the exact opposite true when a B is present?

Page 2: Magnetic Mirroring, MHD Equations, and Resistive Diffusion

Magnetic Mirroring Adiabatic invariants Slow change of ambient parameters: Action ∫p dq (e.g. Energy/frequency) is conserved Apply this to gyromotion: E = (1/2) mv⊥²; Ω = eB/m Then as B slowly changes, mv⊥²/(B/m) = m²v⊥²/B = p⊥²/B is conserved As B increases, p⊥ increases and so, to conserve energy, p// must decrease. This can be expressed as a mirror force F = - (p⊥²/2m) (∇B/B), This force causes particles to be trapped in loops with high field strengths at the ends. Note that a magnetic compression also acts as a reflecting wall; this will help us understand particle acceleration later.

Plasma physics in principle • Solve equations of motion with initial E and B: md²ri/dt² = qi (E + [dri/dt] × B) • Then use the resulting ri and dri/dt to get charge density ρ(r) and current density j(r) • Then obtain the self-consistent E and B through Maxwell’s equations: ∇.E = ρ ∇×B = (4π/c)(j + ∂E/∂t) • “Lather, rinse, repeat”

Plasma physics in principle • Requires the solution of ~ 10²⁷ coupled equations of motion • Not a practical method!

MHD Equations • Replace ~ 10²⁷ coupled equations of motion by “averaged” fluid equations • Neglect displacement current (plasma responds very quickly to charge separation); then body force F = (1/c) j × B = (1/4π) (∇×B) × B

Complete set of MHD Equations Continuity: ∂ρ/∂t + ∇.(ρv) = 0 Momentum: ρ dv/dt = -∇p + (1/4π) (∇×B) × B - ρg Energy: ?? (can use polytrope: d(p/ργ)/dt = 0) Induction: ∂B/∂t = ∇×(v × B) These are 4 equations for the 4 unknowns (ρ, p, v, B)

Force – Free Fields Equation of motion is ρ dv/dt = - ∇p + (1/4π) (∇×B)×B Define the plasma β = ratio of terms on RHS = p/(B²/8π) For typical solar corona, p = 2nkT ~ 2(10¹⁰)(1.38 × 10⁻¹⁶)(10⁷) ~ 10 B ~ 100 → β ~ 10⁻³ So second term on RHS dominates, and in steady-state j must be very nearly parallel to B, i.e. (∇×B)×B ≅ 0

Force – Free Fields (∇×B)×B = 0 Solutions: (1) B = 0 (trivial) (2) ∇×B = 0 (current-free – “potential” field) (3) Linear case: (∇×B) = αB (4) Full case: (∇×B) = α(r)B Note that taking the divergence of (∇×B) = α(r)B gives 0 = ∇α.B + α ∇.B, so that B.∇α = 0, i.e., α is constant on a field line.

Resistive Diffusion Consider the Maxwell equation ∇×E = - (1/c) ∂B/∂t, together with Ohm’s law Elocal = E + (v/c) × B = ηj = (ηc/4π) ∇× B Combined, these give: ∂B/∂t = ∇×(v ×B) - (ηc²/4π) ∇×(∇×B), i.e. ∂B/∂t = ∇×(v ×B) + D ∇²B, where D = ηc²/4π is the resistive diffusion coefficient.

Resistive Diffusion ∂B/∂t = ∇×(v ×B) + D ∇²B The magnetic flux through a given contour S is given by Φ = ∫∫S B. dS. The change in this flux is given by dΦ/dt = ∫∫S ∂B/∂t - ∫Γ E. dΓ, where the second term is due to Faraday’s law. If the electric field is generated due to cross-field fluid motions, then, using Stokes’ theorem ∫Γ E. dΓ = ∫∫S ∇×E. dS = ∫∫S ∇×(v ×B). dS we see that dΦ/dt = ∫∫S [∂B/∂t - ∇×(v ×B)] dS = ∫∫S D ∇²B dS. Thus, if D=0, the field is frozen in to the plasma; the flux through an area stays constant as the area deforms due to fluid motions. If, on the other hand, D ≠ 0, then the flux can change (and as a result the energy in the magnetic field can be released).

Page 3: Resistive Diffusion in Solar Loops, Kinetic Theory, and Vlasov Equation

Resistive Diffusion ∂B/∂t = ∇×(v ×B) + D ∇²B The ratio of the two terms on the RHS: |∇×(v ×B)|/ D |∇²B| ~ vL/D ~ 4πvL/ηc² is known as the magnetic Reynolds number S. For S >> 1, the plasma is essentially diffusion-free, for S << 1 the dynamics are driven by resistive diffusion. For a flare loop, V ~ VA ~ 10⁸ cm s⁻¹, L ~ 10⁹ cm and η ~ 10⁻⁷ T⁻³/² ~ 10⁻¹⁷. This S ~ 10¹⁴, and the plasma should be almost perfectly frozen in. The timescale for energy release should be of order L²/D ~ 4πL²/ηc² (this is of order the timescale for resistive decay of current in an inductor of inductance L/c² and resistance R = ηL/L² = η/L). For solar values, this is 10¹⁵ s ~ 10⁷ years!

Summary to Date • Solar loops are big (they have a high inductance) • Solar loops are good conductors • Solar loops have a low ratio of gas to magnetic pressure β So: The plasma in solar loops is tied to the magnetic field, and the motion of this field determines the motion of the plasma trapped on it.

Also: It is very difficult to release energy from such a high-conductivity, high-inductance system!

The Vlasov Equation Note that we have still prescribed E and B. A proper solution of the plasma equations requires that E and B be obtained self-consistently from the particle densities and currents. The equation that accounts for this is called the Vlasov equation.

Phase-space Distribution Function This is defined as the number of particles per unit volume of space per unit volume of velocity space: At time t, number of particles in elementary volume of space, with velocities in range v → v + dv = f(r,v,t) d³r d³v f(r,v,t) has units cm⁻³ (cm s⁻¹)⁻³

The Boltzmann Equation This equation expresses the fact that the net gain or loss of particles in phase space is due to collisional depletion: Df/Dt ≡ ∂f/∂t + v.∇f + a.∇vf = (∂f/∂t)c The Boltzmann equation takes into account the self-consistent evolution of the E and B fields through the appearance of the acceleration term a.

The Vlasov Equation This is a special case of the Boltzmann equation, with no collisional depletion term: ∂f/∂t + v.∇f + a.∇vf = 0, i.e., ∂f/∂t + v.∇f + (q/m) (E + (v/c) × B).∇vf = 0.

The Electrostatic Vlasov Equation Setting B = 0, we obtain, in one dimension for simplicity, with q = -e (electrons) ∂f/∂t + v ∂f/∂x - (eE/m) ∂f/∂v = 0. Perturb this around a uniform density, equilibrium (E = 0) state fo = ngo: ∂g1/∂t + v ∂g1/∂x - (eE1/m) ∂go/∂v = 0. Also consider Poisson’s equation (∇.E = 4πρ): ∂E1/∂x = 4πρ = - 4πne ∫g1 dv

The Electrostatic Vlasov Equation Now consider modes of the form g ~ exp(i[kx-ωt]) Then the Vlasov equation becomes -iωg1 + ivkg1 – (eE1/m) dgo/dv = 0 (ω – kv)g1 = (ieE1/m) dgo/dv and Poisson’s equation is ikE1 = - 4πne ∫g1 dv Combining, ikE1 = - i(4πne²/m) E1 ∫dgo/dv dv/(ω – kv) Simplifying, and defining the plasma frequency through ωpe² = 4πne²/m, 1 - (ωpe²/k²) ∫dgo/dv dv/(v – ω/k) = 0. This is the dispersion relation for electrostatic plasma waves.

Page 4: Dispersion Relations and Two-Stream Instability

The Electrostatic Vlasov Equation Integrating by parts, we obtain an alternative form: 1 - (ωpe²/ω²) ∫go dv/(1 - kv/ω)² = 0. For a cold plasma, go = δ(v), so that we obtain 1 - (ωpe²/ω²) = 0, i.e., ω = ωpe

The Electrostatic Vlasov Equation For a warm plasma, we expand the denominator to get 1 - (ωpe²/ω²) ∫go dv[1 + 2kv/ω + 3k²v²/ω² + …] = 0 i.e. 1 - (ωpe²/ω²) [1 + 3k²<v>²/ω² + …] = 0, where <v>² = kBT/m is the average thermal speed. This gives the dispersion relation ω² = ωpe² + 3 (kBT/m) k² (cf. ω² = ωpe² + c²k² for EM waves)

Dispersion relations Electrostatic waves in a warm plasma: ω² = ωpe² + 3 (kBTe/m) k² Ion-acoustic waves (includes motion of ions): ω = kcs; cs = [kB(Te + Ti)/mi]¹/² (note electrons effectively provide quasi-neutrality) Upper hybrid waves (includes B): ω² = ωpe² + Ωe²; Ωe = eB/me

Dispersion relations Alfvén waves: ω² = k²VA²/[1 + (VA²/c²)] Magnetoacoustic waves: ω⁴ - ω²k²(cs² + VA²) + cs²VA²k⁴cos²θ = 0 (θ = angle of propagation to magnetic field) etc., etc.

Two-Stream Instability 1 - (ωpe²/ω²) ∫go dv/(1 - kv/ω)² = 0. For two streams, go = ½ [δ(v-U) + δ(v+U)], so that (ωpe²/[ω-kU]²) + (ωpe²/[ω+kU]²) = 1. This is a quadratic in ω²: ω⁴ – 2(ωpe² + k²U²)ω² – 2 ωpe²k²U² + k⁴U⁴ = 0, with solution ω² = (ωpe² + k²U²) ± ωpe (ωpe² + 4k²U²)¹/² There are solutions with ω² negative and so imaginary (exponentially growing) solutions.

Two-Stream Instability Distribution with two maxima (one at zero, one at the velocity of the “beam”) is susceptible to the two-stream instability. This generates a large amplitude of plasma waves and affects the energetics of the particles.

Two-Stream Instability This can also happen due to an “overtaking” instability – fast particles arrive at a location earlier than slower ones and so create a local maximum in f.

Summary High energy solar physics is concerned with the physics of plasma, which is a highly interacting system of particles and waves. “Plasma physics is complicated” (J.C. Brown & D.F. Smith, 1980)