# Presentation: 803631

## 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)
-  = 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
**v**D = c (**E**  **B**)/B2

## E  B drift

## Drifts

**Exercise:**
- What if E > B?

## Drifts

**v**D = c (**E**  **B**)/B2
**E** is “equivalent electric field”
- Examples:
**E** is actual electric field: **v**D = c (**E**  **B**)/ B2 (independent of sign of q)
- Pressure gradient: q**E** = -p: **v**D = -cp  **B**/qB2 (dependent on sign of q)
- Gravitational field: m**g** = q**E**: **v**D = (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?*

## 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**:
- md2**r**i/dt2 = qi (**E** + [d**r**i/dt]  **B**)
- Then use the resulting **r**i and d**r**i/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** = (4/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 = **(1/4)** (********B) **** B**

## Complete set of MHD Equations

*Continuity:* /t + ****.(**v**) = 0
*Momentum:*  d**v**/dt = -****p + (1/4) (******B**)  **B** - **g**
*Energy: ??* (can use polytrope: d(p/γ)/dt = 0)
*Induction:* **B**/t = ****(**v**  **B**)
- These are 4 equations for the 4 unknowns
- (, p, **v**, **B)**

## Force – Free Fields

- Equation of motion is
-  d**v**/dt = - ****p + (1/4) (******B**)**B**
- Define the *plasma **β* = ratio of terms on RHS = p/(B2/8)
- 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:
**B** = 0 (trivial)
******B** = 0 (current-free – “potential” field)
- Linear case: (******B**) = α**B**
- 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
**E**local = **E** + (**v**/c)  **B** = ηj = (ηc/4) **** **B**
- Combined, these give:
- **B**/t = ****(**v** **B**) - (ηc2/4) ****(******B**),
- i.e.
- **B**/t = ****(**v** **B**) + D ****2**B**,
- where D = ηc2/4 is the *resistive diffusion coefficient*.

## Resistive Diffusion

- **B**/t = ****(**v** **B**) + D ****2**B**
- The magnetic flux through a given contour S is given by
-  = S **B**. d**S**.
- The change in this flux is given by
- d/dt = S **B**/t  -  **E. **d******,**
- 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
-  **E**. d****** = **S ******E**. d**S** = S ****(**v** **B**). d**S**
- we see that
- d/dt = S [**B**/t - ****(**v** **B**)] d**S **= S D ****2**B** d**S.**
- 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).

- d**Γ**

**E**

**B**

- d**S**

**Φ**

## Resistive Diffusion

- **B**/t = ****(**v** **B**) + D ****2**B**
- The ratio of the two terms on the RHS:
****(**v** **B**)/ D ****2**B******** ~ **vL/D ~ 4vL/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 ~ 4L2/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** + d**v** = f(**r**,**v**,t) d3**r** d3**v**
- 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** = 4):
- E1/x = 4ρ = - 4ne 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 = - 4ne g1 dv
- Combining,
- ikE1= - i(4ne2/m) E1 dgo/dv dv/(ω – kv)
- Simplifying, and defining the *plasma frequency* through ωpe2 = 4ne2/m,
- 1- (ωpe2/k2) dgo/dv dv/(v – ω/k) = 0.
- This is the *dispersion relation* for electrostatic plasma waves.

## 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 = [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.

- U

- -U

- g

- v

## 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)