1. ESS 154/200C Lecture 8 MHD including waves

2. Date Day Topic Instructor Due
ESS 200C Space Plasma Physics ESS 154 Solar Terrestrial Physics M/W/F 10:00 – 11:15 AM Geology Instructors: C.T. Russell (Tel. x-53188; Office: Slichter 6869) R.J. Strangeway (Tel. x-66247; Office: Slichter 6869)
Date Day Topic Instructor Due
1/4 M A Brief History of Solar Terrestrial Physics CTR
1/6 W Upper Atmosphere / Ionosphere CTR
1/8 F The Sun: Core to Chromosphere CTR
1/11 M The Corona, Solar Cycle, Solar Activity
Coronal Mass Ejections, and Flares CTR PS1
1/13 W The Solar Wind and Heliosphere, Part 1 CTR
1/15 F The Solar Wind and Heliosphere, Part 2 CTR
1/20 W Physics of Plasmas RJS PS2
1/22 F MHD including Waves RJS
1/25 M Solar Wind Interactions: Magnetized Planets YM PS3
1/27 W Solar Wind Interactions: Unmagnetized Planets YM
1/29 F Collisionless Shocks CTR
2/1 M Mid-Term PS4
2/3 W Solar Wind Magnetosphere Coupling I CTR
2/5 F Solar Wind Magnetosphere Coupling II;
The Inner Magnetosphere I CTR
2/8 M The Inner Magnetosphere II CTR PS5
2/10 W Planetary Magnetospheres CTR
2/12 F The Auroral Ionosphere RJS
2/17 W Waves in Plasmas 1 RJS PS6
2/19 F Waves in Plasmas 2 RJS
2/26 F Review CTR/RJS PS7
2/29 M Final

3. 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
The number of particles in a phase space volume is
where f is called the distribution function.
The density of particles of species “s” (number per unit volume)
The average velocity (bulk flow velocity)

4. 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 Maxwellian distribution
where

5. 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 3-dimensional proton distribution: if kT = 1 keV, get Eaverage = 1.5 keV, and define vthemal = (2kT/mp)1/2 = 440 km/s
for a 1 keV beam with no thermal spread
kT = 0 and V = 440 km/s

6. 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

7. What makes an ionized gas a plasma?
The electrostatic potential of an isolated charge q:
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 q 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

8. Solution is shielded potential:
The Debye length ( ) is
where n is the electron number density and now e is the electron charge.
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.

9. 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

10. The frequency of this oscillation is the plasma frequency
Because mion>> me

11. Useful formulas:

12. A note on conservation laws
Consider a quantity that can be moved from place to place.
Let be the flux of this quantity – 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 s 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 preceding form means that the quantity whose density is s is conserved.

13. Expansion of conservation law to phase space (Boltzmann equation)
Define as the density of particles in the phase space defined by r and v
By analogy to previous discussion, conservation of particles in the phase space means
provided v and a are independent of r and v, respectively. True for Lorentz force:
We have used “suffix notation,” where a subscript i, j, or k indicates a component of a vector, repeated indices mean summation over all three components (e.g. A·B = AiBi), and ijk is the permutation operator (AB = ijk AjBk).
On integrating over velocity space we get the generalized moment equation:
where (v) is some function of velocity.

14. Magnetohydrodynamics (MHD)
The average properties are governed by the basic conservation laws for mass, momentum and energy in a fluid.
Continuity equation ( = 1)
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 dV is a volume element (e.g. dxdydz) where dA is a surface element bounding the volume.

15. Momentum equation ( = mv)
where is the charge density, is the current density, and the last term is the density of non-electromagnetic forces.
Assumes collisions only change momentum via loss of particles.
The operator is called the convective derivative and gives the total time derivative resulting from intrinsic time changes and spatial motion.
The term means that the fluid transports momentum with it.
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.

16. 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

17. Energy equation ( = 1/2mv2, add species and define T = Te + Ti)
where q 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
q adds three unknowns to our set of equations. It is usually treated by making approximations so it can be handled by the other variables.
If it is assumed that then all the terms in the energy equation have already been defined – the equations are closed
After some algebra, and making use of the continuity and momentum equations:
where Cs is the sound speed, cp and cv are the specific heats, and  = (N + 2)/N = 5/3 for N = 3.
Alternatively:

18. Maxwell’s equations Generalized Ohm’s law doesn’t help because
There are 14 unknowns in this set of equations: E, B, j, u, , P
We have 11 equations [continuity, momentum (3), energy, Maxwell’s equations (6)]
Need one more vector relation
Generalized Ohm’s law
Multiply the momentum equations for each individual species by qs/ms and add
where and s is the electrical conductivity (related to electron-ion collision frequency)

19. Generalized Ohm’s Law
Usually assumed only leading terms on RHS remain:
If collision frequency is small, and we get the frozen-in condition
Sometimes referred to as the ideal Ohm’s law, and MHD that uses this approximation is known as Ideal MHD

20. Frozen in flux Combining Faraday’s law and Ampere’ law with gives
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.
On time scales much shorter than D
The electric field vanishes in the frame moving with the fluid.

21. Frozen-in Theorem – 1 Rate of change of line element: dr' r2'
r1' = r1 + u1dt
r1'
u1dt
r2' = r2 + u2dt
u2dt dr r1 r2
Taylor Expansion:

22. Frozen-in Theorem – 2 Rate of change of magnetic field:
Mass Conservation:
Therefore, if dr and B/ are initially parallel, they remain parallel. I.e., the magnetic field is frozen to the fluid.

23. Frozen-in Theorem – 3 Rate of change of area element:
Change in magnitude:
To first order:
Therefore:
Change in direction of the normal:
By Taylor expansion:
Or more generally:

24. Frozen-in Theorem – 4
With:
Rate of change of magnetic flux through the surface:

25. Magnetic pressure and tension since
a magnetic pressure analogous to the plasma pressure (P)
a “cold” plasma has and a “warm” plasma has
In equilibrium and slow flows (I.e., ignore u in the momentum equation) pressure gradients form currents:

26. cancels the parallel component
The second term in can be written as a sum of two terms
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 anti-parallel to the radius of curvature (RC) of the field line. Note that n is directed outward.

27. 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 k. (In the organ pipe the frequency is f and  = 2f. The wave number k = 2/).

28. The exponent gives the phase of the wave
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

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

30. 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.

31. MHD Waves – Linearized Harmonic Perturbations
Assume uniform B0, perfect conductivity, with equilibrium pressure P0 and mass density 0
Variables with subscript “0” are zero-order and do not change
Un-subscripted variables are first-order and do change
First order perturbations are assumed to vary as
E is the wave electric field amplitude
Hence:

32. Geometry:
Assume B0 defines the z-axis of a right-handed triad (also defines parallel direction)
The component of the wave vector perpendicular to B0 defines the x-direction
– Linearized MHD mass, momentum, and energy equations:
       terms are second order and therefore ignored, to first order

33. From mass and energy:
– Substituting in the momentum equation:
– Taking parallel component:
– For parallel propagation, , we again get sound waves:

34. For other wave modes need to evaluate
From Ampere’s law, without displacement current
From Faraday’s law and the frozen-in condition:
Hence
Defining the Alfvén speed as
Substituting into the momentum equation:

35. Parallel component (already discussed):
Component along wave vector:
Curl:
Taking the parallel component of this equation, since
where is the unit vector along the ambient magnetic field
– Combining 2nd and 3rd equation:

36. Alfvén (shear-mode):
Compressional modes:
Both equations must be satisfied
For the Alfvén mode – no compression It can also be shown that the magnetic field perturbation is transverse to the ambient magnetic field – the magnetic field twists or shears:
For the compressional modes – no parallel vorticity For the mode with (the fast mode) and are in phase, i.e., magnetic pressure and thermal pressure add. For the slow mode the pressures are in anti-phase

37. Alfvén (shear-mode) dispersion relation:
Compressional modes dispersion relation:
Compressional mode parallel propagation:
Compressional mode perpendicular propagation (fast mode only):

38. Example, VA=2CS VA F S I Phase Velocities
Arbitrary angle between k and B0
Example, VA=2CS VA F S I
Phase Velocities