Single Particle Motion in a Magnetized Plasma

Transcription

1 Single Particle Motion in a Magnetized Plasma Aurora observed from the Space Shuttle Bounce Motion At Earth, pitch angles are defined by the velocity direction of particles at the magnetic equator, therefore: Particles with 90 pitch angle (i.e. all energy is in V ) are trapped at the equator. Particles with 0 or 180 pitch angle are moving straight along the field line. Particles with too much parallel speed will mirror or bounce at very low altitudes and thus will not remain trapped since they will collide with atmosphere and be lost from the system. Hence there is a minimum pitch angle for trapped particles, defined as the loss cone -- it is this loss or precipitation of particles into the atmosphere that creates the aurora! EAS 4360/6360 3:14

2 Single Particle Motion in a Magnetized Plasma Bounce Motion A few examples of magnetically trapped plasmas: After space storms, plasma particles can be accelerated to extremely high energies (millions of ev!), and are then trapped in the Earth s magnetic field in intense zones of radiation: Van Allen Belts The turbulent magnetic field of the sun forms loops of intensified B-field strength, trapping the plasma in the Sun s atmosphere and forming prominences. Solar Prominences (from NASA SOHO) EAS 4360/6360 3:15

3 Single Particle Motion in a Magnetized Plasma Bounce Motion A few examples of magnetically trapped plasmas: Courtesy of the Swedish Solar Observatory Solar flare observed by TRACE EAS 4360/6360 3:16

4 Homework/Reading/Last thoughts Reading Assignment: Finish though the end of Chapter 2 in Baumjohann Treumann Note in the homework, the questions as for plasma frequency, f pe, which is in Hz. ω pe is in radians/s and needs to be converted to Hz. (f = ω/2π) Upcoming -- Drift motion, finish up plasma motion and onto Solar activity. EAS 4360/6360 3:17

5 The Plasma State What is a plasma? Basic plasma properties: Qualitative & Quantitative Examples of plasmas Single particle motion in a magnetized plasma Gyromotion Bounce motion Drift motion Generating electric currents in plasmas Fluid approximation for plasmas EAS 4360/6360 4:1

6 Drift Motion Guiding Center -- The center of the orbit of a charged particle defined by the gyroradius (r g ) proton electron B Guiding Center EAS 4360/6360 4:2

7 Drift Motion Qualitative Assume a force on the charge perpendicular to the B-field proton B F * Force is charge independent, i.e. gravity EAS 4360/6360 4:3

8 Drift Motion Qualitative Assume a force on the charge perpendicular to the B-field proton V F F B * Force is charge independent, i.e. gravity EAS 4360/6360 4:4

9 Drift Motion Qualitative Assume a force on the charge perpendicular to the B-field electron B F * Force is charge independent, i.e. gravity EAS 4360/6360 4:5

10 Drift Motion Qualitative Assume a force on the charge perpendicular to the B-field V F electron F B * Force is charge independent, i.e. gravity EAS 4360/6360 4:6

11 Drift Motion Qualitative Assume a force on the charge perpendicular to the B-field V F electron proton V F F B * Force is charge independent, i.e. gravity EAS 4360/6360 4:7

12 Drift Motion Qualitative Oppositely drifting ions and electrons establishes currents! V F electron proton V F j ( ), since n e n i ( ) j = e n i v i n e v e j = en e v i v e EAS 4360/6360 4:8

13 Drift Motion General Force Drifts: Forces perpendicular to the magnetic field will cause charged particles to drift in a direction perpendicular to both the force and the B-field. The drift motion is superimposed on the gyromotion, which is why describing it in terms of the guiding center is a useful simplification. Force on a Gyrating Particle v F = 1 ω g $ F m B ' & ) = 1 % B( q F B B 2 V F EAS 4360/6360 4:9

14 Drift Motion: E x B drift What about forces that are not charge independent? F = q E From our generic equation, the drift velocity would be in the same direction for both + and - charges. Therefore, no currents are generated by this drift motion. Force on a Gyrating Particle v F = 1 ω g $ F m B ' & ) = 1 % B( q F B B 2 V F EAS 4360/6360 4:10

15 Drift Motion: E x B drift For and Electric Field perpendicular to B: F = q E E Ion v F = 1 q v E = 1 q F B B 2 qe B B 2 = E B B 2 B Electron Modified from B. & T. EAS 4360/6360 4:11 V E For a fun interactive demo, check out:

16 Whiteboard Notes

17 Drift Motion: B drift What happens when a charged particle experience an increasing magnetic field during its orbit? r g = mv qb EAS 4360/6360 4:12

18 Drift Motion: B drift What happens when a charged particle experience an increasing magnetic field during its orbit? r g = mv qb v F = 1 q v = 1 q F B B 2, F = µ B µ B B B 2 = 1 2 mv 2 B B qb 3 EAS 4360/6360 4:13

19 Drift Motion: Magnetic curvature drift What happens when a charged particle moves along a curved field? B 2 R F R = mv c 2 R c v F = 1 q v R = 1 q F B B 2 mv 2 R c B R c 2 B 2 Total magnetic drift motion: v B = v R + v = (v v 2 ) B B ω g B 2 F R EAS 4360/6360 4:14

20 Drift Motion: Summary There are several other drifts that occur from perpendicular forces such as gravity, and slowly time varying E-fields EAS 4360/6360 4:15

21 Drift Motion: Creating Currents For drift motion that acts differently for + and - charged particles, associated currents are generated by the differential flux of oppositely charged particles, i.e. For E x B drift, the + and - charged particles drift in the same direction, so no associated currents. EAS 4360/6360 4:16

22 Drift Motion: Creating Currents Example: Earth s Magnetosphere EAS 4360/6360 4:17

23 Fluid Approximation for Plasmas Describing the plasma population as a whole: Single Particle Motion -- Individual particle dynamics calculated for all particles in the system, computationally $$$. Magnetohydrodynamics (MHD) -- Single conducting fluid, only bulk (or average) parameters are tracked, assumes plasma maintains local equilibria, which works for low frequency waves in highly conductive fluids in magnetic fields. Multi-fluid -- Separates electrons and ion species into distinct but simultaneously tracked fluids, coupled through electric and magnetic fields. Similar to MHD, but can track differential mass effects and higher frequency waves. Kinetic Theory -- Statistical approach, tracks the evolution of particle distribution functions. EAS 4360/6360 4:18

24 Things to Review Plasma Criterion Debye Length/Sphere, Plasma Parameter, Plasma Frequency Single Particle Motion Lorentz Force Law, Gyromtion, Bounce Motion, Drift Motion, Adiabatic Invariant, Pitch Angle Net Plasma Motion Currents, Guiding Center, Approaches for describing plasma population as a whole: MHD, Multi-fluid, Kinetic General Maxwell s Equations, Div, Grad, Curl, Cross products EAS 4360/6360 4:19