Observational Signatures of Chaotic Scattering In the Earth's Magnetotail

By Brian Lawrence Mavity

Illinois State University Physics Department
Presented at the Argonne Symposium for Undergraduate Research
November, 1998


The earth's magnetosphere is the region of space where the earth1s magnetic field dominates the physics. The magnetosphere consists of a low density gas of charged particles, i.e. a plasma. Due to the solar wind, the magnetosphere is compressed on the sunward, or day, side and elongated on the night side to form a "tail". This region, called the magnetotail, extends for several hundred earth radii. Particles in the plasma sheet at the center of the tail, can become energized and accelerated through the plasma sheet boundary layer (PSBL) between the plasma sheet and the tail lobes, into the auroral region. Particles entering the auroral region are accelerated into the ionosphere and upper atmosphere, colliding with atoms to produce visible light in the polar regions. These dynamic, colorful displays of light in the night sky are called auroras. The whole process is a magetospheric or auroral substorm. During substorms, ground based and satellite communication can be disrupted and the electric power grid can feel strong perturbations.

One way to study magnetotail plasma is to investigate the motion of individual charged particles. Associated with each particle is a quantity known as the magnetic moment, which is nearly constant except near a small region at the equatorial plane of the magnetotail. In this region, a particle1s magnetic moment can undergo large jumps which are associated with the chaotic nature of the particle motion. We define a parameter , where rg is the gyroradius of the particles orbit, and Rc is the radius of curvature of the magnetic field. The k parameter can be used to describe types of orbits. There are three main behavior types which are exhibited. Adiabatic behavior occurs when k >> 1. It is characterized by helical orbits about magnetic field lines and by the conservation of the magnetic moment. When k < 1, the magnetic moment is not conserved and the particle will oscillate about the plasma sheet center. This is known as current sheet behavior. Intermediate behavior is our area of interest and occurs when k ~ 1. In this case, we find that the magnetic moment is scattered according to a 3-branch pattern: particles with large pitch angle are nearly adiabatic, those with small pitch angles have current sheet behavior, while internediate particles scatter chaotically with potentially large decreases in magnetic moment. We are particularly interested in observational evidence for this 3-branch behavior.

Our research involves a numerical simulation of the motion of many particles as they progress through the plasma sheet These particles are then collected by a "virtual satellite" and the ion velocity distribution function is computed. We then search for any signatures of the three-branch behavior in the distribution function graphs, and analyze its source by tracing individual particles. We find some beam-like strucutres that appear to be related to the intermediate branch. Such signatures will be useful in analyzing data from the GEOTAIL spacecraft.


Earth's Magnetosphere

Charged Particles in Magnetic Fields:

  • Lorentz Force: F=q vxB
  • Newton's 2nd Law: F=ma=mr"
  • Simplest Case: Uniform B
    • cyclotron (gyro) radius: equation
    • cyclotron (gyro) frequency: equation
  • equation
  • Helical Orbit: equation
  • Guiding Center Approximation and Adiabatic Invariance [Alfven 1950,Northrup,1963]
    • separate length scales:
      • short: gyro motion
      • long: motion of "guiding center"
      • small parameter:equation
      • graph
        • average over short lengthscale
        • adiabatic invariants
      • conserved on fast (gyro) timescales
      • associated with fast oscillatory motion
      • magnetic moment:equation
        • very useful for inner magnetosphere
        • equation

Kappa Parameter

  • Definition: equation
  • r sub c= field line curvature radius
  • phi= gyro-radius of particle
  • Meaning
    • k>>1: adiabatic motion
    • k<1: current sheet motion
    • k~1: complex motion (what we're interested in)

Example: Truncated Linear Current Sheet

  • Model:equation
  • Field:graph

Intermediate Regime (k~0.7 - 3)

  • Breakdown of adiabatic and current sheet approximations Predicts 3-branch behavior:
  1. Large alpha naught: equation => adiabatic
  2. Small alpha naught: equation=> µ increase
  3. Intermediate : phase dependent µ increase or decrease
    graph graph

Field Reversal Particle Dynamics

  • Field Reversal Orbit types:
    [Chen and Palmadesso,1986; Buchner and Zelenyi, 1986,1989; Chen, 1992]
    1. Transient (Speiser, resonant)
      graphk>=1 graphk<1
    2. Regular (trapped in CS)
    3. Quasi-trapped (chaotic, "cucumber")
      graphk>=1 graphk<1

Research Project


  • Is there an observable effect of this 3-branch behavior?


  • Look for features in modeled ion velocity distribution function.
  • If present: compare to spacecraft observations.


  • Follow 10,000 ions through model current sheet magnetic field, ending at "virtual detector" at spacecraft position.
  • Calculate Distribution function.

Related Graphs

T=10,000 km,
bn=.2, Bz0=4 nT,
Hmax=4.75 keV
Current Sheet

log(µ ratio) kslice=5

f(v) for kappa>.8

CS Crossings f(v) correlation

µ Ratio f(v) correlation


  • Simulation predicts small beams:
    • "Islands" in distribution function are of iterest.
    • They represent beams in the plasma.
    • They are formed from transient (single-crossing) orbits.
    • They correspond to high µ ratio: branch 2.
  • Simulation predicts ridges and valleys for higher bn:
    • Ridges and valleys represent a variation of f in pitch angle => pitch angle scattering.
    • Ridges are single orbit transient: branch 2.
  • Signature of 3-branch behavior
    • Branch 2: Beams
    • Branch 3: between the beams are low µ ratio orbits: multiple-crossing, chaotic orbits.
    • Next step: vary model parameters to see how distribution changes