Thursday, May 31, 2012

Commemorating Hannes Alfven's Achievements (2)

We continue now looking at more of Hannes Alfven's achievements with this 70th anniversary year of his research letter on Alfven waves. Two quick ones first:

1) Alfven postulated that there must be a galactic magnetic field, otherwise cosmic ray distributions would not be limited to galactic dimensions.  Most thinking at the time rebelled at this, under the conviction that if space was a vacuum it couldn't possibly sustain or carry the electric current needed to generate a magnetic field. Much later, the galactic magnetic field was verified.

2) Alfven warned that the concept of a "frozen in magnetic field" for a plasma, had to be treated very cautiously. In particular, the concept was based on idealized conditions and assumptions not always fulfilled in a real plasma. With his studies of the aurora, he became convinced that the frozen in concept could be very misleading.

Only later, was the concept tied to a quantitative basis via the magnetic Reynolds number (R_m).  If R_M >> 1 then any magnetic diffusion can be ignored and we have the frozen in condition. R_m is expressed as:

R_m = L V(A)/ n

where L is a typical scale length, V(A) is the Alfven velocity and n the magnetic diffusivity.

Thus, using the computed value of R_m for a given plasma, one could calculate if the field was frozen in or not.

3) Another major contribution of Alfven's was the guiding center approximation and adiabatic invariance. Before Alfven's guiding center approximation, the computation of electron orbits in the Earth's dipole magnetic field was an almost superhuman task, requiring enormous time and computing resources.

Alfven's genius was to separate the motion of the trapped particle into a gyration transverse to the local magnetic field and a drift of the center of this gyration, which he called 'the guiding center".

If we consider a charged particle (say of charge q) in a uniform and constant magnetic field (B). the governing equation of motion for charged particles is:

m (dv/dt) = q(v X B)

The motion here is such that v is always perpendicular to the force acting on the particle so:
v ⊥ F, implying circular motion.

Readers can pick up from this point and get an extensive insight into the guiding center approximation and also obtaining one of the adiabatic invariants - from this earlier blog:

Note in particular how the magnetic moment u, of a gyrating particle is what is being called an adiabatic invariant (this is always a property that does not change when the magnetic field changes very slowly on the time scale of the gyration). Since then, two more adiabatic invariants have been found, and have since become an indispensable tool in plasma physics.

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