where Tr = -[??] x P is the transpose of P x [??] and d/d[tau] = [partial derivative]/[partial derivative]t + u x [nabla] denotes the convective derivative, [OMEGA] = [sigma][B.sub.0]/mc is the gyrofrequency
, where [sigma] is the electron charge, m is the mass of particle, c is the speed of light, and [B.sub.0] is the magnitude of the constant ambient field assumed to be oriented in the x-direction and approximately equal to B.
are the electron-ion and electron-neutral collision frequencies, respectively ; [k.sub.0] = [omega]/c, c is a light speed in the free space, [omega] = 2[pi]f, f is an operating frequency, u = [[omega].sup.2.sub.H]/[[omega].sup.2] and v = [[omega].sup.2.sub.p]/[[omega].sup.2] are non-dimensional magneto-ionic parameters, [[omega].sub.H] = e[H.sub.0]/mc and [[omega].sub.p] = 4[pi][Ne.sup.2]/m[[omega].sup.2] are the electron gyrofrequency
and plasma frequency, respectively, N is an electron density, e and m are the charge and mass of an electron, m = [H.sub.0]/[H.sub.0] is the unite vector along the direction of an external magnetic field locating in the YOZ plane.
where [[omega].sub.L] = [[omega].sub.H] cos [[theta].sub.H] is the magnetic angular gyrofrequency corresponding to the longitudinal (parallel) component of the geomagnetic field, [[theta].sub.H] is the angle between the direction of wave propagation and the geomagnetic field, [[omega].sub.H] = [B.sub.0]([q.sub.e]/[m.sub.e]), and [B.sub.0] is the amplitude of magnetic induction field and where the + and - signs refer to the ordinary and extraordinary waves, respectively.
wave, [[omega].sub.p] = [square root of N[q.sup.2.sub.e/[m.sub.e][[epsilon].sub.0]] the plasma frequency, and N the profile of electron density); [Y.sub.L] = Y cos [[theta].sub.H], [Y.sub.T] = Y sin [[theta].sub.H] ([[theta].sub.H] being the angle between the wave vector and the geomagnetic field), and Y = [[omega].sub.H]/[omega] (being [[omega].sub.H] = [B.sub.0]([q.sub.e]/[m.sub.e]) the angular gyrofrequency, and [B.sub.0] the amplitude of magnetic induction field); Z = v/to (v being the collision frequency).
The magnetic gyrofrequency [[omega].sub.H] is a function of the height h above the ground and the geomagnetic colatitude [lambda] :
This extent of the wave-particle resonance is estimated as being within a spatial wave number range [+ or -][k.sub.[parallel]]/2 where [k.sub.[parallel]] = [[omega].sub.B]/[[upsilon].sub.[parallel]] for gyrofrequency
[[omega].sub.B] and particle velocity [[upsilon].sub.[parallel]] along field B.
The electrons spin around the magnetic field lines, emitting radio waves both on the electron's gyrofrequency
and on many of its harmonic frequencies.