(a) A whistler-mode impulse generated by lightning travels along a field-aligned…

PHYSICS · Subjective · Fermi University Physics · Plasma Physics · Waves in plasmas · Laboratory, industrial and space plasmas

Plasma Physics, Ch Waves in a non-uniform plasma: rays, reflection, mode conversion and radiofrequency heating

Problem

(a) A whistler-mode impulse generated by lightning travels along a field-aligned density duct whose travel time to a ground receiver follows

t = \mathcal{D}/\sqrt{f}

, with dispersion constant

\mathcal{D} = 45\,\mathrm{s\,Hz^{1/2}}

for this duct (a result of the cold-plasma dispersion relation for whistlers, quoted here). Using the ray-tracing idea that a wave packet moves along

\partial\omega/\partial\mathbf{k}

, not along

\mathbf{k}

, explain in one or two sentences why the received tone descends in pitch, and compute the delay between the

5\,\mathrm{kHz}

and

15\,\mathrm{kHz}

components. (b) In a separate tokamak experiment, a hydrogen minority (charge

+e

, mass

m_p

) in a deuterium plasma at

B = 2.0\,\mathrm{T}

gyrates with initial perpendicular speed

v_{\perp,1} = 1.0\times10^5\,\mathrm{m\,s^{-1}}

. An RF field at the hydrogen cyclotron frequency kicks the perpendicular speed to

v_{\perp,2} = 1.2\times10^5\,\mathrm{m\,s^{-1}}

on a single pass through the resonance layer. Recall that the magnetic moment

\mu = mv_\perp^2/2B

is an adiabatic invariant only when the field seen by the particle varies slowly compared with its gyration. Find the RF frequency, and compute

\mu

before and after the kick to show it is not conserved.