Table of Waves

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Contents

Alfven

ω < < Ωi
\epsilon \simeq ( \begin{array}{ccc} 1+\gamma & 0 \\ 0 & 1+\gamma \\ \end{array} ), \gamma = \frac{\omega_{pi}^2}{\Omega_i^2}
The two solutions are:
n2 = 1 + γ Ordinary mode -- fast, compressional, E_x\simeq 0,E_z\simeq 0
n_{\parallel}^2=1+\gammaExtrordinary mode -- slow, shear, E_y\simeq 0,E_z\simeq 0 )

Magnetosonic

Upper/Lower Hybrid

For \perp propagation, the dispersion relation can be factored:

n_{\perp}^2=\frac{R L}{S} = \frac{(\omega^2+\omega \Omega_e +\Omega_e \Omega_i -\omega_{pe}^2) (\omega^2-\omega \Omega_e +\Omega_e \Omega_i -\omega_{pe}^2) }{(\omega^2-\omega_{LH}^2)(\omega^2-\omega_{UH}^2)}

The hybrid resonances are given by:

\omega_{LH}^2=\omega_{pe}^2+\Omega_{e}^2
\frac{1}{\omega_{UH}^2}=\frac{1}{\Omega_i^2+\omega_{pi}^2}+\frac{1}{|\Omega_i \Omega_e|}

Ωi < < ω < < Ωe
Electrostatic: k \cdot \epsilon \cdot k =0 \rightarrow k_x^2 S+k_z^2 P=0 \rightarrow k_{\perp}^2(1+\frac{\omega_{pe}^2}{\Omega_e^2})+k_{\parallel}^2(1-\frac{\omega_{pe}^2}{\omega^2})=0


Ion cyclotron

ω < < Ωe
\gamma=\frac{\omega_{pi}^2}{\Omega_i^2}>>1

R\simeq\frac{\gamma \Omega_i}{\Omega_i+\omega}
L\simeq\frac{\gamma \Omega_i}{\Omega_i-\omega}
n_{\perp}^2=\frac{(R-n_{\parallel}^2)(L-n_{\parallel}^2)}{S-n_{\parallel}^2} From this the limiting solutions when n_{\perp}\rightarrow 0, n_{\parallel}\rightarrow 0 can be obtained. Note the Alfven resonance when n_{\parallel}\rightarrow S


Ion Acoustic

-Electrostatic
-P = 0

-Longitudinal (E = Ez)

-v_i<<\frac{\omega}{k}<<v_e

1+\frac{1}{k^2 \lambda_d^2}-\frac{\omega_{pi}^2}{\omega^2}(1+3 \frac{k^2}{\omega^2}\frac{T_i}{M_i})=0


Electromagnetic

Whistler

n^2=1-\frac{\omega_{pe}^2}{\omega(\omega+\Omega_e cos\theta} \simeq \frac{\omega_{pe}^2}{\omega |\Omega_e cos\theta|} for n2 > > 1 and ω < < Ωecosθ

Some properties of whistlers:

  • v_g \propto \sqrt{\omega}\propto k This means that higher frequency whistlers propagate faster. Thus the descending tones heard by zee germans.
  • ψmax = 19.28o Whistlers propagate parallel to the B field
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