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Two-stream instability:

  • Version 1 (cf. {cite}Crouseilles08):

$$ f_0 = (1+\varepsilon \cos(k_xx))0.5 /\sqrt{2\pi} (\exp(-0.5 (v-v_0)^2)+ \exp(-0.5(v+v_0)^2)) $$

  • Version 2 (cf. {cite}Filbet03):

$$ f_0 = \frac{2}{7\sqrt{2\pi}}\left( 1+ 5v^2 \right) \left( 1+ \alpha \left( (\cos(2kx)+\cos(3kx) \right)/1.2 + \cos(kx) \right) \mathrm{e}^{-v^2/2} $$