Local WKB dispersion relation for the Vlasov–Maxwell equations

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Abstract
A formalism for analyzing systems of integral equations, based on the Wentzel–Kramers–Brillouin (WKB) approximation, is applied to the Vlasov–Maxwell integral equations in an arbitrary-β, spatially inhomogenous plasma model. It is shown that when treating frequencies comparable with and larger than the cyclotron frequency, relevant new terms must be accounted for to treat waves that depend upon local spatial gradients. For a specific model, the response for very short wavelength and high frequency is shown to reduce to the straight-line orbit approximation when the WKB rules are correctly followed.

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