A hybrid fluid-gyrokinetic derivation of the kinetic modifications to the ballooning-interchange eigenmode equation is presented which removes the large and small wave frequency restrictions of Tang and Catto. The derivation employs a long-thin ordering and an eikonal ansatz, but otherwise allows arbitrary asymmetry, beta, pressure anisotropy, and unperturbed parallel and perpendicular electrostatic field.
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Lee et al. (1981) studied this question.