A closed-form framework for predicting complex dispersion relations and instability growth rates in magnetized plasmas

We develop an explicit closed-form formulation for fluid plasma instabilities in magnetic-field geometries for near-bi-Maxwellian plasmas. Starting from the continuity, momentum, and anisotropic pressure equations, we derive a generalized closed-form expression for the imaginary part of the frequency, expressed in terms of spatial gradients of density, flow velocity, anisotropic pressures, and magnetic fields. The results demonstrate how electromagnetic wave behavior depends explicitly on geometric effects and pressure anisotropy arising from their intrinsic coupling, clarifying instability criteria that were not captured in previous fluid models. The analysis further reveals that non-plane wave effects can independently modify wave dynamics, even in the absence of spatial inhomogeneities. Taken together, these findings provide an analytic generalization that incorporates effects overlooked in conventional fluid formulations. Comparison with numerical solutions confirms the predictive capability of the analytic framework, demonstrating its ability to avoid the iterative and computational overhead required in traditional approaches. To demonstrate practical utility, we present complex dispersion relations and representative instability growth rates and criteria for plasma waves in nonuniform conditions, including circularly polarized, ordinary, extraordinary, whistler, and Alfvén modes, as well as electromagnetic modes in unmagnetized plasmas. The resulting formalism provides a precise and robust analytic framework for predicting plasma wave dispersion and instability properties, offering both direct physical interpretation and practical evaluation capability across fusion plasmas, space environments, and laboratory devices.