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ABSTRACT This paper formulates a velocity moment-based quasi-linear theory that combines the impacts of weakly unstable proton–cyclotron- (or, equivalently, electromagnetic ion cyclotron) and proton-mirror instabilities on the solar wind plasma initially characterized by an excessive perpendicular proton temperature anisotropy. The present formalism is an alternative to the existing model in that the weakly unstable modes are characterized by analytical formalism that involves the assumption of weak growth rate and/or fluid-theoretical dispersion relation, in place of numerical root-finding method based on the transcendental plasma dispersion function. This results in an efficient numerical platform for analyzing the quasi-linear development of the saidmore »Free, publicly-accessible full text available December 8, 2022