The axion is a well-motivated hypothetical particle capable of resolving both the strong CP problem and the dark matter mystery, with recent post-inflationary cosmological simulations favoring masses above 40 {}eV. Plasma haloscopes serve as a promising experimental approach to reach theoretically preferred sensitivities in this mass range. ALPHA, hosted at Yale Wright Laboratory, is an international collaboration developing plasma haloscopes to search for QCD dark matter axions. In this letter we present the detailed design and sensitivity projection for the first phase of the ALPHA experiment, which will search the mass range from 10 GHz to 20 GHz (~40 {}eV to 80 {}eV). This search will make use of wire-array plasma resonators to decouple the physical size from the resonant frequency, a limitation typically faced by traditional microwave cavities, allowing broadband sensitivity approaching KSVZ coupling strengths.