We investigate the effects of the magnetostatic (B) field topology on the plasma behavior in a 2D collisionless simulation setup that represents an axial-azimuthal cross-section of a Hall thruster. The influence of the B-field topology is assessed in terms of two principal design properties of the field in a typical Hall thruster, i.e., the field’s peak intensity along the axial direction, and the field’s axial distribution. The effects of the field’s intensity is investigated for three propellants – xenon, krypton, and argon. Whereas, the effects of the axial profile of the magnetic field is studied only for the xenon propellant as an example. We primarily aim to understand how the changes in the B-field topology affects the spectra of the resolved instabilities as well as the electrons’ transport characteristics and the contributions of various momentum terms to transport. The numerical observations on the instabilities’ characteristics are compared against the relevant existing theories to determine the extent to which the simulated and the theoretically predicted characteristics are consistent across the studied parameter space. It is, most notably, found that modes related to the ion acoustic instability are dominantly present across the simulation cases. The ion transit time instability additionally develops at the highest B-field intensities as a long-wavelength structure. The main influence of the axial profile of the B field on the plasma discharge is observed to be in terms of the electrons’ transport characteristics. Where possible, the insights from the simulations are discussed with respect to the relevant experimental observations available in the literature.