The three-dimensional quasi-geostrophic system comprises equations crucial for simulating atmospheric and oceanic circulation. Within this framework, we focus on a specific class of 3D quasi-geostrophic systems incorporating viscosity, studied across various spatial domains. As typical in fluid mechanics, ensuring the well-posedness of these models is both physically pertinent and mathematically intriguing. This study delves into exploring the physical characteristics of the quasi-geostrophic system, employing contemporary mathematical tools such as regularity theory, asymptotic analysis, and spectrum theory. This is a thesis defense. Zoom Link: https://utexas.zoom.us/j/96484267098