The thermally driven motion of a fluid contained in a rotating annulus is investigated by numerical integration of the Navier-Stokes equations as an initial value problem. Four distinct regimes of hydro-dynamical flow can exist in the annulus system. This paper will consider the nature and computational requirements of the axisymmetric state for its own sake and partly as a prelude to a quantitative study of the more complex irregular regime. Calculations were made for two flows whose parameters, with the exception of the rotation rates, are identical, and whose upper surfaces are free. How the axisymmetric state varies with the Rossby and Taylor parameters will be discussed in Part 2. The solutions show that the flow forms a direct circulation with countercurrents on both side walls, and with a strong flow from the base of the hot wall, across the interior, up toward the top of the cold wall. The thermal boundary layers form in small and isolated regions near the top of the cold wall and base of the hot wall. The fluid and container effect most of their heat exchange through these discrete regions. The isotherms slope up toward the cold wall and a large region of constant temperature exists near the fluid surface. The higher rotation rate makes the isotherms ware vertical and, as a consequence, the Nusselt number is inversely proportional to the first power of the rotation rate. The upper three-fourths of the fluid flows in the same zonal direction as the rotation, while the remainder flows in the opposite direction. Although the fluid interior is essentially geostrophic, the nonlinear terms do make a significant contribution to the vorticity balance. The angular momentum has a single sink region; this occurs at the top of the cold inner cylinder, and the fluid ignores the potential maximum source at the (hot) outer cylinder. The contributions of the sidewall boundary layers to the energy transformation oppose each other; this leaves the interior region of the fluid as a significant source of energy. Application of Eady's criterion for baxoclinic instability when applied to the solutions, shows one flow to be stable, the other unstable. This conclusion agrees with observation. Contours of the transient fields show the predominately isothermal evolution of the flow towards a steady state. The close association of the sidewall countercurrents to the sidewall boundary layers appears at all stages of development. Changing the number of grid points used and repeating the calculations demonstrated the accuracy of the solutions.