A series of experiments were performed in a rotating cylindrical tank over a wide range of rotation rates in which convective turbulence was generated by a bottom-mounted heated plate in both homogeneous and stratified fluids. Measurements were made of the turbulent velocities in all three axes over the full depth of the chamber, and of the temperatures at the mid-depth near the centre of the tank. For even small rotation rates, the measurements showed that the turbulent velocities were weakly affected by rotation at all depths, but as the rotation rate increased, the deviation from the non-rotational scaling slowly and progressively increased until eventually the turbulent velocities were fully rotationally controlled. The results indicated that there was no sudden transition of the turbulent field from the non-rotational state (a function only of the surface buoyancy flux B and the depth z) to the rotational state (where the strength of the turbulent field is a function of only B and the Coriolis parameter f). Rather the transition was a smooth asymptotic one from one state to the other. Nevertheless, it was possible to parametrize this transition by a single value of the turbulent or small scale Rossby number, defined by Ro=(B/f(3)z(2))(1/3). Our measurements suggested a critical value of Ro(c) approximate to 0.1, below which the turbulence was fully rotationally controlled and which was equivalent to a critical depth z(c)=(35+/-15)(B/f(3))(1/2). Using typical oceanic values for B and f, the oceanic turbulence driven by surface cooling events becomes rotationally controlled only for depths greater than about IO km, a depth which is greater than that of the bulk of the world's oceans. Thus, convective turbulence actively being generated by cooling of the ocean surface is best described by non-rotating turbulent velocity and length scales and is a function only of the surface buoyancy flux and the depth.