It is shown that in the Earth's core, where the geodynamo is at work (and is supplied with energy by the prevailing unstable density stratification), a buoyancy instability of a local character exists which is highly supercritical. This instability results in fully developed turbulence dominated by small scale vortices. The influence of the Earth's rotation and of the magnetic field produced by the geodynamo makes this small scale turbulence highly anisotropic. A qualitative picture of this local anisotropic turbulence is devised and the main parameters characterizing it are estimated. Expressions for the turbulent diffusivity are developed and discussed.