The solar tachocline is a differentially rotating shear layer that appears to be largely located in a stably stratified region of the solar interior, immediately below the convective envelope. Turbulence could be generated in this layer by overshooting convection and by shear instabilities. In order to study the nature of such turbulence, we introduce a three-dimensional nonlinear model of stably stratified fluid motions in a thin rotating spherical shell. Numerical solutions of the freely evolving system are presented for a variety of random initial conditions and parameter values. The velocity field is decomposed into a Rossby-wave (r-mode) component and a gravity-wave (g-mode) component, and their evolution and interaction is investigated. The two components are closely coupled when the rotation is rapid, with oscillatory exchanges of energy and wave modes that propagate toward the equator. Upscale kinetic energy transfer is produced by nonlinear interactions among r-modes and also among g-modes when the stratification is strong.