Most geophysical flows encompass turbulence and internal and/or Rossby waves. We demonstrate that these two different classes of waves cause remarkably similar anomalies in the turbulent transport. While all scales in both types of flows contribute to the momentum diffusion, the vertical (diapycnal) scalar diffusion in stratified flows and lateral diffusion in β-plane turbulence can be carried out only by turbulent eddies whose size is smaller than the thresholds of turbulence anisotropization. Beyond these thresholds, both flows become dominated by waves that provide no contribution to the scalar diffusion. Stably stratified flows exhibit enhanced isopycnal diffusion of both momentum and scalar. These results shed new light on the Osborn mixing model, diapycnal and isopycnal viscosity and diffusivity, absence of the critical gradient Richardson number, and large scale meridional transport.