On the Nature of Turbulence in a Stratified Fluid. Part I: The Energetics of Mixing

The definition of the flux Richardson number Rf is generalized to be the ratio of the turbulent buoyancy flux b to the net turbulent mechanical energy m available from all sources. For mechanically energized turbulence where turbulence kinetic energy is used to sustain an upward buoyancy flux (b > 0), it is shown the magnitude of Rf is quantitatively determined by the location of the event in the FrT,–ReT where FrT and –ReT are the local instantaneous overturn Froude number and Reynolds number. In this parameter space the value of Rf varies between 0 and 0.20 for a fluid with Prandtl number greater than one, and between 0 and 0.15 for a fluid with a Prandtl number less than one. For turbulence sustained by a negative buoyancy flux (b < 0), such as penetrative convection in a cooling surface layer, it is shown that the flux Richardson number Rf is a fraction of depth below the surface Rf−1 varies between 0.55 at the surface and –∞ towards the base of the surface layer where the buoyancy flux vanishes. The result may again be interpreted in terms of location in the FrT–ReT diagram. Finally, it is shown that once the value of Rf is known the vertical buoyancy flux may he evaluated directly without recourse to a turbulence model.