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

Abstract

The definition of the flux Richardson number R_f 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 R_f is quantitatively determined by the location of the event in the Fr_T,–Re_T where Fr_T and –Re_T are the local instantaneous overturn Froude number and Reynolds number. In this parameter space the value of R_f 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 R_f is a fraction of depth below the surface R_f⁻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 Fr_T–Re_T diagram. Finally, it is shown that once the value of R_f is known the vertical buoyancy flux may he evaluated directly without recourse to a turbulence model.