Lagrangian Vertical Spreading and Its Relation to Diapycnal Diffusivity

Abstract The present study provides a theoretical linkage between the Lagrangian dispersion diffusivity and the diapycnal diffusivity in the context of vertical mixing, although previous studies have demonstrated their equivalence under the assumptions of stationary, homogeneous, and stratified turbulence. This is achieved in a new coordinate in which the fluid density is adiabatically sorted in the vertical direction. In the density-sorted coordinate, 1) the vertical motion of Lagrangian particles is solely subjected to irreversible diffusion process; 2) relations between Lagrangian dispersion diffusivity, diapycnal diffusivity, and the generalized Osborn diffusivity are exact; and 3) a generalization of the classical Munk balance between vertical advection and diffusion is also illustrated in an exact sense. Since the adiabatic sorting of the fluid does not require the turbulence to be statistically stationary, homogeneous, and stably stratified, the present solution eliminated these requirements and is thus more general than previous studies. Upon this, a new Lagrangian diagnostic is proposed to quantify the local, instantaneous, and irreversible mixing. Applications are demonstrated in a turbulent scenario of internal wave breaking induced by current–topography interaction, in which the turbulence is intermittent, nonstationary, and inhomogeneous.