Scaling analysis and simulation of strongly stratified turbulent flows

Abstract

Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re ≫ 1 and horizontal Froude number F_h ≪ 1 are presented. The results are interpreted on the basis of a scaling analysis of the governing equations. The analysis suggests that there are two different strongly stratified regimes according to the parameter $\mathcal{R} \,{=}\, \hbox{\it Re} F^2_h$ . When $\mathcal{R} \,{\gg}\, 1$ , viscous forces are unimportant and l_v scales as l_v ∼ U/N (U is a characteristic horizontal velocity and N is the Brunt–Väisälä frequency) so that the dynamics of the flow is inherently three-dimensional but strongly anisotropic. When $\mathcal{R} \,{\ll}\, 1$ , vertical viscous shearing is important so that $l_v \,{\sim}\, l_h/\hbox{\it Re}^{1/2}$ (l_h is a characteristic horizontal length scale). The parameter $\cal R$ is further shown to be related to the buoyancy Reynolds number and proportional to (l_O/η)^4/3, where l_O is the Ozmidov length scale and η the Kolmogorov length scale. This implies that there are simultaneously two distinct ranges in strongly stratified turbulence when $\mathcal{R} \,{\gg}\, 1$ : the scales larger than l_O are strongly influenced by the stratification while those between l_O and η are weakly affected by stratification. The direct numerical simulations with forced large-scale horizontal two-dimensional motions and uniform stratification cover a wide Re and F_h range and support the main parameter controlling strongly stratified turbulence being $\cal R$ . The numerical results are in good agreement with the scaling laws for the vertical length scale. Thin horizontal layers are observed independently of the value of $\cal R$ but they tend to be smooth for $\cal R$ < 1, while for $\cal R$ > 1 small-scale three-dimensional turbulent disturbances are increasingly superimposed. The dissipation of kinetic energy is mostly due to vertical shearing for $\cal R$ < 1 but tends to isotropy as $\cal R$ increases above unity. When $\mathcal{R}$ < 1, the horizontal and vertical energy spectra are very steep while, when $\cal R$ > 1, the horizontal spectra of kinetic and potential energy exhibit an approximate k^−5/3_h-power-law range and a clear forward energy cascade is observed.