Turbulence in Stratified Shear Flows: Implications for Interpreting Shear-induced Mixing in the Ocean

E. C. Itsweire Division of Ocean Sciences, National Science Foundation, Washington, D.C.

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J. R. Koseff Department of Civil Engineering, Stanford University, Stanford, California

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D. A. Briggs Department of Mechanical Engineering, Stanford University, Stanford, California

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J. H. Ferziger Department of Mechanical Engineering, Stanford University, Stanford, California

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Abstract

Direct numerical simulations of the time evolution of homogeneous stably stratified turbulent sheer flows have been performed for several Richardson numbers Ri and Reynolds numbers Rλ. The results show excellent agreement with length scale models developed from laboratory experiments to characterize oceanic turbulence. When the Richardson number Ri is less than the stationary value Ris, the turbulence intensity grows at all scales; the growth rate is a function of Ri. The size of the vertical density inversions also increases. When Ri ≥ Ri, the largest turbulent eddies become vertically constrained by buoyancy when the Ellison (turbulence) scale LEand the Ozmidov (buoyancy) scale LO are equal. At this point the mixing is most efficient and the flux Richardson number or mixing efficiency is Rf ≈ 0.20 for the stationary Richardson number Ris = 0.21. The vertical mass flux becomes countergradient when ε ≈ 19vN2, and vertical density overturns are suppressed in few than half of a Brunt-Väisälä period. The results of the simulations have also been recast in terms of the hydrodynamic phase diagram introduced for fossil turbulence models. In this framework, buoyancy control of the energy-containing scales begins when ε ≈ 4DCN2. This value is in good agreement with indirect laboratory observations and field observations. Careful examination et the individual components of the velocity and scalar dissipation tensors reveals that, for fully developed, stably stratified shear flows, these tensors are far from isotropic, implying that the isotropic formulas often used to calculate the dissipation rates ε and χ in the oceanic thermocline could underestimate these rates by factors of 2 to 4. Finally, the validity of the steady-state models used to estimate vertical eddy diffusivities in the thermocline is discussed.

Abstract

Direct numerical simulations of the time evolution of homogeneous stably stratified turbulent sheer flows have been performed for several Richardson numbers Ri and Reynolds numbers Rλ. The results show excellent agreement with length scale models developed from laboratory experiments to characterize oceanic turbulence. When the Richardson number Ri is less than the stationary value Ris, the turbulence intensity grows at all scales; the growth rate is a function of Ri. The size of the vertical density inversions also increases. When Ri ≥ Ri, the largest turbulent eddies become vertically constrained by buoyancy when the Ellison (turbulence) scale LEand the Ozmidov (buoyancy) scale LO are equal. At this point the mixing is most efficient and the flux Richardson number or mixing efficiency is Rf ≈ 0.20 for the stationary Richardson number Ris = 0.21. The vertical mass flux becomes countergradient when ε ≈ 19vN2, and vertical density overturns are suppressed in few than half of a Brunt-Väisälä period. The results of the simulations have also been recast in terms of the hydrodynamic phase diagram introduced for fossil turbulence models. In this framework, buoyancy control of the energy-containing scales begins when ε ≈ 4DCN2. This value is in good agreement with indirect laboratory observations and field observations. Careful examination et the individual components of the velocity and scalar dissipation tensors reveals that, for fully developed, stably stratified shear flows, these tensors are far from isotropic, implying that the isotropic formulas often used to calculate the dissipation rates ε and χ in the oceanic thermocline could underestimate these rates by factors of 2 to 4. Finally, the validity of the steady-state models used to estimate vertical eddy diffusivities in the thermocline is discussed.

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