Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T06:20:50.714Z Has data issue: false hasContentIssue false
  • English
  • Français

The tidally induced bottom boundary layer in a rotating frame: similarity of turbulence

Published online by Cambridge University Press:  25 November 2008

KEI SAKAMOTO  and
KAZUNORI AKITOMO
KEI SAKAMOTO
Affiliation:
Oceanographic Research Department, Meteorological Research Institute, Tsukuba, Japan Center for Climate System Research, University of Tokyo, Kashiwa, Japan
KAZUNORI AKITOMO
Affiliation:
Department of Geophysics, Graduate School of Science, Kyoto University, Kyoto, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

To investigate turbulent properties of the tidally induced bottom boundary layer (TBBL) in a rotating frame, we performed three-dimensional numerical experiments under unstratified conditions, varying the temporal Rossby number Rot = |σ*/f*|, where σ* and f* are the tidal frequency and the Coriolis parameter, respectively. The vertical profiles of the time-averaged currents and stresses showed good similarity and coincided well with the turbulent Ekman layer, when they were normalized by the modified ‘outer’ scales, the frictional velocity u*τ, T* = 1/|f* + σ*| and δ* = u*τ/|f* + σ*| for the velocity, time and length scales (σ* is positive when the tidal ellipse rotates anticlockwise). This means that the similarity in turbulent statistics is universally applicable to the TBBL in the world's ocean except near the equator. Although strong inertial waves contaminated the perturbation field when Rot ~ 1 and masked the similarity, the apparent diffusivity κ*ap estimated by tracer experiments again showed similarity, since the inertial waves did not affect the mixing process in the present experiments. Thus, κ*ap can be represented in terms of the three external parameters: the latitude (f*), the tidal frequency (σ*) and the tidal amplitude (u*τ). The obtained scaling of u*τ δ* = u*τ2/|f*+σ*| for κ*ap suggests that effective mixing may occur when Rot ~ 1, i.e. near the critical latitude.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 615 , 25 November 2008 , pp. 1 - 25
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aelbrecht, D., D'Hieres, G. C. & Renouard, D. 1999 Experimental study of the Ekman layer instability in steady or oscillating flows. Continental Shelf Res. 19, 1851–1867.CrossRefGoogle Scholar
Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991 a An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 1. Experiments. J. Fluid Mech. 225, 395–422.CrossRefGoogle Scholar
Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991 b An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 2. Numerical simulations. J. Fluid Mech. 225, 423–444.CrossRefGoogle Scholar
Akitomo, K. 1999 Open-ocean deep convection due to thermobaricity 2. Numerical experiments. J. Geophys. Res. 104, 5235–5249.CrossRefGoogle Scholar
Basu, S., Porté-Agel, F., Foufoula-Georgiou, E., Vinuesa, J.-F. & Pahlow, M. 2006 Revisiting the local scaling hypothesis in stably stratified atmospheric boundary-layer turbulence: An integration of field and laboratory measurements with large-eddy simulations. Boundary-Layer Met. 119, 473–500.CrossRefGoogle Scholar
Coleman, G. N. 1999 Similarity statistics from a direct numerical simulation of the neutrally stratified planetary boundary layer. J. Atmos. Sci. 56, 891–900.2.0.CO;2>CrossRefGoogle Scholar
Coleman, G. N., Ferziger, J. H. & Spalart, P. R. 1990 A numerical study of the turbulent Ekman layer. J. Fluid Mech. 213, 313–348.CrossRefGoogle Scholar
Coleman, G. N., Ferziger, J. H. & Spalart, P. R. 1992 Direct simulation of the stably stratified turbulent Ekman layer. J. Fluid Mech. 244, 677–712.CrossRefGoogle Scholar
Coles, D. 1956 The law of the wake in the turbulent boundary layer. J. Fluid Mech. 1, 191–226.CrossRefGoogle Scholar
Costamagna, P., Vittori, G. & Blondeaux, P. 2003 Coherent structures in oscillatory boundary layers. J. Fluid Mech. 474, 1–33.CrossRefGoogle Scholar
Craig, P. D. 1989 A model of diurnally forced vertical current structure near 30° latitude. Continental Shelf Res. 9, 965–980.CrossRefGoogle Scholar
Csanady, G. T. 1967 On the “resistance law” of a turbulent Ekman layer. J. Atmos. Sci. 24, 467–471.2.0.CO;2>CrossRefGoogle Scholar
Fahrbach, E., Harms, S., Rohardt, G., Schröder, M. & Woodgate, R. A. 2001 Flow of bottom water in the northwestern Weddell Sea. J. Geophys. Res. 106, 2761–2778.CrossRefGoogle Scholar
Fahrbach, E., Rohardt, G., Scheele, N., Schröder, M., Strass, V. & Wisotzki, A. 1995 Formation and discharge of deep and bottom water in the northwestern Weddell Sea. J. Mar. Res. 53, 515–538.CrossRefGoogle Scholar
Faller, A. J. & Kaylor, R. E. 1966 A numerical study of the instability of the laminar Ekman boundary layer. J. Atmos. Sci. 23, 466–480.2.0.CO;2>CrossRefGoogle Scholar
Foldvik, A., Gammelsrød, T., Østerhus, S., Fahrbach, E., Rohardt, G., Schröder, M., Nicholls, K. W., Padman, L. & Woodgate, R. A. 2004 Ice shelf water overflow and bottom water formation in the southern Weddell Sea. J. Geophys. Res. 109, C02015.Google Scholar
Foldvik, A., Middleton, J. H., Foster, T. D. 1990 The tides of the southern Weddell Sea. Deep-Sea Res. 37, 1345–1362.CrossRefGoogle Scholar
Foster, T. D. & Carmack, E. C. 1976 Frontal zone mixing and Antarctic Bottom Water formation in the southern Weddell Sea. Deep-Sea Res. 23, 301–317.Google Scholar
Foster, T. D., Foldvik, A. & Middleton, J. H. 1987 Mixing and bottom water formation in the shelf break region of the southern Weddell Sea. Deep-Sea Res. 34, 1771–1794.CrossRefGoogle Scholar
Friedlander, S. 1980 An Introduction to the Mathematical Theory of Geophysical Fluid Dynamics. Elsevier.Google Scholar
Furevik, T. & Foldvik, A. 1996 Stability at M(2) critical latitude in the Barents Sea. J. Geophys. Res. 101, 8823–8837.CrossRefGoogle Scholar
Gordon, A. L. 1998 Western Weddell Sea thermohaline stratification. Ocean, ice and atmosphere: Interactions at the Antarctic continental margin. Antarct. Res. Ser. 75, 215–240.Google Scholar
Grant, W. D. & Madsen, O. S. 1986 The continental-shelf bottom boundary layer. Annu. Rev. Fluid Mech. 18, 265–305.CrossRefGoogle Scholar
Iida, O., Kasagi, N. & Nagano, Y. 2002 Direct numerical simulation of turbulent channel flow under stable density stratification. Intl J. Heat Mass Transfer 45, 1693–1703.CrossRefGoogle Scholar
Jacobs, S. S. 2004 Bottom water production and its links with the thermohaline circulation. Antarct. Sci. 16, 427–437.CrossRefGoogle Scholar
Kulikov, E. A., Rabinovich, A. B. 2004 Barotropic and baroclinic tidal currents on the Mackenzie shelf break in the southeastern Beaufort Sea. J. Geophys. Res. 109, C05020.Google Scholar
LeBlond, P. H. & Mysak, L. A. 1978 Waves in the Ocean. Elsevier.Google Scholar
Makinson, K. 2002 Modeling tidal current profiles and vertical mixing beneath Filchner–Ronne Ice Shelf, Antarctica. J. Phys. Oceanogr. 32, 202–215.2.0.CO;2>CrossRefGoogle Scholar
Makinson, K., Schröder, M. & Østerhus, S. 2006 Effect of critical latitude and seasonal stratification on tidal current profiles along Ronne Ice Front, Antarctica. J. Geophys. Res. 111, C03022.Google Scholar
Matsuno, T. 1996 Numerical integrations of the primitive equations by a simulated backward difference method. J. Met. Soc. Japan 44, 76–84.CrossRefGoogle Scholar
Mellor, G. L. & Yamada, T. 1982 Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. Space Phys. 20, 851–875.CrossRefGoogle Scholar
Muench, R. D. & Gordon, A. L. 1995 Circulation and transport of water along the western Weddell Sea margin. J. Geophys. Res. 100, 18503–18515.CrossRefGoogle Scholar
Nakano, H. & Suginohara, N. 2002 Effects of Bottom Boundary Layer parameterization on reproducing deep and bottom waters in a world ocean model J. Phys. Oceanogr. 32, 1209–1227.2.0.CO;2>CrossRefGoogle Scholar
Nieuwstadt, F. T. M. 1984 The turbulent structure of the stable, nocturnal boundary layer. J. Atmos. Sci. 41, 2202–2216.2.0.CO;2>CrossRefGoogle Scholar
Nøst, E. 1994 Calculating tidal current profiles from vertically integrated models near the critical latitude in the Barents Sea. J. Geophys. Res. 99, 7885–7901.CrossRefGoogle Scholar
Orsi, A. H., Johnson, G. C. & Bullister, J. L. 1999 Circulation, mixing and production of Antarctic Bottom Water. Progr. Oceanogr. 43, 55–109.CrossRefGoogle Scholar
Pereira, F. P., Beckmann, A. & Hellmer, H. H. 2002 Tidal mixing in the southern Weddell Sea: Results from a three-dimensional model. J. Phys. Oceanogr. 32, 2151–2170.2.0.CO;2>CrossRefGoogle Scholar
Prandle, D. 1982 The vertical structure of tidal currents. Geophys. Astrophys. Fluid Dyn. 22, 29–49.CrossRefGoogle Scholar
Robertson, R. 2001 a Internal tides and baroclinicity in the southern Weddell Sea 1. Model description. J. Geophys. Res. 106, 27001–27016.CrossRefGoogle Scholar
Robertson, R. 2001 b Internal tides and baroclinicity in the southern Weddell Sea 2. Effects of the critical latitude and stratification. J. Geophys. Res. 106, 27017–27034.CrossRefGoogle Scholar
Robertson, R., Padman, L. & Egbert, G. D. 1998 Tides in the Weddell Sea. Antarct. Res. Ser. 75, 341–369.Google Scholar
Sakamoto, K. & Akitomo, K. 2006 Instabilities of the tidally induced bottom boundary layer in the rotating frame and their mixing effect. Dyn. Atmos. Oceans 41, 191–211.CrossRefGoogle Scholar
Sakamoto, K. & Akitomo, K. 2008 The tidally induced bottom boundary layer in a rotating frame: Development of the turbulent mixed layer under stratification. J. Fluid Mech. (in press).CrossRefGoogle Scholar
Schmitz, W. J. 1995 On the interbasin-scale thermohaline circulation. Rev. Geophys. 33, 151–173.CrossRefGoogle Scholar
Soulsby, R. L. 1983 The bottom boundary layer of shelf seas. In Physical Oceanography of Coastal and Shelf Seas (ed. Johns, B.), pp. 189–266. Elsevier.CrossRefGoogle Scholar
Spalart, P. R. 1989 Theoretical and numerical study of a three-dimensional turbulent boundary layer. J. Fluid Mech. 205, 319–340.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Werner, S. R., Beardsley, R. C., Lentz, S. J., Hebert, D. L., Oakey, N. S. 2003 Observations and modelling of the tidal bottom boundary layer on the souther flank of Georges Bank. J. Geophys. Res. 108, 8005.Google Scholar
White, M. 1994 Tidal and subtidal variability in the sloping benthic boundary layer. J. Geophys. Res. 99, 7851–7864.CrossRefGoogle Scholar
Zikanov, O., Slinn, D. N. & Dhanak, M. R. 2003 Large-eddy simulations of the wind-induced turbulent Ekman layer. J. Fluid Mech. 495, 343–368.CrossRefGoogle Scholar