Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T17:18:48.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral Analysis of the Efficiency of Vertical Mixing in theDeep Ocean due to Interaction of Tidal Currents with a Ridge Running down a ContinentalSlope

Published online by Cambridge University Press:  17 July 2014

R. N. Ibragimov  and
A. Tartakovsky
R. N. Ibragimov*
Affiliation:
Pacific Northwest National Laboratory, Richland, WA 99352, USA Lead Mathematician, Applied Statistics Lab, GE Global Research 1 Research Circle Niskayuna, NY 12309
A. Tartakovsky
Affiliation:
Pacific Northwest National Laboratory, Richland, WA 99352, USA School of Geosciences, Department of Mathematics and Statistics University of South Florida, Tampa, FL
*
⋆Corresponding author. E-mail: [email protected]
Get access

Abstract

Efficiency of mixing, resulting from the reflection of an internal wave field imposed onthe oscillatory background flow with a three-dimensional bottom topography, isinvestigated using a linear approximation. The radiating wave field is associated with thespectrum of the linear model, which consists of those mode numbers n and slope valuesα, forwhich the solution represents the internal waves of frequencies ω =nω0 radiating upwrad ofthe topography, where ω0 is the fundamental frequency at whichinternal waves are generated at the topography. The effects of the bottom topography andthe earth’s rotation on the spectrum is analyzed analytically and numerically in thevicinity of the critical slope

αn,θc = arcsin (n 2ω02-f 2 / N 2-f 2) 1/2

which is a slope with the same angle to the horizontal as the internal wavecharacteristic. In this notation, θ is latitude, f is the Coriolis parameterand N is thebuoyancy frequency, which is assumed to be a constant, which corresponds to the uniformstratification.

Keywords

internal wavesocean mixingeffects of rotationflows over topography
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 119 - 137
Copyright
© EDP Sciences, 2014

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

Appenzeller, C.H., Davies, C.H., Norton, W.A.. Fragmentation of stratospheric intrusions. J. Geophys. Res., 101 (1996), 1435–1456. CrossRefGoogle Scholar
Armi, L.. Effects of variation in eddy diffusivity on property distributions in the oceans. J. Mar. Res., 37 (1997), 515–530. Google Scholar
P.G. Baines. Topographic Effects in Stratified Flows. CambridgeUniversity Press. 1971.
Baines, P.G.. On internal tide generation models. Deep Sea Res., 29 (1982), 307-338. CrossRefGoogle Scholar
Balmforth, N.J., Ierley, G.R., Young, W.R.. Tidal conversion by subcritical topography. J. Phys. Oceanogr., 32 (2002), 2900-2914. 2.0.CO;2>CrossRefGoogle Scholar
Bell, T.H.. Lee waves in stratified flows with simple harmonic time dependence. J. Fluid Mech., 67 (1975), 705–722. CrossRefGoogle Scholar
R. Bedard, M. Previsic, G. Hagerman. North American Ocean Energy Status March 2007.Electric Power Research Institute (EPRI) Tidal Power (TP), Volume 8, G. 2007.
G.D. Egbert, R. Ray. Signi cant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data. Nature, 2000.
N. Fraser. Surfing an oil rig. Energy Rev., (1999), 20–4 February/March.
Garrett, C.. Mixing with latitude. Nature, 422 (2003), 477–478. CrossRefGoogle ScholarPubMed
Garrett, C., MacCready, P., Rhines, P.B.. Boundary mixing and arrested Ekman layers: Rotating, stratified flow near a sloping boundary. Annu. Rev. Fluid Mech., 25 (1993), 291-323. CrossRefGoogle Scholar
Grimshaw, R., Smyth, N.. Resonant flow of a stratified fluid over topography. J. Fluid Mech., 169 (1986), 429–464. CrossRefGoogle Scholar
A. Gill. Atmosphere-Ocean Dynamics. New York, etc., Academic Press., A., 1983.
Ibragimov, R.N.. Generation of internal tides by an oscillating background flow along a corrugated slope. Physica Scripta, 78 (2008), 065801. CrossRefGoogle Scholar
Ibragimov, R., Yilmaz, N., Bakhtiyarov, A.. Experimental mixing parameterization due to multiphase fluid-structure interaction. Mech. Res. Comm., 38 (2011), 261-266. CrossRefGoogle Scholar
Ibragimov, R.N.. Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications. Phys. Fluids, 23 (2011), 123102. CrossRefGoogle Scholar
Ibragimov, R.N., Vatchev, V.. Approximation of the Garrett-Munk internal wave spectrum. Phys. Let. A, 376 (2011), 94-101. CrossRefGoogle Scholar
Ibragimov, R.N., Jefferson, G., Carminati, J.. Invariant and approximately invariant solutions of non-linear internal gravity waves forming a column of stratified fluid affected the Earth’s rotation. Int. J. Non-Linear Mech., 51 (2013), 28-44. CrossRefGoogle Scholar
J.C. Kaimal, J.J. Finnigan. Atmospheric Boundary Layer Flows. Their Structure and Measurement. Oxford University Press, 1994 London.
L.H. Kantha, C.A. Clayson. Small Scale Processes in Geophysical Fluid Flows. New York, etc., Academic Press, International Geophysics Series, V. 67, 2000.
S. Khatiwala. Generation of internal tides in an ocean of finite depth: analytical and numerical calculations. Deep See Res., I, 50 2003, 3-21.
P.K. Kundu. Fluid Mechanics. Academic Press, Inc. 1990.
Kunze, E., Garrett, C.. Internal tide generation in the deep ocean. nnu. Rev. Fluid Mech, 39 (2007), 57-87. Google Scholar
Lam, F., Mass, L., Gerkema, T.. Spatial structure of tidal and residual currents as observed over the shelf break in the Bay of Biscay. Deep-See Res., I, 51 (2004) 10751096. Google Scholar
Legg, S., Adcroft, A.. Internal wave breaking at concave and convex continental slopes. J. Phys. Oceanogr., 33 (2003), 2224-2247. 2.0.CO;2>CrossRefGoogle Scholar
Legg, S.. Internal tides generated on a corrugated continental slope. Part 2: Along-slope barotropic forcing. J. Phys. Oceanogr., 34 (2004), no. 8, 1824-1838. 2.0.CO;2>CrossRefGoogle Scholar
Llewellyn Smith, S.G., Young, W.R.. Conversion of the barotropic tide. J. Phys.Oceanogr., 32 (2002), 1554-1566. 2.0.CO;2>CrossRefGoogle Scholar
Long, R.R.. Finite amplitude disturbances in the flow of inviscid rotating and stratified fluids over an obstacle. Annu. Rev. Fluid Mech., 4 (1972), 69–92. CrossRefGoogle Scholar
MacCready, P., Pawlak, G.. Stratified flow along a corrugated slope: Separation Drag and wave drag. J. Phys. Oceanogr., 31 (2001), 2824-2838. 2.0.CO;2>CrossRefGoogle Scholar
J.W. Miles.Waves and wave drag in stratified flows. Applied Mechanics: Proc. 12th Int.Cong. Appl. Mech., Springer, 1969.
P. Muller, A. Naratov. The internal wave action model (IWAM). Proceedings, Aha Huliko’a Hawaiian Winter Workshop, School of Ocean and Earth Science and Technology, Special Publication, 2003.
Munk, W., Wunsch, C.. Abyssal recipes II: energetics of tidal and wind mixing. Deep Sea Res., 45 (1998), 1977-2010. CrossRefGoogle Scholar
J.C. Nappo. An introduction to atmospheric gravity waves. Academic Press, San Diego, 2002.
Nash, J.D., Moum, J.M.. Internal hydraulic flows on the continental shelf: High drag states over a small bank. Geophys. Res., 106 (2001), 4593-4612. CrossRefGoogle Scholar
Needler, G.T.. Dispersion in the ocean by physical, geochemical and biological processes. Phil. Trans. R. Soc. London A 319 (1986), 177-187. CrossRefGoogle Scholar
Polzin, K.L., Toole, J.M., Ledwell, J.R.. Spatial variability of turbulent mixing in the abyssal ocean. Science, 276 (1997), 93-96. CrossRefGoogle ScholarPubMed
Queney, P.. The problem of air flow over mountains: A summary of theoretical studies. Bull. Am. Meteorol. Soc. 29, 1948.
R.S. Scorer. Environmental Aerodynamics, Halsted Press, N.-Y., 1978.
Thorpe, S.A.. The cross-slope transport of momentum by internal waves generated by alongslope currents over topography. J. Phys. Oceanogr., 26 (1996), 191-204. 2.0.CO;2>CrossRefGoogle Scholar
Thorpe, S.A.. The generation of internal waves by flow over rough topography of continental slope. Proc. Roy. Soc. London A, 493 (1992), 115-130. CrossRefGoogle Scholar
Trowbridge, J.H., Lentz, S.J.. Asymmetric behavior of an oceanic boundary layer above a sloping bottom. J. Pjys. Oceanogr., 21 (1991), 1171-1185. 2.0.CO;2>CrossRefGoogle Scholar
U.S. Department of Energy, 2009: Wind & Hydropower Technologies Program. http://www1.eere.energy.gov/windandhydro/hydrokinetic/. Accessed April 2009.
G.N. Watson. A Treatise on the Theory of Bessel Functions. 2nd ed., Cambridge University Press, 1966.
Wunsch, C., Ferrari, R.. Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech., 36 (2004), 281-314. CrossRefGoogle Scholar
Wurtele, M.G., Sharman, R.D., Datta, A.. Atmospheric lee waves”. Annu. Rev. Fluid Mech., 28 (1996), 429-476.CrossRefGoogle Scholar