Horizontal Dispersion in Gyres-Internal Wave Flow Field in a Rotating Circular Lake
Publication: Journal of Hydrologic Engineering
Volume 15, Issue 8
Abstract
A combined stream function is derived for investigating horizontal dispersion of pollutants in a moderate-sized circular rotating basin when internal wave and gyre flow field coexist in the upper mixing layer. The combined flow field is analogous to wave-current flow in which the steady part is the depthwise velocity profile of circumferential gyre that is simulated using a depthwise cosine function. The periodic components of the Kelvin-type internal wave (assuming a small Burger number) are derived up to the second order in which the variation in the wave amplitude in the radial direction is governed by the modified Bessel function of the first kind. The combined vertical diffusivity is estimated by adding its wave-induced and current-induced components. The resultant closed-form solution of the wave-current longitudinal dispersion coefficient (WCLDC) shows additional wave-induced dispersion over the dispersion due to codirectional gyres. The celerity for the combined flow shows a consistently higher magnitude than does the wave-only celerity. As the lack of data on combined gyre-internal wave flow did permit validation of the model, the analysis was applied to a few lakes using the data available for Kelvin waves superimposed on hypothetical gyre flow. For higher wave amplitudes, the combined flow structure shows enhanced wave influence over flow depth that result in higher WCLDC.
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Acknowledgments
This paper forms CWR reference 2307.
References
Antenucci, J. P., and Imberger, J. (2001). “Energetics of long internal gravity waves in large lakes.” Limnol. Oceanogr., 46, 1760–1773.
Aref, H. (1984). “Stirring by chaotic advection.” J. Fluid Mech., 143, 1–22.
Baddour, R. E., and Song, S. W. (1998). “The rotational flow of finite amplitude periodic water waves on shear current.” Appl. Ocean Res., 20(3), 163–171.
Binson, J. (1997). “Chaotic mixing by internal inertia—Gravity waves.” Phys. Fluids, 9, 945–962.
Chappelear, J. E. (1961). “Direct numerical calculation of wave properties.” J. Geophys. Res., 66, 501–508.
Cox, S. M., Drazin, P. G., Ryrie, S. C., and Slater, K. (1990). “Chaotic advection of irrotational flows and of waves in fluids.” J. Fluid Mech., 214, 517–534.
Craik, A. D. D. (1982). “The drift velocity of water waves.” J. Fluid Mech., 116, 187–205.
Crawford, G. B., and Collier, E. R. W. (2007). “Long-term observations of deepwater renewal in Crater Lake, Oregon.” Hydrobiologia, 574, 47–68.
Csanady, G. T. (1963). “Turbulent diffusion in Lake Huron.” J. Fluid Mech., 17, 360–384.
Csanady, G. T. (1966). “Accelerated diffusion in the skewed shear flow of lake currents.” J. Geophys. Res., 71, 411–420.
Dean, R. G. (1965). “Stream function representation of nonlinear ocean waves.” J. Geophys. Res., 70, 4561–4572.
Dean, R. G., and Dalrymple, R. A. (1994). Water wave mechanics for engineers and scientists, Prentice-Hall/World Scientific, Singapore.
de Lima, J. L. M. P., Souza, C. C. S., and Singh, V. P. (2008). “Granulometric characterization of sediments transported by surface runoff generated by moving storms.” Nonlinear Processes Geophys., 15(6), 999–1011.
Deng, Z. Q., Bengtsson, L., Singh, V. P., and Adrian, D. D. (2002). “Longitudinal dispersion coefficient in single-channel streams.” J. Hydraul. Eng., 128(10), 901–916.
Deng, Z. Q., de Lima, J. L. M. P., and Singh, V. P. (2005). “Fractional kinetic model for first flush of stormwater pollutants.” J. Environ. Eng., 131(2), 232–241.
de Vriend, H. J., Zyserman, J., Nicholson, J., Roelvink, J. A., Pechon, P., and Southgate, H. N. (1993). “Medium-term 2DH coastal area modeling.” Coast. Eng. Japan, 21, 193–224.
Eckhardt, B., and Yao, D. (1993). “Local Lyapunov exponents in chaotic systems.” Physica D, 65, 100–108.
Elder, J. W. (1959). “The dispersion of marked fluid in turbulent shear flow.” J. Fluid Mech., 5, 544–560.
Emery, K. O., and Csanady, G. T. (1973). “Surface circulation of lakes and nearly land-locked seas.” Proc. Natl. Acad. Sci. U.S.A., 70, 93–97.
Endoh, S. (1986). “Diagnostic study on the vertical circulation and the maintenance mechanism of the cyclonic gyre in Lake Biwa.” J. Geophys. Res., 91, 869–876.
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J., and Brooks, N. H. (1979). Mixing in inland and coastal waters, Academic, N.Y., 483.
Gill, A. E. (1982). Atmosphere–Ocean dynamics, Academic, San Diego, 662.
Gilmour, A. E., and Heath, R. A. (1989). “Barotropic and baroclinic waves in Lake Taupo.” N.Z.J. Mar. Freshwater Res., 23, 189–194.
Gómez-Giraldo, A., Imberger, J., and Antenucci, J. P. (2006). “Spatial structure of the dominant basin-scale internal waves in Lake Kinneret.” Limnol. Oceanogr., 51, 229–246.
Huang, Z., and Mei, C. C. (2003). “Effects of surface waves on turbulent current.” J. Fluid Mech., 497(24), 253–287.
Imberger, J. (1998). “Flux paths in a stratified lake: A review.” Physical processes in lakes and oceans: Coastal and estuarine studies, J. Imberger, ed., Vol. 54, American Geophysical Union, Washington, D.C., 1–17.
Ivey, G. N., and Maxworthy, T. (1992). “Mixing driven by internal Kelvin waves in large lakes and the coastal oceans. Proc. 11th Australasian Fluid Mechanics Conf., Univ. of Tasmania, Hobart, Australia, 303–306.
Kemp, P. H., and Simons, R. R. (1982). “The interaction between waves and a turbulent current: Waves propagating with the current.” J. Fluid Mech., 116, 227–250.
Kemp, P. H., and Simons, R. R. (1983). “The interaction between waves and a turbulent current: Waves propagating against the current.” J. Fluid Mech., 130, 73–89.
Klopman, G. (1994). “Part II: Vertical structure of the flow due to waves and currents.” Rep. No. H840.30, WL Delft Hydraulics, The Netherlands.
Law, A. W. K. (2000). “Taylor dispersion of contaminant due to surface waves.” J. Hydraul. Res., 38, 41–48.
Lawrence, G. A., Ashley, K. I., Yonemitsu, N., and Ellis, J. R. (1995). “Natural dispersion in a small lake.” Limnol. Oceanogr., 40, 1519–1526.
List, E. J., Gartrell, G., and Winant, C. D. (1990). “Diffusion and dispersion in coastal waters.” J. Hydraul. Eng., 116(10), 1158–1179.
Macintyre, S., Flynn, K. M., Jellison, R., and Romero, J. R. (1999). “Boundary mixing and nutrient fluxes in Mono Lake, California.” Limnol. Oceanogr., 44, 512–529.
Margaretha, H. (2005). “Mathematical modelling of wave-current interaction in a hydrodynamic laboratory basin.” Ph.D. thesis, Univ. of Twente, The Netherlands, 116.
Monismith, S. G. (1985). “Wind-forced motions in stratified lakes and their effect on mixed layer shear.” Limnol. Oceanogr., 30, 771–783.
Murthy, C. R. (1975). “Dispersion of floatables in lake currents.” J. Phys. Oceanogr., 5, 193–195.
Okubo, A. (1971). “Oceanic diffusion diagrams.” Deep-Sea Res., 18, 789–802.
Ostrovsky, I., Yacobi, Y. Z., Walline, P., and Kalikhman, I. (1996). “Seiche-induced mixing: Its impact on lake productivity.” Deep-Sea Res., 41, 323–332.
Ottino, J. M. (1989). The kinematics of mixing: Stretching, chaos and transport, Cambridge University Press, Cambridge, U.K.
Ou, H. W., and Bennett, J. R. (1979). “A theory of the mean flow driven by long internal waves in a rotating basin, with application to Lake Kinneret.” J. Phys. Oceanogr., 9, 1112–1125.
Pearson, J. M., Guymer, I., West, J. R., and Coates, L. E. (2002). “Effect of wave height on cross-shore solute mixing.” J. Waterway, Port, Coastal, Ocean Eng., 128(1), 10–20.
Quay, P. D., Broeker, W. S., Hesslein, R. H., and Schindler, D. W. (1980). “Vertical diffusion rates determined by tritium tracer experiments in the thermocline and hypolimnion of two lakes.” Limnol. Oceanogr., 25, 201–218.
Ridderinkhof, H., and Zimmerman, J. T. F. (1992). “Chaotic stirring in a tidal system.” Science, 258, 1107–1111.
Schallenberg, M., James, M., Wes, I. H., and Howard-Williams, C. (1999). “External forcing by wind and turbid inflows on a deep glacial lake and implications for primary production.” N.Z.J. Mar. Freshwater Res., 33, 311–331.
Schladow, S. G., and Thompson, K. L. (2003). “Winter thermal structure of Lake Tahoe.” MS thesis, Dept. Civil and Environmental Engineering, Univ. of California, Davis, Calif.
Schwab, D. J., O’Connor, W. P., and Mellor, G. L. (1995). “On the net cyclonic circulation in large stratified lakes.” J. Phys. Oceanogr., 25, 1516–1520.
Shimizu, K., Imberger, J., and Kumagai, M. (2007). “Horizontal structure and excitation of primary motion in a strongly stratified lake.” Limnol. Oceanogr., 52, 2641–2655.
Singh, V. P. (1997). Kinematic-wave modeling in water resources: Environmental hydrology, Wiley, N.Y.
Spigel, R. H., and Imberger, J. (1980). “The classification of mixed-layer dynamics in lakes of small to medium size.” J. Phys. Oceanogr., 10, 1104–1121.
Spigel, R. H., and Imberger, J. (1987). “Mixing processes relevant to phytoplankton dynamics in lakes.” N.Z.J. Mar. Freshw. Res., 21, 361–377.
Stocker, R., and Imberger, J. (2003). “Energy partitioning and horizontal dispersion in a stratified rotating lake.” J. Phys. Oceanogr., 33, 512–529.
Taylor, G. I. (1954). “The dispersion of matter in turbulent flow through a pipe.” Proc. R. Soc., London, 223A, 446–468.
Teng, B., Zhao, M., and Bai, W. (2001). “Wave diffraction in a current over a local shoal.” Coast. Eng., 42(2), 163–172.
Umeyama, M. (2005). “Reynolds stresses and velocity distributions in a wave-current coexisting environment.” J. Waterway, Port, Coastal, Ocean Eng., 131(5), 203–212.
Wang, Y., Hutter, K., and Bauerle, E. (2000). “Wind-induced baroclinic response of Lake Constance.” Ann. Geophys., 18, 1488–14501.
Yasuda, H. (1984). “Longitudinal dispersion of matter due to the shear effect of steady and oscillatory currents.” J. Fluid Mech., 148, 383–403.
Zimmerman, J. T. F. (1986). “The tidal whirlpool: A review of horizontal dispersion by tidal and residual currents.” Netherlands J. Sea Res., 20, 133–154.
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Published In
Journal of Hydrologic Engineering
Volume 15 • Issue 8 • August 2010
Pages: 597 - 611
Copyright
© 2010 ASCE.
History
Received: Aug 5, 2008
Published online: May 15, 2009
Accepted: Sep 8, 2009
Published in print: Aug 2010
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