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ON THE TOPOGRAPHY-DRIVEN VORTICITY PRODUCTION IN SHALLOW LAKES

Part of: Incompressible viscous fluids Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  03 May 2019

BALÁZS SÁNDOR Open the ORCID record for BALÁZS SÁNDOR [Opens in a new window] ,
PÉTER TORMA Open the ORCID record for PÉTER TORMA [Opens in a new window] ,
K. GÁBOR SZABÓ Open the ORCID record for K. GÁBOR SZABÓ [Opens in a new window]  and
HONG ZHANG Open the ORCID record for HONG ZHANG [Opens in a new window]
BALÁZS SÁNDOR*
Affiliation:
Department of Hydraulic and Water Resources Engineering, Faculty of Civil Engineering, Budapest University of Technology and Economics, H-1111 Budapest, Hungary email sandor.balazs@epito.bme.hu, torma.peter@epito.bme.hu, szabo.gabor@epito.bme.hu Griffith School of Engineering, Griffith University, Queensland 4222, Australia email hong.zhang@griffith.edu.au
PÉTER TORMA
Affiliation:
Department of Hydraulic and Water Resources Engineering, Faculty of Civil Engineering, Budapest University of Technology and Economics, H-1111 Budapest, Hungary email sandor.balazs@epito.bme.hu, torma.peter@epito.bme.hu, szabo.gabor@epito.bme.hu
K. GÁBOR SZABÓ
Affiliation:
Department of Hydraulic and Water Resources Engineering, Faculty of Civil Engineering, Budapest University of Technology and Economics, H-1111 Budapest, Hungary email sandor.balazs@epito.bme.hu, torma.peter@epito.bme.hu, szabo.gabor@epito.bme.hu
HONG ZHANG
Affiliation:
Griffith School of Engineering, Griffith University, Queensland 4222, Australia email hong.zhang@griffith.edu.au
*
✉sandor.balazs@epito.bme.hu
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Abstract

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We analyse the vorticity production of lake-scale circulation in wind-induced shallow flows using a linear elliptic partial differential equation. The linear equation is derived from the vorticity form of the shallow-water equation using a linear bed friction formula. The features of the wind-induced steady-state flow are analysed in a circular basin with topography as a concave paraboloid, having a quadratic pile in the middle of the basin. In our study, the size of the pile varies by a size parameter. The vorticity production due to the gradient in the topography (and the distance of the boundary) makes the streamlines parallel to topographical contours, and beyond a critical size parameter, it results in a secondary vortex pair. We compare qualitatively and quantitatively the steady-state circulation patterns and vortex evolution of the flow fields calculated by our linear vorticity model and the full, nonlinear shallow-water equations. From these results, we hypothesize that the steady-state topographical vorticity production in lake-scale wind-induced circulations can be described by the equilibrium of the wind friction field and the bed friction field. Moreover, the latter can also be considered as a linear function of the velocity vector field, and hence the problem can be described by a linear equation.

Keywords

vorticity equilibriumshallow waterlinear circulation modellarge-scale environmental flow

MSC classification

Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 76D17: Viscous vortex flows
Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 2 , April 2019 , pp. 148 - 160
Copyright
© 2019 Australian Mathematical Society 

References

Abbot, M. B. and Basco, D. R., Computational fluid dynamics, an introduction for engineers (Longman Group, Harlow, 1989).Google Scholar
Andradóttir, H. O. and Mortamet, M.-L., “Impact of wind on storm-water pond hydraulics”, J. Hydraul. Eng. 142 (2016) Id 04016034; doi:10.1061/(ASCE)HY.1943-7900.0001150.Google Scholar
Borthwick, A. G. L. and Kaar, E. T., “Shallow flow modelling using curvilinear depth-averaged stream function and vorticity transport equations”, Int. J. Numer. Methods Fluids 17 (1993) 417–445; doi:10.1002/fld.1650170506.Google Scholar
Chen, Z. M. and Price, W. G., “Bifurcating periodic solutions of wind-driven circulation equations”, J. Math. Anal. Appl. 304 (2005) 783–796; doi:10.1016/j.jmaa.2004.09.062.Google Scholar
Chubarenko, B. W., Wang, Y., Chubarenko, I. and Hutter, C., “Wind-driven current simulations around the Island Mainau (Lake Constance)”, Ecol. Modell. 138 (2001) 55–73; doi:10.1016/S0304-3800(00)00393-8.Google Scholar
Csanady, G. T., “The arrested topographic wave”, J. Phys. Oceanogr. 8 (1978) 47–62; 10.1175/1520-0485(1978)008¡0047:TATW¿2.0.CO;2.Google Scholar
Da, C., Shen, B., Yan, P. C., Ma, D. and Song, J., “The shallow water equation and the vorticity equation for a change in height of the topography”, PLoS ONE 12 (2017) e0178184; doi:10.1371/journal.pone.0178184.Google Scholar
Danish Hydraulic Institute, MIKE 21 flow model FM hydrodynamic and transport module(Danish Hydraulic Institute for Water and Environment, Horsholm, 2011).Google Scholar
Dippner, J. W., “Vorticity analysis of transient shallow water eddy fields at the river plume front of the River Elbe in the German Bight”, J. Mar. Syst. 14 (1998) 117–133; doi:10.1016/S0924-7963(97)00008-0.Google Scholar
Ferziger, J. H. and Peric, M., Computational methods for fluid dynamics (Springer, Berlin, 2002).Google Scholar
Hansen, E. A. and Arneborg, L., “The use of a discrete Vortex model for shallow water flow around islands and coastal structures”, Coast. Eng. 32 (1997) 223–246; doi:10.1016/S0378-3839(97)81751-6.Google Scholar
Huang, J. C. K. and Saylor, J. H., “Vorticity waves in a shallow basin”, Dyn. Atmos. Ocean. 6 (1982) 177–196; doi:10.1016/0377-0265(82)90023-9.Google Scholar
Jamart, B. M. and Ozer, J., “Comparison of 2-D and 3-D models of the steady wind-driven circulation in shallow waters”, Coast. Eng. 11 (1987) 393–413; doi:10.1016/0378-3839(87)90020-2.Google Scholar
Jenter, H. L. and Madsen, O. S., “Bottom stress in wind-driven depth-averaged coastal flows”, J. Phys. Oceanogr. 19 (1989) 962–974; 10.1175/1520-0485(1989)019¡0962:BSIWDD¿2.0.CO;2.Google Scholar
Józsa, J., “On the internal boundary layer related wind stress curl and its role in generating shallow lake circulations”, J. Hydrol. Hydromech. 62 (2014) 16–23; doi:10.2478/johh-2014-0004.Google Scholar
Józsa, J., Krámer, T., Napoli, E. and Lipari, G., “Sensitivity of wind-induced shallow lake circulation patterns on changes in lakeshore land use”, EGU Gen. Assem. 2006 8 (2006) 1–2; EGU06-A-00786.Google Scholar
Kimura, N., Wu, C., Hoopes, J. A. and Tai, A., “Diurnal dynamics in a small shallow lake under spatially nonuniform wind and weak stratification”, J. Hydraul. Eng. 142 (2016); Id 04016047; doi:10.1061/(ASCE)HY.1943-7900.0001190.Google Scholar
Krámer, T. and Józsa, J., “An adaptively refined, finite-volume model of wind-induced currents in Lake Neusiedl”, Period. Polytech. 49 (2005) 111–136; https://pp.bme.hu/ci/article/view/575.Google Scholar
Krámer, T., Józsa, J. and Torma, P., “Large-scale mixing of water imported into a shallow lake”, Proc. 3rd Int. Symp. on Shallow Flows, Iowa City, IO, USA, 4–6 June 2012, (eds Constantinescu, G. and Fernando, H. J.), available at http://real.mtak.hu/17101/.Google Scholar
Laval, B. and Imberger, J., “Modelling circulation in lakes: Spatial and temporal variations”, Limnol. Oceanogr. 48 (2003) 983–994; doi:10.4319/lo.2003.48.3.0983.Google Scholar
Laval, B., Imberger, J. and Findikakis, A. N., “Dynamics of a large tropical lake: Lake Maracaibo”, Aquat. Sci. 67 (2005) 337–349; doi:10.1007/s00027-005-0778-1.Google Scholar
Li, Y., Zhang, Q., Yao, J. and Werner, A. D., “Hydrodynamic and hydrological modelling of the Poyang Lake Catchment System in China”, J. Hydrol. Eng. 19 (2014) 607–616; doi:10.1061/(ASCE)HE.1943-5584.0000835.Google Scholar
Liu, S., Ye, Q., Wu, S. and Stive, M. J. F., “Horizontal circulation patterns in a large shallow lake: Taihu Lake, China”, Water 10 (2018) 792; doi:10.3390/w10060792.Google Scholar
Park, M. J. and Wang, D. P., “Tidal vorticity over isolated topographic features”, Cont. Shelf Res. 14 (1994) 1583–1599; doi:10.1016/0278-4343(94)90091-4.Google Scholar
Rubbert, S. and Köngeter, J., “Measurements and three-dimensional simulations of flow in a shallow reservoir subject to small-scale wind field inhomogeneities induced by sheltering”, Aquat. Sci. 67 (2005) 104–121; doi:10.1007/s00027-004-0719-4.Google Scholar
Salsa, S., Partial differential equations in action (Springer, Cham, 2015).Google Scholar
Schoen, J. H., Stretch, D. D. and Tirok, K., “Wind-driven circulation patterns in a shallow estuarine lake: St Lucia, South Africa”, Estuar. Coast. Shelf Sci. 146 (2014) 49–59; doi:10.1016/j.ecss.2014.05.007.Google Scholar
Schwab, D. J., “Simulation and forecasting of Lake Erie storm surges”, Month. Weath. Rev. 106 (1978) 1476–1487; 10.1175/1520-0493(1978)106¡1476:SAFOLE¿2.0.CO;2.Google Scholar
Schwab, D. J., “An inverse method for determining wind stress from water-level fluctuations”, Dyn. Atmos. Oceans 6 (1982) 251–278; doi:10.1016/0377-0265(82)90032-X.Google Scholar
Schwab, D. J. and Beletsky, D., “Relative effects of wind stress curl, topograpy, and stratification on large-scale circulation in Lake Michigan”, J. Geophys. Res. 108 (2003); C2 3044; doi:10.1029/2001JC001066.Google Scholar
Shilo, E., “Wind spatial variability and topographic wave frequency”, J. Phys. Oceanogr. 38 (2008) 2085–2096; doi:10.1175/2008JPO3886.1.Google Scholar
Shilo, E., Ashkenazy, Y., Rimmer, A., Assouline, S., Katsafados, P. and Mahrer, Y., “Effect of wind variability on topographic waves: Lake Kinneret case”, J. Geophys. Res. 112 (2007); C1 2024; doi:10.1029/2007JC004336.Google Scholar
Simons, T. J., “Circulation models of lakes and inland seas”, in: Canadian bulletin of fisheries and aquatic sciences (Dept. of Fisheries and Oceans, Ottawa, 1980).Google Scholar
Simons, T. J., “Reliability of circulation models”, J. Phys. Oceanogr. 15 (1985) 1191–1204; 10.1175/1520-0485(1985)015¡1191:ROCM¿2.0.CO;2.Google Scholar
Torma, P. and Wu, C. H., “Temperature and circulation dynamics in a small and shallow lake: effects of weak stratification and littoral submerged macrophytes”, Water 11 (2019) 128; doi:10.3390/w11010128.Google Scholar
Wu, J., “Wind–stress coefficients over sea surface from breeze to hurricane”, J. Geophys. Res. 87 (1982) 9704; doi:10.1029/JC087iC12p09704.Google Scholar
Zimmerman, J. T. F., “Topographic generation of residual circulation by oscillatory (tidal) currents”, Geophys. Astrophys. Fluid Dyn. 11(1) (1978) 35–47; doi:10.1080/03091927808242650.Google Scholar