Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-19T14:17:29.254Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear and time-dependent equivalent-barotropic flows

Published online by Cambridge University Press:  30 May 2019

Luis Zavala Sansón Open the ORCID record for Luis Zavala Sansón [Opens in a new window]
Luis Zavala Sansón*
Affiliation:
Departamento de Oceanografía Física, CICESE, Carretera Ensenada-Tijuana 3918, 22860 Ensenada, Baja California, México
*
†Email address for correspondence: lzavala@cicese.mx
Get access Rights & Permissions [Opens in a new window]

Abstract

Some oceanic and atmospheric flows may be modelled as equivalent-barotropic systems, in which the horizontal fluid velocity varies in magnitude at different vertical levels while keeping the same direction. The governing equations at a specific level are identical to those of a homogeneous flow over an equivalent depth, determined by a pre-defined vertical structure. The idea was proposed by Charney (J. Met., vol. 6 (6), 1949, pp. 371–385) for modelling a barotropic atmosphere. More recently, steady, linear formulations have been used to study oceanic flows. In this paper, the nonlinear, time-dependent model with variable topography is examined. To include nonlinear terms, we assume suitable approximations and evaluate the associated error in the dynamical vorticity equation. The model is solved numerically to investigate the equivalent-barotropic dynamics in comparison with a purely barotropic flow. We consider three problems in which the behaviour of homogeneous flows has been well established either experimentally, analytically or observationally in past studies. First, the nonlinear evolution of cyclonic vortices around a topographic seamount is examined. It is found that the vortex drift induced by the mountain is modified according to the vertical structure of the flow. When the vertical structure is abrupt, the model effectively isolates the surface flow from both inviscid and viscous topographic effects (due to the shape of the bottom and Ekman friction, respectively). Second, the wind-driven flow in a closed basin with variable topography is studied (for a flat bottom this is the well-known Stommel problem). For a zonally uniform, negative wind-stress curl in the homogeneous case, a large-scale, anticyclonic gyre is formed and displaced southward due to topographic effects at the western slope of the basin. The flow reaches a steady state due to the balance between topographic, $\unicode[STIX]{x1D6FD}$, wind-stress and bottom friction effects. However, in the equivalent-barotropic simulations with abrupt vertical structure, such an equilibrium cannot be reached because the forcing effects at the surface are enhanced, while bottom friction effects are reduced. As a result, the unsteady flow is decomposed as a set of planetary waves. A third problem consists of performing simulations of the wind-driven flow over realistic bottom topography in the Gulf of Mexico. The formation of the so-called Campeche gyre is explored. It is found that such circulation may be consistent with the equivalent-barotropic dynamics.

JFM classification

Geophysical and Geological Flows: Ocean circulation Geophysical and Geological Flows: Shallow water flows Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 871 , 25 July 2019 , pp. 925 - 951
Copyright
© 2019 Cambridge University Press 

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

Adem, J. 1956 A series solution for the barotropic vorticity equation and its application in the study of atmospheric vortices. Tellus 8 (3), 364–372.Google Scholar
Bretherton, F. P. & Haidvogel, D. B. 1976 Two-dimensional turbulence above topography. J. Fluid Mech. 78 (1), 129–154.Google Scholar
Charney, J. G. 1949 On a physical basis for numerical prediction of large-scale motions in the atmosphere. J. Met. 6 (6), 371–385.Google Scholar
Flór, J.-B. & Eames, I. 2002 Dynamics of monopolar vortices on a topographic beta-plane. J. Fluid Mech. 456, 353–376.Google Scholar
van Geffen, J. H. G. M. & Davies, P. A. 2000 A monopolar vortex encounters an isolated topographic feature on a 𝛽-plane. Dyn. Atmos. Oceans 32 (1), 1–26.Google Scholar
Ghaffari, P., Isachsen, P. E. & LaCasce, J. H. 2013 Topographic effects on current variability in the Caspian Sea. J. Geophys. Res.: Oceans 118 (12), 7107–7116.Google Scholar
Gille, S.1995 Dynamics of the Antarctic Circumpolar Current. Evidence for topographic effects from altimeter data and numerical model output. Tech. Rep. MIT.Google Scholar
Gille, S. T. 2003 Float observations of the Southern Ocean. Part I: estimating mean fields, bottom velocities, and topographic steering. J. Phys. Oceanogr. 33 (6), 1167–1181.Google Scholar
González Vera, A. S. & Zavala Sansón, L. 2015 The evolution of a continuously forced shear flow in a closed rectangular domain. Phys. Fluids 27 (3), 034106.Google Scholar
Grimshaw, R., Tang, Y. & Broutman, D. 1994 The effect of vortex stretching on the evolution of barotropic eddies over a topographic slope. Geophys. Astrophys. Fluid Dyn. 76 (1–4), 43–71.Google Scholar
Gutiérrez de Velasco, G. & Winant, C. D. 1996 Seasonal patterns of wind stress and wind stress curl over the Gulf of Mexico. J. Geophys. Res.: Oceans 101 (C8), 18127–18140.Google Scholar
van Heijst, G. J. F. & Clercx, H. J. H. 2009 Laboratory modeling of geophysical vortices. Annu. Rev. Fluid Mech. 41, 143–164.Google Scholar
Hughes, C. W. 2005 Nonlinear vorticity balance of the Antarctic Circumpolar Current. J. Geophys. Res. Oceans 110, C110008.Google Scholar
Hughes, C. W. & De Cuevas, B. A. 2001 Why western boundary currents in realistic oceans are inviscid: a link between form stress and bottom pressure torques. J. Phys. Oceanogr. 31 (10), 2871–2885.Google Scholar
Killworth, P. D. 1992 An equivalent-barotropic mode in the fine resolution antarctic model. J. Phys. Oceanogr. 22 (11), 1379–1387.Google Scholar
Krupitsky, A., Kamenkovich, V. M., Naik, N. & Cane, M. A. 1996 A linear equivalent barotropic model of the antarctic circumpolar current with realistic coastlines and bottom topography. J. Phys. Oceanogr. 26 (9), 1803–1824.Google Scholar
LaCasce, J. H. & Isachsen, P. E. 2010 The linear models of the ACC. Prog. Oceanogr. 84 (3–4), 139–157.Google Scholar
Ohlmann, J. C., Niiler, P. P., Fox, C. A. & Leben, R. R. 2001 Eddy energy and shelf interactions in the Gulf of Mexico. J. Geophys. Res.: Oceans 106 (C2), 2605–2620.Google Scholar
Orlandi, P. & van Heijst, G. F. 1992 Numerical simulation of tripolar vortices in 2D flow. Fluid Dyn. Res. 9 (4), 179–206.Google Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer Science and Business Media.Google Scholar
Perez-Brunius, P., García-Carrillo, P., Dubranna, J., Sheinbaum, J. & Candela, J. 2013 Direct observations of the upper layer circulation in the southern Gulf of Mexico. Deep-Sea Res. I 85, 182–194.Google Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.Google Scholar
Stommel, H. 1948 The westward intensification of wind-driven ocean currents. Eos Trans. AGU 29 (2), 202–206.Google Scholar
Sutyrin, G. G. 1994 Long-lived planetary vortices and their evolution: conservative intermediate geostrophic model. Chaos 4 (2), 203–212.Google Scholar
Vázquez de la Cerda, A. M., Reid, R. O., DiMarco, S. F. & Jochens, A. E. 2005 Bay of Campeche circulation: an update. Geophys. Monograph – AGU 161, 279.Google Scholar
Zavala Sansón, L., Barbosa Aguiar, A. C. & van Heijst, G. J. F. 2012 Horizontal and vertical motions of barotropic vortices over a submarine mountain. J. Fluid Mech. 695, 173–198.Google Scholar
Zavala Sansón, L., González-Villanueva, A. & Flores, L. M. 2010 Evolution and decay of a rotating flow over random topography. J. Fluid Mech. 642, 159–180.Google Scholar
Zavala Sansón, L. & van Heijst, G. J. F. 2000 Interaction of barotropic vortices with coastal topography: laboratory experiments and numerical simulations. J. Phys. Oceanogr. 30 (9), 2141–2162.Google Scholar
Zavala Sansón, L. & van Heijst, G. J. F. 2002 Ekman effects in a rotating flow over bottom topography. J. Fluid Mech. 471, 239–255.Google Scholar
Zavala Sansón, L. & van Heijst, G. J. F. 2014 Laboratory experiments on flows over bottom topography. In Modeling Atmospheric and Oceanic Flows: Insights from Laboratory Experiments and Numerical Simulations (ed. von Larcher, T. & Williams, P. D.), Wiley.Google Scholar
Zavala Sansón, L., Pérez-Brunius, P. & Sheinbaum, J. 2017 Surface relative dispersion in the southwestern Gulf of Mexico. J. Phys. Oceanogr. 47 (2), 387–403.Google Scholar
Zavala Sansón, L., Sheinbaum, J. & Pérez-Brunius, P. 2018 Single-particle statistics in the southern Gulf of Mexico. Geofís. Intl 57 (2), 139–150.Google Scholar