Hostname: page-component-76c49bb84f-lvrpm Total loading time: 0 Render date: 2025-07-08T21:11:39.570Z Has data issue: false hasContentIssue false
  • English
  • Français

Stuart-type vortices on a rotating sphere

Published online by Cambridge University Press:  28 February 2019

A. Constantin Open the ORCID record for A. Constantin [Opens in a new window]  and
V. S. Krishnamurthy
A. Constantin*
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
V. S. Krishnamurthy
Affiliation:
Erwin Schrödinger International Institute for Mathematics and Physics, Boltzmanngasse 9, 1090 Vienna, Austria
*
†Email address for correspondence: adrian.constantin@univie.ac.at
Get access Rights & Permissions [Opens in a new window]

Abstract

Stuart vortices are among the few known smooth explicit solutions of the planar Euler equations with a nonlinear vorticity, and they have a counterpart for inviscid flow on the surface of a fixed sphere. By means of a perturbative approach we adapt the method used to investigate Stuart vortices on a fixed sphere to provide insight into some large-scale shallow-water flows on a rotating sphere that model the dynamics of ocean gyres.

JFM classification

Geophysical and Geological Flows: Shallow water flows Vortex Flows: Vortex dynamics

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 865 , 25 April 2019 , pp. 1072 - 1084
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.)

Article purchase

Temporarily unavailable

References

Amitai, Y., Lehahn, Y., Lazar, A. & Heifetz, E. 2010 Surface circulation of the eastern Mediterranean Levantine basin: insights from analyzing 14 years of satellite altimetry data. J. Geophys. Res. 115, C10058.10.1029/2010JC006147Google Scholar
Brézis, H. 2011 Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer.Google Scholar
Constantin, A. 2012 An exact solution for equatorially trapped waves. J. Geophys. Res. Oceans 117, C05029.10.1029/2012JC007879Google Scholar
Constantin, A. & Johnson, R. S. 2015 The dynamics of waves interacting with the equatorial undercurrent. Geophys. Astrophys. Fluid Dyn. 109, 311–358.Google Scholar
Constantin, A. & Johnson, R. S. 2016a An exact, steady, purely azimuthal equatorial flow with a free surface. J. Phys. Oceanogr. 46, 1935–1945.10.1175/JPO-D-15-0205.1Google Scholar
Constantin, A. & Johnson, R. S. 2016b An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current. J. Phys. Oceanogr. 46, 3585–3594.Google Scholar
Constantin, A. & Johnson, R. S. 2017 Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates. Proc. R. Soc. Lond. A 473, 20170063.Google Scholar
Crowdy, D. G. 1997 General solutions to the 2D Liouville equation. Intl J. Engng Sci. 35, 141–149.Google Scholar
Crowdy, D. G. 2004 Stuart vortices on a sphere. J. Fluid Mech. 398, 381–402.10.1017/S0022112003007043Google Scholar
Crowdy, D. G. 2006 Point vortex motion on the surface of a sphere with impenetrable boundaries. Phys. Fluids 18, 036602.10.1063/1.2183627Google Scholar
Crowdy, D. G. & Marshall, J. 2007 Green’s functions for Laplace’s equation in multiply connected domains. IMA J. Appl. Maths 72, 278–301.Google Scholar
Cushman-Roisin, B. & Beckers, J.-M. 2011 Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects. Academic Press.Google Scholar
Dellar, P. J. 2011 Variations on a beta-plane: derivation of non-traditional beta-plane equations from Hamilton’s principle on a sphere. J. Fluid Mech. 674, 174–195.10.1017/S0022112010006464Google Scholar
Fofonoff, N. P. 1954 Steady flow in a frictionless homogeneous ocean. J. Mar. Res. 13, 254–262.Google Scholar
Gilbarg, D. & Trudinger, N. S. 2001 Elliptic Partial Differential Equations of Second Order. Springer.Google Scholar
Gustafsson, B. 1990 On the convexity of a solution of Liouville’s equation. Duke Math. J. 60, 303–311.Google Scholar
Henrici, P. 1986 Applied and Computational Complex Analysis, vol. 3. John Wiley.Google Scholar
Henry, D. 2013 An exact solution for equatorial geophysical water waves with an underlying current. Eur. J. Mech. (B/Fluids) 38, 18–21.Google Scholar
Henry, D. 2016 Equatorially trapped nonlinear water waves in a 𝛽-plane approximation with centripetal forces. J. Fluid Mech. 804, R1.10.1017/jfm.2016.544Google Scholar
Kidambi, R. & Newton, P. 2000 Point vortex motion on a sphere with solid boundaries. Phys. Fluids 12, 581–588.10.1063/1.870263Google Scholar
Kostianov, A. G., Nihoul, J. C. J. & Rodionov, V. B. 2004 Physical Oceanography of Frontal Zones in the Subarctic Seas. Elsevier.Google Scholar
Marshall, J. & Plumb, R. A. 2016 Atmosphere, Ocean and Climate Dynamics: An Introductory Text. Academic Press.Google Scholar
Newns, W. F. 1967 Functional dependence. Am. Math. Mon. 74, 911–920.10.1080/00029890.1967.12000050Google Scholar
Newton, P. K. 2001 The N-vortex Problem. Springer.10.1007/978-1-4684-9290-3Google Scholar
Paldor, N. 2015 Shallow Water Waves on the Rotating Earth. Springer.Google Scholar
Plueddemann, A. J., Krishfield, R., Takizawa, T., Hatakeyama, K. & Honjo, S. 1998 Upper ocean velocities in the Beaufort Gyre. Geophys. Res. Lett. 25, 183–186.10.1029/97GL53638Google Scholar
Ponce, A. C. 2014 Elliptic PDEs, Measures and Capacities. From the Poisson Equations to Nonlinear Thomas–Fermi Problems. EMS Tracts in Mathematics, 23. European Mathematical Society (EMS).Google Scholar
Riffenburgh, B. 2007 Encyclopedia of the Antarctic. Routledge.Google Scholar
Stewart, A. L. & Dellar, P. J. 2010 Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane. J. Fluid Mech. 651, 387–413.10.1017/S0022112009993922Google Scholar
Stuart, J. T. 1967 On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, 417–440.Google Scholar
Talley, L. D., Pickard, G. L., Emery, W. J. & Swift, J. H. 2011 Descriptive Physical Oceanography: An Introduction. Elsevier.10.1016/B978-0-7506-4552-2.10001-0Google Scholar
Vallis, G. K. 2006 Atmosphere and Ocean Fluid Dynamics. Cambridge University Press.10.1017/CBO9780511790447Google Scholar
Viudez, A. & Dritschel, D. G. 2015 Vertical velocity in mesoscale geophysical flows. J. Fluid Mech. 483, 199–223.10.1017/S0022112003004191Google Scholar
Wunsch, C. 2015 Modern Observational Physical Oceanography. Princeton University Press.Google Scholar