We discuss some aspects of mesoscale ocean flows in which vorticity plays a key role. In addition to wave-current interactions on the continental shelf, where Earth’s rotation is negligible, we also look into the interplay of wind forcing and the Coriolis force that drives large-scale ocean eddies.
Clamond, D. 2012. Note on the velocity and related fields of steady irrotational two-dimensional surface gravity waves. Philosophical Transactions of the Royal Society A 370:1,572–1,586, https://doi.org/10.1098/rsta.2011.0470.
Constantin, A. 2006. The trajectories of particles in Stokes waves. Inventiones Mathematicae 166:523–535, https://doi.org/10.1007/s00222-006-0002-5.
Constantin, A. 2011a. Nonlinear water waves with applications to wave-current interactions and tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, https://doi.org/10.1137/1.9781611971873.
Constantin, A. 2011b. Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train. European Journal of Mechanics – B/Fluids 30:12–16, https://doi.org/10.1016/j.euromechflu.2010.09.008.
Constantin, A. 2012. An exact solution for equatorially trapped waves. Journal of Geophysical Research 117, C05029, https://doi.org/10.1029/2012JC007879.
Constantin, A., M. Ehrnström, and E. Wahlen. 2007. Symmetry of steady periodic gravity water waves with vorticity. Duke Mathematical Journal 140:591–603.
Constantin, A., and J. Escher. 2004. Symmetry of steady periodic surface water waves with vorticity. Journal of Fluid Mechanics 498:171–181,
Constantin, A., and R.S. Johnson. 2015. The dynamics of waves interacting with the Equatorial Undercurrent. Geophysical & Astrophysical Fluid Dynamics 109:311–358, https://doi.org/10.1080/03091929.2015.1066785.
Constantin, A., and R.S. Johnson. 2016a. An exact, steady, purely azimuthal equatorial flow with a free surface. Journal of Physical Oceanography 46:1,935–1,945, https://doi.org/10.1175/JPO-D-15-0205.1.
Constantin, A., and R.S. Johnson. 2016b. An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current. Journal of Physical Oceanography 46:3,585–3,594, https://doi.org/10.1175/JPO-D-16-0121.1.
Constantin, A., and R.S. Johnson. 2017a. A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific equatorial undercurrent and thermocline. Physics of Fluids 29, 056604, https://doi.org/10.1063/1.4984001.
Constantin, A., and R.S. Johnson. 2017b. Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates. Proceedings of the Royal Society A 473, https://doi.org/10.1098/rspa.2017.0063.
Constantin, A., and R.S. Johnson. 2018. Steady large-scale ocean flows in spherical coordinates. Oceanography 31(3):42–50, https://doi.org/10.5670/oceanog.2018.308.
Constantin, A., K. Kalimeris, and O. Scherzer. 2015. A penalization method for calculating the flow beneath traveling water waves of large amplitude. SIAM Journal of Applied Mathematics 75:1,513–1,535, https://doi.org/10.1137/14096966X.
Constantin, A., and S.G. Monismith. 2017. Gerstner waves in the presence of mean currents and rotation. Journal of Fluid Mechanics 820:511–528, https://doi.org/10.1017/jfm.2017.223.
Constantin, A., and W. Strauss. 2010. Pressure beneath a Stokes wave. Communications on Pure and Applied Mathematics 63:533–557, https://doi.org/10.1002/cpa.20299.
Constantin, A., W. Strauss, and E. Varvaruca. 2016. Global bifurcation of steady gravity water waves with critical layers. Acta Mathematica 217:195–262, https://doi.org/10.1007/s11511-017-0144-x.
Constantin, A., and E. Varvaruca. 2011. Steady periodic water waves with constant vorticity: Regularity and local bifurcation. Archive for Rational Mechanics and Analysis 199:33–67, https://doi.org/10.1007/s00205-010-0314-x.
daSilva, A.F.T., and D.H. Peregrine. 1988. Steep, steady surface waves on water of finite depth with constant vorticity. Journal of Fluid Mechanics 195:281–302, https://doi.org/10.1017/S0022112088002423.
Dellar, P.J. 2011. Variations on a beta-plane: Derivation of non-traditional beta-plane equations from Hamilton’s principle on a sphere. Journal of Fluid Mechanics 674:174–195, https://doi.org/10.1017/S0022112010006464.
Ekman, V.W. 1905. On the influence of the Earth’s rotation on ocean currents. Arkiv för matematik, astronomi och fysik 2:1–52.
Ewing, J.A. 1990. Wind, wave and current data for the design of ships and offshore structures. Marine Structures 3:421–459, https://doi.org/10.1016/0951-8339(90)90001-8.
Goddijn-Murphy, L., D.K. Woolf, and M.C. Easton. 2013. Current patterns in the Inner Sound (Pentland Firth) from underway ADCP data. Journal of Atmospheric and Oceanic Technology 30:96–111, https://doi.org/10.1175/JTECH-D-11-00223.1.
Haller, G. 2005. An objective definition of a vortex. Journal of Fluid Mechanics 525:1–26, https://doi.org/10.1017/S0022112004002526.
Haller, G., A Hadjighasem, M. Farazmand, and F. Huhn. 2016. Defining coherent vortices objectively from the vorticity. Journal of Fluid Mechanics 795:136–173, https://doi.org/10.1017/jfm.2016.151.
Henry, D. 2013. Dispersion relations for steady periodic water waves with an isolated layer of vorticity at the surface. Nonlinear Analysis: Real World Applications 14:1,034–1,043, https://doi.org/10.1016/j.nonrwa.2012.08.015.
Henry, D. 2016. Equatorially trapped nonlinear water waves in a β-plane approximation with centripetal forces. Journal of Fluid Mechanics 804, R1, https://doi.org/10.1017/jfm.2016.544.
Ionescu-Kruse, D., and C.I. Martin. 2018. Local stability for an exact steady purely azimuthal equatorial flow. Journal of Mathematical Fluid Mechanics 20:27–34, https://doi.org/10.1007/s00021-016-0311-4.
Ko, J., and W. Strauss. 2008. Effect of vorticity on steady water waves. Journal of Fluid Mechanics 608:197–215, https://doi.org/10.1017/S0022112008002371.
Liu, C., A. Köhl, Z. Liu, F. Wang, and D. Stammer. 2016. Deep-reaching thermocline mixing in the equatorial pacific cold tongue. Nature Communications 7, 11576, https://doi.org/10.1038/ncomms11576.
Martin, C.I. 2014. Dispersion relations for rotational gravity water flows having two jumps in the vorticity distribution. Journal of Mathematical Analysis and Applications 418:595–611, https://doi.org/10.1016/j.jmaa.2014.04.014.
Martin, C.I. 2015. Dispersion relations for gravity water flows with two rotational layers. European Journal of Mechanics – B/Fluids 50:9–18, https://doi.org/10.1016/j.euromechflu.2014.10.005.
Martin, C.I. 2017a. On periodic geophysical water flows with discontinuous vorticity in the equatorial f-plane approximation. Philosophical Transactions of the Royal Society A 376, https://doi.org/10.1098/rsta.2017.0096.
Martin, C.I. 2017b. On the existence of free-surface azimuthal equatorial flows. Applicable Analysis 96:1,207–1,214, https://doi.org/10.1080/00036811.2016.1180370.
Martin, C.I. 2017c. Two-dimensionality of gravity water flows governed by the equatorial f-plane approximation. Annali di Mathematica Pura ed Applicata 196:2,253–2,260, https://doi.org/10.1007/s10231-017-0663-2.
Martin, C.I., and B.-V. Matioc. 2016. Gravity water flows with discontinuous vorticity and stagnation points. Communications in Mathematical Sciences 14:415–441, https://doi.org/10.4310/CMS.2016.v14.n2.a5.
Milne-Thomson, L.M. 1960. Theoretical Hydrodynamics. The Macmillan Co. New York, 768 pp.
Münchow, A., T.J. Weingartner, and L.W. Cooper. 1999. The summer hydrography and surface circulation of the East Siberian Shelf Sea. Journal of Physical Oceanography 29:2,167–2,182, https://doi.org/10.1175/1520-0485(1999)029<2167:TSHASC>2.0.CO;2.
Peregrine, D.H., and I.G. Jonsson. 1983. Interaction of Waves and Currents. Report of the U.S. Army Coastal Engineering Research Center 83-6, 88 pp.
Ribeiro, R., P.A. Milewski, and A. Nachbin. 2017. Flow structure beneath rotational water waves with stagnation points. Journal of Fluid Mechanics 812:792–814, https://doi.org/10.1017/jfm.2016.820.
Smyth, W.D., J.N. Moum, and J.D. Nash. 2011. Narrowband oscillations in the upper equatorial ocean. Part II: Properties of shear instabilities. Journal of Physical Oceanography 41:412–428, https://doi.org/10.1175/2010JPO4451.1.
Swan, C., I.P. Cummins, and R.L. James. 2001. An experimental study of two-dimensional surface water waves propagating on depth-varying currents. Journal of Fluid Mechanics 428:273–304, https://doi.org/10.1017/S0022112000002457.
Talley, L.D., G.L. Pickard, W.J. Emery, and J.H. Swift. 2011. Descriptive Physical Oceanography: An Introduction. Academic Press, 560 pp.
Thomas, G.P. 1990. Wave-current interactions: An experimental and numerical study. Journal of Fluid Mechanics 216:505–536, https://doi.org/10.1017/S0022112090000519.
Toggweiler, J.R., and H. Bjornsson. 2000. Drake Passage and palaeoclimate. Journal of Quaternary Science 15:319–328, https://doi.org/10.1002/1099-1417(200005)15:4<319::AID-JQS545>3.0.CO;2-C.
Umeyama, M. 2012. Eulerian-Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry. Philosophical Transactions of the Royal Society A 370:1,687–1,702, https://doi.org/10.1098/rsta.2011.0450.
Vallis, G.K. 2006. Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, 745 pp.
Wahlen, E. 2009. Steady water waves with a critical layer. Journal of Differential Equations 246:2,468–2,483, https://doi.org/10.1016/j.jde.2008.10.005.
Wahlen, E. 2014. Non-existence of three-dimensional travelling water waves with constant non-zero vorticity. Journal of Fluid Mechanics 746, https://doi.org/10.1017/jfm.2014.131.
Walton, D.W.H., ed. 2013. Antarctica: Global Science from a Frozen Continent. Cambridge University Press, Cambridge, 352 pp.
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