{!— ALTMETRIC INDEX TAGS - MJF --}
Oceanography The Official Magazine of
The Oceanography Society
Volume 31 Issue 03

View Issue TOC
Volume 31, No. 3
Pages 14 - 21

The Value of Asymptotic Theories in Physical Oceanography

Robin Stanley Johnson
Article Abstract

Physical oceanography is an exciting, fruitful and important field of study, particularly relevant to the current discourse on, and the effects of, climate change. However, the tried and tested techniques of (and wealth of knowledge available from) classical fluid mechanics seem to have been sidelined, in favor of an emphasis on modeling and numerical methods. In this article, we make the case for returning to fundamental ideas, explaining the essentials of this approach in the context of the Euler (or Navier-Stokes) equation written in a rotating, spherical coordinate system. We support our contention that this is the way forward by presenting (descriptively only) a number of examples that show what can be done, and suggesting that much more is possible. Indeed, we argue that this is the route to be taken before recourse to other, more ad hoc, methods. We will use this approach to provide new insight (and new results) related to the Pacific Equatorial Undercurrent, the Antarctic Circumpolar Current (including the role of exact solutions), and large gyres.

Citation

Johnson, R.S. 2018. The value of asymptotic theories in physical oceanography. Oceanography 31(3):14–21, https://doi.org/10.5670/oceanog.2018.304.

References

Apel, J.R. 1987. Principles of Ocean Physics. Academic Press, London, 634 pp.

Bell, M.J. M.J. Martin, and N.K. Nichols. 2016. Application of data assimilation to ocean and climate prediction. Pp. 3–10 in UK Success Stories in Industrial Mathematics. P. Aston, A. Mulholland, and K. Tant, eds, Springer International Publishing, Switzerland.

Boccaletti, G., R.P. Pacanowski, S.G.H. Philander, and A.V. Fedorov. 2004. The thermal structure of the upper ocean. Journal of Physical Oceanography 34:888–902, https://doi.org/​10.1175/​1520-0485(2004)034​<0888:TTSOTU>2.0.CO;2.

Chang, K.W., and F.A. Howes. 1984. Nonlinear Singular Perturbation Phenomena: Theory and Applications. Springer-Verlag, Berlin.

Cole, J.D. 1968. Perturbation Methods in Applied Mathematics. Blaisdell, Waltham, MA.

Constantin, A. 2012. An exact solution for equatorially trapped waves. Journal of Geophysical Research 117, C05029, https://doi.org/​10.1029/​2012JC007879.

Constantin, A. 2014. Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves. Journal of Physical Oceanography 44:781–789, https://doi.org/10.1175/JPO-D-13-0174.1.

Constantin, A., and J. Escher. 2011. Analyticity of periodic traveling free surface water waves with vorticity. Annals of Mathematics 73(1):559–568, http://doi.org/10.4007/annals.2011.173.1.12.

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. Current and future prospects for the application of systematic theoretical methods to the study of problems in physical oceanography. Physics Letters A 380:3,007–3,012, https://doi.org/10.1016/​j.physleta.2016.07.036.

Constantin, A., and R.S. Johnson. 2016b. 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. 2016c. 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, 20170063, https://doi.org/10.1098/rspa.2017.0063.

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. 2004. Exact steady periodic water waves with vorticity. Communications on Pure and Applied Mathematics 57:481–527, https://doi.org/10.1002/cpa.3046.

Davies, A.M. 2013. Modelling storm surge current structure. Pp. 55–81 in Offshore and Coastal Modelling. P.P.G. Dyke, A.O. Moscardini, and E.H. Robson, eds, Wiley, New York, NY.

Dingle, R.B. 1973. Asymptotic Expansions: Their Derivation and Interpretation. Academic Press, London.

Ekman, V.W. 1905. On the influence of the Earth’s rotation on ocean currents. Arkiv för matematik, astronomi och fysik 2:1–52.

Faghmous, J.H., and K. Vipin. 2014. A big data guide to understanding climate change: The case for theory-guided data science. Big Data 2:155–163, https://doi.org/10.1089/big.2014.0026.

Firing, Y.L., T.K. Chereskin, and M.R. Mazloff. 2011. Vertical structure and transport of the Antarctic Circumpolar Current in Drake Passage from direct velocity observations. Journal of Geophysical Research 116, C08015, https://doi.org/​10.1029/2011JC006999.

Gallego, B., P. Cessi, and J.C. McWilliams. 2004. The Antarctic Circumpolar Current in equilibrium. Journal of Physical Oceanography 34:1,571–1,587, https://doi.org/10.1175/1520-0485(2004)034​<1571:TACCIE>2.0.CO;2.

Garrison, T. 2014. Essentials of Oceanography, 7th ed. National Geographic Society/Cengage Learning, Stamford, CT.

Gill, A. 1982. Atmosphere-Ocean Dynamics. Academic Press, New York, NY, 662 pp.

Hardy, G.H. 1949. Divergent Series. Clarendon, Oxford, UK.

Henry, D. 2013. An exact solution for equatorial geophysical water waves with an underlying current. European Journal of Mechanics - B/Fluids 38:18–21, https://doi.org/10.1016/​j.euromechflu.2012.10.001.

Henry, D. 2016. Equatorially trapped nonlinear water waves in a beta-plane approximation with centripetal forces. Journal of Fluid Mechanics 804, R1, https://doi.org/10.1017/jfm.2016.544.

Hinch, E.J. 1991. Perturbation Methods. Cambridge University Press, Cambridge, UK, 176 pp.

Holmes, M.H. 1995. Introduction to Perturbation Methods. Springer-Verlag, New York, NY, 438 pp.

Ivchenko, V.O., and K.J. Richards. 1996. The dynamics of the Antarctic Circumpolar Current. Journal of Physical Oceanography 26:753–774, https://doi.org/10.1175/1520-0485(1996)026​<0753:TDOTAC>2.0.CO;2.

Johnson, G.C., M.J. McPhaden, and E. Firing. 2001. Equatorial Pacific ocean horizontal velocity, divergence, and upwelling. Journal of Physical Oceanography 31:839–849, https://doi.org/​10.1175/​1520-0485(2001)031​<0839:EPOHVD>​2.0.CO;2.

Johnson, R.S. 2004. Singular Perturbation Theory: Mathematical and Analytical Techniques with Applications to Engineering. Springer-Verlag, New York, NY, 292 pp., https://doi.org/10.1007/b100957.

Johnson, R.S. 2015. An ocean undercurrent, a thermocline, a free surface, with waves: A problem in classical fluid mechanics. Journal of Nonlinear Mathematical Physics 22:475–493, https://doi.org/​10.1080/14029251.2015.1113042.

Johnson, R.S. 2017. Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography. Philosophical Transactions of the Royal Society A 376, 20170092, https://doi.org/10.1098/rsta.2017.0092.

Kessler, W.S. 2005. Intraseasonal variability in the oceans. Pp. 175–222 in Intraseasonal Variability of the Atmosphere-Ocean System. W.K.M. Lau and D.E. Wallser, eds, Springer, New York, NY.

Kevorkian, J., and J.D. Cole. 1996. Multiple Scale and Singular Perturbation Methods. Applied Mathematical Sciences, vol. 114. Springer-Verlag, Berlin.

Lagerloef, G.S.E., G.T. Mitchum, R.B. Lukas, and P.N. Niiler. 1999. Tropical Pacific near-​surface currents estimated from altimeter, wind, and drifter data. Journal of Geophysical Research 104:23,313–23,326, https://doi.org/​10.1029/1999JC900197.

LeBlond, P.H., and L.A. Mysak. 1978. Waves in the Ocean. Elsevier, Amsterdam, 602 pp.

Maslowe, S.A. 1986. Critical layers in shear flows. Annual Review of Fluid Mechanics 18:405–432, https://doi.org/10.1146/annurev.fl.18.010186.002201.

McCreary, J.P. 1985. Modeling equatorial ocean circulation. Annual Review of Fluid Mechanics 17:359–409, https://doi.org/10.1146/annurev.fl.17.010185.002043.

Olbers, D., D. Borowski, C. Völker, and J.-O. Wölff. 2004. The dynamical balance, transport and circulation of the Antarctic Circumpolar Current. Antarctic Science 16:439–470, https://doi.org/​10.1017/​S0954102004002251.

Proehl, J.A., M.J. McPhaden, and L.M. Rothstein. 1986. A numerical approach to equatorial oceanic wave-mean flow interactions. Pp. 111–126 in Advanced Physical Oceanographic Modelling. J.J. O’Brien, ed., D. Reidel Publishing, Dordrecht, The Netherlands.

Rintoul, S.R., C. Hughes, and D. Olbers. 2001. The Antarctic Circumpolar Current system. Pp. 271–302 in Ocean Circulation and Climate: Observing and Modelling the Global Ocean. G. Siedler, J. Church, and J. Gould, eds, Academic Press, New York, NY.

Segar, D.A. 2012. Introduction to Ocean Science, http://www.reefimages.com/oceansci.php.

Smith, D.R. 1985. Singular-Perturbation Theory: An Introduction with Applications. Cambridge University Press, Cambridge, UK, 520 pp.

Smith, K.S., and J. Marshall. 2009. Evidence for enhanced eddy mixing at middepth in the Southern Ocean. Journal of Physical Oceanography 39:50–69, https://doi.org/​10.1175/2008JPO3880.1.

Stewart, A.L., R. Ferrari, and A.F. Thompson. 2014. On the importance of surface forcing in conceptual models of the deep ocean. Journal of Physical Oceanography 44:891–899, https://doi.org/10.1175/JPO-D-13-0206.1.

Toland, J.F. 1996. Stokes waves. Topological Methods in Nonlinear Analysis 7:1–48.

Tomczak, M., and J.S. Godfrey. 2003. Regional Oceanography: An Introduction, 2nd ed. Daya Publishing, Delhi, 401 pp.

Van Dyke, M. 1975. Perturbation Methods in Fluid Mechanics (Annotated Edition). Parabolic Press, Stanford, CA, 271 pp.

Wallcraft, A.J., A.B. Kara, H.E. Hurlburt, E.P. Chassignet, and G.H. Halliwell. 2008. Value of the bulk heat flux parameterizations for ocean SST predictions. Journal of Marine Systems 74:241–258, https://doi.org/10.1016/​j.jmarsys.2008.01.009.

Wang, B., X.L. Zou, and J. Zhu. 2000. Data assimilation and its applications. Proceedings of the National Academy of Sciences of the United States of America 97:11,143–11,144, https://doi.org/10.1073/pnas.97.21.11143.

Wu, J. 1975. Wind-induced drift currents. Journal of Fluid Mechanics 68:49–70, https://doi.org/10.1017/S0022112075000687.

Yelland, M., and P.K. Taylor. 1996. Wind stress measurements from the open ocean. Journal of Physical Oceanography 26:541–558, https://doi.org/10.1175/1520-0485(1996)026​<0541:WSMFTO>2.0.CO;2.