Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2015, Article ID 613683, 9 pages
http://dx.doi.org/10.1155/2015/613683
Review Article

The Ellipsoidal Vortex: A Novel Approach to Geophysical Turbulence

ISMAR-CNR, Arsenale-Tesa 104, Castello 2737/F, 30122 Venice, Italy

Received 8 October 2014; Accepted 14 January 2015

Academic Editor: Touvia Miloh

Copyright © 2015 William J. McKiver. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. B. Chelton, M. G. Schlax, and R. M. Samelson, “Global observations of nonlinear mesoscale eddies,” Progress in Oceanography, vol. 91, no. 2, pp. 167–216, 2011. View at Publisher · View at Google Scholar · View at Scopus
  2. Z. Zhang, W. Wang, and B. Qiu, “Oceanic mass transport by mesoscale eddies,” Science, vol. 345, pp. 322–324, 2014. View at Publisher · View at Google Scholar
  3. J. C. McWilliams, J. B. Weiss, and I. Yavneh, “Anisotropy and coherent vortex structures in planetary turbulence,” Science, vol. 264, no. 5157, pp. 410–413, 1994. View at Publisher · View at Google Scholar · View at Scopus
  4. K. Shaper Smith and G. K. Vallis, “The scales and equilibration of Midocean Eddies: freely evolving flow,” Journal of Physical Oceanography, vol. 31, no. 2, pp. 554–571, 2001. View at Publisher · View at Google Scholar · View at Scopus
  5. W. J. Mckiver and D. G. Dritschel, “Balance in non-hydrostatic rotating stratified turbulence,” Journal of Fluid Mechanics, vol. 596, pp. 201–219, 2008. View at Publisher · View at Google Scholar · View at Scopus
  6. G. Kirchhoff, Vorlesungen über Matematische Physic: Mechanik Teubner, Leipzig,1876.
  7. A. E. Love, “On the stability of certain vortex motions,” Proceedings of the London Mathematical Society, vol. 25, pp. 18–42, 1893. View at Google Scholar · View at MathSciNet
  8. V. V. Zhmur and K. K. Pankratov, “Dynamics of a semi-ellipsoidal subsurface vortex in a nonuniform flow,” Okeanologia, vol. 29, pp. 150–154, 1989. View at Google Scholar
  9. V. V. Zhmur and K. K. Pankratov, “Dynamics of mesoscale eddy formation in the field currents of large intensive vortex,” Okeanologia, vol. 30, pp. 124–129, 1990. View at Google Scholar
  10. S. P. Meacham, “Quasigeostrophic, ellipsoidal vortices in a stratified fluid,” Dynamics of Atmospheres and Oceans, vol. 16, no. 3-4, pp. 189–223, 1992. View at Publisher · View at Google Scholar · View at Scopus
  11. J. G. Charney, “On the scale of atmospheric motions,” Geofysiske Publikasjoner, vol. 17, no. 2, pp. 3–17, 1948. View at Google Scholar
  12. C. MacClaurin, A Treatise on Fluxions, vol. 1-2, Printed by T. W. and T. Ruddimans, Edinburgh, UK, 1742.
  13. P. S. Laplace, Laplace, Pierre-Simon (1749-1827). Théorie du Mouvement et de la Figure Elliptique Des Planètes. Paris: Imprimerie de Ph.-D. Pierres, 1784.
  14. I. Todhunter, History of the Mathematical Theories of Attraction and the Figure of the Earth, Constable, London, UK, 1873.
  15. S. Chandrasekhar, Ellipsoidal Figures of Equilibrium, Dover, 1969.
  16. G. Dassios, Ellipsoidal Harmonics: Theory and Applications, Cambridge University Press, Cambridge, UK, 2012.
  17. W. J. McKiver and D. G. Dritschel, “The motion of a fluid ellipsoid in a general linear background flow,” Journal of Fluid Mechanics, vol. 474, pp. 147–173, 2003. View at Publisher · View at Google Scholar · View at Scopus
  18. D. G. Dritschel, J. N. Reinaud, and W. J. McKiver, “The quasi-geostrophic ellipsoidal vortex model,” Journal of Fluid Mechanics, vol. 505, pp. 201–223, 2004. View at Publisher · View at Google Scholar · View at Scopus
  19. J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, NY, USA, 1979.
  20. J. G. Charney, “Geostrophic turbulence,” Journal of the Atmospheric Sciences, vol. 28, no. 6, pp. 1087–1095, 1971. View at Publisher · View at Google Scholar
  21. B. J. Hoskins, M. E. McIntyre, and A. W. Robertson, “On the use and significance of isentropic potential vorticity maps.,” Quarterly Journal—Royal Meteorological Society, vol. 111, no. 470, pp. 877–946, 1985. View at Publisher · View at Google Scholar · View at Scopus
  22. W. R. Holland, “Quasigeostrophic modelling of Eddy-resolved ocean circulation,” in Advanced Physical Oceanographic Numerical Modelling, J. J. O’Brien, Ed., vol. 186 of NATO ASI Series C: Mathematical and Physical, pp. 203–231, Springer, Dordrecht, Netherlands, 1986. View at Publisher · View at Google Scholar
  23. B. Deremble, G. Lapeyre, and M. Ghil, “Atmospheric dynamics triggered by an oceanic SST front in a moist quasigeostrophic model,” Journal of the Atmospheric Sciences, vol. 69, no. 5, pp. 1617–1632, 2012. View at Publisher · View at Google Scholar · View at Scopus
  24. V. V. Zhmur and A. F. Shchepetkin, “Evolution of an ellipsoidal vortex in a stratified ocean. Survivability of the vortex in a flow with vertical shear,” Izvestiâ Akademii Nauk SSSR. Fizika Atmosfery i Okeana, vol. 27, pp. 492–503, 1991. View at Google Scholar
  25. R. A. Lyttleton, The Stability of Rotating Liquid Masses, Cambridge University, Paris, France, 1953. View at MathSciNet
  26. E. W. Hobson, Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, 1931.
  27. S. P. Meacham, K. K. Pankratov, A. F. Shchepetkin, and V. V. Zhmur, “The interaction of ellipsoidal vortices with background shear flows in a stratified fluid,” Dynamics of Atmospheres and Oceans, vol. 21, no. 2-3, pp. 167–212, 1994. View at Publisher · View at Google Scholar · View at Scopus
  28. S. P. Meacham, P. J. Morrison, and G. R. Flierl, “Hamiltonian moment reduction for describing vortices in shear,” Physics of Fluids, vol. 9, no. 8, pp. 2310–2328, 1997. View at Publisher · View at Google Scholar · View at Scopus
  29. H. Hashimoto, T. Shimonishi, and T. Miyazaki, “Quasigeostrophic ellipsoidal vortices in a two-dimensional strain field,” Journal of the Physical Society of Japan, vol. 68, no. 12, pp. 3863–3880, 1999. View at Publisher · View at Google Scholar · View at Scopus
  30. J. N. Reinaud, D. G. Dritschel, and C. R. Koudella, “The shape of vortices in quasi-geostrophic turbulence,” Journal of Fluid Mechanics, no. 474, pp. 175–192, 2003. View at Publisher · View at Google Scholar · View at Scopus
  31. D. G. Dritschel, R. K. Scott, and J. N. Reinaud, “The stability of quasi-geostrophic ellipsoidal vortices,” Journal of Fluid Mechanics, vol. 536, pp. 401–421, 2005. View at Publisher · View at Google Scholar · View at Scopus
  32. T. Miyazaki, K. Ueno, and T. Shimonishi, “Quasigeostrophic, tilted spheroidal vortices,” Journal of the Physical Society of Japan, vol. 68, no. 8, pp. 2592–2601, 1999. View at Publisher · View at Google Scholar · View at Scopus
  33. W. J. McKiver and D. G. Dritschel, “The stability of a quasi-geostrophic ellipsoidal vortex in a background shear flow,” Journal of Fluid Mechanics, vol. 560, pp. 1–17, 2006. View at Publisher · View at Google Scholar · View at Scopus
  34. J. N. Reinaud and D. G. Dritschel, “The merger of vertically offset quasi-geostrophic vortices,” Journal of Fluid Mechanics, vol. 469, pp. 287–315, 2002. View at Publisher · View at Google Scholar · View at Scopus
  35. V. V. Zhmur and K. K. Pankratov, “Distant interaction of an ensemble of quasigeostrophic ellipsoidal eddies: Hamiltonian formulation,” Atmospheric and Oceanic Physics, vol. 26, pp. 972–981, 1990. View at Google Scholar
  36. T. Miyazaki, Y. Furuichi, and N. Takahashi, “Quasigeostrophic ellipsoidal vortex model,” Journal of the Physical Society of Japan, vol. 70, no. 7, pp. 1942–1953, 2001. View at Publisher · View at Google Scholar · View at Scopus
  37. Y. Li, H. Taira, N. Takahashi, and T. Miyazaki, “Refinements on the quasi-geostrophic ellipsoidal vortex model,” Physics of Fluids, vol. 18, no. 7, Article ID 076604, 2006. View at Publisher · View at Google Scholar · View at Scopus
  38. N. Martinsen-Burrell, K. Julien, M. R. Petersen, and J. B. Weiss, “Merger and alignment in a reduced model for three-dimensional quasigeostrophic ellipsoidal vortices,” Physics of Fluids, vol. 18, no. 5, Article ID 057101, 2006. View at Publisher · View at Google Scholar · View at Scopus
  39. J. N. Reinaud and D. G. Dritschel, “The critical merger distance between two co-rotating quasi-geostrophic vortices,” Journal of Fluid Mechanics, vol. 522, pp. 357–381, 2005. View at Publisher · View at Google Scholar · View at Scopus
  40. T. Miyazaki, M. Yamamoto, and S. Fujishima, “Counter-rotating quasigeostrophic ellipsoidal vortex pair,” Journal of the Physical Society of Japan, vol. 72, no. 8, pp. 1948–1962, 2003. View at Publisher · View at Google Scholar · View at Scopus
  41. K. V. Koshel, E. A. Ryzhov, and V. V. Zhmur, “Diffusion-affected passive scalar transport in an ellipsoidal vortex in a shear flow,” Nonlinear Processes in Geophysics, vol. 20, no. 4, pp. 437–444, 2013. View at Publisher · View at Google Scholar · View at Scopus
  42. Y.-K. Tsang and D. G. Dritschel, “Ellipsoidal vortices in rotating stratified fluids: beyond the quasi-geostrophic approximation,” Journal of Fluid Mechanics, vol. 762, pp. 196–231, 2015. View at Publisher · View at Google Scholar
  43. W. J. McKiver and D. G. Dritschel, “Balanced solutions for an ellipsoidal vortex in a rotating stratified flow,” Journal of Fluid Mechanics, submitted.