Abstract
The aim of this paper is to provide, in a \(\beta \)-plane approximation with centripetal forces, an explicit three-dimensional nonlinear solution for geophysical waves propagating at an arbitrary latitude, in the presence of a constant underlying background current. This solution is linearly unstable when the steepness of the wave exceeds a specific threshold.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bayly, B.J.: Three-dimensional instabilities in quasi-two-dimensional inviscid flows. In: Miksad, R.W., et al. (eds.) Nonlinear Wave Interactions in Fluids, pp. 71–77. ASME, New York (1987)
Chu, J., Ionescu-Kruse, D., Yang, Y.: Exact solution and instability for geophysical waves at arbitrary latitude. Discrete Contin. Dyn. Syst. (in press)
Constantin, A.: Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, vol. 81. SIAM, Philadelphia (2011)
Constantin, A.: An exact solution for equatorially trapped waves. J. Geophys. Res. Oceans 117, C05029 (2012)
Constantin, A.: Some three-dimensional nonlinear equatorial flows. J. Phys. Oceanogr. 43, 165–175 (2013)
Constantin, A.: Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves. J. Phys. Oceanogr. 44, 781–789 (2014)
Constantin, A., Germain, P.: Instability of some equatorially trapped waves. J. Geophys. Res. Oceans 118, 2802–2810 (2013)
Constantin, A., Johnson, R.S.: The dynamics of waves interacting with the equatorial undercurrent. Geophys. Astrophys. Fluid Dyn. 109, 311–358 (2015)
Constantin, A., Johnson, R.S.: An exact, steady, purely azimuthal equatorial flow with a free surface. J. Phys. Oceanogr. 46, 1935–1945 (2016)
Constantin, A., Johnson, R.S.: An exact, steady, purely azimuthal flow as a model for the antarctic circumpolar current. J. Phys. Oceanogr. 46, 3585–3594 (2016)
Constantin, A., Johnson, R.S.: A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the pacific equatorial undercurrent and thermocline. Phys. Fluids 29, 056604 (2017)
Constantin, A., Monismith, S.G.: Gerstner waves in the presence of mean currents and rotation. J. Fluid Mech. 820, 511–528 (2017)
Cushman-Roisin, B., Beckers, J.M.: Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects. Academic Press, Waltham (2011)
Fan, L., Gao, H.: Instability of equatorial edge waves in the background flow. Proc. Am. Math. Soc. 145, 765–778 (2017)
Fan, L., Gao, H., Xiao, Q.: An exact solution for geophysical trapped waves in the presence of an underlying current. Dyn. Partial Differ. Equ. 15, 201–214 (2018)
Friedlander, S., Vishik, M.M.: Instability criteria for the flow of an inviscid incompressible fluid. Phys. Rev. Lett. 66, 2204–2206 (1991)
Genoud, F., Henry, D.: Instability of equatorial water waves with an underlying current. J. Math. Fluid Mech. 16, 661–667 (2014)
Gill, A.E.: Atmosphere–Ocean Dynamics. Elsevier, Amsterdam (1982)
Henry, D.: An exact solution for equatorial geophysical water waves with an underlying current. Eur. J. Mech. B Fluids 38, 18–21 (2013)
Henry, D.: Exact equatorial water waves in the \(f\)-plane. Nonlinear Anal. Real World Appl. 28, 284–289 (2016)
Henry, D.: Equatorially trapped nonlinear water waves in a \(\beta \)-plane approximation with centripetal forces. J. Fluid Mech. 804(R1), 11 (2016)
Henry, D.: A modified equatorial \(\beta \)-plane approximation modelling nonlinear wave–current interactions. J. Differ. Equ. 263, 2554–2566 (2017)
Henry, D.: On three-dimensional Gerstner-like equatorial water waves. Philos. Trans. R. Soc. A 376(2111), 20170088 (2018)
Henry, D., Hsu, H.-C.: Instability of equatorial water waves in the \(f\)-plane. Discrete Contin. Dyn. Syst. 35, 909–916 (2015)
Henry, D., Hsu, H.-C.: Instability of internal equatorial water waves. J. Differ. Equ. 258, 1015–1024 (2015)
Hsu, H.-C.: An exact solution for equatorial waves. Monatsh. Math. 176, 143–152 (2015)
Ionescu-Kruse, D.: An exact solution for geophysical edge waves in the \(f\)-plane approximation. Nonlinear Anal. Real World Appl. 24, 190–195 (2015)
Ionescu-Kruse, D.: An exact solution for geophysical edge waves in the \(\beta \)-plane approximation. J. Math. Fluid Mech. 17, 699–706 (2015)
Ionescu-Kruse, D.: Instability of equatorially trapped waves in stratified water. Ann. Mat. Pura Appl. 195, 585–599 (2016)
Ionescu-Kruse, D.: Instability of Pollard’s exact solution for geophysical ocean flows. Phys. Fluids 28, 086601 (2016)
Ionescu-Kruse, D.: On the short-wavelength stabilities of some geophysical flows. Philos. Trans. R. Soc. A 376(2111), 20170090 (2018)
Ionescu-Kruse, D.: A three-dimensional autonomous nonlinear dynamical system modelling equatorial ocean flows. J. Differ. Equ. 264, 4650–4668 (2018)
Johnson, R.S.: Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography. Philos. Trans. R. Soc. A 376(2111), 20170092 (2018)
Leblanc, S.: Local stability of Gerstner’s waves. J. Fluid Mech. 506, 245–254 (2004)
Lifschitz, A., Hameiri, E.: Local stability conditions in fluid mechanics. Phys. Fluids 3, 2644–2651 (1991)
Matioc, A.-V.: An exact solution for geophysical equatorial edge waves over a sloping beach. J. Phys. A 45, 365501 (2012)
Matioc, A.-V.: Exact geophysical waves in stratified fluids. Appl. Anal. 92, 2254–2261 (2013)
Pollard, R.T.: Surface waves with rotation: an exact solution. J. Geophys. Res. 75, 5895–5898 (1970)
Vallis, G.K.: Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, Cambridge (2006)
Acknowledgements
We would like to show our thanks to the anonymous referees for their valuable suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by A. Constantin
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Jifeng Chu was supported by the National Natural Science Foundation of China (Grant No. 11671118 and No. 11871273). Yanjuan Yang was supported by the Fundamental Research Funds for the Central Universities (Grant No. 2017B715X14).
Rights and permissions
About this article
Cite this article
Chu, J., Ionescu-Kruse, D. & Yang, Y. Exact Solution and Instability for Geophysical Waves with Centripetal Forces and at Arbitrary Latitude. J. Math. Fluid Mech. 21, 19 (2019). https://doi.org/10.1007/s00021-019-0423-8
Accepted:
Published:
DOI: https://doi.org/10.1007/s00021-019-0423-8