Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-18T21:52:19.091Z Has data issue: false hasContentIssue false
  • English
  • Français

Resonant interactions between Rossby modes in a straight coast and a channel

Published online by Cambridge University Press:  14 May 2021

Federico Graef Open the ORCID record for Federico Graef [Opens in a new window]  and
Rigoberto F. García
Federico Graef*
Affiliation:
Centro de Investigación Científica y de Educación Superior de Ensenada, Baja California (CICESE), 22860, Mexico
Rigoberto F. García
Affiliation:
Cooperative Institute for Marine and Atmospheric Studies, University of Miami and NOAA/Atlantic Oceanographic and Meteorological Laboratory, Miami, FL, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the possibility of having resonant interactions between three Rossby modes on a coast or channel of arbitrary orientation. A Rossby mode comprises two propagating Rossby waves (RWs) to satisfy the no normal flow through the boundary(ies). In each geometry, we state the conditions, degrees of freedom and RWs of the primary two modes that could force a third mode. We discuss differences between zonal and non-zonal orientations. Resonant interactions are only possible if all RWs participate in the zonal case, while only three RWs participate in the non-zonal case. The non-zonality reduces the degrees of freedom of the resonance conditions, and the solutions are more restrictive for more meridional orientations. In particular, there are no solutions if the coast or channel is meridional. For the non-zonal coast, we find a family of solutions for given periods $T_1$ and $T_2$ of the primary modes. Using multiple scales, we obtain a uniformly valid solution of the quasi-geostrophic potential vorticity equation (QGPVE), with the resonant modes exchanging energy in space. There are no degrees of freedom for the non-zonal channel, and we develop a graphical method to seek resonant solutions, finding some. We provide a bounded solution of the QGPVE in case the primary modes excite one RW, not a channel mode, and the modes do not exchange energy either in time or space. Regarding possible oceanographic applications, we show solutions for the Hawaiian Ridge and inquire if there are solutions in the Mozambique Channel, Tasman Sea, Denmark Strait and the English Channel.

JFM classification

Geophysical and Geological Flows: Quasi-geostrophic flows Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 918 , 10 July 2021 , A34
Check for updates
Copyright
© The Author(s), 2021. Published by 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.)

References

REFERENCES

Craik, A.D.D. 1988 Wave Interactions and Fluid Flows. Cambridge University Press.Google Scholar
Firing, E., Qiu, B. & Miao, W. 1999 Time-dependent island rule and its application to the time-varying North Hawaiian Ridge Current. J. Phys. Oceanogr. 29 (10), 2671–2688.2.0.CO;2>CrossRefGoogle Scholar
Flierl, G.R. 1977 Simple applications of McWilliams’ “A note on a consistent quasigeostrophic model in a multiply connected domain”. Dyn. Atmos. Oceans 1 (5), 443–453.CrossRefGoogle Scholar
García, R. & Graef, F. 1998 The nonlinear self-interaction of a baroclinic Rossby mode in a channel and a gulf. Dyn. Atmos. Oceans 28, 139–155.CrossRefGoogle Scholar
Graef, F. 1993 First order resonance in the reflection of baroclinic Rossby waves. J. Fluid Mech. 251, 515–532.CrossRefGoogle Scholar
Graef, F. 2016 Free and forced Rossby normal modes in a rectangular gulf of arbitrary orientation. Dyn. Atmos. Oceans 75, 46–57.CrossRefGoogle Scholar
Graef, F. 2017 A note on free and forced Rossby wave solutions: the case of a straight coast and a channel. Dyn. Atmos. Oceans 77, 43–53.CrossRefGoogle Scholar
Graef, F. & Magaard, L. 1994 Reflection of nonlinear baroclinic Rossby waves and the driving of secondary mean flows. J. Phys. Oceanogr. 24, 1867–1894.2.0.CO;2>CrossRefGoogle Scholar
Kenyon, K. 1964 Nonlinear energy transfer in a Rossby wave spectrum. In Summer Study Program in Geophysical Fluid Dynamics: Student Lectures, pp. 69–83. Woods Hole Oceanographic Institution.Google Scholar
LaCasce, J.H. & Pedlosky, J. 2004 The instability of Rossby basin modes and the oceanic eddy field. J. Phys. Oceanogr. 34 (9), 2027–2041.2.0.CO;2>CrossRefGoogle Scholar
Longuet-Higgins, M.S. & Gill, A.E. 1967 Resonant interactions between planetary waves. Proc. R. Soc. Lond. A 299 (1456), 120–144.Google Scholar
Magaard, L. 1983 On the potential energy of baroclinic Rossby waves in the North Pacific. J. Phys. Oceanogr. 13 (1), 38–42.2.0.CO;2>CrossRefGoogle Scholar
Mysak, L.A. 1978 Resonant interactions between topographic planetary waves in a continuously stratified fluid. J. Fluid Mech. 84 (4), 769–793.CrossRefGoogle Scholar
Mysak, L.A. & Magaard, L. 1983 Rossby wave driven Eulerian mean flows along non-zonal barriers, with application to the Hawaiian Ridge. J. Phys. Oceanogr. 13 (9), 1716–1725.2.0.CO;2>CrossRefGoogle Scholar
Oh, I.S. & Magaard, L. 1984 Rossby wave-induced secondary flows near barriers, with application to the Hawaiian Ridge. J. Phys. Oceanogr. 14 (9), 1510–1513.2.0.CO;2>CrossRefGoogle Scholar
Pedlosky, J. 2013 Geophysical Fluid Dynamics. Springer Science & Business Media.Google Scholar
Pinardi, N. & Milliff, R.F. 1989 A note on consistent quasi-geostrophic boundary conditions in partially open, simply and multiply connected domains. Dyn. Atmos. Oceans 14, 65–76.CrossRefGoogle Scholar
Plumb, R.A. 1977 The stability of small amplitude Rossby waves in a channel. J. Fluid Mech. 80 (4), 705–720.CrossRefGoogle Scholar
Price, J.M., Van Woert, M. & Vitousek, M. 1994 On the possibility of a ridge current along the Hawaiian Islands. J. Geophys. Res. Oceans 99 (C7), 14101–14111.CrossRefGoogle Scholar
Qiu, B., Koh, D.A., Lumpkin, C. & Flament, P. 1997 Existence and formation mechanism of the North Hawaiian Ridge Current. J. Phys. Oceanogr. 27 (3), 431–444.2.0.CO;2>CrossRefGoogle Scholar
Serrano, D., Graef, F. & Pares-Sierra, A. 1995 La auto-interacción alineal de un modo normal de Rossby en un océano rectangular. Atmósfera 8 (4), 169–189.Google Scholar
Stern, M.E. 1961 Nonlinear interaction of planetary waves. In Woods Hole Oceanogr. Inst. Gontrib., vol. 1063. Woods Hole Oceanographic Institution.Google Scholar
Sun, L.C., Price, J.M., Magaard, L. & Roden, G. 1988 The North Hawaiian Ridge Current: a comparison between an analytical theory and some prior observations. J. Phys. Oceanogr. 18 (2), 384–388.2.0.CO;2>CrossRefGoogle Scholar
Vanneste, J. 1995 Explosive resonant interaction of baroclinic Rossby waves and stability of multilayer quasi-geostrophic flow. J. Fluid Mech. 291, 83–107.CrossRefGoogle Scholar
White, W. 1983 A narrow boundary current along the eastern side of the Hawaiian Ridge; the North Hawaiian Ridge Current. J. Phys. Oceanogr. 13 (9), 1726–1731.2.0.CO;2>CrossRefGoogle Scholar