### Abstract

There are two types of normal modes in a uniform channel: (1) Those defined at a constant along-channel wavenumber k, for which the eigenvalues are the frequencies ω_{a} (k) and the eigenmodes are Kelvin, Poincaré, and Rossby waves, and (2) Those defined at a fixed ω, for which the eigenvalues are k_{a} (ω) and the eigenmodes are Kelvin and Poincaré/Rossby waves. The first set is useful in the initial value problem, whereas the second one is applied to the study of forced solutions and the wave-scattering phenomena. Orthogonality conditions are derived for both types of normal modes and shown to be related to the existence of adjoint systems whose normal modes coincide with those of the direct system. The corresponding inner products are similar, but not exactly equal, to the energy density and flux, respectively. They do not define a metric, and, consequently, there is not a property of the wave field equal to the sum of the real nonnegative contributions of each mode (except in the first case and without friction). A numerical approximation that has the required orthogonality properties is presented and used to give examples of forced solutions, such as Taylor's problem with wind forcing and topography.

Original language | English (US) |
---|---|

Article number | 1999JC900107 |

Pages (from-to) | 15479-15494 |

Number of pages | 16 |

Journal | Journal of Geophysical Research: Oceans |

Volume | 104 |

Issue number | C7 |

State | Published - Jul 15 1999 |

Externally published | Yes |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Geophysics
- Forestry
- Oceanography
- Aquatic Science
- Ecology
- Water Science and Technology
- Soil Science
- Geochemistry and Petrology
- Earth-Surface Processes
- Atmospheric Science
- Earth and Planetary Sciences (miscellaneous)
- Space and Planetary Science
- Palaeontology

### Cite this

*Journal of Geophysical Research: Oceans*,

*104*(C7), 15479-15494. [1999JC900107].

}

*Journal of Geophysical Research: Oceans*, vol. 104, no. C7, 1999JC900107, pp. 15479-15494.

**Ocean channel modes.** / Ripa, P.; Zavala-Garay, J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Ocean channel modes

AU - Ripa, P.

AU - Zavala-Garay, J.

PY - 1999/7/15

Y1 - 1999/7/15

N2 - There are two types of normal modes in a uniform channel: (1) Those defined at a constant along-channel wavenumber k, for which the eigenvalues are the frequencies ωa (k) and the eigenmodes are Kelvin, Poincaré, and Rossby waves, and (2) Those defined at a fixed ω, for which the eigenvalues are ka (ω) and the eigenmodes are Kelvin and Poincaré/Rossby waves. The first set is useful in the initial value problem, whereas the second one is applied to the study of forced solutions and the wave-scattering phenomena. Orthogonality conditions are derived for both types of normal modes and shown to be related to the existence of adjoint systems whose normal modes coincide with those of the direct system. The corresponding inner products are similar, but not exactly equal, to the energy density and flux, respectively. They do not define a metric, and, consequently, there is not a property of the wave field equal to the sum of the real nonnegative contributions of each mode (except in the first case and without friction). A numerical approximation that has the required orthogonality properties is presented and used to give examples of forced solutions, such as Taylor's problem with wind forcing and topography.

AB - There are two types of normal modes in a uniform channel: (1) Those defined at a constant along-channel wavenumber k, for which the eigenvalues are the frequencies ωa (k) and the eigenmodes are Kelvin, Poincaré, and Rossby waves, and (2) Those defined at a fixed ω, for which the eigenvalues are ka (ω) and the eigenmodes are Kelvin and Poincaré/Rossby waves. The first set is useful in the initial value problem, whereas the second one is applied to the study of forced solutions and the wave-scattering phenomena. Orthogonality conditions are derived for both types of normal modes and shown to be related to the existence of adjoint systems whose normal modes coincide with those of the direct system. The corresponding inner products are similar, but not exactly equal, to the energy density and flux, respectively. They do not define a metric, and, consequently, there is not a property of the wave field equal to the sum of the real nonnegative contributions of each mode (except in the first case and without friction). A numerical approximation that has the required orthogonality properties is presented and used to give examples of forced solutions, such as Taylor's problem with wind forcing and topography.

UR - http://www.scopus.com/inward/record.url?scp=16444378258&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=16444378258&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:16444378258

VL - 104

SP - 15479

EP - 15494

JO - Journal of Geophysical Research

JF - Journal of Geophysical Research

SN - 0148-0227

IS - C7

M1 - 1999JC900107

ER -