Short CommunicationFree and forced Rossby normal modes in a rectangular gulf of arbitrary orientation

- Rossby modes are oscillatory, linear or exponential in the coordinate along the gulf.
- For realistic oceanic parameters most of the Rossby gulf modes are exponential.
- The dominant mode of the response when forcing at the mouth is the resonant mode.
- The resonant mode is around the annual frequency for a wide range of gulfs.
- In general less modes get excited if the gulf is zonally oriented.

A free Rossby normal mode in a rectangular gulf of arbitrary orientation is constructed by considering the reflection of a Rossby mode in a channel at the head of the gulf. Therefore, it is the superposition of four Rossby waves in an otherwise unbounded ocean with the same frequency and wavenumbers perpendicular to the gulf axis whose difference is equal to 2mπ/W, where m is a positive integer and W the gulf's width. The lower (or higher) modes with small m (or large m) are oscillatory (evanescent) in the coordinate along the gulf; these are elucidated geometrically. However for oceanographically realistic parameter values, most of the modes are evanescent.When the gulf is forced at the mouth with a single Fourier component, the response is in general an infinite sum of modes that are needed to match the value of the streamfunction at the gulf's entrance. The dominant mode of the response is the resonant one, which corresponds to forcing with a frequency ω and wavenumber normal to the gulf axis η appropriate to a gulf mode: η =− β sin α/(2ω) ± Mπ/W, where α is the angle between the gulf's axis and the eastern direction (+ve clockwise) and M the resonant's mode number. For zonal gulfs ω drops out of the resonance condition.For the special cases η = 0 in which the free surface goes up and down at the mouth with no flow through it, or a flow with a sinusoidal profile, resonant modes can get excited for very specific frequencies (only for non-zonal gulfs in the η = 0 case). The resonant mode is around the annual frequency for a wide range of gulf orientations α ∈ [40°, 130°] or α ∈ [220°, 310°] and gulf widths between 150 and 200 km; these include the Gulf of California and the Adriatic Sea. If η is imaginary, i.e. a flow with an exponential profile, there is no resonance. In general less modes get excited if the gulf is zonally oriented.