Nonlinear shallow-water oscillations in a parabolic channel: Exact solutions and trajectory analyses. (English) Zbl 0883.76019
A new exact solution of the nonlinear shallow-water equations is presented. The solution corresponds to divergent and non-divergent free oscillations in an infinite straight channel of parabolic cross-section on the rotating Earth. It provides a description of the one-dimensional subclass of shallow-water flows in paraboloidal basins in which the velocity field varies linearly and the free-surface displacement varies quadratically with the spatial coordinates. In contrast to the previous exact solutions describing divergent oscillations in circular and elliptic paraboloidal basins, the oscillation frequency of the divergent oscillation in the parabolic channel is found to depend, in part, on the amplitudes of the relative vorticity and free-surface curvature.
MSC:
76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
76U05 | General theory of rotating fluids |
Keywords:
rotating Earth; oscillation frequency; divergent oscillation; relative vorticity; free-surface curvatureReferences:
[1] | DOI: 10.1017/S0022112065000952 · Zbl 0131.23701 · doi:10.1017/S0022112065000952 |
[2] | DOI: 10.1017/S0022112063001270 · Zbl 0122.21202 · doi:10.1017/S0022112063001270 |
[3] | DOI: 10.1175/1520-0477(1994)075 2.0.CO;2 · doi:10.1175/1520-0477(1994)075 2.0.CO;2 |
[4] | DOI: 10.1017/S0022112081001882 · Zbl 0462.76023 · doi:10.1017/S0022112081001882 |
[5] | DOI: 10.1017/S0022112063001282 · doi:10.1017/S0022112063001282 |
[6] | Cushman-Roisin, J. Geophys. Res. 90 pp 11756– (1985) |
[7] | Goldsbrough, Proc. R. Soc. Lond. 130 pp 157– (1931) · JFM 56.1253.05 · doi:10.1098/rspa.1930.0197 |
[8] | Cushman-Roisin, Tellus 39A pp 235– (1987) |
[9] | Cushman-Roisin, Ocean Modelling 59 pp 5– (1984) · Zbl 0513.76038 |
[10] | DOI: 10.1017/S0022112058000331 · Zbl 0080.19504 · doi:10.1017/S0022112058000331 |
[11] | DOI: 10.1007/BF01396491 · Zbl 0438.65029 · doi:10.1007/BF01396491 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.