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Chaotic Tides

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  • 1 Netherlands Institute for Sea Research, Texel, Netherlands
  • | 2 Korteweg-de Vries Instituut, Universiteit van Amsterdam, Amsterdam, Netherlands
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Abstract

Persistent reports exist that the tides in coastal basins are often accompanied by regular or irregular oscillations of periods ranging from minutes up to several hours. A conceptual model relating the two is proposed here. It employs an almost-enclosed basin, connected to an open (tidal) sea by a narrow strait. Such a basin is a Helmholtz resonator, which is dominated by the “pumping” mode. Its response is governed by an ordinary differential equation that is forced by the tide, damped by friction and wave radiation, and whose restoring term is nonlinear due to the sloping bottom. When forced resonantly by a single frequency tide, due to this nonlinearity, the basin may exhibit multiple equilibria. Its response can either be amplified or choked, depending on the precise initial conditions. The presence of a second forcing term may, on a slow timescale, kick the system irregularly from the amplified into the choked regime, yielding a chaotic response. This may happen when either two nearby frequencies, for example, a combination of semidiurnal lunar and solar tides, are near resonance (and the frequency difference provides the beat), or when a small-amplitude, resonant perturbation is modulated by a large-amplitude, low-frequency tide. The aforementioned observations of irregular tides are discussed in the light of analytical and numerical results obtained with this model for these two regimes.

Corresponding author address: Dr. Leo Maas, Netherlands Institute for Sea Research, P.O. Box 59, 1790 AB Texel, Netherlands. Email: maas@nioz.nl

Abstract

Persistent reports exist that the tides in coastal basins are often accompanied by regular or irregular oscillations of periods ranging from minutes up to several hours. A conceptual model relating the two is proposed here. It employs an almost-enclosed basin, connected to an open (tidal) sea by a narrow strait. Such a basin is a Helmholtz resonator, which is dominated by the “pumping” mode. Its response is governed by an ordinary differential equation that is forced by the tide, damped by friction and wave radiation, and whose restoring term is nonlinear due to the sloping bottom. When forced resonantly by a single frequency tide, due to this nonlinearity, the basin may exhibit multiple equilibria. Its response can either be amplified or choked, depending on the precise initial conditions. The presence of a second forcing term may, on a slow timescale, kick the system irregularly from the amplified into the choked regime, yielding a chaotic response. This may happen when either two nearby frequencies, for example, a combination of semidiurnal lunar and solar tides, are near resonance (and the frequency difference provides the beat), or when a small-amplitude, resonant perturbation is modulated by a large-amplitude, low-frequency tide. The aforementioned observations of irregular tides are discussed in the light of analytical and numerical results obtained with this model for these two regimes.

Corresponding author address: Dr. Leo Maas, Netherlands Institute for Sea Research, P.O. Box 59, 1790 AB Texel, Netherlands. Email: maas@nioz.nl

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