## Abstract

We introduce and study a toy model which captures some essential features of wave radiation by slow ( or balanced) motion in the atmosphere and the ocean. Inspired by the widely studied five-component model due to Lorenz, the model describes the coupling of a nonlinear pendulum with linear waves. The waves obey a one-dimensional linear Klein - Gordon equation, so their dispersion relation is identical to that of inertia-gravity waves in a rotating shallow-water fluid. The model is Hamiltonian. We examine two physically relevant asymptotic regimes in which there is some time-scale separation between the slow pendulum motion and the fast waves: in regime (i), the time-scale separation breaks down for waves with asymptotically large wavelengths; in regime (ii), the time-scale separation holds for all wavelengths. We study the generation of waves in each regime using distinct asymptotic methods. In regime ( i), long waves are excited resonantly in a manner that is analogous to the Lighthill radiation of sound waves in weakly compressible flows, and to the radiation of gravitational waves by slow mass motion in general relativity. Matched asymptotics provides the functional form of the waves radiated, and leads, at higher order, to a closed model describing the pendulum dynamics while accounting for the dissipative effect of wave radiation. In regime ( ii), an exponentially accurate slow manifold can be defined, and the waves radiated are exponentially small. They are captured using an exponential-asymptotic technique combining complex-time matching with Borel summation. The asymptotic results obtained in each regime are tested against numerical simulations of the model.

Original language | English |
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Pages (from-to) | 783-807 |

Number of pages | 25 |

Journal | Siam Journal on Applied Dynamical Systems |

Volume | 5 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2006 |

## Keywords

- slow manifold
- wave radiation
- inertia-gravity wave
- exponential asymptotics
- ROTATING SHALLOW-WATER
- SLOW QUASIMANIFOLD
- MANIFOLD
- DYNAMICS