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Modulational instability of co-propagating internal wavetrains under rotation.

A J Whitfield1, E R Johnson1

  • 1Department of Mathematics, University College London, London WC1E 6BT, United Kingdom.

Chaos (Woodbury, N.Y.)
|March 2, 2015
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Summary

Earth's rotation stabilizes long internal waves, unlike in non-rotating systems. This study analyzes the coupled nonlinear Schrödinger equation for the Ostrovsky equation, revealing stabilization conditions for wavetrain pairs.

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Area of Science:

  • Fluid dynamics
  • Oceanography
  • Nonlinear wave theory

Background:

  • Long internal waves in non-rotating frames are described by the Korteweg-de Vries (KdV) equation.
  • KdV theory predicts stability for isolated monochromatic wavetrains.
  • Coupled nonlinear Schrödinger equation (CNLS) analysis reveals instability in systems of two co-propagating KdV wavetrains.

Purpose of the Study:

  • To investigate the effect of Earth's rotation on the stability of co-propagating internal wavetrains.
  • To derive the CNLS equation for the rotation-modified KdV, or Ostrovsky, equation.
  • To identify conditions under which rotation stabilizes these wave systems.

Main Methods:

  • Derivation of the CNLS equation for the Ostrovsky equation.
  • Analysis of the stability of wavetrain pairs within this framework.
  • Focus on cases with different wavenumbers but identical linear group speeds.

Main Results:

  • Earth's rotation can stabilize pairs of co-propagating internal wavetrains.
  • Stabilization occurs when both waves' wavelengths exceed that of the fastest linear wave.
  • The derived CNLS for the Ostrovsky equation governs this rotational stabilization.

Conclusions:

  • Rotation introduces a stabilizing effect on long internal waves that is absent in non-rotating systems.
  • The Ostrovsky equation framework provides a means to study this rotational stabilization.
  • Understanding these stability dynamics is crucial for modeling internal wave propagation in oceanic environments.