Journal of Nonlinear Science

, Volume 7, Issue 1, pp 9–44 | Cite as

A one-dimensional model for dispersive wave turbulence

  • A. J. Majda
  • D. W. McLaughlin
  • E. G. Tabak


A family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. These models have an explicitly solvable weak turbulence theory which is developed here, with Kolmogorov-type wave number spectra exhibiting interesting dependence on parameters in the equations. These predictions of weak turbulence theory are compared with numerical solutions with damping and driving that exhibit a statistical inertial scaling range over as much as two decades in wave number.

It is established that the quasi-Gaussian random phase hypothesis of weak turbulence theory is an excellent approximation in the numerical statistical steady state. Nevertheless, the predictions of weak turbulence theory fail and yield a much flatter (|k|−1/3) spectrum compared with the steeper (|k|−3/4) spectrum observed in the numerical statistical steady state. The reasons for the failure of weak turbulence theory in this context are elucidated here. Finally, an inertial range closure and scaling theory is developed which successfully predicts the inertial range exponents observed in the numerical statistical steady states.

Key words

turbulence cascades inertial range 

AMSC codes

76F99 60G10 60H15 


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Copyright information

© Springer-Verlag New York Inc 1997

Authors and Affiliations

  • A. J. Majda
    • 1
  • D. W. McLaughlin
    • 1
  • E. G. Tabak
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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