### Abstract

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 as 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.

Original language | English (US) |
---|---|

Pages (from-to) | 9-44 |

Number of pages | 36 |

Journal | Journal of Nonlinear Science |

Volume | 7 |

Issue number | 1 |

State | Published - Jan 1997 |

### Fingerprint

### Keywords

- Cascades
- Inertial range
- Turbulence

### ASJC Scopus subject areas

- Computational Mechanics
- Mechanics of Materials
- Mathematics(all)
- Applied Mathematics
- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

**A One-Dimensional Model for Dispersive Wave Turbulence.** / Majda, A. J.; McLaughlin, D. W.; Tabak, E. G.

Research output: Contribution to journal › Article

*Journal of Nonlinear Science*, vol. 7, no. 1, pp. 9-44.

}

TY - JOUR

T1 - A One-Dimensional Model for Dispersive Wave Turbulence

AU - Majda, A. J.

AU - McLaughlin, D. W.

AU - Tabak, E. G.

PY - 1997/1

Y1 - 1997/1

N2 - 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 as 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.

AB - 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 as 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.

KW - Cascades

KW - Inertial range

KW - Turbulence

UR - http://www.scopus.com/inward/record.url?scp=0004511516&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0004511516&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0004511516

VL - 7

SP - 9

EP - 44

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 1

ER -