### Abstract

Steady scale-invariant solutions of a kinetic equation describing the statistics of oceanic internal gravity waves based on wave turbulence theory are investigated. It is shown in the nonrotating scale-invariant limit that the collision integral in the kinetic equation diverges for almost all spectral power-law exponents. These divergences come from resonant interactions with the smallest horizontal wavenumbers and/or the largest horizontal wavenumbers with extreme scale separations. A small domain is identified in which the scale-invariant collision integral converges and numerically find a convergent power-law solution. This numerical solution is close to the Garrett-Munk spectrum. Power-law exponents that potentially permit a balance between the infrared and ultraviolet divergences are investigated. The balanced exponents are generalizations of an exact solution of the scale-invariant kinetic equation, the Pelinovsky-Raevsky spectrum. A small but finite Coriolis parameter representing the effects of rotation is introduced into the kinetic equation to determine solutions over the divergent part of the domain using rigorous asymptotic arguments. This gives rise to the induced diffusion regime. The derivation of the kinetic equation is based on an assumption of weak nonlinearity. Dominance of the nonlocal interactions puts the self-consistency of the kinetic equation at risk. However, these weakly nonlinear stationary states are consistent with much of the observational evidence.

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

Pages (from-to) | 2605-2623 |

Number of pages | 19 |

Journal | Journal of Physical Oceanography |

Volume | 40 |

Issue number | 12 |

DOIs | |

State | Published - 2010 |

### Fingerprint

### Keywords

- Internal waves
- Oceanic
- Spectral analysis
- Waves

### ASJC Scopus subject areas

- Oceanography

### Cite this

*Journal of Physical Oceanography*,

*40*(12), 2605-2623. https://doi.org/10.1175/2010JPO4132.1

**Oceanic internal-wave field : Theory of scale-invariant spectra.** / Lvov, Yuri V.; Polzin, Kurt L.; Tabak, Esteban; Yokoyama, Naoto.

Research output: Contribution to journal › Article

*Journal of Physical Oceanography*, vol. 40, no. 12, pp. 2605-2623. https://doi.org/10.1175/2010JPO4132.1

}

TY - JOUR

T1 - Oceanic internal-wave field

T2 - Theory of scale-invariant spectra

AU - Lvov, Yuri V.

AU - Polzin, Kurt L.

AU - Tabak, Esteban

AU - Yokoyama, Naoto

PY - 2010

Y1 - 2010

N2 - Steady scale-invariant solutions of a kinetic equation describing the statistics of oceanic internal gravity waves based on wave turbulence theory are investigated. It is shown in the nonrotating scale-invariant limit that the collision integral in the kinetic equation diverges for almost all spectral power-law exponents. These divergences come from resonant interactions with the smallest horizontal wavenumbers and/or the largest horizontal wavenumbers with extreme scale separations. A small domain is identified in which the scale-invariant collision integral converges and numerically find a convergent power-law solution. This numerical solution is close to the Garrett-Munk spectrum. Power-law exponents that potentially permit a balance between the infrared and ultraviolet divergences are investigated. The balanced exponents are generalizations of an exact solution of the scale-invariant kinetic equation, the Pelinovsky-Raevsky spectrum. A small but finite Coriolis parameter representing the effects of rotation is introduced into the kinetic equation to determine solutions over the divergent part of the domain using rigorous asymptotic arguments. This gives rise to the induced diffusion regime. The derivation of the kinetic equation is based on an assumption of weak nonlinearity. Dominance of the nonlocal interactions puts the self-consistency of the kinetic equation at risk. However, these weakly nonlinear stationary states are consistent with much of the observational evidence.

AB - Steady scale-invariant solutions of a kinetic equation describing the statistics of oceanic internal gravity waves based on wave turbulence theory are investigated. It is shown in the nonrotating scale-invariant limit that the collision integral in the kinetic equation diverges for almost all spectral power-law exponents. These divergences come from resonant interactions with the smallest horizontal wavenumbers and/or the largest horizontal wavenumbers with extreme scale separations. A small domain is identified in which the scale-invariant collision integral converges and numerically find a convergent power-law solution. This numerical solution is close to the Garrett-Munk spectrum. Power-law exponents that potentially permit a balance between the infrared and ultraviolet divergences are investigated. The balanced exponents are generalizations of an exact solution of the scale-invariant kinetic equation, the Pelinovsky-Raevsky spectrum. A small but finite Coriolis parameter representing the effects of rotation is introduced into the kinetic equation to determine solutions over the divergent part of the domain using rigorous asymptotic arguments. This gives rise to the induced diffusion regime. The derivation of the kinetic equation is based on an assumption of weak nonlinearity. Dominance of the nonlocal interactions puts the self-consistency of the kinetic equation at risk. However, these weakly nonlinear stationary states are consistent with much of the observational evidence.

KW - Internal waves

KW - Oceanic

KW - Spectral analysis

KW - Waves

UR - http://www.scopus.com/inward/record.url?scp=79251514292&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79251514292&partnerID=8YFLogxK

U2 - 10.1175/2010JPO4132.1

DO - 10.1175/2010JPO4132.1

M3 - Article

AN - SCOPUS:79251514292

VL - 40

SP - 2605

EP - 2623

JO - Journal of Physical Oceanography

JF - Journal of Physical Oceanography

SN - 0022-3670

IS - 12

ER -