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Cited by 1,177Fluid Dynamics and Turbulent FlowsOceanographic and Atmospheric ProcessesOcean Waves and Remote Sensing

The DOI matches its Crossref record; 29 of the 31 references it cites were found in a registry or an index.

Waves and turbulence on a beta-plane

Rhines, Peter B.·1975·Journal of Fluid Mechanics
DOI 10.1017/s0022112075001504
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Abstract

Two-dimensional eddies in a homogeneous fluid at large Reynolds number, if closely packed, are known to evolve towards larger scales. In the presence of a restoring force, the geophysical beta-effect, this cascade produces a field of waves without loss of energy, and the turbulent migration of the dominant scale nearly ceases at a wavenumber k β = (β/2 U ) ½ independent of the initial conditions other than U , the r.m.s. particle speed, and β, the northward gradient of the Coriolis frequency. The conversion of turbulence into waves yields, in addition, more narrowly peaked wavenumber spectra and less fine-structure in the spatial maps, while smoothly distributing the energy about physical space. The theory is discussed, using known integral constraints and similarity solutions, model equations, weak-interaction wave theory (which provides the terminus for the cascade) and other linearized instability theory. Computer experiments with both finite-difference and spectral codes are reported. The central quantity is the cascade rate, defined as \[ T = 2\int_0^{\infty} kF(k)dk/U^3\langle k\rangle , \] where F is the nonlinear transfer spectrum and 〈 k 〉 the mean wavenumber of the energy spectrum. (In unforced inviscid flow T is simply U −1 d 〈 k 〉 −1 / dt , or the rate at which the dominant scale expands in time t .) T is shown to have a mean value of 3·0 × 10 −2 for pure two-dimensional turbulence, but this decreases by a factor of five at the transition to wave motion. We infer from weak-interaction theory even smaller values for k [Lt ] k β . After passing through a state of propagating waves, the homogeneous cascade tends towards a flow of alternating zonal jets which, we suggest, are almost perfectly steady. When the energy is intermittent in space, however, model equations show that the cascade is halted simply by the spreading of energy about space, and then the end state of a zonal flow is probably not achieved. The geophysical application is that the cascade of pure turbulence to large scales is defeated by wave propagation, helping to explain why the energy-containing eddies in the ocean and atmosphere, though significantly nonlinear, fail to reach the size of their respective domains, and are much smaller. For typical ocean flows, $k_{\beta}^{-1} = 70\,{\rm km} $ , while for the atmosphere, $k_{\beta}^{-1} = 1000\,{\rm km}$ . In addition the cascade generates, by itself, zonal flow (or more generally, flow along geostrophic contours).

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Show all 31 references

Research works printed but not linked: 4

  1. Batchelor, G. K. 1953 Homogeneous Turbulence. Cambridge University Press.
  2. Kenyon, K. 1967 Discussion.Proc. Roy. Soc. A299,141.
  3. Taylor, G. I. 1917 Observations and speculations on the nature of turbulent motion. Scientific Papers , vol. 2 (ed. G. K. Batchelor ),p.69.Cambridge University Press.
  4. Lorenz, E. N. 1972 Barotropic instability of Rossby wave motion J. Atmos. Sci. 29,258. · DOI

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Metadata

Authors
Rhines, P. B.
Affiliations
Woods Hole Oceanographic Institution (US)
Corresponding author
Rhines, Peter B.
Venue
Journal of Fluid Mechanics
Publisher
Cambridge University Press (CUP)
Published
1975-06-10
Volume
69
Issue
3
Pages
417-443
ISSN
  • 0022-1120 (ISSN-L)
  • 1469-7645
DOI
10.1017/s0022112075001504
OpenAlex
W2099743374
Cited by
1,177
References
31
Citations by year
2026: 15 · 2025: 27 · 2024: 50 · 2023: 38 · 2022: 43 · 2021: 44 · 2020: 48 · 2019: 71 · 2018: 42 · 2017: 38 · 2016: 28 · 2015: 39 · 2014: 36 · 2013: 38 · 2012: 28
FWCI
26.81
Type
journal article
Language
en
Open access
closed
Licence
www.cambridge.org/core/terms (unspecified, 11250 d)
Locations
Journal of Fluid Mechanics (publishedVersion)
Topics
  • Physical Sciences › Engineering › Computational Mechanics › Fluid Dynamics and Turbulent Flows
  • Physical Sciences › Earth and Planetary Sciences › Oceanography › Oceanographic and Atmospheric Processes
  • Physical Sciences › Earth and Planetary Sciences › Oceanography › Ocean Waves and Remote Sensing
Keywords
beta-plane, two-dimensional turbulence, energy cascade, wave propagation, zonal jets, energy-containing eddies, integral constraints, self-similar solutions, model equations, weakly nonlinear waves

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