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

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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Crossref · 10.1017/s0022112075001504

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  • Based on the Rhines scale LR ∼ (uRMS/β)1/2, an increase of the jets width during the transition is expected, since the RMS eddy velocity increases.

    “Based on the Rhines scale L R ∼ ( u RMS /β ) 1 / 2 (Rhines 1975), where we recall that u RMS takes into account all the components of the flow (equation (3.1)), we do expect an increase of the jets width during the transition, since the RMS eddy velocity increases.”

    From Zonal jets at the laboratory scale: hysteresis and Rossby waves resonance (2021) · cites this paper as Rhines 1975

  • The Rhines scale has often been associated with the width of zonal jets.

    “Originally defined in Rhines (1975), it has often been associated with the width of zonal jets.”

    From Zonal jets at the laboratory scale: hysteresis and Rossby waves resonance (2021) · cites this paper as Rhines 1975

  • The Rhines scale L_R is the scale at which the inertial term equates the β term; below it the advective term dominates and above it the β-term is dominant.

    “Under this scale, the advective term dominates and above it the β -term is dominant.”

    From Zonal jets at the laboratory scale: hysteresis and Rossby waves resonance (2021) · cites this paper as Rhines 1975

  • The latitudinal difference in jet and vortex structure is likely driven by the topographical beta effect, which is stronger near the equator and promotes axisymmetric jets, while at higher latitudes the beta effect decreases and the system approaches a rotating plane layer where giant vortices are favored.

    “This is likely driven by the topographical β effect, due to the spherical shell geometry, which is stronger near the equator, promoting strong axisymmetric jets, while, at higher latitudes, the β -effect decreases and the system approaches a rotating plane layer where the formation of giant vortices is favored”

    From Deep rotating convection generates the polar hexagon on Saturn (2020) · cites this paper as [38]

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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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