Two-dimensional turbulence on a sphere

Following Fjørtoft (Tellus, vol. 5, 1953, pp. 225–230) we undertake a spectral analysis of a non-divergent flow on a sphere. It is shown that the spherical harmonic energy spectrum is invariant under rotations of the polar axis of the spherical harmonic system and argued that a constraint of isotropy would not simplify the analysis but only exclude low-order modes. The spectral energy equation is derived and it is shown that the viscous term has a slightly different form than given in previous studies. The relations involving energy transfer within a triad of modes, which Fjørtoft (Tellus, vol. 5, 1953, pp. 225–230) derived under the condition that energy transfer is restricted to three modes, are derived under general conditions. These relations show that there are two types of interaction within a triad. The first type is where the middle mode acts as a source for the two other modes and the second type is where it acts as a sink. The inequality indicating cascade directions which was derived by Gkioulekas & Tung (J. Fluid Mech., vol. 576, 2007, pp. 173–189) in Fourier space under the assumptions of narrow band forcing and stationarity is derived in spherical harmonic space under the assumption of dominance of first type interactions. The double cascade theory of Kraichnan (Phys. Fluids, vol. 10, 1967, pp. 1417–1423) is discussed in the light of the derived equations and it is hypothesised that in flows with limited scale separation the two cascades may, to a large extent, be produced by the same triad interactions. Finally, we conclude that the spherical geometry is the optimal test ground for exploration of two-dimensional turbulence by means of simulations.

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Two-dimensional turbulence on the surface of a sphere

Journal of Fluid Mechanics ◽

10.1017/s0022112078001615 ◽

1978 ◽

Vol 87 (2) ◽

pp. 305-319 ◽

Cited By ~ 10

Author(s):

Cha-Mei Tang ◽

Steven A. Orszag

Keyword(s):

Energy Transfer ◽

Spectral Methods ◽

Large Scale ◽

Spherical Geometry ◽

Numerical Results ◽

Two Dimensional ◽

Atmospheric Flow ◽

Constants Of Motion ◽

Cartesian Geometry ◽

Back Transfer

Large-scale atmospheric flow shares certain attributes with two-dimensional turbulence. In this paper, we study the effect of spherical geometry on two-dimensional turbulence.Energy transfer is multi-component in spherical geometry in contrast to energy transfer among triads of wave vectors in Cartesian geometry. It follows that energy transfer is more local in spherical than in Cartesian geometry. Enstrophy transfer to higher wavenumbers in spherical geometry is less than enstrophy transfer to higher wavenumbers in Cartesian geometry. Since both energy and enstrophy are inviscid constants of motion, the back transfer of energy is also less in spherical than in Cartesian geometry. Therefore, with a finite viscosity, enstrophy decays more slowly in spherical geometry than in Cartesian geometry. Here these conjectures are tested numerically by spectral methods. The numerical results agree well with the conjectures.

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Energy transfer in rotating turbulence

Journal of Fluid Mechanics ◽

10.1017/s002211209700493x ◽

1997 ◽

Vol 337 ◽

pp. 303-332 ◽

Cited By ~ 186

Author(s):

CLAUDE CAMBON ◽

N. N. MANSOUR ◽

F. S. GODEFERD

Keyword(s):

Energy Transfer ◽

Rotation Axis ◽

Physical Space ◽

Nonlinear Effects ◽

Spectral Energy ◽

Rossby Number ◽

Dimensional Manifold ◽

Two Dimensional ◽

Weakly Nonlinear ◽

Plane Normal

The influence of rotation on the spectral energy transfer of homogeneous turbulence is investigated in this paper. Given the fact that linear dynamics, e.g. the inertial waves regime found in an RDT (rapid distortion theory) analysis, cannot affect a homogeneous isotropic turbulent flow, the study of nonlinear dynamics is of prime importance in the case of rotating flows. Previous theoretical (including both weakly nonlinear and EDQNM theories), experimental and DNS (direct numerical simulation) results are collected here and compared in order to give a self-consistent picture of the nonlinear effects of rotation on turbulence.The inhibition of the energy cascade, which is linked to a reduction of the dissipation rate, is shown to be related to a damping of the energy transfer due to rotation. A model for this effect is quantified by a model equation for the derivative-skewness factor, which only involves a micro-Rossby number Roω=ω′/(2Ω) – ratio of r.m.s. vorticity and background vorticity – as the relevant rotation parameter, in accordance with DNS and EDQNM results.In addition, anisotropy is shown also to develop through nonlinear interactions modified by rotation, in an intermediate range of Rossby numbers (RoL<1 and Roω>1), which is characterized by a macro-Rossby number RoL based on an integral lengthscale L and the micro-Rossby number previously defined. This anisotropy is mainly an angular drain of spectral energy which tends to concentrate energy in the wave-plane normal to the rotation axis, which is exactly both the slow and the two-dimensional manifold. In addition, a polarization of the energy distribution in this slow two-dimensional manifold enhances horizontal (normal to the rotation axis) velocity components, and underlies the anisotropic structure of the integral length-scales. Finally a generalized EDQNM (eddy damped quasi-normal Markovian) model is used to predict the underlying spectral transfer structure and all the subsequent developments of classic anisotropy indicators in physical space. The results from the model are compared to recent LES results and are shown to agree well. While the EDQNM2 model was developed to simulate ‘strong’ turbulence, it is shown that it has a strong formal analogy with recent weakly nonlinear approaches to wave turbulence.

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Spectra and energy transfer in stably stratified turbulence

Journal of Fluid Mechanics ◽

10.1017/s0022112089002648 ◽

1989 ◽

Vol 207 ◽

pp. 419-452 ◽

Cited By ~ 18

Author(s):

E. C. Itsweire ◽

K. N. Helland

Keyword(s):

Energy Transfer ◽

Spectral Characteristics ◽

Water Channel ◽

Three Dimensional ◽

Density Fluctuations ◽

Buoyancy Flux ◽

Spectral Energy ◽

Small Scale ◽

Two Dimensional ◽

Buoyancy Forces

The influence of stabilizing buoyancy forces on the spectral characteristics and spectral energy transfer of grid-generated turbulence was studied in a ten-layer closed-loop stratified water channel. The results are compared to the limiting ideal cases of the three-dimensional turbulence and two-dimensional turbulence theories. The velocity power spectra evolve from a classical isotropic shape to a shape of almost k−2 after the suppression of the net vertical mixing. This final spectral shape is rather different from the k−3 to k−4 predicted by the theory of two-dimensional turbulence and could result from the interaction between small-scale internal waves and quasi-two-dimensional turbulent structures as well as some Doppler shift of advected waves. Several lengthscales are derived from the cospectra of the vertical velocity and density fluctuations and compared with the buoyancy, overturning and viscous lengthscales measured in previous studies, e.g. Stillinger, Helland & Van Atta (1983) and Itsweire, Helland & Van Atta (1986). The smallest turbulent scale, defined when the buoyancy flux goes to zero, can be related to the peak of the cospectra of the buoyancy flux. This new relationship can be used to provide a measure of the smallest turbulent scale in cases where the buoyancy flux never goes to zero, i.e. a growing turbulent stratified shear flow. Finally, the one-dimensional energy transfer term computed from the bispectra shows evidence of a reverse energy cascade from the small scales to the large scales far from the grid where buoyancy forces dominate inertial forces. The observed reverse energy transfer could be produced by the development of quasi-two-dimensional eddies as the original three-dimensional turbulence collapses.

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