Dimension Measurements for Geostrophic Turbulence

Guckenheimer, John; Buzyna, George

doi:10.1103/physrevlett.51.1438

Cited by 99 publications

(32 citation statements)

References 10 publications

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“…While the large scale structure of the flow was highly regular, the fluctuations were shown by them to be not consistent with low-dimensional dynamics. The failure to compute reliable dimension estimators for structural vacillation had previously been reported in two independent experimental studies by Read et al (1992) and Guckenheimer & Buzyna (1983).…”

Section: Transitions On the M = 3 Branchmentioning

confidence: 99%

Direct numerical simulations of bifurcations in an air-filled rotating baroclinic annulus

et al. 2006

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Three-dimensional Direct Numerical Simulation (DNS) on the nonlinear dynamics and a route to chaos in a rotating fluid subjected to lateral heating is presented here and discussed in the context of laboratory experiments in the baroclinic annulus. Following two previous preliminary studies by Randriamampianina (2002, 2003), the fluid used is air rather than a liquid as used in all other previous work. This study investigated a bifurcation sequence from the axisymmetric flow to a number of complex flows.The transition sequence, on increase of the rotation rate, from the axisymmetric solution via a steady, fully-developed baroclinic wave to chaotic flow followed a variant of the classical quasi-periodic bifurcation route, starting with a subcritical Hopf and associated saddle-node bifurcation. This was followed by a sequence of two supercritical Hopf-type bifurcations, first to an amplitude vacillation, then to a three-frequency quasi-periodic modulated amplitude vacillation (MAV), and finally to a chaotic MAV. In the context of the baroclinic annulus this sequence is unusual as the vacillation is usually found on decrease of the rotation rate from the steady wave flow.Further transitions of a steady wave with a higher wave number pointed to the possibility that a barotropic instability of the side wall boundary layers and the subsequent breakdown of these barotropic vortices may play a role in the transition to structural vacillation and, ultimately, geostrophic turbulence.

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Section: Transitions On the M = 3 Branchmentioning

confidence: 99%

Direct numerical simulations of bifurcations in an air-filled rotating baroclinic annulus

et al. 2006

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“…As mentioned in Section 1, there exists a wide body of research geared toward measuring intrinsic dimensionality [28][29][30][31][32][33][34] (see Ref. 35 for a review). Here, we have emphasized the RD of the data as a function of the sampling density.…”

Section: Relative Dimensionalitymentioning

confidence: 99%

“…In addition to estimating entropy, there exists a wide body of research developed to estimate the dimensionality of a data set [28][29][30][31][32][33][34] Images from a given class were randomly divided into two groups: Group T containing the to-be-matched "target" samples, and group N containing the samples from the population. Patches of size r ϫ r pixels were then extracted from the images in a nonoverlapping fashion.…”

Section: Introductionmentioning

confidence: 99%

Estimates of the information content and dimensionality of natural scenes from proximity distributions

Chandler

Field

2007

J. Opt. Soc. Am. A

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Natural scenes, like most all natural data sets, show considerable redundancy. Although many forms of redundancy have been investigated (e.g., pixel distributions, power spectra, contour relationships, etc.), estimates of the true entropy of natural scenes have been largely considered intractable. We describe a technique for estimating the entropy and relative dimensionality of image patches based on a function we call the proximity distribution (a nearest-neighbor technique). The advantage of this function over simple statistics such as the power spectrum is that the proximity distribution is dependent on all forms of redundancy. We demonstrate that this function can be used to estimate the entropy (redundancy) of 3 ϫ 3 patches of known entropy as well as 8 ϫ 8 patches of Gaussian white noise, natural scenes, and noise with the same power spectrum as natural scenes. The techniques are based on assumptions regarding the intrinsic dimensionality of the data, and although the estimates depend on an extrapolation model for images larger than 3 ϫ 3, we argue that this approach provides the best current estimates of the entropy and compressibility of natural-scene patches and that it provides insights into the efficiency of any coding strategy that aims to reduce redundancy. We show that the sample of 8 ϫ 8 patches of natural scenes used in this study has less than half the entropy of 8 ϫ 8 white noise and less than 60% of the entropy of noise with the same power spectrum. In addition, given a finite number of samples ͑Ͻ2 20 ͒ drawn randomly from the space of 8 ϫ 8 patches, the subspace of 8 ϫ 8 natural-scene patches shows a dimensionality that depends on the sampling density and that for low densities is significantly lower dimensional than the space of 8 ϫ 8 patches of white noise and noise with the same power spectrum.

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“…[ l ] and taking logareceived much attention (1)(2)(3)(4)(5)(6)(7)(8). Of special interest is the fractal rithms (12) gives dimensionality d. It is now recognized that for practical purposes the Balatoni-RCnyi generalized dimension (9) dp, q 2 0 , subsumes, for different q's, all currently used definitions of d (see, however, ref.…”

Section: Traduit Par Le Journal]mentioning

confidence: 99%

An efficient algorithm for estimating dimensionalities

Somorjai¹,

Ali²

1988

Can. J. Chem.

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, 979 (1988). We present an algorithm for estimating dimensionality. It is based on a multivariate k nearest neighbor (k-NN) density estimator and shows explicitly that the dependence of k-NN distances on k does not follow pure power law behavior. We achieve very high efficiency (S2000 points for up to 20 dimensions) by abandoning the distribution free feature of our algorithm; it is calibrated for two distributions and tested on a number of strange attractors, the discretized Mackay-Glass system, and trajectories of 3, 10, and 20 coupled harmonic oscillators. 979 (1988). On prtsente un algorithme permettant d'Cvaluer la dimensionnalitC. I1 est bast sur un Cvaluateur de densitt a plusieurs variables sur k voisins les plus proches (k-VP) et il dtmontre explicitement que la dependance des distances k-VP sur k ne suit pas une loi des puissances pure. On a rtalist une trks grande efficacitt (S2000 points jusqu'a 20 dimensions) lorsqu'on a abandonnt la caracttristique non paramCtrique de notre algorithme; celui-ci est calibre' pour deux distributions et on I'a CvaluC sur un certain nombre d'attracteurs ttranges, sur le systtme de Mackay-Glass discrttist ainsi que sur les trajectoires de 3, 10 et 20 oscillateurs harmoniques couplCs. On dCrnontre les dangers qui dtcoulent du plongement dans des espaces de dimensionnalitt plus Clevte.

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Dimension Measurements for Geostrophic Turbulence

Cited by 99 publications

References 10 publications

Direct numerical simulations of bifurcations in an air-filled rotating baroclinic annulus

Direct numerical simulations of bifurcations in an air-filled rotating baroclinic annulus

Estimates of the information content and dimensionality of natural scenes from proximity distributions

An efficient algorithm for estimating dimensionalities

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