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| The classical picture of turbulence | |
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| The ubiquitous nature of turbulence | |
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| The experiments of Taylor and Benard | |
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| Flow over a cylinder | |
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| Reynolds' experiment | |
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| Common themes | |
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| The ubiquitous nature of turbulence | |
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| Different scales in a turbulent flow: a glimpse at the energy cascade of Kolmogorov and Richardson | |
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| The closure problem of turbulence | |
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| Is there a 'theory of turbulence'? | |
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| The interaction of theory, computation, and experiment | |
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| The equations of fluid mechanics | |
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| The Navier-Stokes equation | |
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| Newton's second law applied to a fluid | |
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| The convective derivative | |
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| Integral versions of the momentum equation | |
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| The rate of dissipation of energy in a viscous fluid | |
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| Relating pressure to velocity | |
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| Vorticity dynamics | |
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| Vorticity and angular momentum | |
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| The vorticity equation | |
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| Kelvin's theorem | |
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| Tracking vorticity distributions | |
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| A definition of turbulence | |
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| The origins and nature of turbulence | |
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| The nature of chaos | |
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| From non-linearity to chaos | |
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| More on bifurcations | |
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| The arrow of time | |
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| Some elementary properties of freely evolving turbulence | |
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| Various stages of development | |
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| The rate of destruction of energy in fully developed turbulence | |
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| How much does the turbulence remember? | |
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| The need for a statistical approach and different methods of taking averages | |
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| Velocity correlations, structure functions and the energy spectrum | |
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| Is the asymptotic state universal? Kolmogorov's theory | |
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| The probability distribution of the velocity field | |
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| Turbulent shear flows and simple closure models | |
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| The exchange of energy between the mean flow and the turbulence | |
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| Reynolds stresses and the closure problem of turbulence | |
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| The eddy-viscosity theories of Boussinesq and Prandtl | |
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| The transfer of energy from the mean flow to the turbulence | |
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| A glimpse at the k-[epsilon] model | |
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| Wall-bounded shear flows and the log-law of the wall | |
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| Turbulent flow in a channel and the log-law of the wall | |
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| Inactive motion--a problem for the log-law? | |
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| Turbulence profiles in channel flow | |
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| The log-law for a rough wall | |
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| The structure of a turbulent boundary layer | |
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| Coherent structures | |
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| Spectra and structure functions near the wall | |
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| Free shear flows | |
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| Planar jets and wakes | |
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| The round jet | |
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| Homogeneous shear flow | |
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| The governing equations | |
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| The asymptotic state | |
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| Heat transfer in wall-bounded shear flows--the log-law revisited | |
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| Turbulent heat transfer near a surface and the log-law for temperature | |
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| The effect of stratification on the log-law--the atmospheric boundary layer | |
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| More on one-point closure models | |
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| A second look at the k-[epsilon] model | |
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| The Reynolds stress model | |
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| Large eddy simulation: a rival for one-point closures? | |
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| The phenomenology of Taylor, Richardson, and Kolmogorov | |
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| Richardson revisited | |
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| Time and length-scales in turbulence | |
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| The energy cascade pictured as the stretching of turbulent eddies | |
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| The dynamic properties of turbulent eddies | |
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| Kolmogorov revisited | |
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| Dynamics of the small scales | |
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| Turbulence induced fluctuations of a passive scalar | |
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| The intensification of vorticity and the stretching of material lines | |
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| Enstrophy production, the skewness factor, and scale invariance | |
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| Sheets or tubes? | |
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| Examples of concentrated vortex sheets and tubes | |
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| Are there singularities in the vorticity field? | |
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| The stretching of material line elements | |
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| The interplay of the strain and vorticity fields | |
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| Turbulent diffusion by continuous movements | |
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| Taylor diffusion of a single particle | |
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| Richardson's law for the relative diffusion of two particles | |
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| The influence of mean shear on turbulent dispersion | |
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| Why turbulence is never Gaussian | |
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| The experimental evidence and its interpretation | |
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| A glimpse at closure schemes which assume near-Gaussian statistics | |
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| Closure | |
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| The statistical equations for a passive scalar in isotropic turbulence: Yaglom's four-thirds Law and Corrsin's integral | |
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| Freely decaying, homogeneous turbulence | |
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| Isotropic turbulence (In real space) | |
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| Introduction: exploring isotropic turbulence in real space | |
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| Deterministic cartoons versus statistical phenomenology | |
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| The strengths and weaknesses of Fourier space | |
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| An overview of this chapter | |
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| The governing equations of isotropic turbulence | |
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| Kinematics | |
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| Dynamics | |
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| Overcoming the closure problem | |
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| The dynamics of the large scales | |
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| Loitsyansky's integral | |
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| Kolmogorov's decay laws | |
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| Landau's angular momentum | |
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| Batchelor's pressure forces | |
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| The Saffman-Birkhoff spectrum | |
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| A reappraisal of the long-range pressure forces in E-k[superscript 4] turbulence | |
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| The characteristic signature of eddies of different shape | |
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| Townsend's model eddy | |
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| Other model eddies | |
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| Intermittency in the inertial-range eddies | |
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| A problem for Kolmogorov's theory? | |
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| The [beta]-model of intermittency | |
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| The log-normal model of intermittency | |
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| The distribution of energy and vorticity across the different eddy sizes | |
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| A 'real-space' function which represents, approximately, the distribution of energy | |
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| Cascade dynamics in real space | |
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| A 'real-space' function which represents, approximately, the distribution of enstrophy | |
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| A footnote: can we capture Richardson's vision with our mathematical analysis? | |
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| Turbulence composed of Townsend's model eddy | |
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| The role of numerical simulations | |
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| What is DNS or LES? | |
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| Direct numerical simulation | |
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| Large eddy simulations | |
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| On the dangers of periodicity | |
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| Structure in chaos | |
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| Tubes, sheets, and cascades | |
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| On the taxonomy of worms | |
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| Structure and intermittency | |
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| Shear flows | |
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| Postscript | |
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| Isotropic turbulence (in spectral space) | |
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| Kinematics in spectral space | |
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| The Fourier transform and its properties | |
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| The Fourier transform as a filter | |
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| The autocorrelation function | |
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| The transform of the correlation tensor and the three-dimensional energy spectrum | |
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| One-dimensional energy spectra | |
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| Relating the energy spectrum to the second-order structure function | |
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| A footnote: singularities in the spectrum arising from anisotropy | |
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| Another footnote: the transform of the velocity field | |
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| Definitely the last footnote: what do E(k) and E[subscript 1](k) really represent? | |
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| Dynamics in spectral space | |
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| An evolution equation for E(k) | |
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| Closure in spectral space | |
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| Quasi-Normal type closure schemes (Part 2) | |
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| Special topics | |
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| The influence of rotation, stratification, and magnetic fields on turbulence | |
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| The importance of body forces in geophysics and astrophysics | |
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| The influence of rapid rotation and stable stratification | |
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| The Coriolis force | |
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| The Taylor-Proudman theorem | |
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| Properties of inertial waves | |
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| Turbulence in rapidly rotating systems | |
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| Turbulence with moderate rotation | |
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| From rotation to stratification (or from cigars to pancakes) | |
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| The influence of magnetic fields I--the MHD equations | |
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| The interaction of moving conductors and magnetic fields: a qualitative overview | |
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| From Maxwell's equations to the governing equations of MHD | |
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| Simplifying features of low magnetic Reynolds number MHD | |
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| Simple properties of high magnetic Reynolds number MHD | |
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| The influence of magnetic fields II--MHD turbulence | |
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| The growth of anisotropy in MHD turbulence | |
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| The evolution of eddies at low magnetic Reynolds number | |
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| The Landau invariant for homogeneous MHD turbulence | |
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| Decay laws at low magnetic Reynolds number | |
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| Turbulence at high magnetic Reynolds number | |
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| The combined effects of Coriolis and Lorentz forces | |
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| The shaping of eddies by Coriolis and magnetic forces | |
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| Turbulence in the core of the earth | |
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| Turbulence near the surface of the sun | |
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| Two-dimensional turbulence | |
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| The classical picture of two-dimensional turbulence: Batchelor's self-similar spectrum | |
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| What is two-dimensional turbulence? | |
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| What does the turbulence remember? | |
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| Batchelor's self-similar spectrum | |
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| The inverse energy cascade of Batchelor and Kraichnan | |
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| Different scales in two-dimensional turbulence | |
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| The shape of the energy spectrum: the k[superscript -3] law | |
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| Problems with the k[superscript -3] law | |
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| A Richardson-type law for the inertial range | |
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| Coherent vortices: a problem for the classical theory | |
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| The evidence | |
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| The significance | |
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| The governing equations in statistical form | |
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| Correlation functions, structure functions, and the energy spectrum | |
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| The two-dimensional Karman-Howarth equation and its consequences | |
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| Loitsyansky's integral in two dimensions | |
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| Variational principles for predicting the final state in confined domains | |
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| Minimum enstrophy | |
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| Maximum entropy | |
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| Quasi-two-dimensional turbulence: bridging the gap with reality | |
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| The governing equations for shallow-water, rapidly rotating flow | |
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| Karman-Howarth equation for shallow-water, rapidly rotating turbulence | |
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| Epilogue | |
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| Appendices | |
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| Vector identities and an introduction to tensor notation | |
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| Vector identities and theorems | |
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| An introduction to tensor notation | |
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| The properties of isolated vortices: invariants, far-field properties, and long-range interactions | |
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| The far-field velocity induced by an isolated eddy | |
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| The pressure distribution in the far field | |
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| Integral invariants of an isolated eddy | |
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| Long-range interactions between eddies | |
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| Long-range pressure forces in isotropic turbulence | |
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| A dynamic equation for the pressure-induced, long-range correlations | |
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| Experimental evidence for the strength of long-range pressure forces | |