scholarly journals A Generalization of Lorenz’s Model for the Predictability of Flows with Many Scales of Motion

2008 ◽  
Vol 65 (3) ◽  
pp. 1063-1076 ◽  
Author(s):  
R. Rotunno ◽  
C. Snyder
Keyword(s):  
Energy Spectrum ◽  
Large Scale ◽  
Doubling Time ◽  
Two Dimensional ◽  
Synoptic Scale ◽  
Lorenz Model ◽  
Seminal Paper ◽  
Kinetic Energy Spectrum ◽  
Scale Error ◽  
Quasigeostrophic Equations

Abstract In a seminal paper, E. N. Lorenz proposed that flows with many scales of motion in which smaller-scale error spreads to larger scales and in which the error-doubling time decreases with decreasing scale have a finite range of predictability. Although the Lorenz theory of limited predictability is widely understood and accepted, the model upon which the theory is based is less so. The primary objection to the model is that it is based on the two-dimensional vorticity equation (2DV) while simultaneously emphasizing results using a basic turbulent flow with a “−5/3” energy spectrum in the atmospheric synoptic scale instead of those using a more theoretically and observationally consistent “−3” spectrum. The present work generalizes the Lorenz model so that it may apply to the surface quasigeostrophic equations (SQGs), which are mathematically very similar to 2DV but are known to have a −5/3 kinetic energy spectrum downscale from a large-scale forcing. This generalized Lorenz model is applied here to both 2DV (with a −3 spectrum) and SQG (with a −5/3 spectrum), producing examples of flows with unlimited and limited predictability, respectively. Comparative analysis of the two models allows for the identification of the distinctive attributes of a many-scaled flow with limited predictability.

The mixing layer and its coherence examined from the point of view of two-dimensional turbulence

1988 ◽  
Vol 192 ◽  
pp. 511-534 ◽  
Author(s):  
Marcel Lesieur ◽  
Chantal Staquet ◽  
Pascal Le Roy ◽  
Pierre Comte
Keyword(s):  
Energy Spectrum ◽  
Kinetic Energy ◽  
White Noise ◽  
Mixing Layer ◽  
Finite Size ◽  
Spanwise Direction ◽  
Computational Domain ◽  
Two Dimensional ◽  
Infinite Domain ◽  
Kinetic Energy Spectrum

A two-dimensional numerical large-eddy simulation of a temporal mixing layer submitted to a white-noise perturbation is performed. It is shown that the first pairing of vortices having the same sign is responsible for the formation of a continuous spatial longitudinal energy spectrum of slope between k−4 and k−3. After two successive pairings this spectral range extends to more than 1 decade. The vorticity thickness, averaged over several calculations differing by the initial white-noise realization, is shown to grow linearly, and eventually saturates. This saturation is associated with the finite size of the computational domain.We then examine the predictability of the mixing layer, considering the growth of decorrelation between pairs of flows differing slightly at the first roll-up. The inverse cascade of error through the kinetic energy spectrum is displayed. The error rate is shown to grow exponentially, and saturates together with the levelling-off of the vorticity thickness growth. Extrapolation of these results leads to the conclusion that, in an infinite domain, the two fields would become completely decorrelated. It turns out that the two-dimensional mixing layer is an example of flow that is unpredictable and possesses a broadband kinetic energy spectrum, though composed mainly of spatially coherent structures.It is finally shown how this two-dimensional predictability analysis can be associated with the growth of a particular spanwise perturbation developing on a Kelvin-Helmholtz billow: this is done in the framework of a one-mode spectral truncation in the spanwise direction. Within this analogy, the loss of two-dimensional predictability would correspond to a return to three-dimensionality and a loss of coherence. We indicate also how a new coherent structure could then be recreated, using an eddy-viscosity assumption and the linear instability of the mean inflexional shear.

GRANULAR TURBULENCE IN TWO DIMENSIONS: MICROSCALE REYNOLDS NUMBER AND FINAL CONDENSED STATES

2012 ◽  
Vol 23 (04) ◽  
pp. 1250032 ◽  
Author(s):  
MASAHARU ISOBE
Keyword(s):  
Reynolds Number ◽  
Energy Spectrum ◽  
Large Scale ◽  
Strong Relationship ◽  
Two Dimensions ◽  
Condensed State ◽  
Einstein Condensation ◽  
Two Dimensional ◽  
Developed Turbulence ◽  
Granular Gas

Granular gases from the viewpoint of "two-dimensional turbulence" are investigated. In the quasi-elastic and thermodynamic limit, we obtained clear evidence for an enstrophy (square of vorticity) cascade and -3 exponent in the Kraichnan–Leith–Bachelor energy spectrum by performing large-scale (N ~ 16.8 million number of disks) event-driven molecular dynamics simulations. In these calculations, the enstrophy dissipation rate showed a strong relationship with the evolution of the exponent in the energy spectrum. The growth of the Reynolds number based on the microscale confirmed that the enstrophy cascade regime was that of fully developed turbulence. Moreover, a condensed state resembling Bose–Einstein condensation in decaying two-dimensional Navier–Stokes turbulence also appeared as the final attractor of the evolving granular gas in the long time limit.

Geostrophic Turbulence

1998 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Large Scale ◽  
Bottom Topography ◽  
Single Layer ◽  
Two Dimensional ◽  
Homogeneous Fluid ◽  
Coriolis Parameter ◽  
Strongly Nonlinear ◽  
Geostrophic Turbulence ◽  
Large Scale Flow ◽  
Quasigeostrophic Equations

Strongly nonlinear, rapidly rotating, stably stratified flow is called geostrophic turbulence. This subject, which blends ideas from chapters 2,4, and 5, is relevant to the large-scale flow in the Earth’s oceans and atmosphere. The quasigeostrophic equations form the basis of the study of geostrophic turbulence. We view the quasigeostrophic equations as a generalization of the vorticity equation for two-dimensional turbulence to include the important effects of stratification, bottom topography, and varying Coriolis parameter. Thus the theory of geostrophic turbulence represents an extension of the theory of two dimensional turbulence. However, its richer physics and greater applicability to real geophysical flows make geostrophic turbulence a much more interesting and important subject. This chapter offers a very brief introduction to the theory of geostrophic turbulence. We illustrate the principal ideas by separately considering the effects of bottom topography, varying Coriolis parameter, and density stratification on highly nonlinear, quasigeostrophic flow. We make no attempt at a comprehensive review. In every case, the theory of geostrophic turbulence relies almost solely on two now-familiar components: a conservation principle that energy and potential vorticity are (nearly) conserved and an irreversibility principle in the form of an appealing assumption that breaks the time-reversal symmetry of the exact (inviscid) dynamics. This irreversibility assumption takes a great many superficially dissimilar forms, fostering the misleading impression of a great many competing explanations for the same phenomena. However, broadminded analysis inevitably reveals that these competing explanations are virtually equivalent. We begin by considering the quasigeostrophic flow of a single layer of homogeneous fluid over a bumpy bottom. No case better illustrates how diverse forms of the irreversibility principle lead to the same conclusions.

Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence?

1999 ◽  
Vol 388 ◽  
pp. 259-288 ◽  
Author(s):  
ERIK LINDBORG
Keyword(s):  
Energy Spectrum ◽  
Structure Function ◽  
Structure Functions ◽  
Separation Distance ◽  
Inertial Range ◽  
Order Structure ◽  
Two Dimensional ◽  
Third Order ◽  
Kinetic Energy Spectrum ◽  
The Third

The statistical features of turbulence can be studied either through spectral quantities, such as the kinetic energy spectrum, or through structure functions, which are statistical moments of the difference between velocities at two points separated by a variable distance. In this paper structure function relations for two-dimensional turbulence are derived and compared with calculations based on wind data from 5754 airplane flights, reported in the MOZAIC data set. For the third-order structure function two relations are derived, showing that this function is generally positive in the two-dimensional case, contrary to the three-dimensional case. In the energy inertial range the third-order structure function grows linearly with separation distance and in the enstrophy inertial range it grows cubically with separation distance. A Fourier analysis shows that the linear growth is a reflection of a constant negative spectral energy flux, and the cubic growth is a reflection of a constant positive spectral enstrophy flux. Various relations between second-order structure functions and spectral quantities are also derived. The measured second-order structure functions can be divided into two different types of terms, one of the form r2/3, giving a k−5/3-range and another, including a logarithmic dependence, giving a k−3-range in the energy spectrum. The structure functions agree better with the two-dimensional isotropic relation for larger separations than for smaller separations. The flatness factor is found to grow very fast for separations of the order of some kilometres. The third-order structure function is accurately measured in the interval [30, 300] km and is found to be positive. The average enstrophy flux is measured as Πω≈1.8×10−13 s−3 and the constant in the k−3-law is measured as [Kscr ]≈0.19. It is argued that the k−3-range can be explained by two-dimensional turbulence and can be interpreted as an enstrophy inertial range, while the k−5/3-range can probably not be explained by two-dimensional turbulence and should not be interpreted as a two-dimensional energy inertial range.

Hybrid Eulerian–Lagrangian simulations for polymer–turbulence interactions

2013 ◽  
Vol 717 ◽  
pp. 535-575 ◽  
Author(s):  
Takeshi Watanabe ◽  
Toshiyuki Gotoh
Keyword(s):  
Energy Spectrum ◽  
Energy Dissipation ◽  
Kinetic Energy ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Polymer Concentration ◽  
Hybrid Approach ◽  
Weissenberg Number ◽  
Polymer Additives ◽  
Kinetic Energy Spectrum

AbstractThe effects of polymer additives on decaying isotropic turbulence are numerically investigated using a hybrid approach consisting of Brownian dynamics simulations for an enormous number of dumbbells (of the order of 10 billion,$O(1{0}^{10} )$) and direct numerical simulations of turbulence making full use of large-scale parallel computations. Reduction of the energy dissipation rate and modification of the kinetic energy spectrum in the dissipation range scale were observed when the reaction term due to the polymer additives was incorporated into the equation of motion for the solvent fluid. An increase in the polymer concentration or Weissenberg number${W}_{i} $yielded significant modifications of the turbulence statistics at small scales, such as a suppression of the local energy dissipation fluctuations. A power-law decay of the kinetic energy spectrum$E(k, t)\sim {k}^{- 4. 7} $was observed in the wavenumber range below the Kolmogorov length scale when${W}_{i} = 25$. The generation of intense vortices was suppressed by the polymer additives, consistent with previous studies using the constitutive equations. The field structures of the trace of the polymer stress depended on the intensity of its fluctuation: sheet-like structures were observed for the intermediate intensity region and filamentary structures were observed for the intense region. The results obtained with few polymers and large replicas could approximate those with many polymers and smaller replicas as far as the large-scale statistics were concerned.

Estimation of the Variability of Mesoscale Energy Spectra with Three Years of COSMO-DE Analyses

2019 ◽  
Vol 76 (2) ◽  
pp. 627-637 ◽  
Author(s):  
Tobias Selz ◽  
Lotte Bierdel ◽  
George C. Craig
Keyword(s):  
Energy Spectrum ◽  
Kinetic Energy ◽  
Potential Vorticity ◽  
Large Scale ◽  
Small Scale ◽  
Spectral Slope ◽  
Dynamical Mechanism ◽  
Kinetic Energy Spectrum ◽  
Strong Source ◽  
Highly Correlated

Abstract Research on the mesoscale kinetic energy spectrum over the past few decades has focused on finding a dynamical mechanism that gives rise to a universal spectral slope. Here we investigate the variability of the spectrum using 3 years of kilometer-resolution analyses from COSMO configured for Germany (COSMO-DE). It is shown that the mesoscale kinetic energy spectrum is highly variable in time but that a minimum in variability is found on scales around 100 km. The high variability found on the small-scale end of the spectrum (around 10 km) is positively correlated with the precipitation rate where convection is a strong source of variance. On the other hand, variability on the large-scale end (around 1000 km) is correlated with the potential vorticity, as expected for geostrophically balanced flows. Accordingly, precipitation at small scales is more highly correlated with divergent kinetic energy, and potential vorticity at large scales is more highly correlated with rotational kinetic energy. The presented findings suggest that the spectral slope and amplitude on the mesoscale range are governed by an ever-changing combination of the upscale and downscale impacts of these large- and small-scale dynamical processes rather than by a universal, intrinsically mesoscale dynamical mechanism.

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