Many Scales Of Motion Research Articles

The physics of climate variability and climate change

The climate system is a forced, dissipative, nonlinear, complex and heterogeneous system that is out of thermodynamic equilibrium. The system exhibits natural variability on many scales of motion, in time as well as space, and it is subject to various external forcings, natural as well as anthropogenic. This paper reviews the observational evidence on climate phenomena and the governing equations of planetary-scale flow, as well as presenting the key concept of a hierarchy of models as used in the climate sciences. Recent advances in the application of dynamical systems theory, on the one hand, and of nonequilibrium statistical physics, on the other, are brought together for the first time and shown to complement each other in helping understand and predict the system's behavior. These complementary points of view permit a self-consistent handling of subgrid-scale phenomena as stochastic processes, as well as a unified handling of natural climate variability and forced climate change, along with a treatment of the crucial issues of climate sensitivity, response, and predictability.

A New Generic Method for Quantifying the Scale Predictability of the Fractal Atmosphere: Applications to Model Verification

Abstract The authors revisit the issue regarding the predictability of a flow that possesses many scales of motion raised by Lorenz in 1969 and apply the general systems theory developed by Selvam in 1990 to error diagnostics and the predictability of the fractal atmosphere. They then introduce a new generic method to quantify the scale predictability of the fractal atmosphere following the assumptions of the intrinsic inverse power law and the upscale cascade of error. The eddies (of all scales) are extracted against the instant zonal mean, and the ratio of noise (i.e., the domain-averaged square of error amplitudes) to signal (i.e., the domain-averaged square of total eddy amplitudes), referred to as noise-to-signal ratio (NSR), is defined as a measure of forecast skill. The time limit of predictability for any wavenumber can be determined by the criterion or by the criterion , where is the golden ratio and m is a scale index. The NSR is flow adaptive, bias aware, and stable in variation (in a logarithm transformation), and it offers unique advantages for model verification, allowing evaluation of different model variables, regimes, and scales in a consistent manner. In particular, an important advantage of this NSR method over the widely used anomaly correlation coefficient (ACC) method is that it could detect the successive scale predictability of different wavenumbers without the need to explicitly perform scale decomposition. As a demonstration, this new NSR method is used to examine the scale predictability of the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS) 500-hPa geopotential height.

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

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.

A KARHUNEN-LOÉVE ANALYSIS OF TURBULENT THERMAL CONVECTION

SUMMARY The Karhunen-Ldve (K-L) procedure is applied to a turbulent thermal convection database which is generated numerically through integration of the Boussinesq equation in a periodic box with stress-free boundary conditions using a Fourier collocation spectral method. This procedure generates a complete set of mutually orthogonal functions in terms of which the turbulent flow fluctuation field is represented optimally in the mean square sense. A study is performed ranging from the direct projection of the database onto the set, resulting in a considerable data compression, to developing a system of dynamical equations employing the set as a basis for approximating the Boussinesq equation. In the latter a new strategy is proposed and tested for the treatment of the mean component of the turbulent flow. Finally, the direct projection and the dynamical equations are used to study the effects of truncation on the representation of the turbulent flow. A turbulent flow involves many scales of motion. This sets the basic requirement that a direct numerical simulation must meet if it is to represent turbulence. All scales must be adequately resolved by the computational mesh. This, together with the requirement that a large sample of the set of all possible motions allowed by the governing equation is necessary to provide an adequate statistical evaluation, results in a vast database to be generated through the simulation for subsequent analysis and assessment. A technique based on the Karhunen-Lokve (K-L) procedure’ has been used extensively in extracting essential features of the flow hidden within the database. These features are then used as a basis in subsequent data compression and model reduction schemes. Some applications range from data compression in turbulent thermal convection’ to model reduction in the Ginzburg-Landau (G-L) equation3 and in a turbulent boundary layer4 The application of this procedure to the stochastic turbulent field was proposed by Lumley’ as a rational and quantitative method, referred to as principal orthogonal decomposition (POD), for extraction of organized structures from the field. New perspectives on the applications of the technique are presented in Reference 6. The technique is based on the decomposition of the turbulent flow fluctuation field into a weighted sum of mutually orthogonal eigenfunctions of the two-point correlation tensor which meet the boundary conditions and appropriate side conditions. This decomposition is unique and optimal in the sense that the random variables appearing as weights in the representation are statistically orthogonal and the mean square error resulting from the truncated representation of the flow is minimized. The energy retained in the truncated representation of the flow is maximized by ordering the eigenfunctions in the representation based on the corresponding

Eddies, Waves, Circulation, and Mixing: Statistical Geofluid Mechanics

The complexity of geophysical flows, from scales of planetary radius down to scales of molecular diffusion, has long posed a fascinating and frustrat­ ing challenge to fluid dynamicists. It is not only a tantalizing theo­ retical question but also one of practical importance. Despite determined study, understanding of the oceans and atmosphere and, especially, the pre­ diction of responses to our trespasses upon these environments remain dangerously suspect. Much of the dynamical difficulty arises from nonlinear coupling across many scales of motion. Occasionally one refers to the atmo­ sphere or the ocean. Just as often it is remarked that these systems are not altogether turbulent if that adjective is taken to connote a condition that is highly chaotic, dissipative, diffusive, or possessed of whatever other attributes that one may assign to turbulence. Examples of wavelike phenomena abound but often are partly obscured by nonlinear interactions. Persistent, coherent, finite-amplitude flow features also are observed. Moreover there is the disturbingly nontrivial problem of distinguishing mean and fluctuating fields. Geophysical flows often are characterized by spectra, both in frequency and in wave number, that are continuous and red in character. Then the definition of a mean field becomes more or less

Oceanic microstructure

Interest in small scale structures of temperature, salt, and density in the sea arose during the last decade from the convergence of several factors. First, it became clear from the study of oceanic circulation models that large‐scale oceanic behavior depends critically on the spatial distribution and magnitude of dissipative processes [Niller, 1969]. In present‐day numerical models, these processes are incorporated parametrically by assumed relations between the dissipative processes and some of the large‐scale variables treated by the models [Robinson, 1965; Bryan, 1968], This method is recognized to be imperfect because the relationships themselves are doubtful at best and the distribution of dissipative processes and their magnitudes can scarcely be estimated. The dissipative processes themselves probably embody many scales of motion in the sea, but they are certain to leave an imprint on the microstructure.

The predictability of a flow which possesses many scales of motion

It is proposed that certain formally deterministic fluid systems which possess many scales of motion are observationally indistinguishable from indeterministic systems; specifically, that two states of the system differing initially by a small “observational error” will evolve into two states differing as greatly as randomly chosen states of the system within a finite time interval, which cannot be lengthened by reducing the amplitude of the initial error. The hypothesis is investigated with a simple mathematical model. An equation whose dependent variables are ensemble averages of the “error energy” in separate scales of motion is derived from the vorticity equation which governs two-dimensional incompressible flow. Solutions of the equation are determined by numerical integration, for cases where the horizontal extent and total energy of the system are comparable to those of the earth's atomsphere. It is found that each scale of motion possesses an intrinsic finite range of predictability, provided that the total energy of the system does not fall off too rapidly with decreasing wave length. With the chosen values of the constants, “cumulus-scale” motions can be predicted about one hour, “synoptic-scale” motions a few days, and the largest scales a few weeks in advance. The applicability of the model to real physical systems, including the earth's atmosphere, is considered. DOI: 10.1111/j.2153-3490.1969.tb00444.x

The predictability of a flow which possesses many scales of motion

The predictability of a flow which possesses many scales of motion