Contribution to the experiment of axiomatizing the theory of turbulent diffusion

Summary The efforts to axiomatize the theory of turbulent diffusion, founded on the theory of Markovian random processes, are based on the principal logical algebraic aspects of the concept of modelling.Dedicated to RNDr. Jan Pícha, CSc., on his 60th Birthday     

A diffusion approach to the statistical analysis of Kamchatka Seismicity

A diffusion approach was used to develop a statistical model of seismicity and to analyze Kamchatka earthquakes in order to detect features in the changes that are typical of random walk processes. We proposed a hypothesis of relationships among events and used an energy criterion to decompose the earth-quake catalog into a set of sequences, with each being a Brownian process with definite spatial, temporal, and energy scales. We constructed statistical distributions for these sequences over the number of their terms and total energies, as well as distributions of the sequences over distance, time, and flight times between events. We discuss non-local properties and memory effects in the random walk under different conditions.     

The langevin model for turbulent diffusion

Abstract

The Lagrangian approach to turbulent diffusion is considered through an improved Langevin model approach. The improved model partially accounts for the interaction between pressure and viscous stresses, an interaction not included in previous studies. The model is used to obtain one and two particle dispersion relations and it permits calculation of Eulerian-Lagrangian scale relations. Turbulence Reynolds number and time span limits for Richardson's law of diffusion are obtained. Non-stationarity of the acceleration between two particles is discussed as it relates to Richardson's law and the conclusions are considered in view of both laboratory and geophysical fluid dynamical data.     

Random model of turbulent diffusion and latent parameters of the process

Zusammenfassung Wir studieren die Folgen eines Modells der unter der Voraussetzung konstruierten turbulenten Diffusion, dass die zuf?llige FunktionX(X ₀,t) eine Markowsche Funktion ist und dass der Mittelwert der Konzentration von der Beimengung der zweiten Kolmogorowschen Gleichung entspricht. Das Verlassen dieser Vorstellung führt zur Schlussfolgerung, dass man den Zustand des „Mikroobjekts” (Teilchens) für jedent-Wert als eine Klasse von Gr?ssenwerten betrachten kann, die es gleichzeitig prinzipiell m?glich ist mit beliebiger Genauigkeit zu messen. Eine Abwendung von der statistischen Relation der Unbestimmtheit wird durch die Existenz von verborgenen Parametern bei den diffusions- und Turbulenzprozessen verursacht. ihr Vorkommen ruft jedoch keinen übergang zum Kausalmodell des Prozesses hervor; die FunktionX(X ₀,t) (Lagrangesche Charakteristik vom Fliessen einer unkompressibeln Flüssigkeit) bleibt immer eine zuf?llige Funktion. Die Klasse der m?glichen Wahrscheinlichkeitswerte des übergangsp bildet jedoch keinen Markowschen Prozess im entsprechenden Phasenraum.

---

---

Address: Boční II, Praha 4-Spořilov.     

On some of the properties of the discrete model of turbulent diffusion

Zusammenfassung Den Prozess der Übertragung passiver Substanz im Felde der turbulenten Strömung untersuchen wir an einem Modell der zufälligen Bewegung. Es besteht eine enge Beziehung zwischen der geometrischen Struktur des gewählten stochastischen Modells und den Invarianten isomorpher Abbildungen, von denen die eine als ein Bewegungsgesetz der Evolution des Systems gilt und die zweite die sog. K-adische Abbildung darstellt. Diese Invarianten werden durch die Ergodizität des Systems, sein Durchmischen (die Durchmischungs-Eigenschaft) und durch die Entropie charakterisiert. Die Existenz des Durchmischungs bei der Bewegung des Systems bildet eine unerlässliche Bedingung für die Erklärung des Begriffes der Irreversibilität und des Relaxations-prozesses.

---

---

Dedicated to 90th Birthday of Professor Frantiek Fiala     

Field theoretical techniques in statistical fluid dynamics: With application to nonlinear wave dynamics

Abstract

A derivation of two-point Markovian closure is presented in classical statistical field theory formalism. It is emphasized that the procedures used in this derivation are equivalent to those employed in the quantum statistical field theory derivation of the Boltzmann equation. Application of these techniques to the study of two-dimensional flow on a β-plane yields a quasi-homogeneous, quasi-stationary transport equation and a renormalized dispersion relation for Rossby waves     

Small scale turbulent diffusion determined with underwater flow visualization

Dye plumes were generated at three depths in the seasonal thermocline between 7 and 11 m, 22 km south of Key West on 21 August 1980 and photographed at about 10 second intervals with an underwater camera system. Eleven pairs of consecutive pictures are analyzed to determine the mean current vertical shear and the width of the plumes by positioning reference points relative to the rod attached to the camera system. The relative distances of reference points are calibrated with the stereophotogrammetric method for one pair. The eddy diffusivity is calculated by use of a model of turbulent diffusion developed byTaylor (1921). Its values range from 5 to 25 cm²s^–1 for the plume widths ranging from 33 to 132 cm. The Richardson number is calculated for each pair of pictures with the vertical density gradient estimated from temperature profiles. Its values are higher than the critical value of 0.25 except for one case. The diffusivity was higher by orders of magnitude than the molecular one and indicates the presence of turbulence together with billow like features of the plumes in spite of high Richardson numbers. This suggests that the billow turbulence might be caused by effects of surface gravity waves and not by the Kelvin-Helmholtz instability.     

On the possibility of describing turbulent diffusion by means of a statistical-mechanical model

Zusammenfassung Die grundsätzliche Schwierigkeit der Konstruktion statistischer Mechanik der turbulenten Strömung besteht darin, dass es keinen allegemeinen Nachweis eines Theorems über die Eindeutigkeit der Navier-Stokes'schen Gleichung gibt. Unsere Bestrebungen wenden sich deshalb zum Studium solcher Bewegungen, die durch lineare Gleichungen ausgedrückt werden. Unter solche gehört auch die Bewegung der im Feld der turbulenten Strömung diffundierenden Teilchen, die am Modell einer zufälligen Teilchenbewegung im Phasenraum (x, p) studiert wird. Dabei wird vorausgesetzt, dass diese Bewegung aus einem gewissen Anfangszustand realisiert wird, dem ein bestimmtes, durch eine Störung des Systems bei dessen Beobachtung bedingtes anfängliches Unbestimmtheits-Gebiet entspricht. Daraus ergibt sich die Ergodizität (genauer gesagt die Quasi-Ergodizität) des Systems, die als Folge einer noch allgemeineren Eigenschaft des Systems mit Verschmierung (Durchmischung) des Anfangszustandes gilt. Vom Gesichtspunkt der Ergodizität und Irreversibilität des Prozesses wird die Struktur der statistischen Mechanik irreversibeler Prozesse studiert und ihre Beziehung zur Struktur der turbulenten Diffusion diskreter Teilchen gesucht.     

Combined hillslope diffusion and sediment transport simulation applied to landscape dynamics modelling

We present a new numerical approach for simulating geomorphic and stratigraphic processes that combines open‐channel flow with non‐uniform sediment transport law and semi‐empirical diffusive mass wasting. It is designed to facilitate modelling of surface processes across multiple space‐ and time‐scales, and under a variety of environmental and tectonic conditions. The physics of open‐channel flow is primarily based on an adapted Lagrangian formulation of shallow‐water equations. The interaction between flow and surface geology is performed by a non‐uniform total‐load sediment transport law. Additional hillslope processes are simulated using a semi‐empirical method based on a diffusion approach. In the implementation, the resolution of flow dynamics is made on a triangulated grid automatically mapped and adaptively remeshed over a regular orthogonal stratigraphic mesh. These new methods reduce computational time while preserving stability and accuracy of the physical solutions. In order to illustrate the potential of this method for landscape and sedimentary system modelling, we present a set of three generic experiments focusing on assessing the influence of contrasting erodibilities on the evolution of an active bedrock landscape. The modelled ridges morphometrics satisfy established relationships for drainage network geometry and slope distribution, and provide quantitative information on the relative impact of hillslope and channel processes, sediment discharge and alluviation. Our results suggest that contrasting erodibility can stimulate autogenic changes in erosion rate and influence the landscape morphology and preservation. This approach offers new opportunities to investigate joint landscape and sedimentary systems response to external perturbations. The possibility to define and track a large number of materials makes the implementation highly suited to model source‐to‐sink problems where material dispersion is the key question that needs to be addressed, such as natural resources exploration and basin analysis. Copyright © 2014 John Wiley & Sons, Ltd.     

3D Lagrangian modelling of saltating particles diffusion in turbulent water flow

A 3D Lagrangian model of the saltation of solid spherical particles on the bed of an open channel flow, accounting for turbulence-induced mechanisms, is proposed and employed as the key tool of the study. The differences between conventional 2D models and a proposed 3D saltation model are discussed and the advantages of the 3D model are highlighted. Particularly, the 3D model includes a special procedure allowing generation of 3D flow velocity fields. This procedure is based on the assumption that the spectra of streamwise, vertical and transverse velocity components are known at any distance from the bed. The 3D model was used to identify and quantify effects of turbulence on particle entrainment and saltation. The analysis of particle trajectories focused on their diffusive nature, clarifying: (i) the effect of particle mobility parameter; (ii) the effect of bed topography; and (iii) the effect of turbulence. Specifically, the results of numerical simulations describing the abovementioned effects on the change in time of the variance are presented. In addition, the change in time of the skewness and kurtosis, which are likely to reflect the turbulence influence on the spread of particles, are also shown. Two different diffusion regimes (local and intermediate) for each of the investigated flow conditions are confidently identified.     

On turbulent diffusion from a steady line source in the surface layer under unstable stratification

Summary Solution of the diffusion equation is obtained for a linear source in the boundary layer with unstable stratification, when the Monin-Obukhov similarity theory is used for the wind profile and coefficient of vertical turbulent exchange. On the basis of the solution obtained numerically calculations are made.     

Anisotropic turbulent thermal diffusion and thermal convection in a rapidly rotating fluid sphere

Estimates of the molecular values of magnetic, viscous and thermal diffusion suggest that the state of the Earth’s core is turbulent and that complete numerical simulation of the geodynamo is not realizable at present. Large eddy simulation of the geodynamo with modelling of the sub-grid scale turbulence must be used. Current geodynamo models effectively model the sub-grid scale turbulence with isotropic diffusivities larger than the molecular values appropriate for the core. In the Braginsky and Meytlis (1990) picture of core turbulence the thermal and viscous diffusivities are enhanced up to the molecular magnetic diffusivity in the directions of the rotation axis and mean magnetic field. We neglect the mean magnetic field herein to isolate the effects of anisotropic thermal diffusion, enhanced or diminished along the rotation axis, and explore the instability of a steady conductive basic state with zero mean flow in the Boussinesq approximation. This state is found to be more stable (less stable) as the thermal diffusion parallel to the rotation axis is increased (decreased), if the transverse thermal diffusion is fixed. To examine the effect of simultaneously varying the diffusion along and transverse to the rotation axis, the Frobenius norm is used to control for the total thermal diffusion. When the Frobenius norm of the thermal diffusion tensor is fixed, it is found that increasing the thermal diffusion parallel to the rotation axis is destabilising. This result suggests that, for a fixed total thermal diffusion, geodynamo codes with anisotropic thermal diffusion may operate at lower modified Rayleigh numbers.     

A model for the simulation of turbulent and eddy diffusion processes at heights of 0–65 km

A generalized turbulent diffusion model has been developed which evaluates the time rate of growth of a simulated cloud of particles released into a turbulent (i.e. diffusive) atmosphere. The general model, in the form of second-order differential equations, computes the three-dimensional size of the cloud as a function of time. Parameters which influence the cloud growth, and which are accounted for in the model equations, are: (1) length scales and velocity magnitudes of the diffusive field, (2) rate of viscous dissipation , (3) vertical stability as characterized by the relative adiabatic lapse rate (1/T)(g/C _p +T/z), and (4) vertical shear in the mean horizontal winds , and , for a given height and of spatial extent equal to that of the diffusing cloud. Sample results for near ground level and for upper stratospheric heights are given. For the atmospheric boundary layer case, the diffusive field is microscale turbulence. In the upper stratospheric case it is considered to be a field of highly interactive and dispersive gravity waves.     

A physically constrained wavelet-aided statistical model for multi-decadal groundwater dynamics predictions

A physically constrained wavelet-aided statistical model (PCWASM) is presented to analyse and predict monthly groundwater dynamics on multi-decadal or longer time scales. The approach retains the simplicity of regression modelling but is constrained by temporal scales of processes responsible for groundwater level variation, including aquifer recharge and pumping. The methodology integrates statistical correlations enhanced with wavelet analysis into established principles of groundwater hydraulics including convolution, superposition and the Cooper–Jacob solution. The systematic approach includes (1) identification of hydrologic trends and correlations using cross-correlation and multi-time scale wavelet analyses; (2) integrating temperature-based evapotranspiration and groundwater pumping stresses and (3) assessing model prediction performances using fixed-block k-fold cross-validation and split calibration-validation methods. The approach is applied at three hydrogeologicaly distinct sites in North Florida in the United States using over 40 years of monthly groundwater levels. The systematic approach identifies two patterns of cross-correlations between groundwater levels and historical rainfall, indicating low-frequency variabilities are critical for long-term predictions. The models performed well for predicting monthly groundwater levels from 7 to 22 years with less than 2.1 ft (0.7 m) errors. Further evaluation by the moving-block bootstrap regression indicates the PCWASM can be a reliable tool for long-term groundwater level predictions. This study provides a parsimonious approach to predict multi-decadal groundwater dynamics with the ability to discern impacts of pumping and climate change on aquifer levels. The PCWASM is computationally efficient and can be implemented using publicly available datasets. Thus, it should provide a versatile tool for managers and researchers for predicting multi-decadal monthly groundwater levels under changing climatic and pumping impacts over a long time period.     

A statistical approach to the earth's shape

Oneapproach to the study of the distribution of the Earth's surface elevation is to look for the log normal curves that best fit the hypsographic curve.A new calculation of the parameters of the log normal curves, obtained using elevation data provided by the National Geophysical Data Center Boulder Co., is presented here.A anomalous flattening is observed for the ellipsoid which best fits the oceanic abyssal plain. The global shape of the oceanic floors seems to be less flattened than the geoid, with a marked asymmetry between the two hemispheres. There is a close correlation between the distribution of continents and the elevation of oceanic floors as a function of latitude.