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.

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Dedicated to 90th Birthday of Professor Frantiek Fiala     

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.

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Address: Boční II, Praha 4-Spořilov.     

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.     

Contribution to the statistical dynamics of turbulent diffusion

Zusammenfassung Bei Studium der turbulenten Diffusion passiver Teilchen gehen wir aus statistischer Dynamik des Prozesses hervor, die auf dem zu einem unendlichen System von Gleichungen führenden Formalismus der Verteilungsfunktionen beruht. Wir behandeln in Kürze das Problem der Abgeschlossenheit, das man sowohl für die Turbulenz-Dynamik, als auch für die turbulente Diffusion übereinstimmend formulieren kann. Verschieden sind nur die Ausgangs-Bewegungsgleichungen. Wir studieren die statistische Struktur des entsprechenden Operators der Übertragung (kinetisches Modell) und suchen den Zusammenhang zwischen dem Modell des Übergangs zum Gleichgewichtszustand für die grundlegende kinetische Gleichung und die Struktur des Übergangs für die Einteilchen-Dichte, Schließlich untersuchen wir ein einfaches stochastisches Modell der in der Übertragungs-Gleichung der Diffusion enthaltenen Korrelations-Funktion.     

On the possibility of quantifying diffusion and horizontal Lorentz forces at the Earth’s core surface

If the main magnetic field at the core surface changes in time only as a result of its advection by fluid motions (i.e. under the frozen-flux (FF) assumption), it should satisfy a number of integral constraints. Additional similar constraints should apply if the flow is further assumed to be tangentially geostrophic or purely toroidal. Those integral constraints have been used in the past to test each of these assumptions. Possible violation of some of the frozen-flux constraints have led to the suggestion that significant diffusion could have occurred below South Africa in the past century. This suggests that diffusion could indeed be quantified with the help of those integrals. Here, we consider this possibility in general terms, and show that when all assumptions are relaxed, the no-longer zero values taken by those integrals could be used, at least in principle, to infer some quantitative information about the amount of diffusion and horizontal Lorentz forces acting everywhere at the core surface. The extent to which those theoretical results could be of any practical use remains to be assessed.     

On the possibility of utilizing grid reports in a local model of short-range forecast of meteorological elements

Summary In the experiment described, we test the possibility of utilizing forecasts of the pressure field, contained in GRID reports, to solve the problem of horizontal boundary conditions of a local model of short-range forecast of meteorological elements. We prove that the assumption of a linear tendency of the prognostic variables in the boundary region yields good results, using the Perkey-Kreitzberg method[1], even if applied to period T=24 hrs. In the Perkey-Kreitzberg method the effect of the horizontal diffusion in the boundary region is suppressed[2]. However, considering the diffusive term apart from the procedure of the method mentioned proves detrimental to the forecast.

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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 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.     

Simulating estuarine nitrous oxide production by means of a dynamic model

We describe a dynamic model developed from a commercially available modeling package (ECoS-III) to simulate estuarine dissolved inorganic nitrogen (DIN) dynamics, and consequent N(2)O production and atmospheric flux on the timescale of tidal cycles. Simulated model state variables were NH(4)(+), NO(3)(-) and N(2)O concentrations, and salinity. Model outputs were evaluated through comparison with summer field data for the Tyne estuary, UK. The model adequately reproduced the observed axial profiles of NH(4)(+), NO(3)(-) and N(2)O concentrations. Nitrification was shown to be the dominant N(2)O source and estimates of the ratios nitrification to DIN load and N(2)O emission to DIN load are considerably lower than the corresponding values adopted in global scale models of estuarine N(2)O emissions based on DIN transformations. Hence our results are consistent with the requirement imposed by atmospheric N(2)O growth rate constraints that the amount of atmospheric N(2)O arising from agriculturally related sources, including estuarine transformations of N, be revised downward.     

A shell model for a large-scale turbulent dynamo

This article addresses the interesting and important problem of large-scale magnetic field generation in turbulent flows, using a self-consistent dynamo model recently developed. The main idea of this model is to consider the induction equation for the large-scale magnetic field, integrated consistently with the turbulent dynamics at smaller scales described by a magnetohydrodynamic shell model. The questions of dynamo action threshold, magnetic field saturation, magnetic field reversals, nature of the dynamo transition and the changes of small-scale turbulence as a consequence of the dynamo onset are discussed. In particular, the stability curve obtained by the model integration is shown in a very wide range of values of the magnetic Prandtl number not yet accessible by direct numerical simulation but more realistic for natural dynamos. Moreover, from our analysis it is shown that the large-scale dynamo transition displays a hysteretic behaviour and therefore a subcritical nature. The model successfully reproduces magnetic polarity reversals, showing the capability to generate persistence times which are increasing for decreasing magnetic diffusivity. Moreover, when the system reaches a statistically stationary dynamo state, where the large-scale magnetic field can abruptly reverse its polarity (magnetic reversal state) or not, keeping the same polarity (steady state), it shows an unmistakable tendency towards the energy equipartition for the turbulence at small scale.     

New petrophysical model describing the pressure dependence of seismic velocity

Seismic data are increasingly applied to predict the characteristics of reservoirs, as their quality improves. Since the change of pressure is a major component in exploitation of reservoirs, a thorough understanding of the influence of pressure on seismic velocity is also important. In this study we introduce the first results of the developed petrophysical model which describes the pressure dependence of seismic velocity. The model is based on the idea that microcracks in rocks open and close under the change of pressure. Laboratory measurements are presented on several sandstone samples. Longitudinal wave velocities were measured at various incremental pressures increased from 0 to 20 MPa. During the measurements, the pulse transmission technique was used and the parameters of the model were determined by using a linearized inversion method. The inversion results proved that the proposed petrophysical model well applies in practice.     

On the possibility of combining functional optimality criteria in a geodetic net

Summary The procedure of computing the optimum plan, satisfying the conditions of the criterion generated by a convex linear combination of two convex functional optimality criteria is discussed. A numerical example of computing the optimum plan of a geodetic positional net combining criteria of L-optimality and D-optimality is given.

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On the possibility of a convective vortex sheet in a fluid heated from below

Summary A theoretical explanation for equatorial accelerations is briefly reviewed. A simple laboratory dishpan experiment is suggested to explore the possibility of the formation in a fluid of what may be termed aconvective vortex sheet, characterized by a sharp zone of horizontal shear in the mean motion field and whose dynamics should tend to produce an equatorial acceleration.     

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.