see url Issues related to the large-scale environmental aspects of particle-laden flows were addressed by considering turbulence modulation arising in high density clay-laden flows, and by focussing on transport processes in the stratosphere and its relevance to climate and weather predictions.
Fundamental aspects of transport of particles formed the topic of the second day of the colloquium. Insights from experimental and computational research were combined to understand the distortion of flow in the neighborhood of embedded particles. Aspects of Lagrangian statistics in turbulence were discussed at length, addressing the dispersion of embedded point particles. Bridging the environ-VI Preface mental and the fundamental aspects of particle-laden flows was the topic of the final day of the colloquium. The Lagrangian dispersion of particles in the context of their environmental setting was presented.
The closing lecture provided a synopsis of transport processes in particle-laden flow in which possibilities of multi-resolution, multi-physics modeling and monitoring were discussed. No citations. About us scite combines deep learning with a network of experts to evaluate the veracity of scientific work.
Funded in part by the National Science Foundation. Contact hi scite. Follow us. For more details, we refer to  or . In order to model suspended sediment transport qs , we describe sediment concentration c throughout the water column, i. Equations are due to . At this reference height, a reference concentration can be imposed as a boundary condition.
Various reference levels and concentrations exist for rivers, nearshore and laboratory conditions. Those often applied are [17, 14, 5, 21]. In this case, the reference equation of  equation 12 is used, with a reference height of 1 percent of the water depth, corresponding with the minimum reference height proposed in .
Both the gradient and the quantity of suspended sediment are largest close to the reference height. Therefore, concentration values are calculated on a grid with a quadratic point distribution on the vertical axis, such that more points are located closer to the reference height and fewer points are present higher in the water column. We started each simulation with a sinusoidal bed-form with an amplitude of 0. Next, we investigated the initial growth rate and the fastest growing sand wavelength FGM. Table 1 shows some basic values used in the simulations Suspended sediment transport 35 and the characteristics of the simulations are given in Table 2.
Where possible, typical values for sand waves in the North Sea are used. Table 1. For simulation 1, we included suspended sediment in the reference computation. Figure 3 b shows a comparison between the reference simulation and simulation 1.
The growth rate is shown for a range of wavelengths. Most remarkable is the increase of the growth rate by a factor of approximately This was unexpected as suspended sediment is assumed to be of minor importance in these circumstances. The FGM for simulation 1 is m, 80m less than in the reference simulation. However, these small variations in velocity are enough for the suspended sediment to be entrained and to settle again within one tidal cycle.
Parameters in Table 1.
More details see Fig 6 upper. Growth rate for simulation 2 solid , compared to simulation 1 dashed. For simulation characteristics, see Table 1. Suspended sediment transport 37 Fig. Comparison between simulation 1 upper and 2 lower decreased to 0. This height is used as the lowest measurable height for suspended sediment in shallow seas [10, 6]. Note that the growth rate, compared to the situation without suspended sediment, is still larger. Table 3. The growth rate of the FGM remains of the same order of magnitude.
Due to these constant values, Av might be overestimated near the bed, which is corrected for by the partial slip boundary condition. Only during minor storms suspended sediment was detected. Maximal values were around 2. Although the sediment concentration predicted in the model seems to be in a comparable order of magnitude, transport rates are too large. The most likely cause is the high entrainment of sediment into the water column.
As w turned out to be around an order of magnitude smaller then ws during most of the tide, this term was neglected in the sediment continuity equation equation 7. In that case w should be incorporated and might increase the amount of suspended sediment during a part of the tidal cycle on certain locations on the sand waves, leading to further growth or decay of the sand waves. However,  stated that this leads to unrealistically high reference levels in water depths of tens of meters.
Both heights are tested in simulations 1 and 2. The authors are indebted to Jan Ribberink for his suggestions. References  Besio, G. A note on tidally generated sand waves. Fluid Dynamics, ,  Blondeaux, P. Oceans, C, doi Flow and sediment transport induces by tide propagation:2 the wavy bottom case. The relation between currents and seasonal sand wave variability as observed with ferry-mounted adcp. Entrainment of bed sediment into suspension. Hydraulic Engg, ,  Grasmeijer, B.
Aqua Publications  Green, M. Storm sediment transport: observations from the British North Sea shelf. Continental Shelf Res. Tidal-induced large-scale regular bed form patterns in a three-dimensional shallow water model. Linear instability mechanisms for sand wave formation. Fluid Mech. Examination of reference concentration under waves and currents on the inner shelf. Marine Geology, 10 3 ,  Nemeth, A. Coastal Engineering, 53,  Passchier, S. Sand wave simulations on large domains. Grain size dependency in the occurence of sand waves.
Ocean Dynamics, DOI Sediment transport, part ii: Suspended load transport. Hydraulic Engineering, 11,  Van Rijn, L. Principles of sediment transport in rivers, estuaries and coastal seas, vol.
Grain size sorting over sand waves. Data-analysis of bed concentration of suspended sediment. Hydraulic Engg, , Breugem and W. It appears that in this situation, downward going particles are indeed found in sweeps Q4 , whereas upward going particles are preferentially concentrated in both Q1 and Q2 events. In the fully developed situation on the other hand, upward going particles are preferentially concentrated in ejections, while downward going ones are found in both Q3 and Q4 events, with a relatively increased frequency in Q3, and a decreased one in Q4.
Therefore, much research already has been done. The drift velocity results from averaging the relative velocity in the Stokes drag term of this equation. Turbophoresis is neglected, because for almost neutrally buoyant particles, it is counterbalanced by the pressure gradient working on the particles. Equation 1 physically means that the mean particle velocity i. Simonin et al. Nevertheless, some DNS results show preferential concentration for similar particles , but this seems to be mainly due to the initial conditions .
These particles can either get trapped inside the vortex, leading to an upward drift velocity, or remain outside the vortex, leading to a downward drift that enhances their apparent settling velocity, but that does not change their slip velocity. The combination of these two situations leads to a zero drift velocity for these kind of particles in homogeneous isotropic turbulence.
We inject polystyrene particles near the free surface and perform measurements at either 16 or 75 water depths from the injection point. The complete experimental setup is described in the next section. In section 4, we determine the conditionally averaged drift velocity in the vicinity of a vortex head, followed by our conclusions in section 5. The walls and bottom were made of glass in order to have a hydraulically smooth boundary. The position of the nozzle was varied from 80 cm to cm from the measurement section, i.
In the latter situations, the vertical particle velocities were zero up to experimental accuracy, which means that a fully developed situation exists. Apart from that, the statistics of that situation 46 W. The volumetric sediment concentration that was introduced was 1. A 45 mm x 45 mm measurement section was located at a distance of A PTV algorithm , which uses the displacement of the centroid of the particles to determine the particle velocities, was applied to the image with only sediment particles.
By subtracting the image of the sediment particles from the original image, an image containing only the tracer particles was obtained. This leads to x vectors with distance of 0. This seems adequate for transport process as these are goverend mostly by the large scale structure. The latter result has been found before by for example Kiger and Pan . From theory, it is expected that the vertical drift is equal to the settling velocity, but in the data, the drift is smaller a maximum of 0. There are two reasons for this discrepancy. The settling case is shown above, the fully developed situation below.
Each contour line indicates a doubled probability density. In the fully developed situation on the other hand, upward moving particles are preferentially concentrated in ejections Q2 , whereas downward moving particles are concentrated preferentially in inward interactions Q3. The settling case is shown in the upper row, the fully developed situation in the lower row. Each contour line indicates a doubled intensity and negative values are indicated with dashed lines. Both the preferential concentration of upgoing particles in ejections and of downgoing particles in inward interactions Q3 can clearly be seen in their data.
Conditional averaging is a widely used tool in turbulence research. Unfortunately, statistical convergence is quite slow, because only part of the data can be used to determine conditional averages. In LSE, a conditional average is approximated from two-point correlations. Because of homogeneity in the streamwise direction, the standard deviation and correlation function do not change with x and therefore, only a reference height y0 has to be chosen. Swirling strength does not detect the direction of the rotation, this in contrast to vorticity.
Yet it is known that both vortices in the direction of and opposite to the mean shear are encountered in boundary layer turbulence . In the fully developed situation on the other hand, the contribution to the drift velocity becomes higher for vortices higher in the water column when its intensity is not changed , and it is located below the vortex 1 1 0. Only every other vector is shown. Left for the settling case, right for the fully developed situation. It appears that the concerned structure consists of a vortex head with a sweep upstream and above of it.
A model for this kind of structure could be the type B eddies of Perry and Marusic , which consists of an spanwise oscillating vortex tube, inclined 45 degrees in the streamwise direction and rotating with the mean shear. They proposed this structures in order to obtain a better comparison with experimental data without claiming their existence. In the fully developed situation on the other hand, it seems that hairpin vortices are responsible for the drift velocity structure. We varied the distance between the introduction of sediment and the measurement location.
A spatial view of these structures shows that mainly the structures located above a spanwise vortex head rotating with the shear, are important for this downward transport. A possible eddy that could display this kind of behavior is the type B eddy from Perry and Marusic . The ejections are clearly related to hairpin vortex structures.
In this situation, the number of upward and downward moving particles is approximately equal. Particles from the near the bottom, where the largest concentration exists, are transport upwards by an ejection related to a hairpin vortex. These hairpin vortices travel in Sediment transport by coherent structures 53 Fig. Physical mechanism of particle transport in the fully developed situation. The image shows a hairpin vortex packet and two typical particles in a frame moving with the hairpin vortex packet convective velocity. From there, it might remain at the same vertical location , be transported further upwards by another hairpin vortex packet or be transported down by a sweep similar as what happens in the settling case.
Note that these light particles do not seem to settle down by gravity [also found by 13], but are transported downwards by coherent structures. References  R. Stochastic estimation of conditional structure: a review. Adrian and P. Journal of Fluid Mechanics, —, Uijttewaal  R. Adrian, C. Meinhart, and C. Vortex organization in the outer region of the turbulent boundary layer.
Journal of Fluid Mechanics, —54, Cambridge university press, Ferreira, E. Alves, J. Leal, and A. Cardoso, editors, River Flow, volume 1, pages — Cellino and U. Journal of Hydraulic Engineering, 11 —, Christensen and R. Statistical evidence of hairpin vortex packets in wall turbulence. Eaton and J.
Request PDF on ResearchGate | On Jan 1, , Bernard J. Geurts and others published Particle-Laden Flow: From Geophysical to Kolmogorov Scales. Turbulence and Shear Flow Phenomena-5 Symposium · T. Miyauchi et al. Journal of Turbulence. Volume 10, - Issue. Published online: 5 Nov
Preferential concentration of particles by turbulence. International Journal of Multiphase Flow, 20 Suppl — , Kiger and C. Journal of Fluids Engineering, —, Journal of Turbulence, 3 :1—21, FlowMaster Manual. Maxey and S. Journal of the Atmospheric Sciences, 43 11 —, Mutlu Sumer and R. Nikora and D. Journal of Hydraulic Engineering, 2 —, Perry and I. A wall-wake model for the turbulence structure of boundary layers.
PhD thesis, Stanford University, Sediment transport by coherent structures 55  O. Simonin, E. Deutsch, and J. Squires and H. Preferential concentration of marine particles in isotropic turbulence. Deep-Sea Research Part I, 43 —, Zoeteweij, R. Bastiaans, R. Kieft, and C. Eindhoven University of Technology, April Zhou, R. Adrian, S. Balachandar, and T. The cross-isentropic adiabatic circulation is slow with time scales of the order of the season to several years.
Their distribution is then dependent on the transport and mixing properties. Owing to the separation between fast horizontal adiabatic motion and slow vertical diabatic motion, the potential temperature is often used as a vertical coordinate. It is most often used as a diagnostic of transport and dynamical activity. Observations by in situ instruments and remote sensing show that the stratosphere exhibits well-mixed regions separated by transport or mixing barriers. Particularly, in the winter hemisphere, two dominating barriers are formed at the periphery of the polar vortex and in the sub-tropics that isolate the mid-latitudes from both the polar and tropical regions .
Since vertical, diabatically induced, motion is upward in the tropics and downward in the extra tropics, with the largest descent within the polar vortex, vertical tracer Bernard J. In the UTLS, the barrier associated with the subtropical jets near 30N and 30S in latitude separates the upper troposphere from the lowermost stratosphere on isentropic surfaces crossing the tropopause.
Figure 1 summarizes these processes. Scheme summarizing the Brewer-Dobson meridional circulation in the stratosphere end exchange processes. The scope of theory and modeling is to provide a qualitative and quantitative account of these observed properties. A number of progresses in this matter over the last ten years have been due to the extensive usage of Lagrangian calculations of parcel trajectories based on the analyzed winds provided by the operational meteorological centers.
It is the goal of this presentation to review the ongoing work in this direction. Such dynamics is known to produce transport barriers where tracer gradients intensify. The method applies to a tracer, usually PV, with a mean latitudinal gradient such that the longitude-latitude coordinates can be replaced by the area of embedded tracer contours and an azimuthal coordinate along the contours. The device in 1 is to bind the complex stirring of the passive scalar in the averaging over the contour. It can be shown  that Leq is always larger than the actual length of the tracer contour but that in practice the two quantities scale similarly.
Another measure of atmospheric stirring is provided by the local Lyapunov exponent [6, 7] which estimates the Lagrangian stretching as the separation rate of two close parcels over a given period of time or over a given growth. This minimum is surrounded by a very active mixing region with large stretching where air is brought from and to the mid-latitudes within a few days. The singular vectors of these transformations are the Lyapunov eigenvectors.
The eigenvector pointing to the smallest axis of the ellipse in the transformation indicates the unstable manifold or stable material line for backward time and the stable manifold or unstable material line for forward time [9, 8]. The convergence of the eigenvectors is at least as fast and generally much faster than the Lyapunov exponent .
Consequently, the tracer gradient tends, at any time, to be perpendicular to the local unstable manifold [11, 12]. Since stable and unstable manifolds are parallel in the direction of the wind for a pure shear, the transverse Lyapunov exponent vanishes in this case. As a matter of fact, when the gradient is perpendicular to the shear, it does not intensify at all.
This is visible in Fig. For further discussion, see . Transport and mixing in the stratosphere: the role of Lagrangian studies 61 Fig. This value is essentially the ratio of the vertical shear to the horizontal strain. Owing to this high aspect ratio, the dissipation of a tracer sheet is mainly a product of vertical mixing by small-scale turbulence. Present meteorological analysis provided by weather centers basically resolve the motion that induces stirring and generation of tracer sheets, but small-scale turbulence due to shear instability or gravity wave breaking is unresolved by any large-scale numerical atmo- Transport and mixing in the stratosphere: the role of Lagrangian studies 63 spheric model.
Lagrangian reconstruction methods are often used to explain the spatial and temporal variations of atmospheric tracers. Such methods, sketched in Fig. In most early studies, the reconstructed tracer was PV but recent works focus on reconstructed chemical tracers that can be compared more directly with observations.
This procedure is able to reconstruct the small-scales at time t if transport is dominated by the resolved scales of motion. The thick line shows the corresponding transect in the chemical transport model used to provide the coarse distribution of the tracer. For a more quantitative assessment, see . It is an important requirement, for the distribution of long-lived chemical species that numerical models reproduce quantitatively this circulation. A large class of models of atmospheric chemistry, denoted as chemicaltransport models, rely on the analyzed winds provided by the operational weather centers to advect the chemical compounds horizontally and vertically.
This means that vertical velocities are calculated from the the continuity equation, that is basically from the vertical integration of the horizontal divergence. Such estimate is known to be noisy by nature as the divergent circulation is weak and badly constrained by observations. Most studies indeed rely, by tradition, on 6-hourly winds. However, a 3-hourly dataset, also available from ECMWF reduces considerably the discrepancy with observations.
A chemical-transport model using this dataset is able to reproduce with good accuracy the ozone column at mid-latitude F. Comparison of the age of air between observations and model calculations. The age of an air parcel is the time spent by this parcel in the stratosphere since it entered it at tropical latitudes .
Since an air parcel is a mixture, the mean age of air is the average over all particles within the parcel. The age of air can be measured using gases like SF6 , CH4 and N2 O which are well mixed in the troposphere and are slowly destroyed in the stratosphere or CO2 which is also well mixed in the troposphere and increases with time. The observation curve  is based an aircraft measurements at about 20km. Model calculations are performed using Lagrangian trajectories initialised almost uniformly at 20km integrated backward until they cross the tropopause. The age is averaged over latitude circles.
The four curves are built using wind datasets over the cycle and calculations are done over 15 years repeating this cycle. For parcels which have not left the stratosphere after 15 years, the age is extrapolated as in . Dotted: reconstruction using the ERA winds at 6-hour interval. Dash: reconstruction using the ERA winds at 3-hour intervals with forecasts interleaved with analysis as in .
Gray solid: reconstruction using the ERA winds in the horizontal and heating rates for vertical motion. Black solid: same as previous with a correction on the horizontal isentropic divergence to balance the heating rate. Fig 9 shows that the meridional circulation calculated from such data provides a good agreement with observations. Another important factor impacting the quality of the analysis is the the model itself and the type of assimilation scheme [28, 30].
References  L. Polvani, D. Waugh, R. Plumb, On the subtropical edge of the stratospheric surf zone, J. Nakamura, Two-dimensional mixing, edge formation, and permeability diagnosed in an area coordinate, J. Haynes, E. Part I: stratosphere, J. Part II: troposphere and lower stratosphere, J.
Shuckburgh, P. Haynes, Diagnosing transport and mixing using a tracer-based coordinate system, Phys. Fluids 15 — Pierrehumbert, Large-scale horizontal mixing in planetary atmospheres, Phys. Fluids A 3 5 — Joseph, B. Legras, Relation between kinematic boundaries, stirring, and barriers for the Antarctic polar vortex, J. Koh, B. Legras, Hyperbolic lines and the stratospheric polar vortex, Chaos 12 2 — Haller, G.
Yuan, Lagrangian coherent structures and mixing in twodimensional turbulence, Physica D — Legras, R. Vautard, A guide to Liapunov vectors, in: Predictability, Vol. Klein, B. Hua, G. Lapeyre, Alignment of tracer gradient vectors in 2d turbulence, Physica D — Lapeyre, B. Hua, P. Klein, Dynamics of the orientation of active and passive scalars in two-dimensional turbulence, Phys. Fluids 13 — Shapiro, H. Wernli, N. Bond, R. Waugh, L. Polvani, Climatology of intrusions into the tropical upper troposphere, Geophys.
Legras, E. Shuckburgh, Local diagnostic of mixing and barrier modulation at the tropopause, submitted to Geophys. Plumb, R. Atkinson, M. Schoeberl, L. Lait, P. Newman, M. Lowenstein, D. Toohey, L. Avallone, C. Webster, R. May, Transport of material out of the stratospheric Arctic vortex by Rossby wave breaking, J. Balluch, P. Plumb, J. Elkins, D.
Fahey, K. Boering, G. Dutton, C. Volk, E. Keim, R. Gao, B. Daube, S. Wofsy, M. Lowenstein, J. Podolske, K. Chan, M.
Kelly, P. Newman, L. Lait, Mixing of polar air into middle latitudes as revealed by tracer-tracer scatter plots, J. Legras, I. Pisso, G. Berthet, F. Appenzeller, H. Davies, W. Norton, Fragmentation of stratospheric intrusions, J. Mariotti, M. Moustaoui, B. Legras, H. Teitelbaum, Comparison between vertical ozone soundings and reconstructed potential vorticity maps by contour advection with surgery, J.
Issartel, J. Sutton, H.
Maclean, R. Swinbank, A. Brewer, Evidence for a world circulation provided by the measurements of helium and water vapour distribution in the stratopshere, Quart. Uppala, et al. Waugh, T. Hall, Age of stratospheric air: theory, observations, and models, Rev. Andrews, et al. Scheele, P. Siegmund, P. Schoeberl, A. Douglass, Z. Zhu, S. Pawson, A comparison of the lower stratospheric age spectra derived from a general circulation Transport and mixing in the stratosphere: the role of Lagrangian studies 69 model and two assimilation systems, J.
Monge-Sanz, M. Simmons, S. Numerical modeling of heat and water vapor transport through the interfacial boundary layer into a turbulent atmosphere A. A stochastic numerical model is developed to simulate heat and water vapor transfer from a rough surface through a boundary layer into the fully turbulent atmosphere. The so-called interfacial boundary layer is conceptualized as a semistagnant layer of air in the roughness cavities at the surface into which the smallest eddies penetrate to random approach distances and with random inter-arrival times, carrying away energy, molecules, or any other scalar admixture.
The model makes use of the one-dimensional transient heat conduction equation where the boundary conditions are updated in time and space by random deviates from a general gamma distribution. The algorithm is simple to implement and allows generation of large ensembles for statistical analysis in short periods of time. The simulations were used to compare and contrast earlier results obtained for heat and mass transfer through Earth surface-air interfaces.
For this reason the transport problem is formulated here in terms of heat transport, closely following  and . Under these circumstances turbulent transport of momentum, heat, water vapor, carbon dioxide and dust particles are analogous. However, close to the surface Reynolds analogy ceases to be valid.
This stagnant layer is here referred to as interfacial sub-layer following the usual micro-meteorological convention [3, 12]. The roughness Stanton number Stk must be determined by experiment  and is a function of the type of surface roughness. The general Heat and water vapor transport 73 scalar roughness length z0c becomes z0h for heat transport and z0v for water vapor transport.
Also shown are some results for vegetated areas grass, corn and forest. The equations 5 show the close correspondence between the results obtained by  for transport of water vapor and sensible heat. Figure 1 also shows some results for vegetated areas. The set of equations 5 were developed by  and  through a stochastic analytical model, whereas  developed a more complex, partly numerical and partly analytical model.
First a short review of the theory leading to the basic analytical solution is given in section 2. The structure of the model is discussed in section 3 while the results of the simulations are given in section 4. Finally, the results are compared with previous work, ending with a few conclusions and recommendations. However, the closer to the surface the smaller these eddies become. Ultimately a limit is reached where the Reynolds number becomes too low and no more eddies can be generated. These eddies are randomly swept away into the fully turbulent stream and constantly replaced by new ones.
The procedure is well known in chemical engineering [10, 19] and appears to provide a reasonable physical and statistical picture of the transport processes across the interfacial sub-layer. Note that the roughness length for heat transport zoh does not appear as a parameter in the solution. However, in the interfacial sub-layer the Reynolds analogy is no longer valid as mentioned already in the introduction, so the physical interpretation of the roughness length for heat transport must be treated with caution. The same holds for all other scalar roughness lengths.
The Brutsaert model outlined above is simple because it assumes that the replacement of eddies takes place right at the surface. A more general version of the surface renewal model was already proposed by Harriott [5, 13] and improved later by Thomas and Fan  who considered eddies not only with random arrival times but also with variable approach distances, i. Relation 16 reduces to the Brutsaert inter-arrival time distribution 76 A. Gieske Eqn. If the time t is replaced by the distance z, the same gamma distribution can used to model the approach distances hp. The eddies are assumed to arrive to a distance hp see Figure 2 with a uniform temperature Ta causing an instantaneous temperature drop from T t, z to Ta at distances larger than hp.
In practice, an outer model boundary is assumed with constant temperature Ta at a distance of about 7. The wall itself is assumed to have a constant temperature T0. Conduction of heat takes place from the wall according to the general heat conduction equation 6 after this event until the next event when a fresh eddy arrives at another distance hp. Two random values need to be determined after each event: an inter-arrival time ti and an approach distance hp.
Harriott  did not present a fully analytical solution but rather a mixture of analytical expressions with numerical simulations. The time intervals t1 , t2 ,. After each arrival the existing temperature curve T t, z is truncated at the new approach distance hp , i. The solution between the eddy arrivals is given by the analytical solution of the heat conduction equation 9.
Up to t0 the time evolution of the temperature is governed by the conduction of heat equation. After that the temperature evolves again according to the heat conduction equation and steps 1, 2, 3 and 4 dashed lines mark the successive changes in temperature. This continues until the arrival of a fresh eddy at another approach distance. Heat and water vapor transport 77 3 Numerical model implementation The model proposed here makes use of the transient heat conduction equation 9 and the general gamma distribution In this case the implicit system of equations can be written as a tridiagonal matrix equation that is easily and quickly solved with the Thomas algorithm.
In order to make the model as realistic as possible, a grid was used consisting of elements with properties as summarized in Table 1. A temperature solution array of elements is produced at each time step and an average H was determined for each set of parameters after each simulation run. The Brutsaert model  can be implemented by drawing the arrival rates with gamdev 1, 1 and resetting the entire temperature T array to the lower temperature upon arrival of the eddies. However, this has been ignored in the simulations.
In section 4. Gieske Table 1: Model parameters with their selected values. The analytical solution is given by 13 with the renewal rate s given by It is clear that the stochastic numerical model results compare well with the analytical approach by Brutsaert [1, 3, 4]. It should be noted that both the analytical and numerical model make use of relation 15 with constant C2 having a value of 4.
The average approach distance was assigned the values 0. Some results are illustrated in Figures 4 and 5 below. Heat and water vapor transport 79 Fig. All variations show 80 A. Gieske Fig. The solid line shows the results obtained with the Brutsaert analytical model Equation The most important reviews were made by [1, 3, 10, 20, 21, 22]. It appears that the stagnant interfacial layer thickness as modeled here with the approach distance may perhaps explain the variability in reported experimental results.
The stagnant layer thickness would then be related to the type of surface roughness used in these experiments. Inspection of Fig. However, the value of the constant is probably as high as 9. To a lesser extent it depends on the variance in the distribution. This leads to a tridiagonal matrix equation that is inverted with the Thomas algorithm . The gamma distribution then determines when and to what depth the boundary conditions need to be updated. The procedure to implement the gamma distribution in the model is a generalization of the procedure described in .
The algorithm is simple to implement and makes it possible to generate large ensembles for statistical analysis in a short period of time. In all cases the numerically simulated heat transfer is lower than in the simple analytical stochastic model as developed by . They do not depend on z0h , the scalar roughness length for heat transport.
References  Brutsaert W A theory for local evaporation from rough and smooth surfaces at ground level, Water Resour Res 11 4 :  Brutsaert W Heat and mass transfer to and from surfaces with dense vegetation or similar permeable roughness, Bnd-Layer Met. Chemical Engineering Science Bnd-Layer Met. In: Encyclopedia of hydrological sciences : 5 Volumes. Anderson and J. Freeman and Company. Water Res. Quart J. We demonstrate a novel purely hydrodynamic concept of formation of stromatactic cavities in geological sediments, originated by Hladil a,b.
Then the new concept is introduced, and laboratory experiments described that were designed to validate it. Finally, the result obtained are presented and discussed, and the prospect for the future research is outlined. These are not the surface-related patterns like ripples or dunes. The name Stromatactis was originally used as a biological name Dupont , because these objects were then believed to be remnants of organisms buried in the sediments.
Singular and plural forms are not settled yet. One consistent choice seems to be stromatactis and stromatactites. The other, we prefer, is stromatactum and stromatacta adj. A simple new concept is presented and discussed in this paper. Since the origin of these Bernard J. ST display the following basic features. The chamber is domical in shape.
In contrast, its arched roof is highly irregular, ornamented with many cuspate or digitate protrusions, from large to very small, spanning a range of length scales. The vertical intersection with a plane resembles a fractal curve. Their width : height aspect ratio typically varies from to , say. Typically, they occur in swarms, either interconnected, forming a reticulate network, or isolated, but also individual occurrences can be found. The width of stromatacta can range from millimeters or centimeters, to decimetres, or even metres.
The best fossil examples of ST are from Middle Paleozoic formations, and we have good access to localities in the Barrandian area, for instance. However, their occurrence is broad range, in many parts of the world, where the relevant carbonate deposition facies are present, see Figure 1. The other types can be, for instance, the following: shelter cavities, large inter- and intragranular pores e. It is a more than year puzzle, not fully resolved until now.
The most common version, which found its place also in textbooks, is that ST are cavities that remained after decay of certain organic precursors soft-bodied organisms like sponges, microbial mats, extracellular polymers, etc. Other concept says that ST are cavities after erodible mineral aggregates. Stromatactic patterns formation in geological sediments 87 Fig. Stromatacta shapes. The advantages and disadvantages of the many solutions suggested to the ST problem are discussed in the special geological literature; this discussion is so intriguing and so voluminous that cannot be presented here e.
The universality means that the mechanism must be robust, operating both in the past and in the present, under various circumstances, being less sensitive to evolution of hydrosphere and forms of life. The experimentability means that the concept must have manageable length and time scales, and can therefore be subjected to laboratory tests. The simple and universal way of stromatactic patterns formation could thus consist of a purely physical process. We have suggested that the process is the hydrodynamic process of rapid sedimentation of complex polydisperse mixtures of nonspherical anisometric rough grains of common geological materials, under suitable hydrodynamic conditions, where the stromatacta voids, cavities are formed inside the body of the deposit material, growing below the freely sedimenting dispersion Hladil a,b; Hladil et al.
The second advantage is the fact that we can easily perform the necessary sedimentation experiments in laboratory, where the typical length and time scales are quite manageable Stromatactic patterns formation in geological sediments 89 seconds and minutes, hours. Also the consolidation, durability and secular early diagenetic changes can be imitated in ranges of days to several months. In general, these conditions in the case of ST are presently little known or unknown, as well as the hydrodynamic mechanism leading to the formation of void structures in the sediment deposit.
These two key issues present the object of our current research. In particular, some suitable conditions have already been found, and ST-like structures have been obtained in laboratory experiments. There also are some early hints towards discerning the physics behind the structures formation. Indeed, the new concept is rather simple: the ST are formed by pure sedimentation. Within the framework of the hydromechanic concept, these details are not essential for the ST formation and are not relevant to the basic formation mechanism.
We ask if the unknown hydrodynamic mechanism leading to ST formation on geological temporal and spatial scales in nature can also operate in a laboratory cell. To this end, we prepared our sedimenting mixture very similar to what is believed to be the genuine suspension of the past. This material was then homogenized by stirring and shaking and let settle in a transparent rectangular plexiglas column of volume ca.
The sedimentation process was recorded with a high-speed video, and the resulting 2D images of the near-wall sediment evolution and the patterns formation were analyzed, mainly visually. In addition, the 3D structures of the interior of the sedimentation deposit were investigated, after cutting it when frozen in liquid nitrogen.
The 3D observations were in accord with the 2D observations. The result was positive. The stromatacta-like structures were formed in the middle part of the layer of the polydisperse sediment bed seen vertically , in size of millimeters to centimeters, see Figure 2. Thus, a similar mechanism like that far before was likely Stromatactic patterns formation in geological sediments 91 operating. The result was positive, and the ST structures were obtained, with slightly reduced size.
Certain combinations produced the structures, and certain did not. The results are in Figure 3. It follows that certain proportions among polydispersity, non-sphericity, anisometricity, roughness and abrasiveness of the material is needed to obtain the stromatactic structures. In this way, the important components of the unknown physical mechanism underlying the ST formation can be disclosed. Experiment E3. A - cube C3. The experiments proved that it is possible to generate ST-like cavities in laboratory experiments, even with particle mixtures much simpler than the common geological materials.
Thus, the patterns formation in the sedimentary deposits deserve attention not only as a geological phenomenon, but also as a kind of a hydrodynamic instability in dispersed systems, where the process Stromatactic patterns formation in geological sediments 93 of the particle sedimentation is strongly coupled with the process of building of the void structures in the sediment below. This kind of instability seems to be typical for systems with polydisperse, nonspherical rough particles. The experiments are not exhaustive and represent a preliminary random mapping of the parameter space.
Starting with a simple phenomenology, we can consider the process as a sequence of several overlapping steps: mixing suspension, sedimentation, deposit formation, structure formation and structure duration. It can naturally be parameterized by time, or for convenience, also by the particle concentration. Certain physical aspects relevant for each step are indicated as: I. Sedimentation: polydisperse, nonspherical, textured particles, instabilities: planar waves, coarse graining, convective, lateral Structure duration: compaction, aging, soil mechanics One way of solution leads via increasing complexity of a simple base particle system, following the spirit of experiment E3.
Here, the relevant control parameters related to the particles seem to be these: non-sphericity, polydispersity and surface roughness. Within this three-parameter space, certain niches should exist where the ST-like structures will typically be formed. Quantitative relations can then be obtained between the ST properties dimensions, shape features and the parameters, in form of correlations, with help of dimensional analysis.
The other way goes through systematic reducing complexity of the genuine ST-forming sedimenting systems experiments E1 and E2. At a certain point, the formation ability will be lost since the key ingredients or their suitable proportion disappear. Taking this point as the base state, quantitative relations can be found between the ST properties and the control parameters, expanding the latter beyond the base state. The physical mechanisms should be understood by investigation into the hydrodynamic stability of sedimenting polydisperse mixtures.
The ultimate goal could be reached by coupling the granular rheology aspects with the hydrodynamic stability aspects. Facies  Bathurst RGC Genesis of stromatactis cavities between submarine crusts in Palaeozoic carbonate mud buildups. Vesmir Prague 84 7  Hladil J b The formation of stromatactis-type fenestral structures during the sedimentation of experimental slurries - a possible clue to a year-old puzzle about stromatactis.
Bulletin of Geosciences 80 3  Hladil J, Ruzicka M, Koptikova L Stromatactis cavities in sediments and the role of coarse-grained accessories. And for calculation of particle dispersion, we use the cubic spline interpolation method by Yeung and Pope . The time dependence in this regime is called a ballistic mode. The former corresponds to a time scale which is often called as Lagrangian time scale.
Our methodology is as follows. Particle trajectories are computed by solving 1 using the same time—marching scheme as with the velocity. Single Particle Dispersion in vertical direction at various parameters. If rotation is also imposed, we can observe that  the dispersion plots show more oscillation with rotation, and  the suppression of the vertical dispersion is enhanced. Single Particle Dispersion in horizontal direction at various parameters.
For dispersion in quasigeostrophic turbulence, reader is referred to  and the references therein. References  Yeung, P. Pope J. T J. Fluids 6: — Fluids 22— This model M. Chertkov, A. Pumir and B. Shraiman, Phys. The joint probability distribution functions of the second and third invariants of M , as well as the scaling laws of the average enstrophy, strain and energy transfer are computed by using a semi-classical method of resolution of the model.
These results are compared with direct numerical simulations DNS data. But, although these quantities are very suitable for the study of scaling laws, they do not provide much information on dynamical processes. However, the existence of coherent vortical structures has a profound impact on turbulent motion [2, 3, 4], and the strain determines the local stretching of material lines, and of vorticity itself.
Bernard J. The model proposed by M. The solutions of this system can be formally written in a path integral representation. Because of the large number of degrees of freedom, a straightforward Monte-Carlo approach is unreliable to evaluate these solutions. The formal procedure is equivalent to the saddle-point approximation for computing simple integrals. Semi-classical solutions of this model are presented and compared with direct numerical simulations DNS data [7, 9].