Heat and water vapor transport 77 3 Numerical model implementation The model proposed here makes use of the transient heat conduction equa- tion (9) and the general gamma distribution (16). The one-dimensional flow equation can be solved by the implicit finite difference method as discussed below in section 3.1, whereas the generation of random deviates for arrival times and approach distances is described in section 3.2. 3.1 Finite difference model setup The one-dimensional transient system of equation (9) with boundary con- ditions (10) can easily be written in finite differences [15]. 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 consist- ing of 1000 elements with properties as summarized in Table 1. A temperature solution array of 1000 elements is produced at each time step and an average H was determined for each set of parameters after each simulation run. 3.2 Gamma distribution A basic procedure to generate random deviates with a gamma distribution is given by [16]. However, since this procedure assumes that β =1,amore general procedure gamdev(α, β) was developed which returns random deviates as a function of both α and β. The average approach distance h p,avg was used instead of β as a parameter during the simulation runs. Parameter β is then calculated as h p,avg /α. The Brutsaert model [1] 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. The more general procedure consists of drawing arrival rates as in the Brutsaert model with gamdev(1, 1) and then, after selecting h p,avg and α, drawing an approach distance h p with gamdev(α, β). 4Results Table 1 below shows the basic parameter set with their chosen values. The parameters such as κ, ρ, C p and ν depend to a minor extent on temperature. However, this has been ignored in the simulations. The following parameters were varied during the simulations: the friction velocity u ∗ , the average ap- proach distance h p,avg and the gamma distribution parameter α. In section 4.1 the model validation against the analytical Brutsaert solution is briefly described, after which the simulations with variable approach distance are summarized in section 4.2. 78 A.S.M. Gieske Table 1: Model parameters with their selected values. 4.1 Model validation with the Brutsaert analytical model As mentioned before, the Brutsaert model [1] can be implemented by drawing the arrival rates with gamdev(1, 1) and resetting all temperature values to the constant air temperature at z = L immediately after arrival of the eddies. This offers the opportunity to validate the stochastic numerical model against the analytical solution of the simple case with approach distance zero. The analytical solution is given by (13) with the renewal rate s given by (15). This renewal rate s depends mainly on the friction velocity u ∗ because z 0 is taken as a constant equal to 0.001 m. Figure 3 below shows the roughness Stanton number St k as a function of the surface roughness number Re ∗ with a range from 6 to 200, corresponding to a range in friction velocity from 0.1 to 2 ms −1 . 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 C 2 having a value of 4.84 based on reported experimental values [1]. 4.2 Model simulations with variable approach distances In addition to varying Re ∗ as in Figure 3, α and h p,avg were also changed systematically. Parameter α wasgiventhevalues1,2,4,9,16.Increasein α means a decrease in the variance of the gamma distribution. The average approach distance was assigned the values 0.0001 m, 0.0002 m, 0.0005 m, 0.0010 m, 0.0015 m, 0.0020 m and 0.0040 m and finally, the gamma distribu- tion parameter β was calculated as h p,avg /α. Some results are illustrated in Figures 4 and 5 below. Figure 4 shows the simulation results for the inverse roughness Stanton number St −1 k as a function of the approach distance at Re ∗ =13.34 (u ∗ =0.2 ms −1 ). The curves show a marked increase in the St −1 k value when the approach distances become larger. The curves also indicate that the heat transfer coefficient does not depend strongly on α, especially at low values of the approach distance h p . Heat and water vapor transport 79 Fig. 3. Inverse Stanton number (St −1 ) as a function of surface roughness (Re ∗ )for both the numerical model simulation and the analytical solution by Brutsaert [1]. This seems to be the case for all values of u ∗ . Because it appears that changes in α only have a minor influence on the heat transfer coefficients, a value of α = 1 is chosen to show the general response of St k to Re ∗ and h p . Fig. 4. Simulation results for the inverse roughness Stanton number St −1 k as a function of the approach distance (thickness interfacial boundary layer) at u ∗ = 0.2 ms −1 (Re ∗ =13.3). Figure 5 shows the simulation results for St −1 k versus Re ∗ for α =1. The curves show that a strong decrease of the heat transfer coefficient St (increase in St −1 ) occurs with larger approach distance. All variations show 80 A.S.M. Gieske Fig. 5. The figure shows the inverse roughness Stanton number as a function of Re ∗ for several model approach distances. The shaded bar indicates the range of reported experimental results, for simplicity only shown at Re ∗ = 10 [1, 3, 10, 20, 21, 22]. The solid line shows the results obtained with the Brutsaert analytical model (Equation 20). The black square indicates the offset from the Brutsaert line resulting from the analysis by Trombetti et al. [10]. a decrease from the simple Brutsaert model with h p = 0 (section 4.1). The reported experimental/theoretical results are shown in figure 5 where the solid line shows the results obtained with the Brutsaert analytical model (as in Fig. 3) while the shaded rectangle indicates the range of reported results. These have been indicated for simplicity at Re ∗ = 10 only. The wide range of results appears to be caused partly by the nature of the different experiments, partly by the different definitions and conventions with regard to the Stanton numbers B, St k and the drag coefficient C d (relations 2, 3, 4 and 5). 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. 5 suggests that the approach distance lies on average between 0.0002 and 0.0005 m based on the experimental evidence. The Brutsaert model [1] is St −1 =7.3 Re 1/4 ∗ Pr 1/2 (Brutsaert) (17) where the constant 7.3 is mainly based on the experiments reported by [1]. However, the value of the constant is probably as high as 9.3 based on the review by [10] and therefore it is suggested to adapt relation (20) to the following relation which is also more in accordance with [22] Heat and water vapor transport 81 St −1 =9.3 Re 1/4 ∗ Pr 1/2 (18) This leads to a slightly different relation for kB −1 from the one shown in (5) kB −1 =3.21Re 1/4 ∗ − 2 (19) In summary, the simulations show that the heat transfer from the surface is strongly dependent on the approach distance. To a lesser extent it depends on the variance in the distribution. In all cases the simulated inverse Stanton number is higher than in the simple analytical stochastic model [1] and this model with h p = 0 should therefore be seen as a special case of the more general case with h p ≥ 0. Although there is not enough recent experimental evidence to draw definite conclusions, most of the historical data seems to corroborate this. 5 Discussion The stochastic model proposed here makes use of the transient heat con- duction equation (6) and the general gamma distribution (16). The one- dimensional flow equation can be solved by implicit finite difference methods. This leads to a tridiagonal matrix equation that is inverted with the Thomas algorithm [15]. The gamma distribution then determines when and to what depth the boundary conditions need to be updated. The procedure to imple- ment the gamma distribution in the model is a generalization of the procedure described in [16]. The algorithm is simple to implement and makes it possible to generate large ensembles for statistical analysis in a short period of time. Good correspondence was achieved between the analytical solution of Brutsaert’s model with h p = 0 and the stochastic numerical solution. The simulations with the variable approach distance showed the large influence of the approach distance on the energy transfer. The heat transfer coefficient depends to a lesser extent on the variance in the distribution as modeled with parameter α. In all cases the numerically simulated heat transfer is lower than in the simple analytical stochastic model as developed by [1]. This model with h p = 0 should be seen as a special case of the more general case with h p ≥ 0. Although there is not enough recent experimental evidence to draw definite conclusions, most of the historical data appears to confirm this. The solutions for both the analytical and numerical models depend on the parameters z 0 (surface roughness), u ∗ (friction velocity) and the surface- air temperature difference (T 0 − T a ). They do not depend on z 0h , the scalar roughness length for heat transport. Indeed, as already noted by [3] (and many other authors for that matter) this auxiliary parameter is used merely to facilitate parameterization of the boundary layer; in effect it is redundant. 82 A.S.M. Gieske The uncertainty still surrounding the parameterization of heat and water vapor transfer near the Earth’s surface suggests to verify the Stanton number values for natural environments by experiment. References [1] Brutsaert W (1975) A theory for local evaporation from rough and smooth surfaces at ground level, Water Resour Res 11(4): 543-550 [2] Brutsaert W (1979 Heat and mass transfer to and from surfaces with dense vegetation or similar permeable roughness, Bnd-Layer Met. 16:365-388 [3] Brutsaert W (1982) Evaporation into the atmosphere. Reidel Pub Co, Dordrecht, The Netherlands [4] Brutsaert W (1965) A model for evaporation as a molecular diffusion process into a turbulent atmosphere. J Geophys Res 70(20): 5017-5024 [5] Harriott P (1962a) A random eddy modification of the penetration the- ory. Chemical Engineering Science 17:149-154. [6] Kolmogorov AN (1962) A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J Fluid Mech 13:82-85 [7] Obukhov AM (1971) Turbulence in an atmosphere with a non-uniform temperature. Bnd-Layer Met. 2:7-29 [8] Kays WM, Crawford ME (1993) Convective heat and mass transfer. McGraw-Hill, USA [9] Carslaw HS, Jaeger JC (1986) Conduction of Heat in Solids. Oxford University Press, UK [10] Trombetti F, Caporaloni M, Tampieri F (1978) Bulk transfer velocity to and from natural and artificial surfaces. Bnd-Layer Met. 14: 585-595 [11] Kustas WP, Humes KS, Norman JM, Moran MS (1996) Single- and Dual-Source Modeling of Energy Fluxes with Radiometric Surface Tem- perature. J Appl Meteor 35: 110-121 [12] Su Z (2005) Estimation of the surface energy balance. In: Encyclopedia of hydrological sciences : 5 Volumes. / ed. by M.G. Anderson and J.J. McDonnell. Chichester Wiley & Sons 2:731-752 [13] Harriott P (1962b) A review of Mass Transfer to Interfaces. Can J Chem Eng 4:60-69 [14] Thomas LC, Fan LT (1971) Adaptation of the surface rejuvenation model to turbulent heat and mass transfer at a solid-fluid interface. Ind Eng Chem Fundam 10 (1): 135-139 [15] Wang HF, Anderson MP (1982) Introduction to Groundwater Model- ing, Finite Difference and Finite Element Methods. W.H. Freeman and Company. San Francisco, USA Heat and water vapor transport 83 [16] Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1986) Numerical Recipes, The Art of Scientific Computing. Cambridge University Press [17] Crago R, Hervol N, Crowley R (2005) A complementary evaporation approach to the scalar roughness length. Water Res. Res. 41: W06117 [18] Verhoef A, De Bruin HAR, Van den Hurk BJJM (1997) Some practical notes on the parameter kB-1 for sparse vegetation. J Appl Met 36: 560 [19] Bird RB, Stewart WE, Lightfoot EN (1960) Transport Phenomena. Wiley and Sons, USA [20] Owen PR, Thomson WR (1962) Heat transfer across rough surfaces. Journal Fluid Mech 15: 321-334 [21] Chamberlain (1968) Transport of gases to and from surfaces with bluff and wave-like roughness elements. Quart J. Royal Met Soc 94: 318-332 [22] Dipprey DF, Sabersky RH (1963) Heat and momentum transfer in smooth and rough tubes at various Prandtl numbers. Int Journal Heat Mass Transfer 6:329-353 Stromatactic patterns formation in geological sediments: field observations versus experiments Jindrich Hladil 1 and Marek Ruzicka 2 1 Institute of Geology, Czech Academy of Sciences, Rozvojova 269, 16500 Prague, Czech Republic hladil@gli.cas.cz 2 Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Rozvojova 135, 16502 Prague, Czech Republic ruzicka@icpf.cas.cz Summary. We demonstrate a novel purely hydrodynamic concept of formation of stromatactic cavities in geological sediments, originated by Hladil (2005a,b). First, the characteristic features of these cavities are described, as for their geometry and occurrence in the sedimentary rocks, and the several existing contemporary concepts of their formation are briefly reviewed. 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. Note that the stromatactic patterns are three-dimensional cavities which are formed inside the rapidly thickening suspension/sediment. These are not the surface-related patterns like ripples or dunes. 1 Introduction Here, the problem of the stromatacta origin is formulated in the perspective of the currently existing theories and their weaknesses. The name Stromatactis was originally used as a biological name (Dupont 1881), because these objects were then believed to be remnants of organisms buried in the sediments. Despite the later counter-evidence, this name stromatactis was continuously used for this specific type of filled cavities. 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. stromatactic). A simple new concept is presented and discussed in this paper. 1.1 What are stromatacta? Stromatacta (abbreviated as ST) are, plainly said, petrified ”holes” in sedi- mentary rocks. They are very specific cavities (voids, structures, patterns) oc- curring usually in carbonate sedimentary materials. Since the origin of these Bernard J. Geurts et al. (eds), Particle Laden Flow: From Geophysical to Kolmogorov Scales, 85–94. © 2007 Springer. Printed in the Netherlands. 86 Jindrich Hladil and Marek Ruzicka particular voids is unclear, sedimentologists spent a great effort in studying this phenomenon during the last ca. 120 years, but with a little success to date. One can define ST as cavities occurring usually in sedimentary carbonate material, filled with fine-grained infiltrated internal sediment or with isopach- ous calcite cements. ST display the following basic features. The chamber is domical in shape. It has a smooth, flat, or wavy, well-defined sharp base. 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 3:1 to 6:1, say. Typically, they occur in swarms, either interconnected, forming a reticulate network, or isolated, but also individual occurrences can be found. The horizontally inter- connected structures (usually a series of dish-shaped openings with chimneys) are usually flatter in comparison with other forms. The width of stromatacta can range from millimeters or centimeters, to decimetres, or even metres. The individual ages of their origin with sedimented beds span (with some lacunae in the documentation) a giant part of the geological time-scale, with the first possible occurrences in Paleoproterozoic (billions of years). The best fossil ex- amples 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. By the above properties, summarized in Table 1, ST strongly differ from other types of cavities occurring in natural sedimentary materials. The other types can be, for instance, the following: shelter cavities, large inter- and intra- granular pores (e.g., with shells), voids related to gas bubbles (accumulated under impermeable ”umbrellas”), openings with sheet cracks, and variety of secondary hollowed structures). 1.2 How stromatacta formed? It is a more than 120-year puzzle, not fully resolved until now. Various sug- gestions, speculations, hypothesis and theories have been offered by many authors, to explain the way ST were created. 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. Another concept stems from a selective dissolution of the base material or specifically in conditions of hydro- thermal vents, accompanied with leaching, precipitation, corrosion, etc. Yet another concepts consider ST to be structures formed by non-uniform compac- tion of the sediment (maturation, de-watering), opening of shear fissures by gravitational sliding, over-pressured cracks, gas-hydrate decomposition, etc. Stromatactic patterns formation in geological sediments 87 Fig. 1. 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.g., Ba- thurst 1982; Boulvain 1993; Neuweiler et al. 2001; Aubrecht et al. 2002; Hladil 2005b; Hladil et al. 2006). Striking is the severe contrast between the great diversity of the possible solutions, linked to particular conditions and specific presumptions, and the universality of the ST shape geometry and occurrence in space and time. All the concepts mentioned above share the two following features: • They need many specific assumptions (diversity of dead bodies and/or material-related heterogeneities, up to heterogenetic/polygenetic nature of these cavities) • None can be proved experimentally, here and now 1.3 Hydrodynamic concept The marked discrepancy between the universality of the problem and the par- ticularity of the suggested solutions lead us, more and more, to a new concept of ST formation. To overcome the deficiencies of the existing approaches, the new concept should comply with the following two demands: • It should be based on a universal physical mechanism; [...]... 2 -1 2 Ω2 = 1 N =1 0 N = 10 2 2 Ω2 = 10 2 2 10 Ω2 = 1 10 -2 10 -3 10 Ω2 = 10 -4 Ω2 = 100 Ω2 = 100 10 Ω2 = 1000 -2 10 -1 10 Ω2 = 1000 0 1 10 10 -2 10 -1 t - t0 10 10 Ω2 = 0 N = 100 0 -1 2 Ω2 = 1 N = 1000 2 Ω2 = 1 1 10 10 1 Ω2= 10 -2 10 -3 10 Ω2 = 0 2 Ω2 = 10 2 2 10 10 1 2 10 0 t - t0 -4 Ω2 = 100 Ω2 = 100 10 Ω2 = 1000 -2 10 -1 10 t - t0 0 Ω2 = 1000 1 10 10 -2 10 -1 ... 10 Ω2= 0 N =1 -2 2 99 Ω2 = 100 10 10 Ω2 = 100 Ω2 = 1000 -3 Ω2 = 1000 -4 10 -2 10 -1 10 0 1 10 10 -2 10 -1 t - t0 10 0 -1 10 2 10 1 10 1 2 N = 100 -2 -3 N = 1000 Ω2 = 0 Ω2 = 1 2 10 10 0 1 10 10 t - t0 10 Ω2 = 10 Ω2 = 0 Ω2 = 100 Ω2 = 1 Ω2 = 1000 Ω2 = 1000 -4 10 -2 10 -1 10 t - t0 0 Ω2 = 10 Ω2 = 100 1 10 10 -2 10 -1 10 0 t - t0 Fig 1 Single Particle Dispersion in vertical direction at various... Ruzicka Table 2: Experiments on stromatacta formation Fig 3 Experiment E3 Effect of mixture complexity on its ability to produce void structures, see Table 2 A - cube C3 B - cube C3+C2 C - cube C3+C2 and sphere Y D - cube C3+C2 and sphere Y+B E - cube C3+C2 and matrix S E - cube C3+C2 and matrix L The experiments proved that it is possible to generate ST-like cavities in laboratory experiments, even with... BeauxArts de Belgique, 3- ieme serie, 2( 9-1 0):26 4-2 80 [5] Hladil J (2005a) Stromatactis in glass of water: An experiment simulating formation of particular cavities in limestone sediments (in Czech) Vesmir (Prague) 84 (7) :38 8 -3 94 [6] Hladil J (2005b) The formation of stromatactis-type fenestral structures during the sedimentation of experimental slurries - a possible clue to a 120-year-old puzzle about stromatactis... puzzle about stromatactis Bulletin of Geosciences 80 (3) :19 3- 2 11 [7] Hladil J, Ruzicka M, Koptikova L (2006) Stromatactis cavities in sediments and the role of coarse-grained accessories Bulletin of Geosciences 81(2):12 3- 1 46 [8] Neuweiler F, Bourque PA, Boulvain F (2001) Why is stromatactis so rare in Mesozoic carbonate mud mounds? Terra Nova 13: 33 8 -3 46 Part II Lagrangian statistics, simulation and experiments... vertical vorticity 3 2 skewness S 3 ≡ 3 / 3 3/2 with the vorticity ωi ≡ ijk ∂j uk and the brackets · denoting ensemble averaging, since the third order vorticity correlation 3 3 can distinguish by its sign cyclonic prevalence ( 3 > 0) from anticyclonic prevalence ( 3 < 0) Bartello et al [1] found a clear growth of the vertical vorticity skewness using Large Eddy Simulation (LES) with hyper-viscosity Although... in rotating stratified turbulence Yoshi Kimura1 and Jackson R Herring2 1 2 Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 46 4-8 602 JAPAN kimura@math.nagoya-u.ac.jp National Center for Atmospheric Research, P.O.Box 30 00, Boulder, Colorado 8 030 7 -3 000, USA herring@ucar.edu Diffusion in rotating and stratified fluids is one of the central subjects in geophysical and astrophysical... stromatactis mud-mound in the Pieniny Klippen Belt (Western Carpathians) Facies 47:11 3- 1 26 [2] Bathurst RGC (1982) Genesis of stromatactis cavities between submarine crusts in Palaeozoic carbonate mud buildups J Geol Soc London 139 :16 5-1 81 [3] Boulvain F (19 93) Sedimentologie et diagenese des monticules micritiques ”F2j” du Frasnien de l’Ardenne Service Geologique de Belgique, Papier Professionel 2(260): 1-4 27... and plan to study the effect of large-scale rotations or contractions on the M -dynamics References [1] U Frisch, Turbulence: The legacy of AN Kolmogorov (Cambridge University Press, 1995) [2] E Siggia, J Fluid Mech 107, (1981) 37 5 [3] S Douady, Y Couder and M.E Brachet, Phys Rev Lett 67, (1991) 9 83 [4] J Jimenez, A Wray, P Saffman and R Rogallo, J Fluid Mech 255, (19 93) 65 [5] S Chen, R.E Ecke, G.L Eyink,... divergence-free flow with zero mean-flow in the co-rotating frame of reference In this case it is most convenient to adopt a Cartesian coordinate system (x1 , x2 , x3 ) rotating at constant angular velocity Bernard J Geurts et al (eds), Particle Laden Flow: From Geophysical to Kolmogorov Scales, 115–127 © 2007 Springer Printed in the Netherlands 116 L.J.A van Bokhoven et al Ω We choose Ω = 3 > 0 without . 15: 32 1 -3 34 [21] Chamberlain (1968) Transport of gases to and from surfaces with bluff and wave-like roughness elements. Quart J. Royal Met Soc 94: 31 8 -3 32 [22] Dipprey DF, Sabersky RH (19 63) Heat. formation. Fig. 3. Experiment E3. Effect of mixture complexity on its ability to produce void structures, see Table 2. A - cube C3. B - cube C3+C2. C - cube C3+C2 and sphere Y. D - cube C3+C2 and sphere. (Prague) 84 (7) :38 8 -3 94 [6] Hladil J (2005b) The formation of stromatactis-type fenestral struc- tures during the sedimentation of experimental slurries - a possible clue to a 120-year-old puzzle