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Stochastically generated instantaneous velocity profiles are used to reproduce the outer region of rough-wall turbulent boundary layers in a range of Reynolds numbers extending from the wind tunnel to field conditions. Each profile consists in a sequence of steps, defined by the modal velocities and representing uniform momentum zones (UMZs), separated by velocity jumps representing the internal shear layers. Height-dependent UMZ is described by a minimal set of attributes: thickness, mid-height elevation, and streamwise (modal) and vertical velocities. These are informed by experimental observations and reproducing the statistical behaviour of rough-wall turbulence and attached eddy scaling, consistent with the corresponding experimental datasets. Sets of independently generated profiles are reorganized in the streamwise direction to form a spatially consistent modal velocity field, starting from any randomly selected profile. The operation allows one to stretch or compress the velocity field in space, increases the size of the domain and adjusts the size of the largest emerging structures to the Reynolds number of the simulated flow. By imposing the autocorrelation function of the modal velocity field to be anchored on the experimental measurements, we obtain a physically based spatial resolution, which is employed in the computation of the velocity spectrum, and second-order structure functions. The results reproduce the Kolmogorov inertial range extending from the UMZ and their attached-eddy vertical organization to the very-large-scale motions (VLSMs) introduced with the reordering process. The dynamic role of VLSM is confirmed in the$$-u^{\prime }w^{\prime }$$co-spectra and in their vertical derivative, representing a scale-dependent pressure gradient contribution. 

Abstract While it has been known for some time that reducing fluids have bleached red beds adjacent to fault zones and regionally across the Colorado Plateau, the volumes of fluids expelled along faults have never been quantified. We have developed and applied a suite of one-dimensional hydrologic models to test the hypothesis that internally generated, reducing fluids migrated up sub-basin bounding faults across the Paradox Basin and bleached overlying red beds. The internal fluid driving mechanisms included are mechanical compaction, petroleum and natural gas generation, aquathermal expansion of water, and clay dewatering. The model was calibrated using pressure, temperature, porosity, permeability, and vitrinite reflectance data. Model results indicate that sediment compaction was the most important pressure generation mechanism, producing the majority of internal fluids sourced during basin evolution. Peak fluid migration occurred during the Pennsylvanian–Permian (325–300 Ma) and Cretaceous (95–65 Ma) periods, the latter being concurrent with simulated peak oil/gas generation (87–74 Ma), which likely played a role in the bleaching of red beds. Batch geochemical advection models and mass balance calculations were utilized to estimate the volume of bleaching in an idealized reservoir having a thickness (~100 m) and porosity (0.2) corresponding to bleached reservoirs observed in the Paradox Basin. Bleaching volume calculations show that internal fluid driving mechanisms were likely responsible for fault-related alteration observed within the Wingate, Morrison, and Navajo Formations in four localities across the Paradox Basin in the Colorado Plateau, Utah and Colorado, USA. The volume calculation required that 33%–55% of the total basinal fluids, composed of hydrogen-sulfide and paleo-seawater, migrated into an overlying red bed reservoir (0.5 wt% Fe2O3). 

The statistical properties of uniform momentum zones (UMZs) are extracted from laboratory and field measurements in rough wall turbulent boundary layers to formulate a set of stochastic models for the simulation of instantaneous velocity profiles. A spatiotemporally resolved velocity dataset, covering a field of view of$$8 \times 9\,{\rm m}^2$$, was obtained in the atmospheric surface layer using super-large-scale particle image velocimetry (SLPIV), as part of the Grand-scale Atmospheric Imaging Apparatus (GAIA). Wind tunnel data from a previous study are included for comparison (Heiselet al.,J. Fluid Mech., vol. 887, 2020, R1). The probability density function of UMZ attributes such as their thickness, modal velocity and averaged vertical velocity are built at varying elevations and modelled using log-normal and Gaussian distributions. Inverse transform sampling of the distributions is used to generate synthetic step-like velocity profiles that are spatially and temporally uncorrelated. Results show that in the wide range of wall-normal distances and$$Re_\tau$$up to$$\sim O(10^6)$$investigated here, shear velocity scaling is manifested in the velocity jump across shear interfaces between adjacent UMZs, and attached eddy behaviour is observed in the linear proportionality between UMZ thickness and their wall normal location. These very same characteristics are recovered in the generated instantaneous profiles, using both fully stochastic and data-driven hybrid stochastic (DHS) models, which address, in different ways, the coupling between modal velocities and UMZ thickness. Our method provides a stochastic approach for generating an ensemble of instantaneous velocity profiles, consistent with the structural organisation of UMZs, where the ensemble reproduces the logarithmic mean velocity profile and recovers significant portions of the Reynolds stresses and, thus, of the streamwise and vertical velocity variability.