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The attached-eddy model (AEM) predicts mean velocity and streamwise velocity variance profiles that follow a logarithmic shape in the overlap region of high Reynolds number wall-bounded turbulent flows. Moreover, the AEM coefficients are presumed to attain asymptotically constant values at very high Reynolds numbers. Here, the logarithmic behaviour of the AEM predictions in the near-neutral atmospheric surface layer is examined using sonic anemometer measurements from a 62-m meteorological tower located in the Eastern Snake River Plain, Idaho, US. Utilizing an extensive 210-day dataset, the inertial sublayer (ISL) is first identified by analyzing the measured momentum flux and mean velocity profile. The logarithmic behaviour of the streamwise velocity variance and the associated `-1' scaling of the streamwise velocity energy spectra are then investigated. The findings indicate that the Townsend-Perry coefficient ($A_1$) is influenced by mild non-stationarity that manifests itself as a Reynolds number dependence. After excluding non-stationary runs and requiring a Reynolds number higher than $4 \times 10^7$, the inferred $A_1$ converges to values ranging between 1 and 1.25, consistent with laboratory experiments. Moreover, the independence of the normalized vertical velocity variance from the wall-normal distance in the ISL is further checked and the constant coefficient value agrees with reported laboratory experiments at very high Reynolds numbers as well as many surface layer experiments. Furthermore, nine benchmark cases selected through a restrictive quality control reveal a closer relationship between the `-1' scaling in the streamwise velocity energy spectrum and the logarithmic behaviour of streamwise velocity variance at higher Reynolds numbers, though no direct equivalence between them is observed.

One of the main statistical features of near-neutral atmospheric boundary layer (ABL) turbulence is the positive vertical velocity skewness $Sk_w$ above the roughness sublayer or the buffer region in smooth-walls. The $Sk_w$ variations are receiving renewed interest in many climate-related parameterizations of the ABL given their significance to cloud formation and to testing sub-grid schemes for Large Eddy Simulations (LES). The vertical variations of $Sk_w$ are explored here using high Reynolds number wind tunnel and flume experiments collected above smooth, rough, and permeable-walls in the absence of buoyancy and Coriolis effects. These laboratory experiments form a necessary starting point to probe the canonical structure of $Sk_w$ as they deal with a key limiting case (i.e. near-neutral conditions) that has received much less attention compared to its convective counterpart in atmospheric turbulence studies. Diagnostic models based on cumulant expansions, realizability constraints, and the now-popular constant mass flux approach routinely employed in the convective boundary layer as well as prognostic models based on third-order budgets are used to explain variations in $Sk_w$ for the idealized laboratory conditions. The failure of flux-gradient relations to model $Sk_w$ from the gradients of the vertical velocity variance $\sigma_w^2$ are explained and corrections based on models of energy transport offered. Novel links between the diagnostic and prognostic models are also featured, especially for the inertial term in the third order budget of the vertical velocity fluctuation. The co-spectral properties of $w'/\sigma_w$ versus $w'^2/\sigma_w^2$ are also presented for the first time to assess the dominant scales governing $Sk_w$ in the inner and outer layers, where $w'$ is the fluctuating vertical velocity and $\sigma_w$ is the vertical velocity standard deviation.

We provide empirical evidence that within the inertial sub layer of adiabatic turbulent flows over smooth walls, the skewness of the vertical velocity component $Sk_w$ displays universal behaviour, being constant and constrained within the range $Sk_w \approx 0.1-0.16$, regardless of flow configuration and Reynolds number. A theoretical model is proposed to explain the observed behaviour, including the observed range of variations of $Sk_w$. The model clarifies why $Sk_w$ cannot be predicted from down-gradient closure approximations routinely employed in meteorological and climate models whereby $Sk_w$ impacts cloud formation and dispersion processes. The model also offers an alternative and implementable approach.

Turbulent flows over a large surface area (S) covered by n obstacles experience an overall drag due to the presence of the ground and the protruding obstacles into the flow. The drag partition between the roughness obstacles and the ground is analyzed using an analytical model proposed by Raupach (1992) and is hereafter referred to as R92. The R92 is based on the premise that the wake behind an isolated roughness element can be described by a shelter area A and a shelter volume V. The individual sizes of A and V without any interference from other obstacles can be determined from scaling analysis for the spread of wakes. To upscale from an individual roughness element to n/S elements where wakes may interact, R92 adopted a background stress re-normalizing instead of reducing A or V with each element addition. This work shows that R92's approach only converges to a linear reduction in A and V for small n/S where wakes have low probability of interacting with one another. This probabilistic nature suggests that up-scaling from individual to multiple roughness elements can be re-formulated using stochastic averaging methods proposed here. The two approaches are shown to recover R92 under plausible conditions. Comparisons between R92 and available data on blocks and vegetation-like roughness elements confirm the practical utility of R92 and its potential use in large-scale models provided the relevant parameters accommodate certain features of the roughness element type (cube versus vegetation-like) and, to a lesser extent, their configuration throughout S.

The drag coefficient $C_d$ for a rigid and uniformly distributed rod canopy covering a sloping channel following the instantaneous collapse of a dam was examined using flume experiments. The measurements included space $x$ and time $t$ high resolution images of the water surface $h(x,t)$ for multiple channel bed slopes $S_o$ and water depths behind the dam $H_o$ along with drag estimates provided by sequential load cells. Analysis of the Saint-Venant Equation (SVE) for the front speed using the diffusive wave approximation lead to a front velocity $U_f=\sqrt{\Gamma_h 2 g \phi_v'/(C_d m D)}$, where $\Gamma_h=-\partial h/\partial x$, $g$ is the gravitational acceleration, $\phi_v'=1-\phi_v$ is fluid volume fraction per ground area, $\phi_v=m \pi D^2/4$ is the solid volume fraction per ground area, $m$ is the number of rods per ground area, and $D$ is the rod diameter. An inferred $C_d=0.4$ from the $h(x,t)$ data near the advancing front region, also confirmed by load cell measurements, is much reduced relative to its independently measured steady-uniform flow case. This finding suggests that drag reduction mechanisms associated with transients and flow disturbances are more likely to play a dominant role when compared to conventional sheltering or blocking effects on $C_d$ examined in uniform flow. The increased air volume entrained into the advancing wave front region as determined from an inflow-outflow volume balance partly explains the $C_d$ reduction from unity.

Conventional and recently developed approaches for estimating turbulent scalar fluxes under stable conditions are evaluated. The focus is on methods that do not require fast scalar sensors such as the relaxed eddy accumulation (REA) approach, the disjunct eddy-covariance (DEC) approach, and a novel mixing length parametrization labelled as A22. Using high-frequency measurements collected from two contrasting sites (Utqiagvik, Alaska and Wendell, Idaho "during winter"), it is shown that the REA and A22 models outperform the conventional Monin-Obukhov Similarity Theory (MOST) utilized in Earth System Models. With slow trace gas sensors used in disjunct eddy-covariance (DEC) approaches and the more complex signal filtering associated with REA devices (here simulated using filtered signals from fast-response sensors), A22 outperforms REA and DEC in predicting the observed unfiltered (total) eddy-covariance (EC) fluxes. However, REA and DEC can still capture the observed filtered EC fluxes computed with the filtered scalar signal. This finding motivates the development of a correction, blending the REA and DEC methods, for the underestimated net averaged fluxes to incorporate the effect of sensor filtering. The only needed parameter for this correction is the mean velocity at the instrument height, a surrogate of the advective timescale., Comment: 14 pages, 8 figures, 2 tables

A generalized logarithmic law for high-order moments of passive scalars is proposed for turbulent boundary layers. This law is analogous to the generalized log law that has been proposed for high-order moments of the turbulent longitudinal velocity and is derived by combining the random sweeping decorrelation hypothesis with a spectral model informed by the attached eddy hypothesis. The proposed theory predicts that the high-order moments of passive scalar fluctuations within the inertial sublayer will vary logarithmically with wall-normal distance ($z$). The proposed theory is evaluated using high frequency time-series measurements of temperature and streamwise velocity fluctuations obtained in the first meter of the atmospheric surface layer (ASL) under near-neutral thermal stratification. The logarithmic dependence with $z$ within the inertial sublayer is observed in both the air temperature and velocity moments, with good agreement to the predictions from the proposed theory. Surprisingly, the proposed theory appears to be as, if not more, valid for transported passive scalars than for the longitudinal velocity.

A persistent spatial organization of eddies is identified in the lowest portion of the stably-stratified planetary boundary layer. The analysis uses flow realizations from published large-eddy simulations (Sullivan et al., J Atmos Sci 73(4):1815-1840, 2016) ranging in stability from neutral to nearly z-less stratification. The coherent turbulent structure is well approximated as a series of uniform momentum zones (UMZs) and uniform temperature zones (UTZs) separated by thin layers of intense gradients that are significantly greater than the mean. This pattern yields stairstep-like instantaneous flow profiles whose shape is distinct from the mean profiles that emerge from long-term averaging. However, the scaling of the stairstep organization is closely related to the resulting mean profiles. The differences in velocity and temperature across the thin gradient layers remain proportional to the surface momentum and heat flux conditions regardless of stratification. The vertical thickness of UMZs and UTZs is proportional to height above the surface for neutral and weak stratification, but becomes thinner and less dependent on height as the stability increases. Deviations from the logarithmic mean profiles for velocity and temperature observed under neutral conditions are therefore predominately due to the reduction in zone size with increasing stratification, which is empirically captured by existing Monin-Obukhov similarity relations for momentum and heat. The zone properties are additionally used to explain trends in the turbulent Prandtl number, thus providing a connection between the eddy organization, mean profiles, and turbulent diffusivity in stably stratified conditions., Comment: 35 pages, 12 figures

Drawing on an analogy to critical phenomena, it was shown that the Nikuradse turbulent friction factor ($f_t$) measurements in pipes of radius $R$ and wall roughness $r$ can be collapsed onto a one-dimensional curve expressed as a conveyance law $f_t Re^{1/4}=g_o(\chi)$, where $Re$ is a bulk Reynolds number, $\chi =Re^{3/4}\left({r}/{R}\right)$. The implicit function $g_o(.)$ was conjectured based on matching two asymptotic limits of $f_t$. However, the connection between $g_o(.)$ and the phenomenon it proclaims to represent - turbulent eddies - remains lacking. Using models for the wall-normal velocity spectrum and return-to-isotropy for pressure-strain effects to close a co-spectral density budget, a derivation of $g_o(.)$ is offered. The proposed method explicitly derives the solution of the conveyance law and provides a physical interpretation of $\chi$ as a dimensionless length scale reflecting the competition between viscous sublayer thickness and characteristic height of roughness elements. The application of the proposed method to other published measurements spanning roughness and Reynolds numbers beyond the original Nikuradse range is further discussed.

The inertial subrange of turbulent scales is commonly reflected by a power law signature in ensemble statistics such as the energy spectrum and structure functions - both in theory and from observations. Despite promising findings on the topic of fractal geometries in turbulence, there is no accepted image for the physical flow features corresponding to this statistical signature in the inertial subrange. The present study uses boundary layer turbulence measurements to evaluate the self-similar geometric properties of velocity isosurfaces and investigate their influence on statistics for the velocity signal. The fractal dimension of streamwise velocity isosurfaces, indicating statistical self-similarity in the size of "wrinkles" along each isosurface, is shown to be constant only within the inertial subrange of scales. For the transition between the inertial subrange and production range, it is inferred that the largest wrinkles become increasingly confined by the overall size of large-scale coherent velocity regions such as uniform momentum zones. The self-similarity of isosurfaces yields power law trends in subsequent one-dimensional statistics. For instance, the theoretical 2/3 power law exponent for the structure function can be recovered by considering the collective behavior of numerous isosurface level sets. The results suggest that the physical presence of inertial subrange eddies is manifested in the self-similar wrinkles of isosurfaces., Comment: 30 pages, 11 figures

The pattern formation and spatial spread of infectious populations are investigated using a kernel-based Susceptible-Infectious-Recovered (SIR) model applicable across a wide range of basic reproduction numbers $R_o$. The focus is on the role of geographic spreaders defined here as a portion of the infected population ($\phi$) experiencing high mobility between identical communities. The spatial organization of the infected population and invasive front speeds ($c_{max}$) are determined when the infections are randomly initiated in space within multiple communities. For small but finite $\phi$, scaling analysis in 1-dimension and simulation results in 2-dimensions suggest that $c_{max}\sim (1-\phi) \gamma (R_o-1) \sigma$, where $\gamma$ is the inverse of the infectious duration, and $\sigma^2$ is the variance of the spatial kernel describing mobility of long-distance spreaders across communities. Hence, $c_{max}$ is not significantly affected by the small $\phi$ though reductions in $\phi$ act as retardation factors to the attainment of $c_{max}$. The $\sigma$ determines the spatial organization of infections across communities. When $\sigma >5dr$ (long-distance mobility, where $dr$ is the minimum spatial extent defining adjacent communities), the infectious population will experience a transient but spatially coherent pattern with a wavelength that can be derived from the spreading kernel properties.

The vertical distribution of suspended sediment concentration (SSC) remains a subject of active research given its relevance to a plethora of problems in hydraulics, hydrology, ecology, and water quality control. Much of the classical theories developed over the course of 90 years represent the effects of turbulence on suspended sediments (SS) using an effective mixing length or eddy diffusivity without explicitly accounting for the energetics of turbulent eddies across scales. To address this gap, the turbulent flux of sediments is derived using a co-spectral budget (CSB) model that can be imminently used in SS and other fine particle transport models. The CSB closes the pressure-redistribution effect using a spectral linear Rotta scheme modified to include isotropoziation of production and interactions between turbulent eddies and sediment grains through a modified scale-dependent de-correlation time. The result is a formulation similar in complexity to the widely used Rouse's equation but with all characteristic scales, Reynolds number, and Schmidt number effects derived from well-established spectral shapes of the vertical velocity and accepted constants from turbulence models. Finally, the proposed CSB model can recover Prandtl's and Rouse's equations under restricted conditions.

The coupling between wind-waves and atmospheric surface layer turbulence sets surface drag. This coupling is however usually represented through a roughness length. Originally suggested on purely dimensional grounds, this roughness length does not directly correspond to a measurable physical quantity of the wind-and-wave system. Here, to go beyond this representation, we formalize ideas underlying the Beaufort scale by quantifying the velocity of breaking short waves that are the most coupled to near-surface wind. This velocity increases with wind speed, reflecting the fact that stronger winds can be visually identified by longer (and faster) breakers becoming predominant on the sea surface. A phenomenological turbulence model further shows that this velocity is associated with breaking waves that impede the most the formation of turbulent eddies. Scales of such eddies are then constrained inside a so-called roughness sub-layer. Unlike previous theoretical developments, the proposed breaker velocity is a directly measurable quantity, which could be used to characterize the coupling between wind and waves using remote sensing techniques. This work provides a physical framework for new formulations of air-sea momentum exchange in which the effects of surface currents and slicks on surface drag can also be incorporated. Finally, it provides a long-sought physical explanation for the Beaufort scale: a universal link between wave breaking, wind speed and surface drag., Comment: 21 pages, 9 figures

Anisotropy and multifractality in velocity and temperature time series sampled at multiple heights in the roughness sublayer (RSL) over a boreal mixed-coniferous forest are reported. In particular, a turbulent-stress invariant analysis along with a scalewise version of it are conducted to elucidate the nature of relaxation of large-scale anisotropy to quasi-isotropic states at small scales. As the return to isotropy is linked to nonlinear interactions and correlations between different fluctuating velocity components across scales, we study the velocity and temperature time series by using multifractal detrended fluctuation analysis and multiscale multifractal analysis to assess the effects of thermal stratification and surface roughness on turbulence in the RSL. The findings are compared so as to quantify the anisotropy and multifractality ubiquitous to RSL turbulent flow. As we go up in the RSL, (a) the length scale at which return to isotropy commences increases because of the weakening of the surface effects and (b) the largest scales become increasingly anisotropic. The anisotropy in multifractal exponents for the velocity fluctuations is diminished when we use the extended-self-similarity procedure to extract the multifractal-exponent ratios., Comment: 17 pages, 15 figures

Turbulent flows are out-of-equilibrium because the energy supply at large scales and its dissipation by viscosity at small scales create a net transfer of energy among all scales. Here, the energy cascade is approximated by a combined contribution of a forward drift and diffusion that recover accepted phenomenological theories of turbulence. The fluctuation theorem (FT) is then shown to describe the scale-wise statistics of forward and backward energy transfer and their connection to irreversibility and entropy production. The ensuing turbulence entropy may be used to formulate an extended turbulence thermodynamics.