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Gabriel Katul and Brani Vidakovic

Abstract

The partitioning of turbulent perturbations into a “low-dimensional” active part responsible for much of the turbulent energy and fluxes and a “high-dimensional” passive part that contributes little to turbulent energy and transport dynamics is investigated using atmospheric surface-layer (ASL) measurements. It is shown that such a partitioning scheme can be achieved by transforming the ASL measurements into a domain that concentrates the low-dimensional part into few coefficients and thus permits a global threshold of the remaining coefficients. In this transformation–thresholding approach, Fourier rank reduction and orthonormal wavelet and wavelet packet methods are considered. The efficiencies of these three thresholding methods to extract the events responsible for much of the heat and momentum turbulent fluxes are compared for a wide range of atmospheric stability conditions. The intercomparisons are performed in four ways: (i) compression ratios, (ii) energy conservation, (iii) turbulent flux conservation, and (iv) finescale filtering via departures from Kolmogorov’s K41 power laws. For orthonormal wavelet and wavelet packets analysis, wavelet functions with varying time–frequency localization properties are also considered. The study showed that wavelet and wavelet packet Lorentz thresholding can achieve high compression ratios (98%) with minimal loss in energy (3% loss) and fluxes (4%). However, these compression ratios and energy and flux conservation measures are comparable to the linear Fourier rank reduction method if a Lorentz threshold function is applied to the latter. Finally, it is demonstrated that orthonormal wavelet and wavelet packets thresholding are insensitive to the analyzing wavelet.

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Gabriel G. Katul and Marc B. Parlange

Abstract

Orthonormal wavelet expansions were derived and applied to atmospheric surface-layer turbulence measurements of temperature and vapor concentration under unstable and stable atmospheric stability conditions. These expansions were used to investigate both the statistical and spectral structure of turbulence simultaneously in space and scale using two tracers: temperature and specific humidity. It was found that at small wavenumbers, both temperature and specific humidity Fourier and wavelet spectra exhibit a −1 power law behavior consistent with other atmospheric boundary-layer experiments. The mean values of the energy spectrum obtained from the wavelet analysis are in agreement with the classical Fourier counterparts. The wavelet flatness factors (values up to 10) indicate strong deviation from Gaussian statistics in space for the temperature fluctuations as the wavenumber increases. In contrast, the spatial wavelet flatness factor for the specific humidity exhibits near Gaussian statistics (values up to 4) for all wavenumbers. The wavelet skewness in space indicates that the specific humidity attains a near-isotropic state with increasing wavenumber for both stability conditions. Unlike the specific humidity, the temperature wavelet skewness in space did not decay with increasing wavenumber, indicating the presence of large eddy anisotropy in space. Land surface heating/cooling inhomogeneity appears to affect the local structure of turbulence, and therefore, at small scales temperature behaves as an active scalar when compared to specific humidity. The active role of temperature was also analyzed within the framework of Bolgiano's spectral theory. Deviations from Bolgiano's theory for the temperature spectrum were observed at all wavenumbers with measured energy power law behavior of |1.2|, which is less than the theoretical value of |7/5|. Conditional wavelet analysis was developed and used to investigate the nature of these deviations from Bolgiano's scaling law for the temperature measurements. It was found that by suppressing energy-containing and intermittent events, Bolgiano's scaling law for the temperature spectrum held under stable stability conditions. The effect of different wavelet basis functions on the statistical and spectral description of atmospheric turbulence was also considered.

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Mario Siqueira, Gabriel Katul, and Amilcare Porporato

Abstract

The linkages between soil moisture dynamics and convection triggers, defined here as the first crossing between the boundary layer height (h BL) and lifting condensation level (h LCL), are complicated by a large number of interacting processes occurring over a wide range of space and time scales. To progress on this problem, a soil–plant hydrodynamics model was coupled to a simplified ABL budget to explore the feedback of soil moisture on convection triggers. The soil–plant hydraulics formulation accounted mechanistically for features such as root water uptake, root water redistribution, and midday stomatal closure, all known to affect diurnal cycles of surface fluxes and, consequently, ABL growth. The ABL model considered the convective boundary layer as a slab with a discontinuity at the inversion layer. The model was parameterized using the wealth of data already collected for a maturing Loblolly pine plantation situated in the southeastern United States. A 30-day dry-down simulation was used to investigate the possible feedback mechanisms between soil moisture and convective rainfall triggers. Previous studies, which made use of surface flux measurements to drive an ABL model, have postulated that a negative feedback was possible, which could award the ecosystem with some degree of self-regulation of its water status. According to model simulation results here, this negative feedback is unlikely. However, drastic changes in external water sources to the ABL are needed for triggering convection when soil moisture is depleted. The apparent negative feedback originated from a decoupling between the water vapor sources needed to produce convection triggers and surface water vapor fluxes.

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Gabriel G. Katul and Wei-han Chang

Abstract

Triaxial sonic anemometer velocity measurements vertically arrayed at six levels within and above a pine forest were used to examine the performance of two second-order closure models put forth by Wilson and Shaw and by Wilson. Based on these measurements, it was demonstrated that Wilson’s model reproduced the longitudinal velocity standard deviation σ u better than did Wilson and Shaw’s model. However, Wilson and Shaw’s model reproduced the measured mean velocity 〈u〉 near the forest–atmosphere interface better than Wilson’s model did. The primary mechanisms responsible for discrepancies between modeled and measured 〈u〉 and σ u profiles were investigated.

The conceptual formulations of these two closure models differ in the characteristic length scales and timescales used in the closure parameterizations of the mean turbulent kinetic energy dissipation rate term, the pressure–strain rate term, and the flux-transport term. These characteristic length scales were computed and compared with measured integral length scales inside the canopy. A discussion on how these length scales compare with the mixing layer analogy also is presented.

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Dan Li, Gabriel G. Katul, and Sergej S. Zilitinkevich

Abstract

Two recently proposed turbulence closure schemes are compared against the conventional Mellor–Yamada (MY) model for stably stratified atmospheric flows. The Energy- and Flux-Budget (EFB) approach solves the budgets of turbulent momentum and heat fluxes and turbulent kinetic and potential energies. The Cospectral Budget (CSB) approach is formulated in wavenumber space and integrated across all turbulent scales to obtain flow variables in physical space. Unlike the MY model, which is subject to a “critical gradient Richardson number,” both EFB and CSB models allow turbulence to exist at any gradient Richardson number and predict a saturation of flux Richardson number () at sufficiently large . The CSB approach further predicts the value of and reveals a unique expression linking the Rotta and von Kármán constants. Hence, all constants in the CSB model are nontunable and stability independent. All models agree that the dimensionless sensible heat flux decays with increasing . However, the decay rate and subsequent cutoff in the MY model appear abrupt. The MY model further exhibits an abrupt cutoff in the turbulent stress normalized by vertical velocity variance, while the CSB and EFB models display increasing trends. The EFB model produces a rapid increase in the ratio of turbulent potential energy and vertical velocity variance as is approached, suggesting a strong self-preservation mechanism. Vertical anisotropy in the turbulent kinetic energy is parameterized in different ways in MY and EFB, but this consideration is not required in CSB. Differences between EFB and CSB model predictions originate from how the vertical anisotropy is specified in the EFB model.

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Tirtha Banerjee, Dan Li, Jehn-Yih Juang, and Gabriel Katul

Abstract

A spectral budget model is developed to describe the scaling behavior of the longitudinal turbulent velocity variance with the stability parameter and the normalized height in an idealized stably stratified atmospheric surface layer (ASL), where z is the height from the surface, L is the Obukhov length, and δ is the boundary layer height. The proposed framework employs Kolmogorov’s hypothesis for describing the shape of the longitudinal velocity spectra in the inertial subrange, Heisenberg’s eddy viscosity as a closure for the pressure redistribution and turbulent transfer terms, and the Monin–Obukhov similarity theory (MOST) scaling for linking the mean longitudinal velocity and temperature profiles to ζ. At a given friction velocity , reduces with increasing ζ as expected. The model is consistent with the disputed z-less stratification when the stability correction function for momentum increases with increasing ζ linearly or as a power law with the exponent exceeding unity. For the Businger–Dyer stability correction function for momentum, which varies linearly with ζ, the limit of the z-less onset is . The proposed framework explains why does not follow MOST scaling even when the mean velocity and temperature profiles may follow MOST in the ASL. It also explains how δ ceases to be a scaling variable in more strongly stable (although well-developed turbulent) ranges.

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Dan Li, Gabriel G. Katul, and Sergej S. Zilitinkevich

Abstract

Cospectral budgets are used to link the kinetic and potential energy distributions of turbulent eddies, as measured by their spectra, to macroscopic relations between the turbulent Prandtl number (Prt) and atmospheric stability measures such as the stability parameter ζ, the gradient Richardson number R g, or the flux Richardson number R f in the atmospheric surface layer. The dependence of Prt on ζ, R g, or R f is shown to be primarily controlled by the ratio of Kolmogorov and Kolmogorov–Obukhov–Corrsin phenomenological constants and a constant associated with isotropization of turbulent flux production that can be independently determined using rapid distortion theory in homogeneous turbulence. Changes in scaling laws of the vertical velocity and air temperature spectra are also shown to affect the Prtζ (or PrtR g or PrtR f) relation. Results suggest that departure of Prt from unity under neutral conditions is induced by dissimilarity between momentum and heat in terms of Rotta constants, isotropization constants, and constants in the flux transfer terms. A maximum flux Richardson number R fm predicted from the cospectral budgets method (=0.25) is in good agreement with values in the literature, suggesting that R fm may be tied to the collapse of Kolmogorov spectra instead of laminarization of turbulent flows under stable stratification. The linkages between microscale energy distributions of turbulent eddies and macroscopic relations that are principally determined by dimensional considerations or similarity theories suggest that when these scalewise energy distributions of eddies experience a “transition” to other distributions (e.g., when R f is increased over R fm), dimensional considerations or similarity theories may fail to predict bulk flow properties.

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Khaled Ghannam, Gabriel G. Katul, Elie Bou-Zeid, Tobias Gerken, and Marcelo Chamecki

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The low-wavenumber regime of the spectrum of turbulence commensurate with Townsend’s “attached” eddies is investigated here for the near-neutral atmospheric surface layer (ASL) and the roughness sublayer (RSL) above vegetation canopies. The central thesis corroborates the significance of the imbalance between local production and dissipation of turbulence kinetic energy (TKE) and canopy shear in challenging the classical distance-from-the-wall scaling of canonical turbulent boundary layers. Using five experimental datasets (two vegetation canopy RSL flows, two ASL flows, and one open-channel experiment), this paper explores (i) the existence of a low-wavenumber k −1 scaling law in the (wind) velocity spectra or, equivalently, a logarithmic scaling ln(r) in the velocity structure functions; (ii) phenomenological aspects of these anisotropic scales as a departure from homogeneous and isotropic scales; and (iii) the collapse of experimental data when plotted with different similarity coordinates. The results show that the extent of the k −1 and/or ln(r) scaling for the longitudinal velocity is shorter in the RSL above canopies than in the ASL because of smaller scale separation in the former. Conversely, these scaling laws are absent in the vertical velocity spectra except at large distances from the wall. The analysis reveals that the statistics of the velocity differences Δu and Δw approach a Gaussian-like behavior at large scales and that these eddies are responsible for momentum/energy production corroborated by large positive (negative) excursions in Δu accompanied by negative (positive) ones in Δw. A length scale based on TKE dissipation collapses the velocity structure functions at different heights better than the inertial length scale.

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Gabriel G. Katul, Chris D. Geron, Cheng-I. Hsieh, Brani Vidakovic, and Alex B. Guenther

Abstract

Turbulent velocity, temperature, water vapor concentration, and other scalars were measured at the canopy–atmosphere interface of a 13–14-m-tall uniform pine forest and a 33-m-tall nonuniform hardwood forest. These measurements were used to investigate whether the mixing layer (ML) analogy of predicts eddy sizes and flow characteristics responsible for much of the turbulent stresses and vertical scalar fluxes. For this purpose, wavelet spectra and cospectra were derived and analyzed. It was found that the ML analogy predicts well vertical velocity variances and integral timescales. However, at low wavenumbers, inactive eddy motion signatures were present in horizontal velocity wavelet spectra, suggesting that ML may not be suitable for scaling horizontal velocity perturbations. Momentum and scalar wavelet cospectra of turbulent stresses and scalar fluxes demonstrated that active eddy motion, which was shown by to be the main energy contributor to vertical velocity (w) spectral energy (E w), is also the main scalar flux–transporting eddy motion. Predictions using ML of the peak E w frequency are in excellent agreement with measured wavelet cospectral peaks of vertical fluxes (Kh = 1.5, where K is wavenumber and h is canopy height). Using Lorentz wavelet thresholding of vertical velocity time series, wavelet coefficients associated with active turbulence were identified. It was demonstrated that detection frequency of organized structures, as predicted from Lorentz wavelet filtering, relate to the arrival frequency 〈U〉/h and integral timescale, where 〈U〉 is the mean horizontal velocity at height z = h. The newly proposed wavelet thresholding approach, which relies on a “global” wavelet threshold formulation for the energy in w, provides simultaneous energy–covariance-preserving characterization of “active” turbulence at the canopy–atmosphere interface.

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Gabriel G. Katul, Peter L. Finkelstein, John F. Clarke, and Thomas G. Ellestad

Abstract

The conditional sampling flux measurement technique was evaluated for four scalars (temperature, water vapor, ozone, and carbon dioxide) by comparison with direct eddy correlation measurements at two sites. The empirical constant β relating the turbulent flux to the accumulated concentration difference between updrafts and downdrafts was computed from 10-Hz turbulence measurements. Comparison between the simulated relaxed eddy accumulation flux formulation and the eddy correlation measurements allowed the direct determination of β for all four scalars. The β models previously proposed overpredicted the measured β by about 8%–10%. It was found that a mean β = 0.58 reproduced the eddy correlation measurements independent of the scalar type being analyzed, roughness and atmospheric stability conditions, in agreement with previous studies. The role of energy-containing eddy motion in the deviations between the measured and predicted β was considered using orthonormal wavelet expansion in conjunction with a wavelet shrinkage approach. It was demonstrated that the energy-containing large eddy motion contributed to a reduction in β when compared to the predicted β. Finally, the deadband vertical velocity effects were also considered and found to reduce β exponentially, in agreement with other studies.

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