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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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Gabriel G. Katul and Wei-han Chang

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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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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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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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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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Konstantin Kröniger, Gabriel G. Katul, Frederik De Roo, Peter Brugger, and Matthias Mauder

Abstract

Simulating the influence of heterogeneous surfaces on atmospheric flow using mesoscale models (MSM) remains a challenging task, as the resolution of these models usually prohibits resolving important scales of surface heterogeneity. However, surface heterogeneity impacts fluxes of momentum, heat, or moisture, which act as lower boundary conditions for MSM. Even though several approaches for representing subgrid-scale heterogeneities in MSM exist, many of these approaches rely on Monin–Obukhov similarity theory, preventing those models from resolving all scales of surface heterogeneity. To improve upon these residual heterogeneity scales, a novel heterogeneity parameterization is derived by linking the heterogeneous covariance function in spectral space to an associated homogeneous one. This covariance function approach is subsequently used to derive a parameterization of the aerodynamic resistance to heat transfer of the surface layer. Here, the effect of surface heterogeneity enters as a factor applied to the stability correction functions of the bulk similarity approach. To perform a first comparison of the covariance function approach against the conventional bulk similarity and tile approaches, large-eddy simulations (LESs) of distinct surface heterogeneities are conducted. The aerodynamic resistances from these three parameterizations are subsequently tested against the LES reference by resolving the surface heterogeneities with six different test-MSM grids of varying cell dimension. The results of these comparisons show that the covariance function approach proposed here yields the smallest deviations from the LES reference. In addition, the smallest deviation of the covariance function approach to the reference is observed for the LES with the largest surface heterogeneity, which illustrates the advantage of this novel parameterization.

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

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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, John D. Albertson, Marc B. Parlange, Cheng-I. Hsieh, Paul S. Conklin, John T. Sigmon, and Ken R. Knoerr

Abstract

The statistical structure of the turbulent pressure fluctuations was measured in the dynamic sublayer of a large grass-covered forest clearing by a free air static pressure probe and modeled using Townsend's hypothesis. Townsend's hypothesis states that the eddy motion in the equilibrium layer can be decomposed into an active component, which is only a function of the ground shear stress and height, and an inactive component, which is produced by turbulence in the outer region. It is demonstrated that the inactive eddy motion contributes significantly to the pressure and longitudinal velocity power spectra for wavenumbers much smaller than that corresponding to the height above the ground surface. Because of the importance of this inactive eddy motion contribution, it was possible to derive and validate a scaling law for the pressure power spectrum at low wavenumbers. The root-mean-square pressure was derived from the ground shear stress using simplifications to the Poisson equation that relate the Laplacian of the pressure fluctuations to the divergence of momentum. The theoretically derived and experimentally measured root-mean-square pressure values were in close agreement with other theoretical predictions and numerous laboratory measurements for wall pressure fluctuations. The relation between the root-mean-square pressure and the ground shear stress was also used to determine the similarity constant for the large-scale pressure spectrum. From considerations of the integral representation of the Poisson equation, previous laboratory measurements, and the present data, it was shown that this similarity constant does not vary appreciably with the roughness of the boundary layer. Finally, it was demonstrated that the inactive eddy motion does not contribute to the vertical velocity power spectrum in agreement with Monin and Obukhov surface-layer similarity theory.

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Dennis Baldocchi, Eva Falge, Lianhong Gu, Richard Olson, David Hollinger, Steve Running, Peter Anthoni, Ch. Bernhofer, Kenneth Davis, Robert Evans, Jose Fuentes, Allen Goldstein, Gabriel Katul, Beverly Law, Xuhui Lee, Yadvinder Malhi, Tilden Meyers, William Munger, Walt Oechel, K. T. Paw U, Kim Pilegaard, H. P. Schmid, Riccardo Valentini, Shashi Verma, Timo Vesala, Kell Wilson, and Steve Wofsy

FLUXNET is a global network of micrometeorological flux measurement sites that measure the exchanges of carbon dioxide, water vapor, and energy between the biosphere and atmosphere. At present over 140 sites are operating on a long-term and continuous basis. Vegetation under study includes temperate conifer and broadleaved (deciduous and evergreen) forests, tropical and boreal forests, crops, grasslands, chaparral, wetlands, and tundra. Sites exist on five continents and their latitudinal distribution ranges from 70°N to 30°S.

FLUXNET has several primary functions. First, it provides infrastructure for compiling, archiving, and distributing carbon, water, and energy flux measurement, and meteorological, plant, and soil data to the science community. (Data and site information are available online at the FLUXNET Web site, http://www-eosdis.ornl.gov/FLUXNET/.) Second, the project supports calibration and flux intercomparison activities. This activity ensures that data from the regional networks are intercomparable. And third, FLUXNET supports the synthesis, discussion, and communication of ideas and data by supporting project scientists, workshops, and visiting scientists. The overarching goal is to provide information for validating computations of net primary productivity, evaporation, and energy absorption that are being generated by sensors mounted on the NASA Terra satellite.

Data being compiled by FLUXNET are being used to quantify and compare magnitudes and dynamics of annual ecosystem carbon and water balances, to quantify the response of stand-scale carbon dioxide and water vapor flux densities to controlling biotic and abiotic factors, and to validate a hierarchy of soil–plant–atmosphere trace gas exchange models. Findings so far include 1) net CO2 exchange of temperate broadleaved forests increases by about 5.7 g C m−2 day−1 for each additional day that the growing season is extended; 2) the sensitivity of net ecosystem CO2 exchange to sunlight doubles if the sky is cloudy rather than clear; 3) the spectrum of CO2 flux density exhibits peaks at timescales of days, weeks, and years, and a spectral gap exists at the month timescale; 4) the optimal temperature of net CO2 exchange varies with mean summer temperature; and 5) stand age affects carbon dioxide and water vapor flux densities.

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