Energy- and Flux-Budget Turbulence Closure Model for Stably Stratified Flows. Part II: The Role of Internal Gravity Waves

We advance our prior energy- and flux-budget (EFB) turbulence closure model for stably stratified atmospheric flow and extend it to account for an additional vertical flux of momentum and additional productions of turbulent kinetic energy (TKE), turbulent potential energy (TPE) and turbulent flux of potential temperature due to large-scale internal gravity waves (IGW). For the stationary, homogeneous regime, the first version of the EFB model disregarding large-scale IGW yielded universal dependencies of the flux Richardson number, turbulent Prandtl number, energy ratios, and normalised vertical fluxes of momentum and heat on the gradient Richardson number, Ri. Due to the large-scale IGW, these dependencies lose their universality. The maximal value of the flux Richardson number (universal constant ≈0.2–0.25 in the no-IGW regime) becomes strongly variable. In the vertically homogeneous stratification, it increases with increasing wave energy and can even exceed 1. For heterogeneous stratification, when internal gravity waves propagate towards stronger stratification, the maximal flux Richardson number decreases with increasing wave energy, reaches zero and then becomes negative. In other words, the vertical flux of potential temperature becomes counter-gradient. Internal gravity waves also reduce the anisotropy of turbulence: in contrast to the mean wind shear, which generates only horizontal TKE, internal gravity waves generate both horizontal and vertical TKE. Internal gravity waves also increase the share of TPE in the turbulent total energy (TTE = TKE + TPE). A well-known effect of internal gravity waves is their direct contribution to the vertical transport of momentum. Depending on the direction (downward or upward), internal gravity waves either strengthen or weaken the total vertical flux of momentum. Predictions from the proposed model are consistent with available data from atmospheric and laboratory experiments, direct numerical simulations and large-eddy simulations.     

Energy- and flux-budget (EFB) turbulence closure model for stably stratified flows. Part I: steady-state,homogeneous regimes

We propose a new turbulence closure model based on the budget equations for the key second moments: turbulent kinetic and potential energies: TKE and TPE (comprising the turbulent total energy: TTE = TKE + TPE) and vertical turbulent fluxes of momentum and buoyancy (proportional to potential temperature). Besides the concept of TTE, we take into account the non-gradient correction to the traditional buoyancy flux formulation. The proposed model permits the existence of turbulence at any gradient Richardson number, Ri. Instead of the critical value of Richardson number separating—as is usually assumed—the turbulent and the laminar regimes, the suggested model reveals a transitional interval, , which separates two regimes of essentially different nature but both turbulent: strong turbulence at ; and weak turbulence, capable of transporting momentum but much less efficient in transporting heat, at . Predictions from this model are consistent with available data from atmospheric and laboratory experiments, direct numerical simulation and large-eddy simulation.     

Investigating a Hierarchy of Eulerian Closure Models for Scalar Transfer Inside Forested Canopies

Modelling the transfer of heat, water vapour, and CO₂ between the biosphere and the atmosphere is made difficult by the complex two-way interaction between leaves and their immediate microclimate. When simulating scalar sources and sinks inside canopies on seasonal, inter-annual, or forest development time scales, the so-called well-mixed assumption (WMA) of mean concentration (i.e. vertically constant inside the canopy but dynamically evolving in time) is often employed. The WMA eliminates the need to model how vegetation alters its immediate microclimate, which necessitates formulations that utilize turbulent transport theories. Here, two inter-related questions pertinent to the WMA for modelling scalar sources, sinks, and fluxes at seasonal to inter-annual time scales are explored: (1) if the WMA is to be replaced so as to resolve this two-way interaction, how detailed must the turbulent transport model be? And (2) what are the added predictive skills gained by resolving the two-way interaction vis-à-vis other uncertainties such as seasonal variations in physiological parameters. These two questions are addressed by simulating multi-year mean scalar concentration and eddy-covariance scalar flux measurements collected in a Loblolly pine (P. taeda L.) plantation near Durham, North Carolina, U.S.A. using turbulent transport models ranging from K-theory (or first-order closure) to third-order closure schemes. The multi-layer model calculations with these closure schemes were contrasted with model calculations employing the WMA. These comparisons suggested that (i) among the three scalars, sensible heat flux predictions are most biased with respect to eddy-covariance measurements when using the WMA, (ii) first-order closure schemes are sufficient to reproduce the seasonal to inter-annual variations in scalar fluxes provided the canonical length scale of turbulence is properly specified, (iii) second-order closure models best agree with measured mean scalar concentration (and temperature) profiles inside the canopy as well as scalar fluxes above the canopy, (iv) there are no clear gains in predictive skills when using third-order closure schemes over their second-order closure counterparts. At inter-annual time scales, biases in modelled scalar fluxes incurred by using the WMA exceed those incurred when correcting for the seasonal amplitude in the maximum carboxylation capacity (V _{cmax, 25}) provided its mean value is unbiased. The role of local thermal stratification inside the canopy and possible computational simplifications in decoupling scalar transfer from the generation of the flow statistics are also discussed.

“The tree, tilting its leaves to capture bullets of light; inhaling, exhaling; its many thousand stomata breathing, creating the air”. Ruth Stone, 2002, In the Next Galaxy

     

A Second-Order Closure for Neutrally Stratified Vegetative Canopy Flows

An existing second-order closure model is modified to include the effects on mean and turbulent motions of form and viscous drag in vegetative canopies. The additional physical mechanisms represented by the closure are viscous and pressure drag on canopy elements, their role in momentum absorption, in the creation of fine scale turbulent eddies and in enhancing the total viscous dissipation in the canopy airspace. Viscous dissipation is split into a standard 'isotropic contribution associated with the spectral eddy cascade and a foliage contribution associated with work against pressure and viscous drag on the foliage. Changes in the turbulent time scale that result from these mechanisms are included in the standard parameterisations of third moments and of the eddy cascade contribution to dissipation. The model is tested against a wind- tunnel 'wheat canopy, a corn canopy and a eucalypt canopy, a height range from 50 mm to 12.6 m. Model results show that the parameterisations of foliage interaction used in the closure are sufficiently robust to reproduce second-moment profiles within and above vegetative canopies to a high degree of accuracy without resorting to 'tuning of the model constants. The model also shows the natural emergence of two length scales, one associated with the familiar eddy cascade isotropic contribution to total dissipation and the other associated with the length scales of the canopy elements.     

An Investigation of Higher-Order Closure Models for a Forested Canopy

Simultaneous triaxial sonic anemometer velocity measurements vertically arrayed at six levels within and above a uniform pine forest were used to examine two parameterization schemes for the triple-velocity correlation tensor employed in higher-order closure models. These parameterizations are the gradient-diffusion approximation typically used in second-order closure models, and the full budget for the triple-velocity correlation tensor typically employed in third-order closure models. Both second- and third-order closure models failed to reproduce the measured profiles of the triple-velocity correlation within and above the canopy. However, the Reynolds stress tensor profiles (including velocity variances) deviated greatly from the measurements only within the lower levels of the canopy. It is shown that the Reynolds stresses are most sensitive to the parameterization of the triple-velocity correlation in these lower canopy regions where local turbulent production is negligible and turbulence is mainly sustained by the flux transport term. The failure of the third-order closure model to reproduce the measured third moments in the upper layers of the canopy-top contradicts conclusions from a previous study over shorter vegetation but agrees with another study for a deciduous forest. Whether the third-order closure model failure is due to the zero-fourth-cumulant closure approximation is therefore considered. Comparisons between measured and predicted quadruple velocity correlations suggest that the zero-fourth-cumulant approximation is valid close to the canopy-atmosphere in agreement with recent experiments.     

A Refinement of the McMillen (1988) Recursive Digital Filter for the Analysis of Atmospheric Turbulence

We present a refinement of the recursive digital filter proposed by McMillen (Boundary-Layer Meteorol 43:231–245, 1988), for separating surface-layer turbulence from low-frequency fluctuations affecting the mean flow, especially over complex terrain. In fact, a straightforward application of the filter causes both an amplitude attenuation and a forward phase shift in the filtered signal. As a consequence turbulence fluctuations, evaluated as the difference between the original series and the filtered one, as well as higher-order moments calculated from them, may be affected by serious inaccuracies. The new algorithm (i) produces a rigorous zero-phase filter, (ii) restores the amplitude of the low-frequency signal, and (iii) corrects all filter-induced signal distortions.     

The Analysis of Turbulence Intensity and Reynolds Shear Stress in Wall-Bounded Turbulent Flows at High Reynolds Numbers

We investigated the turbulent intensities and Reynolds shear stress at high Reynolds number $({Re_\tau = 5 \times 10^{6}})$ in the atmosphere surface layer (ASL) through analyzing observations in near-neutral stratified conditions. The results show that with increasing Reynolds number the streamwise turbulent intensity increases linearly, and the peak of the Reynolds shear stress extends to a higher non-dimensional height, which means that the thickness of the logarithmic region increases. Furthermore, our results provide evidence for treating the ASL as a canonical turbulent boundary layer, the results of which can be extended and applied to higher Reynolds number wall turbulence in the ASL.     

An Analytical one-Dimensional Second-Order Closure Model of Turbulence Statistics and the Lagrangian Time Scale Within and Above Plant Canopies of Arbitrary Structure

An analytical one-dimensional second-order closure model is developed to describe the within canopy velocity variances, turbulent intensities, dissipation rates, Lagrangian time scale and Lagrangian far field diffusivities for vegetation canopies of arbitrary structure and density. The model incorporates and extends the model of momentum transfer developed by Massman (1997) and the model of within canopy velocity variances developed by Weil (unpublished) from the second-order closure model of Wilson and Shaw (1977). Model predictions of within and above canopy velocity variances, turbulent intensities, dissipation rates and the Lagrangian time scale are in reasonable agreement with previously measured or estimated values for these parameters. The present model suggests that the Lagrangian time scale and the far field diffusivity could be strongly dependent upon foliage structure and density through the foliage effects on the velocity variances. A simple formulation for the Lagrangian time scale at canopy height is derived from model results. Taken as a whole, the present model may provide a relatively simple way to incorporate turbulence parameters into models of soil/canopy/atmosphere mass transfer.     

Numerical Prediction of the Boundary-Layer Flow Over the Bolund Hill: Assessment of Turbulence Models and Advection Schemes

Boundary-Layer Meteorology - The WindStation software package is applied to simulate the wind field over the Bolund hill. The standard, ReNormalization Group (RNG), realizable, and limited-length...     

Turbulence Spectra and Cospectra Under the Influence of Large Eddies in the Energy Balance EXperiment (EBEX)

During the Energy Balance EXperiment, the patch-by-patch, flood irrigation in a flat cotton field created an underlying surface with heterogeneous soil moisture, leading to a dry (warm)-to-wet (cool) transition within the cotton field under northerly winds. Moreover, the existence of an extremely dry, large bare soil area upstream beyond the cotton field created an even larger step transition from the bare soil region to the cotton field. We investigated the turbulence spectra and cospectra in the atmospheric surface layer (ASL) that was disturbed by large eddies generated over regions upstream and also influenced by horizontal advection. In the morning, the ASL was unstable while in the afternoon a stable internal boundary layer was observed at the site. Therefore, the turbulence data at 2.7 and 8.7 m are interpreted and compared in terms of interactions between large eddies and locally generated turbulence under two atmospheric conditions: the unstable ASL beneath the convective boundary layer (CBL) (hereafter the unstable condition) and the stable ASL beneath the CBL (hereafter the stable condition). We identified the influences of multiple sizes of large eddies on ASL turbulence under both stratifications; these large eddies with multiple sizes were produced over the dry patches and dry, large bare soil areas upstream. As a consequence of the disturbance of large eddies, the broadening, erratic variability, and deviation of spectra and cospectra, relative to those described by Monin–Obukhov similarity theory, are evident in the low- to mid-frequencies. Transfer of momentum, heat, and water vapour by large eddies is distinctly observed from the turbulence cospectra and leads to significant run-to-run variations of residuals of the surface energy balance closure. Our results indicate that these large eddies have greater influences on turbulence at higher levels compared to lower levels, and in the unstable ASL compared to the stable ASL.     

A Numerically Efficient Parametrization of Turbulent Wind-Turbine Flows for Different Thermal Stratifications

The wake characteristics of a wind turbine in a turbulent atmospheric boundary layer under different thermal stratifications are investigated by means of large-eddy simulation with the geophysical flow solver EULAG. The turbulent inflow is based on a method that imposes the spectral energy distribution of a neutral boundary-layer precursor simulation, the turbulence-preserving method. This method is extended herein to make it applicable for different thermal stratification regimes (convective, stable, neutral) by including suitable turbulence assumptions, which are deduced from velocity fields of a diurnal-cycle precursor simulation. The wind-turbine-wake characteristics derived from simulations that include the parametrization result in good agreement with diurnal-cycle-driven wind-turbine simulations. Furthermore, different levels of accuracy are tested in the parametrization assumptions, representing the thermal stratification. These range from three-dimensional matrices of the precursor-simulation wind field to individual values. The resulting wake characteristics are similar, even for the simplest parametrization set-up, making the diurnal-cycle precursor simulation non-essential for the wind-turbine simulations. Therefore, the proposed parametrization results in a computationally fast, simple, and efficient tool for analyzing the effects of different thermal stratifications on wind-turbine wakes by means of large-eddy simulation.     

A Method of Analysis for Turbulent Flows Using the Streamline Coordinate System

A procedure is described for analysing the transport equations for Reynolds stresses written in a streamline coordinate system, starting from the fields of first- and second-order moments of wind velocity measured in a terrain-following system over topography. In the analysis, the equations are split into two parts: the first contains the terms that can be calculated directly from measurements; the second contains third-order moments that are parameterized using suitable models. To evaluate the error associated with both parts, a Monte-Carlo technique that takes into account the experimental errors is proposed. An example of the application of this method for the Reynolds shear stress equation, using wind-tunnel data for non-separating flow over a two-dimensional valley, is reported. The comparison between the measured and modelled parts is fair near the surface, while at higher levels, the modelled part can be shown to miss a correct treatment of the third-order moments. In the frame of this analysis, the use of the correct derivative transformation has been found to be significant even for moderately sloping topography.     

A Hybrid Wind-Farm Parametrization for Mesoscale and Climate Models

We investigate the area-averaged sensible heat flux (\(Q_{H}\)) obtained with a scintillometer along a 3.1-km path length over the city centre of ?ód?, Central Poland. The annual cycle of \(Q_{H}\) peaks in June but is lower by the middle of summer. In winter, due to a large amount of anthropogenic heat input, \(Q_{H}\) remains positive all day long, with positive night-time fluxes also found during months with frequent cold advection, e.g., June 2010. In the diurnal cycle of this flux, several features specific to urban areas are seen: the peak shifts 1–2 h after noon, the heat flux turns from positive to negative 1–2 h after sunset. In ?ód? \(Q_{H}\) was observed during inflow from the north and north-west, i.e. from the city centre. As this area is mostly covered with impervious materials, most of the heat exchanged between the ground and the overlying air is in the form of sensible heat flux. Under the conditions of inflow from the east and south-east, the maximum heat flux is approximately \(100\,\hbox {W}\,\hbox {m}^{-2}\) lower than during the inflow from the city centre, since more vegetation exists to the east and south-east of the scintillometer path. Cold and warm advection are found to be a vital factor in the observed heat-flux variability in the centre of ?ód?.     

Impacts of Steering Flows with Different Timescales on the Track of Typhoon Sanba(2012)

Typhoon Sanba(2012), the strongest tropical cyclone(TC) of the year worldwide, moved northward almost along130° longitude during its lifetime and passed through different background flows from low to high latitudes. The steering flows with different timescales for Sanba are retrieved by using the NCEP reanalysis data with the total wind field separated into: a mean state, an interannual component, an intraseasonal component, and a synoptic component.Our analysis indicates that the intraseasonal timescale wave train(WT) with east–west oriented circulations made the largest contribution to the movement of Sanba. The effects of the environmental steering with different timescales on Sanba's movement are investigated with numerical simulations using the Weather Research and Forecasting(WRF)model. In the control simulation, total fields from the NCEP reanalysis are used as initial and boundary conditions,and the northward motion of Sanba is well captured. In sensitivity experiments, each of the intraseasonal and interannual components is removed one at a time. The steering vectors associated with these timescales can explain their influences on the movement of Sanba in the experiments. Vorticity budget analyses indicate that the horizontal vorticity advection made the largest contribution to the movement of the storm.     

A Theoretical Model for the Study of Convective Turbulence Decay and Comparison with Large-Eddy Simulation Data

The development of a theoretical model fora decaying convective boundary layeris considered. The model relies on thedynamical energy spectrumequation in which the buoyancy andinertial transfer terms are retained,and a closure assumptionmade for both. The parameterization for thebuoyancy term is given providing a factorizationbetween the energy source termand its temporal decay. Regarding the inertialtransfer term a hypothesis ofsuperposition is used to describe theconvective energy source and time variationof velocity correlation separately.The solution of the budget equation for theturbulent kinetic energy spectrum is possible,given the three-dimensional initial energyspectrum. This is doneutilizing a version of the Kristensen et al.(see Boundary-Layer Meteorol. 47, 149–193)model valid for non-isotropic turbulence. During thedecay the locus of the spectralpeak remains at about the sameposition as the heat flux decreases.Comparison of the theoretical modelis performed against large-eddy simulationdata for a decaying convectiveboundary layer.     

Lagrangian Particle Dispersion Models in the Grey Zone of Turbulence: Adaptations to FLEXPART-COSMO for Simulations at 1 km Grid Resolution

Boundary-Layer Meteorology - Lagrangian particle dispersion models (LPDMs) are frequently used for regional-scale inversions of greenhouse gas emissions. However, the turbulence parameterizations...