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Results for “Convective mixing layer”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 271 records · Page 15
All sourcesNASA NTRSDOE OSTI
NASA NTRS2001

Filtered Mass Density Function for Design Simulation of High Speed Airbreathing Propulsion Systems

The objective of this research is to develop and implement new methodology for large eddy simulation of (LES) of high-speed reacting turbulent flows. We have just completed two (2) years of Phase I of this research. This annual report provides a brief and up-to-date summary of our activities during the period: September 1, 2000 through August 31, 2001. In the work within the past year, a methodology termed "velocity-scalar filtered density function" (VSFDF) is developed and implemented for large eddy simulation (LES) of turbulent flows. In this methodology the effects of the unresolved subgrid scales (SGS) are taken into account by considering the joint probability density function (PDF) of all of the components of the velocity and scalar vectors. An exact transport equation is derived for the VSFDF in which the effects of the unresolved SGS convection, SGS velocity-scalar source, and SGS scalar-scalar source terms appear in closed form. The remaining unclosed terms in this equation are modeled. A system of stochastic differential equations (SDEs) which yields statistically equivalent results to the modeled VSFDF transport equation is constructed. These SDEs are solved numerically by a Lagrangian Monte Carlo procedure. The consistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed by comparison with results obtained by an Eulerian LES procedure in which the corresponding transport equations for the first two SGS moments are solved. The unclosed SGS convection, SGS velocity-scalar source, and SGS scalar-scalar source in the Eulerian LES are replaced by corresponding terms from VSFDF equation. The consistency of the results is then analyzed for a case of two dimensional mixing layer.

Drozda, T. G.↗
NASA NTRS2019

The Interaction Between Boundary Layer and Convection Schemes in a WRF Simulation of Post‐Cold‐Frontal Clouds over the ARM East North Atlantic Site

The correct representation of lowlevel midlatitude clouds found in the wake of cold fronts strongly relies on the representation of planetary boundary layer (PBL) and convection processes, which are typically parameterized separately in numerical models. Using the Weather Research and Forecasting Model (WRF), this study investigates how distinct pairs of PBL and convection parameterization schemes represent cloud fraction in the postcoldfrontal region. The simulations focus on the region of the DOEARM Eastern North Atlantic observation site in the Azores Islands in the wake of a cold front that passed on December 25, 2015. Different PBL and convection schemes are combined to create 12 distinct configurations. The main differences between the selected physical parameterizations are the strength of vertical mixing and the entrainment. The simulations produce a wide range of cloud fractions, where some configurations significantly underestimate while others clearly overestimate satellite and surfacebased estimates of cloud fraction. A skill score is used to quantitatively assess the performance of each configuration with respect to groundbased radar data. The key processes that are found to significantly impact the cloud fraction distribution are the strength of the PBL decoupling, the vertical wind shear, entrainment and detrainment rates in shallow convection and the occurrence of drizzle. This indicates that to successfully simulate postcoldfrontal clouds, modeled physics must balance strong internal vertical mixing and weak exchange with the free troposphere. For this case study, cloud fraction was more sensitive to the choice of convection scheme than PBL scheme.

moist convection interaction between parameterizat↗
NASA NTRS1990

Supersonic combustor modeling

The physical phenomena involved when a supersonic flow undergoes chemical reaction are discussed. Detailed physical models of convective and diffusive mixing, and finite rate chemical reaction in supersonic flow are presented. Numerical algorithms used to solve the equations governing these processes are introduced. Computer programs using these algorithms are used to analyze the structure of the reacting mixing layer. It is concluded that, as in subsonic flow, exothermic heat release in unconfined supersonic flows retards fuel/air mixing. Non mixing is shown to be a potential problem in reducing the efficiency of supersonic as well as subsonic combustion. Techniques for enhancing fuel/air mixing and combustion are described.

Riggins, David W.↗
NASA NTRS1992

Vorticity generation by contoured wall injectors

A class of contoured wall fuel injectors was designed to enable shock-enhancement of hypervelocity mixing for supersonic combustion ramjet applications. Previous studies of these geometries left unresolved questions concerning the relative importance of various axial vorticity sources in mixing the injectant with the freestream. The present study is a numerical simulation of two generic fuel injectors which is aimed at elucidating the relative roles of axial vorticity sources including: baroclinic torque through shock-impingement, cross-stream shear, turning of boundary layer vorticity, shock curvature, and diffusive flux. Both the magnitude of the circulation, and the location of vorticity with respect to the mixing interface were considered. Baroclinic torque and cross-stream shear were found to be most important in convectively mixing the injectant with the freestream, with the former providing for deposition of vorticity directly on the fuel/air interface.

Waitz, Ian A.↗
DOE OSTI2024

On the Geometry of the Near-core Magnetic Field in Massive Stars

It is well-known that the cores of massive stars sustain a stellar dynamo with a complex magnetic field configuration. However, the same cannot be said for the field's strength and geometry at the convective–radiative boundary, which are crucial when performing asteroseismic inference. In this Letter, we present 3D magnetohydrodynamic (MHD) simulations of a 7 M ⊙ mid-main-sequence star, with particular attention given to the convective–radiative boundary in the near-core region. Our simulations reveal that the toroidal magnetic field is significantly stronger than the poloidal field in this region, contrary to recent assumptions. Moreover, the rotational shear layer, also important for asteroseismic inference, is specifically confined within the extent of the Brunt–Väisälä frequency peak. These results, which are based on the inferred properties of HD 43317, have widespread implications for asteroseismic studies of rotation, mixing, and magnetism in stars. While we expect our results to be broadly applicable across stars with similar Brunt–Väisälä frequency profiles and stellar masses, we also expect the MHD parameters (e.g., Re m ) and the initial stellar rotation rate to impact the geometry of the field and differential rotation at the convective–radiative interface.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1994

Budget analysis of FIFE 1987 sonde data

Mean thermodynamic profiles are presented for the daytime convective boundary layer evolution for three groups of sunny days in late June and early July, August, and October using data from the First International Satellite Land Surface Climatology Project (ISLSCP) Field Experiment (FIFE) 1987 Kansas experiment. Budget models for the mixed layer evolution and the boundary layer rise are used to estimate the boundary layer top entrainment parameter. A value of A(sub R) = 0.44 +/- 0.21 is found for the mean ratio of the inversion level virtual heat flux to the surface virtual heat flux.

Betts, Alan K.↗
NASA NTRS1986

Nonlinear compressible convection penetrating into stable layers and producing internal gravity waves

Penetrative convection spanning multiple scale heights is studied within a simple stellar envelope consisting of three layers: a convectively unstable middle layer bounded above and below by stably stratified polytropes. Two-dimensional numerical simulations are used to investigate the fully compressible nonlinear motions that ensue. The cellular flows display prominent downward-directd plumes surrounded by broader regions of upflow. Such asymmetry arises because pressure fluctuations accentuate buoyancy driving in the concentrated plumes and can even lead to weak buoyancy braking in the surrounding ascending flows. As the plumes plunge downward into a region of stable stratification, they serve to excite a broad spectrum of internal gravity waves there. The induced waves are not passive, for they feed back upon the plumes by deflecting them sideways, thereby modulating the amplitude of the convection in time even in the unstable layer. The penetrative motions that billow upward into the upper stable zone are distinctly weaker, and they cascade back downward toward the unstable zone over a broad horizontal scale. The strong excitation of gravity waves by the convection has implications for gradual mixing deep within a star.

Hurlburt, Neal E.↗
DOE OSTI2020

Subcloud and Cloud-Base Latent Heat Fluxes during Shallow Cumulus Convection

Doppler and Raman lidar observations of vertical velocity and water vapor mixing ratio are used to probe the physics and statistics of subcloud and cloud-base latent heat fluxes during cumulus convection at the ARM Southern Great Plains (SGP) site in Oklahoma, United States. The statistical results show that latent heat fluxes increase with height from the surface up to ~0.8 Z i (where Z i is the convective boundary layer depth) and then decrease to ~0 at Z i . Peak fluxes aloft exceeding 500 W m −2 are associated with periods of increased cumulus cloud cover and stronger jumps in the mean humidity profile. These entrainment fluxes are much larger than the surface fluxes, indicating substantial drying over the 0–0.8 Z i layer accompanied by moistening aloft as the CBL deepens over the diurnal cycle. We also show that the boundary layer humidity budget is approximately closed by computing the flux divergence across the 0–0.8 Z i layer. Composite subcloud velocity and water vapor anomalies show that clouds are linked to coherent updraft and moisture plumes. The moisture anomaly is Gaussian, most pronounced above 0.8 Z i and systematically wider than the velocity anomaly, which has a narrow central updraft flanked by downdrafts. This size and shape disparity results in downdrafts characterized by a high water vapor mixing ratio and thus a broad joint probability density function (JPDF) of velocity and mixing ratio in the upper CBL. We also show that cloud-base latent heat fluxes can be both positive and negative and that the instantaneous positive fluxes can be very large (~10 000 W m −2 ). However, since cloud fraction tends to be small, the net impact of these fluxes remains modest.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

Development of Composite Photocatalyst Materials that are Highly Selective for Solar Hydrogen Production and their Evaluation in Z-Scheme Reactor Designs

The key technology gap preventing a vertically stacked dual-bed particle suspension reactor from achieving the DOE MYRD&D ultimate cost target for H 2 production remains the lack of materials in particle form factor that exhibit ≥10% solar-to-H 2 energy conversion (STH) efficiency as a suspension. Therefore, our project goals centered around strategies to increase the STH efficiency by enhancing photophysical properties of perovskite oxide particles including increased visible-light absorption, increased selectivity for electrocatalysis of the H 2 evolution reaction (HER) and the O 2 evolution reaction (OER) through development of ultrathin oxide coatings, correlating composition and structure to function, and improving understanding of multiscale transport and kinetic processes.

08 HYDROGEN↗
NASA NTRS1990

Effects of compressibility on the characteristics of free shear layers

A high Reynolds number two-dimensional constant pressure compressible shear layer was formed at the trailing edge of an 0.5 mm-thick splitter plate. Convective Mach numbers of 0.51 and 0.64 were investigated using a two-component coincident LDV for the measurements. For the lower convective Mach number case, the nondimensionalized shear-layer and vorticity thickness growth rates were over 20 percent higher and the momentum thickness growth rate was over 30 percent higher than those of the higher convective Mach number case. The results seen to indicate that both small scale and large scale mixing are reduced with increasing convective Mach number.

Samimy, M.↗
NASA NTRS1999

Ocean-Atmosphere Interaction in Climate Changes

The diagram, which attests the El Nino teleconnection observed by the NASA Scatterometer (NSCAT) in 1997, is an example of the results of our research in air-sea interaction - the core component of our three-part contribution to the Climate Variability Program. We have established an interplay among scientific research, which turns spacebased data into knowledge, a push in instrument technology, which improves observations of climate variability, and an information system, which produces and disseminates new data to support our scientific research. Timothy Liu led the proposal for advanced technology, in response to the NASA Post-2002 Request for Information. The sensor was identified as a possible mission for continuous ocean surface wind measurement at higher spatial resolution, and with the unique capability to measure ocean surface salinity. He is participating in the Instrument Incubator Program to improve the antenna technology, and is initiating a study to integrate the concept on Japanese missions. He and his collaborators have set up a system to produce and disseminate high level (gridded) ocean surface wind/stress data from NSCAT and European missions. The data system is being expanded to produce real-time gridded ocean surface winds from Quikscat, and precipitation and evaporation from the Tropical Rain Measuring Mission. It will form the basis for a spacebased data analysis system which will include momentum, heat and water fluxes. The study on 1997 El Nino teleconnection illustrates our interdisciplinary and multisensor approach to study climate variability. The diagram shows that the collapse of trade wind and the westerly wind anomalies in the central equatorial Pacific led to the equatorial ocean warming. The equatorial wind anomalies are connected to the anomalous cyclonic wind pattern in the northeast Pacific. The anomalous warming along the west coast of the United States is the result of the movement of the pre-existing warm sea surface temperature anomalies with the cyclonic wind anomalies toward the coast. The results led to a new study which identifies decadal ocean variations in the Northeast Pacific. Three studies of oceanic responses to wind forcing caused by the seasonal change of monsoons, the passage of a typhoon, and the 1997 El Nino, were successfully conducted. Besides wind forcing, we continue to examine new techniques for estimating thermal and hydrologic fluxes, through the inverse ocean mixed-layer model, through divergence of atmospheric water transport, and by direct retrieval from radiances observed by microwave radiometers. Greenhouse warming has been linked to water vapor measured by two spaceborne sensors in two studies. In the first study, strong baroclinicity and deep convection were found to transport water vapor to the upper atmosphere and increase greenhouse trapping over the storm tracks of the North Pacific and Atlantic. In another study, the annual cycle of greenhouse warming were related to sea surface temperature (SST) and integrated water vapor, and the latitudinal dependence of the magnitudes and phases of the annual cycles were compared.

Liu, W. Timothy↗
NASA NTRSDate not supplied

Comparing Tropopause-Penetrating Convection Identifications Derived from NEXRAD and GOES over the Contiguous United States

Overshooting cloud tops (OTs) are a result of powerful updrafts that rapidly transport air from the troposphere to the lower stratosphere. Aside from the well-known relationships between strong updrafts and severe weather, the boundaries of updrafts, including the OT, are turbulent, which leads to irreversible mixing of air between the stratosphere and troposphere, altering the composition of both layers. In order to better understand the impact of OTs, knowing precisely when and where they occur is vital. Within the last decade remote-sensing observations from satellites and radars have been used to identify OTs, but the results thus far have not been entirely consistent due to challenges with accurately identifying OTs with these sensor datasets. This study compares OTs identified using NEXRAD reflectivity data from the GridRad dataset with an approach that uses GOES infrared (IR) data. The GridRad method estimates the altitude of echo tops directly from the radar pointing geometry, merging a combination of radars that enables products every 5 mins, with 2 km spatial and 1 km vertical grid spacing. The GOES satellite method uses spatial patterns in tropopause-relative IR brightness temperature to derive an OT likelihood. The study region covers a large part of the contiguous United States during selected active convection dates in 2017 and includes data from GOES-13/16. A large sample of OTs observed by NEXRAD and GOES-13/16 satellites are analyzed to better understand NEXRAD and IR observations of OTs, quantify agreement between the OT detection methods, and demonstrate how the 4x increase in spatial resolution from GOES-13 to GOES-16 impacts OT algorithm detection performance. The results show that for nearly time-matched scenes, with many severe storms, GOES-13 and -16 detection accuracy is quite comparable; however, some missed detections are attributed to poorer GOES-13 image resolution. It was found that GOES-13 OT regions are ~2 K warmer than GOES-16 on average, but such differences exceed 5 K for the most intense OT regions. It is found that false GOES IR-based detections are unavoidable, given that cold cloud pixels significantly colder than the tropopause from nearby or recently decayed OTs can persist within anvils for 15 minutes or more.

John W. Cooney↗
NASA NTRS2002

Convection and Easterly Wave Structure Observed in the Eastern Pacific Warm-Pool during EPIC-2001

During September-October 2001, the East Pacific Investigation of Climate Processes in the Coupled Ocean-Atmosphere System (EPIC-2001) ITCZ field campaign focused on studies of deep convection in the warm-pool region of the East Pacific. In addition to the TAO mooring array, observational platforms deployed during the field phase included the NOAA ship RN Ronald H. Brown, the NSF ship RN Horizon, and the NOAA P-3 and NCAR C-130 aircraft. This study combines C-band Doppler radar, rawinsonde, and surface heat flux data collected aboard the RN Brown to describe ITCZ convective structure and rainfall statistics in the eastern Pacific as a function of 3-5 day easterly wave phase. Three distinct easterly wave passages occurred during EPIC-2001. Wind and thermodynamic data reveal that the wave trough axes exhibited positively correlated U and V winds and a slight westward phase tilt with height. A relatively strong (weak) northeasterly deep tropospheric shear followed the trough (ridge) axis. Temperature and humidity perturbations exhibited mid-to upper level cooling (warming) and drying (moistening) in the northerly (trough and southerly) phase. At low levels warming (cooling) occurred in the northerly (southerly) phase with little change in the relative humidity, though mixed layer mixing ratios were larger during the northerly phase. When composited, radar, sounding, lightning and surface heat flux observations suggest the following systematic behavior as a function of wave phase: approximately zero to one quarter wavelength ahead of (behind) the wave trough in northerly (southerly) flow, larger (smaller) CAPE, lower (higher) CIN, weaker (stronger) tropospheric shear, higher (lower) conditional mean rain rates, higher (lower) lightning flash densities, and more (less) robust convective vertical structure occurred. Latent and sensible heat fluxes reached a minimum in the northerly phase and then increased through the trough, reaching a peak during the ridge phase (leading the peak in CAPE). From a radar echo coverage perspective, larger areas of light rain and slightly larger (10%) area averaged rain rates occurred in the vicinity of, and just behind, the trough axes in southerly flow. Importantly, the transition in convective structure observed across the trough axis when considered with the relatively small change in area mean rain rates suggests the presence of a transition in the vertical structure of diabatic heating across the easterly waves examined. The inferred transition in heating structure is supported by radar diagnosed divergence profiles that exhibit convective (stratiform) characteristics ahead of (behind) the trough.

Peterson, Walter A.↗
NASA NTRS1994

Biogenic nonmethane hydrocarbon emissions estimated from tethered balloon observations

A new technique for estimating surface fluxes of trace gases, the mixed-layer gradient technique, is used to calculate isoprene and terpene emissions from forests. The technique is applied to tethered balloon measurements made over the Amazon forest and a pine-oak forest in Alabama at altitudes up to 300 m. The observations were made during the dry season Amazon Boundary Layer Experiment (ABLE 2A) and the Rural Oxidants in the Southern Environment 1990 experiment (ROSE I). Results from large eddy simulations of scalar transport in the clear convective boundary layer are used to infer fluxes from the balloon profiles. Profiles from the Amazon give a mean daytime emission of 3630 +/- 1400 micrograms isoprene sq m/h, where the uncertainty represents the standard deviation of the mean of eight flux estimates. Twenty profiles from Alabama give emissions of 4470 +/- 3300 micrograms isoprene sq m/h, 1740 +/- 1060 micrograms alpha-pinene sq m/h, and 790 +/- 560 micrograms beta-pinene sq m/h, respectively. These results are in agreement with emissions derived from chemical budgets. The emissions may be overestimated because of uncertainty about how to incorporate the effects of the canopy on the mixed-layer gradients. The large variability in these emission estimates is probably due to the relatively short sampling times of the balloon profiles, though spatially heterogeneous emissions may also play a role. Fluxes derived using this technique are representative of an upwind footprint of several kilometers and are independent of hydrocarbon oxidation rate and mean advection.

Davis, K. J.↗
DOE OSTI2020

Observations Pertaining to Precipitation within the Northeast Pacific Stratocumulus-to-Cumulus Transition

Three genuine stratocumulus-to-cumulus transitions sampled during the Cloud System Evolution over the Trades (CSET) campaign are documented. The focus is on Lagrangian evolution of in situ precipitation, thought to exceed radar/lidar retrieved values because of Mie scattering. Two of the three initial stratocumulus cases are pristine [cloud droplet number concentrations ($N_d$) of ~22 cm –3 ] but occupied boundary layers of different depths, while the third is polluted ($N_d$ ~ 225 cm –3 ). Hourly satellite-derived cloud fraction along Lagrangian trajectories indicate that more quickly deepening boundary layers tend to transition faster, into more intense but more occasional precipitation. These transitions begin either in the morning or late afternoon, suggesting that preceding night processes can precondition or delay the inevitable transition. The precipitation shifts toward larger drop sizes throughout the transition as the boundary layers deepen, with aerosol concentrations only diminishing in two of the three cases. Ultraclean ($N_d$ < 1 cm –3 ) cumulus clouds evolved from pristine stratocumulus cloud with unusually high precipitation rates occupying a shallow, well-mixed boundary layer. Results from a simple one-dimensional evaporation model and from radar/lidar retrievals suggest subcloud evaporation likely increases throughout the transition. This, coupled with larger drop sizes capable of lowering the latent cooling profile, facilitates the transition to more surface-driven convection. The coassociation between boundary layer depth and precipitation does not provide definitive conclusions on the isolated effect of precipitation on the pace of the transition. Differences between the initial conditions of the three examples provide opportunities for further modeling studies.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1982

Features of a reattaching turbulent shear layer subject to an adverse pressure gradient

Experimental data have been obtained in an incompressible turbulent flow over a rearward-facing step with superimposed adverse pressure gradient. Mean velocities, Reynolds stresses and triple-products measured by a laser Doppler velocimeter are presented for two cases of adverse pressure gradient. Mixing lengths, eddy viscosities, production, convection, turbulent diffusion, and dissipation terms are extracted from the data. These data are compared with various mixing length and eddy-viscosity turbulence models. Numerical calculations incorporating the k-epsilon and the algebraic-stress turbulence models are compared with the data. When determining quantities of engineering interest, the modified algebraic-stress model (ASM) is a significant improvement over the unmodified ASM and the unmodified k-epsilon model

Driver, D. M.↗
NASA NTRS1996

Mixing and Transition Control Studied

Considerable progress in understanding nonlinear phenomena in both unbounded and wallbounded shear flow transition has been made through the use of a combination of high- Reynolds-number asymptotic and numerical methods. The objective of this continuing work is to fully understand the nonlinear dynamics so that ultimately (1) an effective means of mixing and transition control can be developed and (2) the source terms in the aeroacoustic noise problem can be modeled more accurately. Two important aspects of the work are that (1) the disturbances evolve from strictly linear instability waves on weakly nonparallel mean flows so that the proper upstream conditions are applied in the nonlinear or wave-interaction streamwise region and (2) the asymptotic formulations lead to parabolic problems so that the question of proper out-flow boundary conditions--still a research issue for direct numerical simulations of convectively unstable shear flows--does not arise. Composite expansion techniques are used to obtain solutions that account for both mean-flow-evolution and nonlinear effects. A previously derived theory for the amplitude evolution of a two-dimensional instability wave in an incompressible mixing layer (which is in quantitative agreement with available experimental data for the first nonlinear saturation stage for a plane-jet shear layer, a circular-jet shear layer, and a mixing layer behind a splitter plate) have been extended to include a wave-interaction stage with a three-dimensional subharmonic. The ultimate wave interaction effects can either give rise to explosive growth or an equilibrium solution, both of which are intimately associated with the nonlinear self-interaction of the three dimensional component. The extended theory is being evaluated numerically. In contrast to the mixing-layer situation, earlier comparisons of theoretical predictions based on asymptotic methods and experiments in wall-bounded shear-flow transition have been somewhat lacking in one aspect or another. The current work strongly suggests that the main weakness is the underlying asymptotic representation of the linear "part" of the problem and not the explicit modeling of the nonlinear/wave-interaction effects. Consequently, the long-wave-length/high-Reynolds-number asymptotic limit for the Blasius boundary-layer stability problem was reexamined, and a new dispersion relationship for the instability waves that is uniformly valid for both the upper- and lower branch regions to the required order of approximation was obtained. A comparison with numerical results, obtained by solving the Orr-Sommerfeld stability problem, shows that the asymptotic formula provides surprisingly good results, even for values of the frequency parameter usually encountered in experimental investigations. This is particularly evident in the dynamically important upper-branch region, where much of the nonlinear interactions in transition experiments are believed to take place. The result is important in that it can be used to greatly improve the accuracy of weakly nonlinear critical-layer-based theories, and a consistent nonlinear theory is currently under evaluation.

Source record↗
NASA NTRS1987

Measurement of heat and moisture fluxes at the top of the rain forest during ABLE

Observations are presented of turbulent heat, moisture, and momentum transport made at two levels, approximately 5 and 10 m above the Amazon rain forest canopy. Data acquired at 10 Hz included variances and some mixed third moments of vertical velocity, temperature, and humidity. Two features of the data appear to question the displacement height hypothesis: (1) The characteristic dissipation length scale in the near-canopy layer varied between 20 m in stable conditions to approximately 150 m during afternoon convective conditions, generally larger scales than would be expected; and (2) No appreciable difference in dissipation scales was seen at the two observed levels. Observed peaks in vertical velocity-temperature cospectra lead to similar length scale estimates for dominant eddies. Heat budgets on selected days show that frequent periods with negative heat flux concurrent with continuing positive moisture flux occur in early afternoon, and this is believed to indicate the patchy nature of canopy-atmosphere coupling. Vertical velocity skewness was observed to be negative on three successive days and exhibited a sharp positive gradient.

Fitzjarrald, David R.↗