Footprint Analysis: A Closed Analytical Solution Based On Height-Dependent Profiles Of Wind Speed And Eddy Viscosity

Based on the theoretical background of existing models for the crosswind-integrated footprint, a new model is presented, which, in contrast to the existing models, describes the normalized footprint by a closed analytical formula. This was made possible by using well-known power profiles for wind speed and eddy viscosity instead of Monin–Obukhov based profiles at a certain stage of model development. However, the major difference between the new model and the existing models is that the so-called shape parameter of vertical plume dispersion, a function of upwind distance in the existing models, is set constant in the new model in order to circumvent a formal inconsistency found in the derivation of the existing models. Due to this inconsistency, the existing models do not generally satisfy the fundamental condition that the cumulative normalized footprint must approach unity for the upwind distance tending towards infinity.     

Numerical analysis of flux footprints for different landscapes

Summary A model for the canopy – planetary boundary layer flow and scalar transport based on E- closure was applied to estimate footprint for CO₂ fluxes over different inhomogeneous landscapes. Hypothetical heterogeneous vegetation patterns – forest with clear-cuts as well as hypothetical heterogeneous relief – a bell-shaped valley and a ridge covered by forest were considered. The distortions of airflow caused by these heterogeneities are shown – the upwind deceleration of the flow at the ridge foot and above valley, acceleration at the crest and the flow separation with the reversed flow pattern at lee slopes of ridge and valley. The disturbances induce changes in scalar flux fields within the atmospheric surface layer comparing to fluxes for homogeneous conditions: at a fixed height the fluxes vary as a function of distance to disturbance. Correspondingly, the flux footprint estimated from model data depends on the location of the point of interest (flux measurement point) and may significantly deviate from that for a flat terrain. It is shown that proposed method could be used for the choice of optimal sensor position for flux measurements over complex terrain as well as for the interpretation of data for existing measurement sites. To illustrate the latter the method was applied for experimental site in Solling, Germany, taking into account the complex topography and vegetation heterogeneities. Results show that in certain situations (summer, neutral stratification, south or north wind) and for a certain sensor location the assumptions of idealized air flow structure could be used for measurement interpretation at this site, though in general, extreme caution should be applied when analytical footprint models are used in the interpretation of flux measurements over complex sites.     

Footprint Analysis For Measurements Over A Heterogeneous Forest

The air flow and vertical distribution of sources/sinks inside aforest canopy have been taken into accountin the analysis of the contribution of sources/sinks to measured fluxes and concentrations above a forest. Thestochastic estimators for concentrations and fluxes are described and their evaluation is performed by simulationof an ensemble of fluid parcel trajectories. The influence of the forest canopy on the footprint is important forobservation levels up to a few times the forest height. The influence of along-wind turbulent diffusion, whichanalytical atmospheric surface layer (ASL) footprint models do not account for, is significant even at higherlevels. The footprint analysis has been performed to deduce the Douglas fir canopy carbon dioxide uptake from eddycovariance flux measurements above a mixed Douglas fir–beech forest during the pre-leaf periods of the beech.The scatter in the results indicates that such an analysis is limited, presumably due to horizontal inhomogenetiesin flow statistics, which were not included in trajectory simulation. The analysis, however, is useful for theestimation of the qualitative effect of the forest canopy on the footprint function.     

Comments on the 'Universality of the Lagrangian Velocity Structure Function Constant (C0) ACROSS DIFFERENT KINDS OF TURBULENCE’

Recently Du ( Boundary-Layer Meteorology 83, 207–219, 1997) estimated the value of the Lagrangian velocity structure constant, C0, in the inertial subrange by comparing experimental diffusion data and simulation results obtained with the one-dimensional form of Thomson's model ( J. Fluid Mech. 180, 529–556, 1987). Du reported that for several different flows (grid turbulence, a wind-tunnel boundary layer and the atmospheric surface layer under neutral stratification) the value of C0 is 3.0±0.5. Here, it is shown that optimal model agreement with experimental diffusion data for the wind-tunnel boundary layer is, in fact, obtained when C0=5.0 ± 0.5. It is also shown that accounting for the skewness of velocity statistics and finite Reynolds number effects does not significantly change this estimate for the value of C0. It is suggested that one-dimensional Lagrangian stochastic models are inconsistent with the supposed universality of C0.     

Determination of Area-Averaged Sensible Heat Fluxes with a Large Aperture Scintillometer over a Heterogeneous Surface – Flevoland Field Experiment

To test the applicability of the scintillation method over a heterogeneous area an experiment was carried out in the summer of 1998 in Flevoland (The Netherlands). In the patchy area only four crops were grown namely sugar beet, potatoes, wheat and onions. From eddy covariance measurements it was found that the heterogeneity was mainly caused by differences in thermal properties. No variations in the aerodynamics roughness length were observed. Two large aperture scintillometers were installed at a height of 11.6 and 20.4 m. A good resemblance was found between the sensible heat fluxes derived from both LAS instruments and the area-averaged fluxes obtained from the in-situ eddy covariance measurements. The slight underestimation of the lower LAS could be assessed using a blending height model and an analytical footprint model. The results also indicated that when scintillometer measurements are made below the blending height the violation to Monin–Obukhov Similarity Theory is small and that reasonable fluxes can be obtained from path-averaged structure parameters.     

Transfer processes in vegetation by electrochemical analog

Electrochemical modelling may provide fast order-of-magnitude estimates of energy and mass transfer in crops and partly fill the gap between rigid, expensive field experiments and oversimplified mathematical modelling. The technique comprises breaking up a flowing electrolyte in a manner analogous — but not necessarily completely similar — to the way the atmospheric flow is broken up by crops and studying the flow of ions in the electrolyte.The effects of variations in free-flow velocity, plant spacing, row spacing and orientation on velocity field, transfer coefficients and eddy diffusivities have been studied in an electrochemical model under conditions relevant to well-ventilated crops with relatively simple canopy structure. A one-dimensional analytical model is developed for the transfer at solid surfaces, with direct proportionality between transfer coefficient and eddy diffusivity. The proportionality constant has values in the model of (3.5±0.5)×10^–3cm^–1 at the ground and (5+-2)×10^–4cm^–1 (for a cylindrical probe) near canopy top. The latter is not too different from the values proposed for foliage surfaces in real canopies, viz., approximately 1×10^–3 cm^–1 (Philip, 1964) and 7×10^–4 cm^–1 (Uchijima, 1966). may therefore become a useful parameter in scaling transfer coefficients from systems with different molecular diffusion properties.A tentative extrapolation of measured data to corresponding values in air is given. The agreement between predicted values and presently available field data is encouraging although more precise field data are required for a final judgment on the validity of the model.This paper is published with the permission of the International Institute of Heat and Mass Transfer; an abbreviated version of the paper is given in the Proceedings, Intl. Seminar on Heat and Mass Transfer in the Environment of Vegetation, Dubrovnik, Aug. 26–30, 1974.     

A Three-Dimensional Backward Lagrangian Footprint Model For A Wide Range Of Boundary-Layer Stratifications

We present a three-dimensional Lagrangian footprint model with the ability to predict the area of influence (footprint) of a measurement within a wide range of boundary-layer stratifications and receptor heights. The model approach uses stochastic backward trajectories of particles and satisfies the well-mixed condition in inhomogeneous turbulence for continuous transitions from stable to convective stratification. We introduce a spin-up procedure of the model and a statistical treatment of particle touchdowns which leads to a significant reduction of CPU time compared to conventional footprint modelling approaches. A comparison with other footprint models (of the analytical and Lagrangian type) suggests that the present backward Lagrangian model provides valid footprint predictions under any stratification and, moreover, for applications that reach across different similarity scaling domains (e.g., surface layer to mixed layer, for use in connection with aircraft measurements or with observations on high towers).     

Experimental evaluation of analytical and Lagrangian surface-layer flux footprint models

Three surface-layer flux footprint models have been evaluated with the results of an SF₆ tracer release experiment specifically designed to test such models. They are a Lagrangian stochastic model, an analytical model, and a simplified derivative of the analytical model. Vertical SF₆ fluxes were measured by eddy correlation at four distances downwind of a near-surface crosswind line source in an area of homogeneous sagebrush. The mean fluxes were calculated for 136 half-hour test periods and compared to the fluxes predicted by the footprint models. All three models gave similar predictions and good characterizations of the footprint over the stability range -0.01 < z ₀/L < 0.005. The predictions of the three models were within the limits of the uncertainty of the experimental measurements in all but a few cases within this stability range. All three models are unconditionally recommended for determining the area defined by the footprint over short vegetative canopies in this range. They are also generally appropriate for estimating flux magnitudes within the limits of experimental uncertainties. Most of the mean differences observed between the measured and predicted fluxes at each of the four towers reflect a tendency for the measured fluxes to be greater than those predicted by the three models. Rigorous verification of the models in strongly stable conditions was complicated by the need to obtain very accurate measurements of small fluxes in only marginally stationary conditions. Verification in strongly unstable conditions was hampered by the limited number of appropriate data.     

An Improved Mellor–Yamada Level-3 Model with Condensation Physics: Its Design and Verification

A computational scheme for an improved Mellor–Yamada(M–Y) Level-3 model with condensation physics is proposedand its performance is examined against large-eddy-simulationdata on radiation fog. The improved M–Y model greatlycorrects several shortcomings of the original M–Y model:the underestimations of the mixed-layer depth and themagnitude of turbulent kinetic energy, and the discrepanciesin the formation and dissipation times of the fog. Inaddition the improved M–Y model can reproduce theoccurrence of Kelvin–Helmholtz instability and periodicoscillations due to its energy cycle. It is shown that theoptimization of both the closure constants and the masterlength scale is required for this improvement.The improved M–Y model has an improvement also in theLevel-2.5 version. Although the performance of theLevel-2.5 version is not so good as that of the Level-3version, the former has the advantage of relatively lowcomputational cost and is popularly used in operationalweather forecasts. Our computational scheme for theimproved M–Y model allows us to switch its hierarchylevels easily according to the purpose.     

Heat transfer to and from vegetated surfaces: An analytical method for the bulk exchange coefficients

The semi-analytical model outlined in a previous study (Massman, 1987) to describe momentum exchange between the atmosphere and vegetated surfaces is extended to include the exchange of heat. The methods employed are based on one-dimensional turbulent diffusivities and use analytical solutions to the steady-state diffusion equation. The model is used to assess the influence that (1) the canopy foliage structure and density, (2) the wind profile structure within the canopy, and (3) the shelter factor can have upon the inverse surface Stanton number (kB ^–1) as well as to explore the consequences of introducing a scalar displacement height which can be different from the momentum displacement height. In general, the triangular foliage area density function gives results which agree more closely to observations than that for constant foliage area density. The intended application of this work and its predecessor (Massman, 1987) is for parameterizing the bulk aerodynamic resistances for heat and momentum exchange for use within large-scale models of plant-atmosphere exchanges.     

CO2 efflux from agricultural soils in Eastern Germany – comparison of a closed chamber system with eddy covariance measurements

Summary In order to quantify the effects of temperature and soil water content on soil respiration, during June and July 2002 CO₂ soil efflux was measured with a closed chamber (non-steady state, flow through) system in the field. The amount of CO₂ emission was highly dependent on the land-use in the observation area, which consisted of meadow soil and brownfield. The CO₂ emission from the brownfield ranged from 0.9 to 5.5µmol CO₂ m^–2s^–1, and that for meadow soil from 1.1 to 12.6µmol CO₂ m^–2s^–1. Soil respiration, as a function of soil temperature (T_soil), relative soil water content (RSWC), soil pH, and the soil carbon/nitrogen ratio (C/N), was analysed by a modified closed non-linear regression model. Between 63% and 81% of the variation of soil CO₂ emission could be explained with changes of T_soil, RSWC, pH, and C/N for the individual chambers on the brownfield.Subsequent analysis involved a comparison of the soil chamber results with eddy covariance (EC) measurements of one week, and included a footprint analysis to account for the influence of the different land use types on the measurements. For this, EC data (143 measurements after quality check) were restricted to those originating from the brownfield area with more than 90% of the flux. For a second comparison, the net ecosystem exchange (NEE) was calculated for different parts of the meadow using the SVAT model PROXEL. Together with the respiration from the brownfield, a weighted average of model NEE was produced using the flux contribution determined by the footprint model. Acceptable agreement (r²=0.69) was found between the modelled data and individual EC measurements, except during situations where the performance of the footprint model was disturbed by internal boundary layer effects.     

Application and numerical experiments with a highly-accurate advection scheme in a regional transfer model

Summary This paper examines a 19-level regional transfer model, EM3. The sensitivity of the results to different transfer schemes and effectiveness of using a high-precision scheme in raising modeling accuracy is examined. The use of a good scheme to improve the ability to simulate horizontal diffusion close to reality is also explored. EM3 was run with different precision schemes, i.e., a high-precision second-order moment conservation prather scheme (SOM for short) and an anti-diffusion Smolarkiewicz scheme (Smolar for short). The model was run 70 hrs. Results indicate that EM3 is much more sensitive to the precision of the schemes. SOM and Smolar, were compared, the former showing very weak numerical diffusion and computational dispersion so that positive solutions were kept. It is shown that i) numerical diffusion in SOM is one order of magnitude less compared to the latter; ii) the central intensity of transferred SO₂ increases by 4–5 times and the average motion trajectory of the concentration center is much improved in SOM; iii) SOM allows EM3 to precisely reflect the real horizontal diffusion. Therefore, choice of a high-precision advection scheme for EM3 will contribute greatly to an increase in its simulation accuracy.With 4 Figures     

A global three-dimensional model of the tropospheric sulfur cycle

The tropospheric part of the atmospheric sulfur cycle has been simulated in a global three-dimensional model. The model treats the emission, transport, chemistry, and removal processes for three sulfur components; DMS (dimethyl sulfide), SO₂ and SO₄ ^2– (sulfate). These processes are resolved using an Eulerian transport model, the MOGUNTIA model, with a horizontal resolution of 10° longitude by 10° latitude and with 10 layers in the vertical between the surface and 100 hPa. Advection takes place by climatological monthly mean winds. Transport processes occurring on smaller space and time scales are parameterized as eddy diffusion except for transport in deep convective clouds which is treated separately. The simulations are broadly consistent with observations of concentrations in air and precipitation in and over polluted regions in Europe and North America. Oxidation of DMS by OH radicals together with a global emission of 16 Tg DMS-S yr^–1 from the oceans result in DMS concentrations consistent with observations in the marine boundary layer. The average turn-over times were estimated to be 3, 1.2–1.8, and 3.2–6.1 days for DMS, SO₂, and SO₄ ^2– respectively.     

Convective heat transfer measurements of plants in a wind tunnel

Heat transfer was studied between intact leaves of various sizes and shapes in vivo under free and forced air conditions. Use of a wind tunnel and a microwave transmitter to heat the leaves facilitated measurements of convective, along with radiative and evaporative, heat losses from plant leaves. Knowledge of input energy, analysis of cooling curves, and established formulae, respectively, formed the basis of the steady-state, unsteady-state, and analytical methods for the determination of heat transfer coefficients.Typical values of steady-state free convection coefficients for Peperomia obtusifolia varied from 1.5 × 10^–4 to 1.9 × 10^–4 cal cm^–2 s^–1 C^–1 as the temperature difference was increased from 5.9 to 9.6°C, whereas the forced convection coefficient was found to be 4.2 × 10^–4 cal cm^–2 s^–1 C^–1 at 122 cm s^–1 wind velocity. For egg-plant, this value was about 9 × 10^–4 cal cm^–2 s^–1 C^–1 at 488 cm s^–1 wind velocity. Convection coefficients as determined under steady-state conditions are compared with those of the unsteady-state and with analytical values for a single leaf and leaves of three different plants. In general, experimental values were found to be higher than the analytical ones.     

A semi-analytical solution for the three-dimensional advection–diffusion equation considering non-local turbulence closure

Atmospheric air pollution turbulent fluxes can be assumed to be proportional to the mean concentration gradient. This assumption, along with the equation of continuity, leads to the advection–diffusion equation. Moreover, large eddies are able to mix scalar quantities in a manner that is counter to the local gradient. In this work we present a semi-analytical solution for the three-dimensional steady-state advection–diffusion equation, considering non-local turbulence closure using the Generalized Integral Advection Diffusion Multilayer Technique (GIADMT). We report some examples of applications of the new solution for two different datasets and for a water tank experiment.     

Improvement Of The Mellor–Yamada Turbulence Closure Model Based On Large-Eddy Simulation Data

On the basis of data constructed with large-eddy simulation (LES), an attempt is made to improve the Mellor–Yamada (M–Y) turbulence closure model. Firstly, stably-stratified and convective planetary boundary layers without moisture are simulated by a LES model to obtain a database for the improvement. Secondly, based on the LES data, closure constants are re-evaluated and a new diagnostic equation for the master length scale L is proposed. The new equation is characterized by allowing L in the surface layer to vary with stability instead of constant kz, where k is the von Kármán constant, and z is height.The non-dimensional eddy-diffusivity coefficients calculated from the modifiedM–Y model are in satisfactory agreement with those from the LES data. It isfound that the modified M–Y model improves the original one largely, and thatthe improvement is achieved by considering buoyancy effects on the pressurecovariances andby using the newly proposed equation for L.     

Chloride and Bromide Loss from Sea-Salt Particles in Southern Ocean Air

Datasets on aerosol composition in Southern Ocean air at Cape Grim and Macquarie Island, and rainwater composition at Cape Grim, have been analysed for sea-salt components in order to test the validity of the multiphase autocatalytic halogen activation process proposed initially by Sander and Crutzen (1996) and developed fully for clean marine air by Vogt et al. (1996). Four distinct datasets from the two locations were analysed. All four datasets provided consistent evidence in support of three predictions of the autocatalytic model: (1) overall Cl- deficits in sea-salt aerosol were small, difficult to quantify against analytical uncertainty and at most a few percent; (2) Br- deficits were large, averaging –30% to –50% on an annual basis, with strong seasonality ranging from about –10% in some winter samples to –80% or more in some summer samples; and (3) the Br- and Cl- deficits were clearly linked to the availability of strong, S-acidity in the aerosol, confirming the importance of acid catalysis to the dehalogenation process.     

Flux Footprints Over an Undulating Surface

The flux footprint probability distribution (FPD) functions for near-surface receptors over an idealised undulating surface are evaluated using a backward Lagrangian stochastic model. The wind and turbulence fields employed to drive the stochastic model are derived from large-eddy simulations, in which the horizontal wind aligns with the surface-elevation-varying direction. The flux FPD for a receptor is affected by flow divergence or convergence, and varies with the receptor’s location. The widest crosswind-integrated FPD (CIFPD) curve with the smallest peak value appears when the receptor is located in the crest area, while the narrowest CIFPD curve with the largest peak value appears when the receptor is located in the windward area. Experiments are designed to highlight the impact of the horizontal homogeneity assumption on the estimation of the FPD. When the receptor is located in the area with surface-wind convergence, the peak value of the CIFPD is larger than its counterpart under assumed horizontally homogeneous flow conditions, with the peak position being closer to the receptor. The case is reversed when the receptor is located in the area with surface-wind divergence. Similar results are obtained when the CIFPD derived from an analytical footprint model (developed under the assumption of horizontally homogeneous flow conditions) is compared with that from the stochastic model over the undulating surface. The analytical model fails to simulate the CIFPD in the local downwind area under weak wind conditions due to the longitudinal wind fluctuation not being considered.     

Mathematical modelling of the annual temperature wave based on monthly mean temperatures,and comparisons between local climate trends at seven Norwegian stations

Summary Based on observed monthly mean temperatures, it is possible to construct a simple mathematical model of the annual variation of daily mean temperature, the annual temperature wave. For periods of 15 years, the model gives a good correlation with the observed monthly values. The model may be used as a tool for the generation of daily mean temperatures for the corresponding period. It is continuous, differentiable and strictly monotonous between the unique maximum and the minimum of the curve. Consequently, climate quantities of interest for each period can be calculated by the means of simple mathematical analyses. The model was tested by reproducing values for quantities such as annual mean temperature, winter mean temperature, summer mean temperature and temperature sums. Model calculated values, fit values calculated directly from observed data well. The model was also tested by comparing results from two different but neighbouring stations. There was a good correlation between the results from the two stations. Long homogenised time series with 130 years of monthly mean temperature from seven Norwegian stations were analysed by means of the model. It was found that the Frost Free Season Length and the Growth Season Length had increased for all stations by 10–20 days/100 years in the period 1871–1990. The Summer Half-year Length, even if it was defined relative to the annual mean temperature, also increased for all stations by 4–9 days/100 years. The Hot Season Length showed positive trends as well, and for the five stations in Southern Norway, the trends were as high as 18–29 days/100 years. The Heat Sum had increased by 6–11% for southern stations and 20–22% for the northern stations. The results indicate that the level as well as the shape of the annual temperature wave changed in the period from 1871 to 1990. Some of the results for the period 1990–1999 diverge substantially from the trends, possibly indicating significant changes in the shape of the annual temperature wave in this last period.