Cylindrical estimates for hypersurfaces moving by convex curvature functions

We prove a complete family of `cylindrical estimates' for solutions of a class of fully non-linear curvature flows, generalising the cylindrical estimate of Huisken-Sinestrari [HS09, Section 5] for the mean curvature flow. More precisely, we show that, for the class of flows considered, an (m + 1)-convex (0 ≤ m ≤ n - 2) solution becomes either strictly m-convex, or its Weingarten map approaches that of a cylinder Rᵐ x Sⁿ⁻ ᵐ at points where the curvature is becoming large. This result complements the convexity estimate proved in [ALM13] for the same class of flows.This research was partially supported by Discovery grant DP120102462 of the Australian Research Council. The second author gratefully acknowledges the support of an Australian Postgraduate Award during the completion of this work

Development and Application of Disjunct Eddy Covariance Techniques for the Measurement and Interpretation of Fluxes of Volatile Organic Compounds from Urban and Rural Canopies.

Two disjunct eddy covariance systems for the measurement of volatile organic compound (VOC) fluxes were developed. The first, disjunct eddy covariance (DEC), was validated against the standard eddy covariance (EC) technique, in a study of CO2 and H2O fluxes from a grassland field (Easter Bush, Edinburgh, Scotland). The comparison convincingly showed fluxes measured by the DEC technique to be comparable to those measured using the EC technique. A second, simplified approach, virtual disjunct eddy covariance (vDEC), was developed and compared against standard DEC during the CityFlux project, where measurements of VOC fluxes were made from Portland Tower in Manchester. Averaged daily fluxes measured by the vDEC system typically ranged between 19 and 90 microg m-2 h-1 for individual VOC species and were comparable to those measured by the DEC system, but were typically 19% higher than the latter. The discrepancies between the two methods were thought to relate to both the reduced response time of the DEC system which attenuated higher frequency flux contributions and the high level of noise in the covariance function which may have led to a systematic overestimation of the flux. The vDEC technique was subsequently deployed on the Telecom Tower in central London to give very detailed flux information on seven VOC species. Individual average fluxes ranged between 5 and 100 microg m-2 h-1 and were well correlated with traffic density. Fluxes of benzene were extrapolated to give an annual emission estimate for the city, which was found to be 1. 8 times lower than that suggested by the National Atmospheric Emission Inventory. Finally, two vDEC systems, one using a high sensitivity (HS) proton transfer reaction mass spectrometer (PTR-MS) and the other a standard model (Std), were used alongside each other to measure biogenic VOC fluxes from macchia vegetation at the Castelporziano nature reserve near Rome, Italy. The two systems compared well, although the HS system appeared to give fluxes with greater amplitude than the Std model. This highlighted the importance of the allocation of correct lag times when using vDEC, particularly at night. Fluxes of isoprene and monoterpenes were compared with the Guenther algorithm of 1995 and showed excellent agreement between the modelled and measured values. The results presented in this study have convincingly demonstrated the capacity of the DEC and vDEC techniques to give very detailed VOC flux information over a range of non-ideal canopies, which can be used to both validate and constrain "bottom-up" style emission inventories

Convexity estimates for hypersurfaces moving by convex curvature functions

We consider the evolution of compact hypersurfaces by fully non-linear, parabolic curvature ows for which the normal speed is given by a smooth, convex, degree one homoge- neous function of the principal curvatures. We prove that solution hypersurfaces on which the speed is initially positive become weakly convex at a singularity of the ow. The result extends the convexity estimate [HS99b] of Huisken and Sinestrari for the mean curvature ow to a large class of speeds, and leads to an analogous description of `type-II\u27 singularities. We remark that many of the speeds considered are positive on larger cones than the positive mean half-space, so that the result in those cases also applies to non-mean-convex initial data

Non-collapsing in fully nonlinear curvature flows

We consider embedded hypersurfaces evolving by fully nonlinear flows in which the normal speed of motion is a homogeneous degree one, concave or convex function of the principal curvatures, and prove a non-collapsing estimate: Precisely, the function which gives the curvature of the largest interior sphere touching the hy- persurface at each point is a subsolution of the linearized flow equation if the speed is concave. If the speed is convex then there is an analogous statement for exterior spheres. In particular, if the hypersurface moves with positive speed and the speed is concave in the principal curvatures, then the curvature of the largest touching inte- rior sphere is bounded by a multiple of the speed as long as the solution exists. The proof uses a maximum principle applied to a function of two points on the evolving hypersurface. We illustrate the techniques required for dealing with such functions in a proof of the known containment principle for flows of hypersurfaces

Eddy-covariance data with low signal-to-noise ratio:time-lag determination, uncertainties and limit of detection

All eddy-covariance flux measurements are associated with random uncertainties which are a combination of sampling error due to natural variability in turbulence and sensor noise. The former is the principal error for systems where the signal-to-noise ratio of the analyser is high, as is usually the case when measuring fluxes of heat, CO2 or H2O. Where signal is limited, which is often the case for measurements of other trace gases and aerosols, instrument uncertainties dominate. Here, we are applying a consistent approach based on auto- and cross-covariance functions to quantify the total random flux error and the random error due to instrument noise separately. As with previous approaches, the random error quantification assumes that the time lag between wind and concentration measurement is known. However, if combined with commonly used automated methods that identify the individual time lag by looking for the maximum in the cross-covariance function of the two entities, analyser noise additionally leads to a systematic bias in the fluxes. Combining data sets from several analysers and using simulations, we show that the method of time-lag determination becomes increasingly important as the magnitude of the instrument error approaches that of the sampling error. The flux bias can be particularly significant for disjunct data, whereas using a prescribed time lag eliminates these effects (provided the time lag does not fluctuate unduly over time). We also demonstrate that when sampling at higher elevations, where low frequency turbulence dominates and covariance peaks are broader, both the probability and magnitude of bias are magnified. We show that the statistical significance of noisy flux data can be increased (limit of detection can be decreased) by appropriate averaging of individual fluxes, but only if systematic biases are avoided by using a prescribed time lag. Finally, we make recommendations for the analysis and reporting of data with low signal-to-noise and their associated errors

Volatile organic compound fluxes and concentrations in London (ClearfLo)

Volatile organic compounds (VOCs) from anthropogenic sources such as fuel combustion or evaporative emissions can directly and indirectly affect human health. Some VOCs, such as benzene and 1,3- butadiene are carcinogens. These and other VOCs contribute to the formation of ozone (O3) and aerosol particles, which have effects on human health and the radiative balance of the atmosphere. Although in the UK VOC emissions are subject to control under European Commission Directive 2008/50/EC and emission reducing echnologies have been implemented, urban air pollution remains a concern. Urban air quality is likely to remain a priority since currently >50% of the global population live in urban areas with trends in urbanization and population migration predicted to increase. The ClearfLo project is a large multi-institutional consortium funded by the UK Natural Environment Research Council (NERC) and provides integrated measurements of meteorology, gas phase and particulate composition of the atmosphere over London. Both long term and IOP measurements were made at street and elevated locations at a range of sites across London and its surroundings during 2011 and 2012. Mixing ratios of a selection of nine VOCs were measured using a high sensitivity proton transfer reaction-mass spectrometer (PTR-MS) at a ground level urban background (North Kensington) and kerbside (Marylebone Road) site during the winter IOP. VOC fluxes were measured by virtually disjunct eddy covariance (vDEC) at an elevated urban site (King’s College Strand) in Aug-Dec 2012. Our results for the first IOP showed that most of the selected compound concentrations depended on traffic emissions, although there was a marked difference between the urban ackground and kerbside sites. We identified some temperature effects on VOC concentrations. We also present the first analyses of VOC flux measurements over London. Preliminary analyses indicate most compounds associated with vehicle emissions closely followed diurnal traffic counts. Fluxes of isoprene and methanol appear to be controlled by light intensity and temperature, consistent with a redominantly biogenic source of these compounds

Seasonal and diurnal trends in concentrations and fluxes of volatile organic compounds in central London

Concentrations and fluxes of seven volatile organic compounds (VOCs) were measured between August and December 2012 at a rooftop site in central London as part of the ClearfLo project (Clean Air for London). VOC concentrations were quantified using a proton transfer reaction mass spectrometer (PTR-MS) and fluxes were calculated using a virtual disjunct eddy covariance technique. The median VOC fluxes, including aromatics, oxygenated compounds and isoprene, ranged from 0.07 to 0.33 mgm&#x100000;^-2 h^-&#x100000;1. Median mixing ratios were 7.3 ppb for methanol and < 1 ppb for the other compounds. Strong relationships were observed between the fluxes and concentrations of some VOCs with traffic density and between the fluxes and concentrations ofisoprene and oxygenated compounds with photosynthetically active radiation (PAR) and temperature. An estimated 50–90% of the fluxes of aromatic VOCs were attributable to traffic activity, which showed little seasonal variation, suggesting that boundary layer effects or possibly advected pollutionmay be the primary causes of increased concentrations of aromatics in winter. Isoprene, methanol and acetaldehyde fluxes and concentrations in August and September showed high correlations with PAR and temperature, when fluxes andconcentrations were largest suggesting that biogenic sources contributed to their fluxes. Modelled biogenic isoprene fluxes from urban vegetation using the Guenther et al. (1995) algorithm agreed well with measured fluxes in August andSeptember. Comparisons of estimated annual benzene emissions from both the London and the National Atmospheric Emissions Inventories agreed well with measured benzene fluxes. Flux footprint analysis indicated emission sourceswere localised and that boundary layer dynamics and source strengths were responsible for temporal and spatial VOC flux and concentration variability during the measurement period