On the Regularity of Optimal Transportation Potentials on Round Spheres

In this paper the regularity of optimal transportation potentials defined on round spheres is investigated. Specifically, this research generalises the calculations done by Loeper, where he showed that the strong (A3) condition of Trudinger and Wang is satisfied on the round sphere, when the cost-function is the geodesic distance squared. In order to generalise Loeper's calculation to a broader class of cost-functions, the (A3) condition is reformulated via a stereographic projection that maps charts of the sphere into Euclidean space. This reformulation subsequently allows one to verify the (A3) condition for any case where the cost-fuction of the associated optimal transportation problem can be expressed as a function of the geodesic distance between points on a round sphere. With this, several examples of such cost-functions are then analysed to see whether or not they satisfy this (A3) condition.Comment: 24 pages, 4 figure

Evaluation of radar reflectivity factor simulations of ice crystal populations from in situ observations for the retrieval of condensed water content in tropical mesoscale convective systems

This study presents the evaluation of a technique to estimate cloud condensed water content (CWC) in tropical convection from airborne cloud radar reflectivity factors at 94 GHz and in situ measurements of particle size distributions (PSDs) and aspect ratios of ice crystal populations. The approach is to calculate from each 5 s mean PSD and flight-level reflectivity the variability of all possible solutions of m(D) relationships fulfilling the condition that the simulated radar reflectivity factor (T-matrix method) matches the measured radar reflectivity factor. For the reflectivity simulations, ice crystals were approximated as oblate spheroids, without using a priori assumptions on the mass–size relationship of ice crystals. The CWC calculations demonstrate that individual CWC values are in the range ±32 % of the retrieved average CWC value over all CWC solutions for the chosen 5 s time intervals. In addition, during the airborne field campaign performed out of Darwin in 2014, as part of the international High Altitude Ice Crystals/High Ice Water Content (HAIC/HIWC) projects, CWCs were measured independently with the new IKP-2 (isokinetic evaporator probe) instrument along with simultaneous particle imagery and radar reflectivity. Retrieved CWCs from the T-matrix radar reflectivity simulations are on average 16 % higher than the direct CWCIKP measurements. The differences between the CWCIKP and averaged retrieved CWCs are found to be primarily a function of the total number concentration of ice crystals. Consequently, a correction term is applied (as a function of total number concentration) that significantly improves the retrieved CWC. After correction, the retrieved CWCs have a median relative error with respect to measured values of only −1 %. Uncertainties in the measurements of total concentration of hydrometeors are investigated in order to calculate their contribution to the relative error of calculated CWC with respect to measured CWCIKP. It is shown that an overestimation of the concentration by about +50 % increases the relative errors of retrieved CWCs by only +29 %, while possible shattering, which impacts only the concentration of small hydrometeors, increases the relative error by about +4 %. Moreover, all cloud events with encountered graupel particles were studied and compared to events without observed graupel particles. Overall, graupel particles seem to have the largest impact on high crystal number-concentration conditions and show relative errors in retrieved CWCs that are higher than for events without graupel particles

The variability of tropical ice cloud properties as a function of the large-scale context from ground-based radar-lidar observations over Darwin, Australia

The high complexity of cloud parameterizations now held in models puts more pressure on observational studies to provide useful means to evaluate them. One approach to the problem put forth in the modelling community is to evaluate under what atmospheric conditions the parameterizations fail to simulate the cloud properties and under what conditions they do a good job. It is the ambition of this paper to characterize the variability of the statistical properties of tropical ice clouds in different tropical "regimes" recently identified in the literature to aid the development of better process-oriented parameterizations in models. For this purpose, the statistical properties of non-precipitating tropical ice clouds over Darwin, Australia are characterized using ground-based radar-lidar observations from the Atmospheric Radiation Measurement (ARM) Program. The ice cloud properties analysed are the frequency of ice cloud occurrence, the morphological properties (cloud top height and thickness), and the microphysical and radiative properties (ice water content, visible extinction, effective radius, and total concentration). The variability of these tropical ice cloud properties is then studied as a function of the large-scale cloud regimes derived from the International Satellite Cloud Climatology Project (ISCCP), the amplitude and phase of the Madden-Julian Oscillation (MJO), and the large-scale atmospheric regime as derived from a long-term record of radiosonde observations over Darwin. The vertical variability of ice cloud occurrence and microphysical properties is largest in all regimes (1.5 order of magnitude for ice water content and extinction, a factor 3 in effective radius, and three orders of magnitude in concentration, typically). 98 % of ice clouds in our dataset are characterized by either a small cloud fraction (smaller than 0.3) or a very large cloud fraction (larger than 0.9). In the ice part of the troposphere three distinct layers characterized by different statistically-dominant microphysical processes are identified. The variability of the ice cloud properties as a function of the large-scale atmospheric regime, cloud regime, and MJO phase is large, producing mean differences of up to a factor 8 in the frequency of ice cloud occurrence between large-scale atmospheric regimes and mean differences of a factor 2 typically in all microphysical properties. Finally, the diurnal cycle of the frequency of occurrence of ice clouds is also very different between regimes and MJO phases, with diurnal amplitudes of the vertically-integrated frequency of ice cloud occurrence ranging from as low as 0.2 (weak diurnal amplitude) to values in excess of 2.0 (very large diurnal amplitude). Modellers should now use these results to check if their model cloud parameterizations are capable of translating a given atmospheric forcing into the correct statistical ice cloud properties

Theory and observations of ice particle evolution in cirrus using Doppler radar: evidence for aggregation

Vertically pointing Doppler radar has been used to study the evolution of ice particles as they sediment through a cirrus cloud. The measured Doppler fall speeds, together with radar-derived estimates for the altitude of cloud top, are used to estimate a characteristic fall time tc for the `average' ice particle. The change in radar reflectivity Z is studied as a function of tc, and is found to increase exponentially with fall time. We use the idea of dynamically scaling particle size distributions to show that this behaviour implies exponential growth of the average particle size, and argue that this exponential growth is a signature of ice crystal aggregation.Comment: accepted to Geophysical Research Letter

A Comparison of Airborne In Situ Cloud Microphysical Measurement with Ground-Based C-Band Radar Observations in Deep Stratiform Regions of African Squall Lines

This study addresses clouds with significant ice water content (IWC) in the stratiform regions downwind of the convective cores of African squall lines in the framework of the French–Indian satellite Megha-Tropiques project, observed in August 2010 next to Niamey (13.5°N, 2°E) in the southwestern part of Niger. The objectives included comparing the IWC–Z reflectivity relationship for precipitation radars in deep stratiform anvils, collocating reflectivity observed from ground radar with the calculated reflectivity from in situ microphysics for all aircraft locations inside the radar range, and interpreting the role of large ice crystals in the reflectivity of centimeter radars through analysis of their microphysical characteristics as ice crystals larger than 5 mm frequently occurred. It was found that, in the range of 20–30 dBZ, IWC and C-band reflectivity are not really correlated. Cloud regions with high IWC caused by important crystal number concentrations can lead to the same reflectivity factor as cloud regions with low IWC formed by a few millimeter-sized ice crystals

Antarctic clouds, supercooled liquid water and mixed phase, investigated with DARDAR: geographical and seasonal variations

Antarctic tropospheric clouds are investigated using the DARDAR (raDAR/liDAR)-MASK products between 60 and 82∘&thinsp;S. The cloud fraction (occurrence frequency) is divided into the supercooled liquid-water-containing cloud (SLC) fraction and its complementary part called the all-ice cloud fraction. A further distinction is made between SLC involving ice (mixed-phase clouds, MPC) or not (USLC, for unglaciated SLC). The low-level (&lt;3&thinsp;km above surface level) SLC fraction is larger over seas (20&thinsp;%–60&thinsp;%), where it varies according to sea ice fraction, than over continental regions (0&thinsp;%–35&thinsp;%). The total SLC fraction is much larger over West Antarctica (10&thinsp;%–40&thinsp;%) than it is over the Antarctic Plateau (0&thinsp;%–10&thinsp;%). In East Antarctica the total SLC fraction – in summer for instance – decreases sharply polewards with increasing surface height (decreasing temperatures) from 40 % at the coast to &lt;5% at 82∘&thinsp;S on the plateau. The geographical distribution of the continental total all-ice fraction is shaped by the interaction of the main low-pressure systems surrounding the continent and the orography, with little association with the sea ice fraction. Opportunistic comparisons with published ground-based supercooled liquid-water observations at the South Pole in 2009 are made with our SLC fractions at 82∘&thinsp;S in terms of seasonal variability, showing good agreement. We demonstrate that the largest impact of sea ice on the low-level SLC fraction (and mostly through the MPC) occurs in autumn and winter (22&thinsp;% and 18&thinsp;% absolute decrease in the fraction between open water and sea ice-covered regions, respectively), while it is almost null in summer and intermediate in spring (11&thinsp;%). Monthly variability of the MPC fraction over seas shows a maximum at the end of summer and a minimum in winter. Conversely, the USLC fraction has a maximum at the beginning of summer. However, monthly evolutions of MPC and USLC fractions do not differ on the continent. This suggests a seasonality in the glaciation process in marine liquid-bearing clouds. From the literature, we identify the pattern of the monthly evolution of the MPC fraction as being similar to that of the aerosols in coastal regions, which is related to marine biological activity. Marine bioaerosols are known to be efficient ice-nucleating particles (INPs). The emission of these INPs into the atmosphere from open waters would add to the temperature and sea ice fraction seasonalities as factors explaining the MPC fraction monthly evolution.</p

A glimpse into the differential topology and geometry of optimal transport

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.Comment: 27 page

Toward understanding of differences in current cloud retrievals of ARM ground-based measurements

Accurate observations of cloud microphysical properties are needed for evaluating and improving the representation of cloud processes in climate models and better estimate of the Earth radiative budget. However, large differences are found in current cloud products retrieved from ground-based remote sensing measurements using various retrieval algorithms. Understanding the differences is an important step to address uncertainties in the cloud retrievals. In this study, an in-depth analysis of nine existing ground-based cloud retrievals using ARM remote sensing measurements is carried out. We place emphasis on boundary layer overcast clouds and high level ice clouds, which are the focus of many current retrieval development efforts due to their radiative importance and relatively simple structure. Large systematic discrepancies in cloud microphysical properties are found in these two types of clouds among the nine cloud retrieval products, particularly for the cloud liquid and ice particle effective radius. Note that the differences among some retrieval products are even larger than the prescribed uncertainties reported by the retrieval algorithm developers. It is shown that most of these large differences have their roots in the retrieval theoretical bases, assumptions, as well as input and constraint parameters. This study suggests the need to further validate current retrieval theories and assumptions and even the development of new retrieval algorithms with more observations under different cloud regimes

SIRTA, a ground-based atmospheric observatory for cloud and aerosol research

Ground-based remote sensing observatories have a crucial role to play in providing data to improve our understanding of atmospheric processes, to test the performance of atmospheric models, and to develop new methods for future space-borne observations. Institut Pierre Simon Laplace, a French research institute in environmental sciences, created the Site Instrumental de Recherche par T&amp;#233;l&amp;#233;d&amp;#233;tection Atmosph&amp;#233;rique (SIRTA), an atmospheric observatory with these goals in mind. Today SIRTA, located 20km south of Paris, operates a suite a state-of-the-art active and passive remote sensing instruments dedicated to routine monitoring of cloud and aerosol properties, and key atmospheric parameters. Detailed description of the state of the atmospheric column is progressively archived and made accessible to the scientific community. This paper describes the SIRTA infrastructure and database, and provides an overview of the scientific research associated with the observatory. Researchers using SIRTA data conduct research on atmospheric processes involving complex interactions between clouds, aerosols and radiative and dynamic processes in the atmospheric column. Atmospheric modellers working with SIRTA observations develop new methods to test their models and innovative analyses to improve parametric representations of sub-grid processes that must be accounted for in the model. SIRTA provides the means to develop data interpretation tools for future active remote sensing missions in space (e.g. CloudSat and CALIPSO). SIRTA observation and research activities take place in networks of atmospheric observatories that allow scientists to access consistent data sets from diverse regions on the globe

The EarthCARE satellite: the next step forward in global measurements of clouds, aerosols, precipitation, and radiation

The collective representation within global models of aerosol, cloud, precipitation, and their radiative properties remains unsatisfactory. They constitute the largest source of uncertainty in predictions of climatic change and hamper the ability of numerical weather prediction models to forecast high-impact weather events. The joint European Space Agency (ESA)–Japan Aerospace Exploration Agency (JAXA) Earth Clouds, Aerosol and Radiation Explorer (EarthCARE) satellite mission, scheduled for launch in 2018, will help to resolve these weaknesses by providing global profiles of cloud, aerosol, precipitation, and associated radiative properties inferred from a combination of measurements made by its collocated active and passive sensors. EarthCARE will improve our understanding of cloud and aerosol processes by extending the invaluable dataset acquired by the A-Train satellites CloudSat, Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO), and Aqua. Specifically, EarthCARE’s cloud profiling radar, with 7 dB more sensitivity than CloudSat, will detect more thin clouds and its Doppler capability will provide novel information on convection, precipitating ice particle, and raindrop fall speeds. EarthCARE’s 355-nm high-spectral-resolution lidar will measure directly and accurately cloud and aerosol extinction and optical depth. Combining this with backscatter and polarization information should lead to an unprecedented ability to identify aerosol type. The multispectral imager will provide a context for, and the ability to construct, the cloud and aerosol distribution in 3D domains around the narrow 2D retrieved cross section. The consistency of the retrievals will be assessed to within a target of ±10 W m–2 on the (10 km)2 scale by comparing the multiview broadband radiometer observations to the top-of-atmosphere fluxes estimated by 3D radiative transfer models acting on retrieved 3D domains