Sobolev Regularity for Monge-Ampere Type Equations

In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly c-convex potentials arising in optimal transportation belong to W2,1+\u3baloc for some \u3ba>0. This generalizes some recents results concerning the regularity of strictly convex Alexandrov solutions of the Monge-Amp\`ere equation with right hand side bounded away from zero and infinity

Ice crystal number concentration estimates from lidar-radar satellite retrievals. Part 2: Controls on the ice crystal number concentration

The ice crystal number concentration (Ni) is a keyproperty of ice clouds, both radiatively and microphysically.Due to sparse in situ measurements of ice cloud properties,the controls on theNihave remained difficult to determine.As more advanced treatments of ice clouds are included inglobal models, it is becoming increasingly necessary to de-velop strong observational constraints on the processes in-volved.This work uses the DARDAR-NiceNiretrieval describedin Part 1 to investigate the controls on theNiat a globalscale. The retrieved clouds are separated by type. The ef-fects of temperature, proxies for in-cloud updraft and aerosolconcentrations are investigated. Variations in the cloud topNi(Ni(top)) consistent with both homogeneous and hetero-geneous nucleation are observed along with differing rela-tionships between aerosol andNi(top)depending on the pre-vailing meteorological situation and aerosol type. Away fromthe cloud top, theNidisplays a different sensitivity to thesecontrolling factors, providing a possible explanation for thelowNisensitivity to temperature and ice nucleating particles(INP) observed in previous in situ studies.This satellite dataset provides a new way of investigat-ing the response of cloud properties to meteorological andaerosol controls. The results presented in this work increaseour confidence in the retrievedNiand will form the basis for further study into the processes influencing ice and mixedphase clouds

Ice crystal number concentration estimates from lidar–radar satellite remote sensing – Part 2: Controls on the ice crystal number concentration

The ice crystal number concentration (Ni) is a key property of ice clouds, both radiatively and microphysically. Due to sparse in situ measurements of ice cloud properties, the controls on the Ni have remained difficult to determine. As more advanced treatments of ice clouds are included in global models, it is becoming increasingly necessary to develop strong observational constraints on the processes involved. This work uses the DARDAR-Nice Ni retrieval described in Part 1 to investigate the controls on the Ni at a global scale. The retrieved clouds are separated by type. The effects of temperature, proxies for in-cloud updraft and aerosol concentrations are investigated. Variations in the cloud top Ni (Ni(top)) consistent with both homogeneous and heterogeneous nucleation are observed along with differing relationships between aerosol and Ni(top) depending on the prevailing meteorological situation and aerosol type. Away from the cloud top, the Ni displays a different sensitivity to these controlling factors, providing a possible explanation for the low Ni sensitivity to temperature and ice nucleating particles (INP) observed in previous in situ studies. This satellite dataset provides a new way of investigating the response of cloud properties to meteorological and aerosol controls. The results presented in this work increase our confidence in the retrieved Ni and will form the basis for further study into the processes influencing ice and mixed phase clouds

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

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

The vertical cloud structure of the West African monsoon: a 4 year climatology using CloudSat and CALIPSO

The West African summer monsoon (WAM) is an important driver of the global climate and locally provides most of the annual rainfall. A solid climatological knowledge of the complex vertical cloud structure is invaluable to forecasters and modelers to improve the understanding of the WAM. In this paper, 4 years of data from the CloudSat profiling radar and CALIPSO are used to create a composite zonal mean vertical cloud and precipitation structure for the WAM. For the first time, the near-coincident vertical radar and lidar profiles allow for the identification of individual cloud types from optically thin cirrus and shallow cumulus to congestus and deep convection. A clear diurnal signal in zonal mean cloud structure is observed for the WAM, with deep convective activity enhanced at night producing extensive anvil and cirrus, while daytime observations show more shallow cloud and congestus. A layer of altocumulus is frequently observed over the Sahara at night and day, extending southward to the coastline, and the majority of this cloud is shown to contain supercooled liquid in the top. The occurrence of deep convective systems and congestus in relation to the position of the African easterly jet is studied, but only the daytime cumulonimbus distribution indicates some influence of the jet position

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

Ice crystal number concentration estimates from lidar–radar satellite remote sensing – Part 2: Controls on the ice crystal number concentration

The ice crystal number concentration (Ni) is a key property of ice clouds, both radiatively and microphysically. Due to sparse in situ measurements of ice cloud properties, the controls on the Ni have remained difficult to determine. As more advanced treatments of ice clouds are included in global models, it is becoming increasingly necessary to develop strong observational constraints on the processes involved.This work uses the DARDAR-Nice Ni retrieval described in Part 1 to investigate the controls on the Ni at a global scale. The retrieved clouds are separated by type. The effects of temperature, proxies for in-cloud updraft and aerosol concentrations are investigated. Variations in the cloud top Ni (Ni(top)) consistent with both homogeneous and heterogeneous nucleation are observed along with differing relationships between aerosol and Ni(top) depending on the prevailing meteorological situation and aerosol type. Away from the cloud top, the Ni displays a different sensitivity to these controlling factors, providing a possible explanation for the low Ni sensitivity to temperature and ice nucleating particles (INP) observed in previous in situ studies.This satellite dataset provides a new way of investigating the response of cloud properties to meteorological and aerosol controls. The results presented in this work increase our confidence in the retrieved Ni and will form the basis for further study into the processes influencing ice and mixed phase clouds.</p

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

Role of thermodynamic and turbulence processes on the fog life cycle during SOFOG3D experiment

In this study, we use a synergy of in situ and remote sensing measurements collected during the SOuthwest FOGs 3D experiment for processes study (SOFOG3D) field campaign in autumn and winter 2019–2020 to analyse the thermodynamic and turbulent processes related to fog formation, evolution, and dissipation across southwestern France. Based on a unique measurement dataset (synergy of cloud radar, microwave radiometer, wind lidar, and weather station data) combined with a fog conceptual model, an analysis of the four deepest fog episodes (two radiation fogs and two advection–radiation fogs) is conducted. The results show that radiation and advection–radiation fogs form under deep and thin temperature inversions, respectively. For both fog categories, the transition period from stable to adiabatic fog and the fog adiabatic phase are driven by vertical mixing associated with an increase in turbulence in the fog layer due to mechanical production (turbulence kinetic energy (TKE) up to 0.4 m2 s−2 and vertical velocity variance (σw2) up to 0.04 m2 s−2) generated by increasing wind and wind shear. Our study reveals that fog liquid water path, fog top height, temperature, radar reflectivity profiles, and fog adiabaticity derived from the conceptual model evolve in a consistent manner to clearly characterise this transition. The dissipation time is observed at night for the advection–radiation fog case studies and after sunrise for the radiation fog case studies. Night-time dissipation is driven by horizontal advection generating mechanical turbulence (TKE at least 0.3 m2 s−2 and σw2 larger than 0.04 m2 s−2). Daytime dissipation is linked to the combination of thermal and mechanical turbulence related to solar heating (near-surface sensible heat flux larger than 10 W m−2) and wind shear, respectively. This study demonstrates the added value of monitoring fog liquid water content and depth (combined with wind, turbulence, and temperature profiles) and diagnostics such as fog liquid water reservoir and adiabaticity to better explain the drivers of the fog life cycle.</p