Dirac-harmonic maps from index theory

We prove existence results for Dirac-harmonic maps using index theoretical tools. They are mainly interesting if the source manifold has dimension 1 or 2 modulo 8. Our solutions are uncoupled in the sense that the underlying map between the source and target manifolds is a harmonic map.Comment: 26 pages, no figur

Retrieving global sources of aerosols from MODIS observations by inverting GOCART model

International audienceKnowledge of the global distribution of tropospheric aerosols is important for studying the effects of aerosols on global climate. Chemical transport models rely on archived meteorological fields, accounting for aerosol sources, transport and removal processes can simulate the global distribution of atmospheric aerosols. However, the accuracy of global aerosol modeling is limited. Uncertainty in location and strength of aerosol emission sources is a major factor in limiting modeling accuracy. This paper describes an effort to develop an algorithm for retrieving global sources of aerosol from satellite observations by inverting the GOCART aerosol transport model. To optimize inversion algorithm performance, the inversion was formulated as a generalized multi-term least-squares-type fitting. This concept uses the principles of statistical optimization and unites diverse retrieval techniques into a single flexible inversion procedure. It is particularly useful for choosing and refining a priori constraints in the retrieval algorithm. For example, it is demonstrated that a priori limitations on the partial derivatives of retrieved characteristics, which are widely used in atmospheric remote sensing, can also be useful in inverse modeling for constraining time and space variability of the retrieved global aerosol emissions. The similarities and differences with the standard "Kalman filter" inverse modeling approach and the "Phillips-Tikhonov-Twomey" constrained inversion widely used in remote sensing are discussed. In order to retain the originally high space and time resolution of the global model in the inversion of a long record of observations, the algorithm was expressed using adjoint operators in a form convenient for practical development of the inversion from codes implementing forward model simulations. The inversion algorithm was implemented using the GOCART aerosol transport model. The numerical tests we conducted showed successful retrievals of global aerosol emissions with a 2°×2.5° resolution by inverting the GOCART output. For achieving satisfactory retrieval from satellite sensors such as MODIS, the emissions were assumed constant within the 24 h diurnal cycle and aerosol differences in chemical composition were neglected. Such additional assumptions were needed to constrain the inversion due to limitations of satellite temporal coverage and sensitivity to aerosol parameters. As a result, the algorithm was defined for the retrieval of emission sources of fine and coarse mode aerosols from the MODIS fine and coarse mode aerosol optical thickness data respectively. Numerical tests showed that such assumptions are justifiable, taking into account the accuracy of the model and observations and that it provides valuable retrievals of the location and the strength of the aerosol emissions. The algorithm was applied to MODIS observations during two weeks in August 2000. The global placement of fine mode aerosol sources retrieved from inverting MODIS observations was coherent with available independent knowledge. This was particularly encouraging since the inverse method did not use any a priori information about the sources and it was initialized under a "zero aerosol emission" assumption. The retrieval reproduced the instantaneous global MODIS observations with a standard deviation in fitting of aerosol optical thickness of ~0.04. The optical thickness during high aerosol loading events was reproduced with a standard deviation of ~48%. Applications of the algorithm for the retrieval of coarse mode aerosol emissions were less successful, mainly due to the currently existing lack of MODIS data over high reflectance desert dust sources. Possibilities for enhancing the global satellite data inversion by using diverse a priori constraints on the retrieval are demonstrated. The potential and limitations of applying our approach for the retrieval of global aerosol sources from aerosol remote sensing are discussed

Links between topography, wind, deflation, lakes and dust: The case of the Bodélé Depression, Chad

The Bodélé Depression, Chad is the planet's largest single source of dust. Deflation from the Bodélé could be seen as a simple coincidence of two key prerequisites: strong surface winds and a large source of suitable sediment. But here we hypothesise that long term links between topography, winds, deflation and dust ensure the maintenance of the dust source such that these two apparently coincidental key ingredients are connected by land-atmosphere processes with topography acting as the overall controlling agent. We use a variety of observational and numerical techniques, including a regional climate model, to show that: 1) contemporary deflation from the Bodélé is delineated by topography and a surface wind stress maximum; 2) the Tibesti and Ennedi mountains play a key role in the generation of the erosive winds in the form of the Bodélé Low Level Jet (LLJ); 3) enhanced deflation from a stronger Bodélé LLJ during drier phases, for example, the Last Glacial Maximum, was probably sufficient to create the shallow lake in which diatoms lived during wetter phases, such as the Holocene pluvial. Winds may therefore have helped to create the depression in which erodible diatom material accumulated. Instead of a simple coincidence of nature, dust from the world's largest source may result from the operation of long term processes on paleo timescales which have led to ideal conditions for dust generation in the world's largest dust source. Similar processes plausibly operate in other dust hotspots in topographic depressions

Retrieving global aerosol sources from satellites using inverse modeling

International audienceUnderstanding aerosol effects on global climate requires knowing the global distribution of tropospheric aerosols. By accounting for aerosol sources, transports, and removal processes, chemical transport models simulate the global aerosol distribution using archived meteorological fields. We develop an algorithm for retrieving global aerosol sources from satellite observations of aerosol distribution by inverting the GOCART aerosol transport model. The inversion is based on a generalized, multi-term least-squares-type fitting, allowing flexible selection and refinement of a priori algorithm constraints. For example, limitations can be placed on retrieved quantity partial derivatives, to constrain global aerosol emission space and time variability in the results. Similarities and differences between commonly used inverse modeling and remote sensing techniques are analyzed. To retain the high space and time resolution of long-period, global observational records, the algorithm is expressed using adjoint operators. Successful global aerosol emission retrievals at 2°×2.5 resolution were obtained by inverting GOCART aerosol transport model output, assuming constant emissions over the diurnal cycle, and neglecting aerosol compositional differences. In addition, fine and coarse mode aerosol emission sources were inverted separately from MODIS fine and coarse mode aerosol optical thickness data, respectively. These assumptions are justified, based on observational coverage and accuracy limitations, producing valuable aerosol source locations and emission strengths. From two weeks of daily MODIS observations during August 2000, the global placement of fine mode aerosol sources agreed with available independent knowledge, even though the inverse method did not use any a priori information about aerosol sources, and was initialized with a "zero aerosol emission" assumption. Retrieving coarse mode aerosol emissions was less successful, mainly because MODIS aerosol data over highly reflecting desert dust sources is lacking. The broader implications of applying our approach are also discussed

The Dirac operator on untrapped surfaces

We establish a sharp extrinsic lower bound for the first eigenvalue of the Dirac operator of an untrapped surface in initial data sets without apparent horizon in terms of the norm of its mean curvature vector. The equality case leads to rigidity results for the constraint equations with spherical boundary as well as uniqueness results for constant mean curvature surfaces in Minkowski space.Comment: 16 page

The effect of harmonized emissions on aerosol properties in global models – an AeroCom experiment

The effects of unified aerosol sources on global aerosol fields simulated by different models are examined in this paper. We compare results from two AeroCom experiments, one with different (ExpA) and one with unified emissions, injection heights, and particle sizes at the source (ExpB). Surprisingly, harmonization of aerosol sources has only a small impact on the simulated diversity for aerosol burden, and consequently optical properties, as the results are largely controlled by model-specific transport, removal, chemistry (leading to the formation of secondary aerosols) and parameterizations of aerosol microphysics (e.g. the split between deposition pathways) and to a lesser extent on the spatial and temporal distributions of the (precursor) emissions. The burdens of black carbon and especially sea salt become more coherent in ExpB only, because the large ExpA diversity for these two species was caused by few outliers. The experiment also indicated that despite prescribing emission fluxes and size distributions, ambiguities in the implementation in individual models can lead to substantial differences. These results indicate the need for a better understanding of aerosol life cycles at process level (including spatial dispersal and interaction with meteorological parameters) in order to obtain more reliable results from global aerosol simulations. This is particularly important as such model results are used to assess the consequences of specific air pollution abatement strategies

Generalized Weierstrass Relations and Frobenius reciprocity

This article investigates local properties of the further generalized Weierstrass relations for a spin manifold $S$ immersed in a higher dimensional spin manifold $M$ from viewpoint of study of submanifold quantum mechanics. We show that kernel of a certain Dirac operator defined over $S$, which we call submanifold Dirac operator, gives the data of the immersion. In the derivation, the simple Frobenius reciprocity of Clifford algebras $S$ and $M$ plays important roles.Comment: 17pages. to be published in Mathematical Physics, Analysis and Geometr

An experimental study of the near wake of a two-dimensional hypersonic blunt body with mass addition

An experimental investigation of the steady, laminar near-wake flow field of a two-dimensional, adiabatic, circular cylinder with surface mass transfer has been made at a free-stream Mach number of 6.0. The pressure and mass-concentration fields associated with the transfer of argon, nitrogen or helium into the near wake were studied for mass transfer from the forward stagnation region, and from the base. For sufficiently low mass transfer rates from the base, for which a recirculating zone exists, the entire near-wake flow field correlates with the momentum flux, not the mass flux, of the injectant, and the mass-concentration field is determined by counter-current diffusion into the reversed flow. For mass addition from the forward stagnation region, the pressure field is undisturbed and the mass-concentration field is nearly uniform in the region of reversed flow. The axial decay of argon mass concentration in the intermediate wake, downstream of the neck, is explained with the aid of an integral solution in the incompressible plane, from which the location of the virtual origin for the asymptotic far-wake solution has been derived as one result

An AeroCom initial assessment – optical properties in aerosol component modules of global models

The AeroCom exercise diagnoses multi-component aerosol modules in global modeling. In an initial assessment simulated global distributions for mass and mid-visible aerosol optical thickness (aot) were compared among 20 different modules. Model diversity was also explored in the context of previous comparisons. For the component combined aot general agreement has improved for the annual global mean. At 0.11 to 0.14, simulated aot values are at the lower end of global averages suggested by remote sensing from ground (AERONET ca. 0.135) and space (satellite composite ca. 0.15). More detailed comparisons, however, reveal that larger differences in regional distribution and significant differences in compositional mixture remain. Of particular concern are large model diversities for contributions by dust and carbonaceous aerosol, because they lead to significant uncertainty in aerosol absorption (aab). Since aot and aab, both, influence the aerosol impact on the radiative energy-balance, the aerosol (direct) forcing uncertainty in modeling is larger than differences in aot might suggest. New diagnostic approaches are proposed to trace model differences in terms of aerosol processing and transport: These include the prescription of common input (e.g. amount, size and injection of aerosol component emissions) and the use of observational capabilities from ground (e.g. measurements networks) or space (e.g. correlations between aerosol and clouds)

Analysis and quantification of the diversities of aerosol life cycles within AeroCom

Simulation results of global aerosol models have been assembled in the framework of the AeroCom intercomparison exercise. In this paper, we analyze the life cycles of dust, sea salt, sulfate, black carbon and particulate organic matter as simulated by sixteen global aerosol models. The diversities among the models for the sources and sinks, burdens, particle sizes, water uptakes, and spatial dispersals have been established. These diversities have large consequences for the calculated radiative forcing and the aerosol concentrations at the surface. The AeroCom all-models-average emissions are dominated by the mass of sea salt (SS), followed by dust (DU), sulfate (SO_4), particulate organic matter (POM), and finally black carbon (BC). Interactive parameterizations of the emissions and contrasting particles sizes of SS and DU lead generally to higher diversities of these species, and for total aerosol. The lower diversity of the emissions of the fine aerosols, BC, POM, and SO_4, is due to the use of similar emission inventories, and does therefore not necessarily indicate a better understanding of their sources. The diversity of SO_4-sources is mainly caused by the disagreement on depositional loss of precursor gases and on chemical production. The diversities of the emissions are passed on to the burdens, but the latter are also strongly affected by the model-specific treatments of transport and aerosol processes. The burdens of dry masses decrease from largest to smallest: DU, SS, SO_4, POM, and BC. The all-models-average residence time is shortest for SS with about half a day, followed by S_O4 and DU with four days, and POM and BC with six and seven days, respectively. The wet deposition rate is controlled by the solubility and increases from DU, BC, POM to SO_4 and SS. It is the dominant sink for SO_4, BC, and POM, and contributes about one third to the total removal rate coefficients of SS and DU species. For SS and DU we find high diversities for the removal rate coefficients and deposition pathways. Models do neither agree on the split between wet and dry deposition, nor on that between sedimentation and turbulent dry Deposition. We diagnose an extremely high diversity for the uptake of ambient water vapor that influences the particle size and thus the sink rate coefficients. Furthermore, we find little agreement among the model results for the partitioning of wet removal into scavenging by convective and stratiform rain. Large differences exist for aerosol dispersal both in the vertical and in the horizontal direction. In some models, a minimum of total aerosol concentration is simulated at the surface. Aerosol dispersal is most pronounced for SO4 and BC and lowest for SS. Diversities are higher for meridional than for vertical dispersal, they are similar for a given species and highest for SS and DU. For these two components we do not find a correlation between vertical and meridional aerosol dispersal. In addition the degree of dispersals of SS and DU is not related to their residence times. SO_4, BC, and POM, however, show increased meridional dispersal in models with larger vertical dispersal, and dispersal is larger for longer simulated residence times