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At least 955 records · Page 53

The Goddard Cumulus Ensemble Model (GCE): Improvements and Applications for Studying Precipitation Processes

Convection is the primary transport process in the Earth's atmosphere. About two-thirds of the Earth's rainfall and severe floods derive from convection. In addition, two-thirds of the global rain falls in the tropics, while the associated latent heat release accounts for three-fourths of the total heat energy for the Earth's atmosphere. Cloud-resolving models (CRMs) have been used to improve our understanding of cloud and precipitation processes and phenomena from micro-scale to cloud-scale and mesoscale as well as their interactions with radiation and surface processes. CRMs use sophisticated and realistic representations of cloud microphysical processes and can reasonably well resolve the time evolution, structure, and life cycles of clouds and cloud systems. CRMs also allow for explicit interaction between clouds, outgoing longwave (cooling) and incoming solar (heating) radiation, and ocean and land surface processes. Observations are required to initialize CRMs and to validate their results. The Goddard Cumulus Ensemble model (GCE) has been developed and improved at NASA/Goddard Space Flight Center over the past three decades. It is amulti-dimensional non-hydrostatic CRM that can simulate clouds and cloud systems in different environments. Early improvements and testing were presented in Tao and Simpson (1993) and Tao et al. (2003a). A review on the application of the GCE to the understanding of precipitation processes can be found in Simpson and Tao (1993) and Tao (2003). In this paper, recent model improvements (microphysics, radiation and land surface processes) are described along with their impact and performance on cloud and precipitation events in different geographic locations via comparisons with observations. In addition, recent advanced applications of the GCE are presented that include understanding the physical processes responsible for diurnal variation, examining the impact of aerosols (cloud condensation nuclei or CCN and ice nuclei or IN) on precipitation processes, utilizing a satellite simulator to improve the microphysics, providing better simulations for satellite-derived latent heating retrieval, and coupling with a general circulation model to improve the representation of precipitation processes.

GCE↗

A Method for Obtaining High Frequency, Global, IR-Based Convective Cloud Tops for Studies of the Tropical Tropopause Layer

Models of varying complexity that simulate water vapor and clouds in the Tropical Tropopause Layer (TTL) show that including convection directly is essential to properly simulating the water vapor and cloud distribution. In boreal winter, for example, simulations without convection yield a water vapor distribution that is too uniform with longitude, as well as cloud incidences that are too low. Two things are important for convective simulations. First, it is important to get the convective cloud top potential temperature correctly, since unrealistically high values (reaching above the cold point tropopause too frequently) will cause excessive hydration of the stratosphere. Second, one must capture the time variation as well, since hydration by convection depends on the local relative humidity (temperature), which has substantial variation on synoptic time scales in the TTL. This paper describes a method for obtaining high frequency (3-hourly) global convective cloud top distributions which can be used in trajectory models. The method uses rainfall thresholds, standard IR (infra-red) brightness temperatures, meteorological temperature analyses, and physically realistic and documented corrections to IR brightness temperatures to derive cloud top altitudes and potential temperatures. The cloud top altitudes compare well with combined CLOUDSAT and CALIPSO data, both in time-averaged overall vertical and horizontal distributions and in individual cases (correlations of .65-.7). Results from the method are compared to convective distributions currently used by global models. In general, the method shows that models underestimate convective cloud top altitudes.

tropopause↗

Cloud Macroscopic Organization: Order Emerging from Randomness

Clouds play a central role in many aspects of the climate system and their forms and shapes are remarkably diverse. Appropriate representation of clouds in climate models is a major challenge because cloud processes span at least eight orders of magnitude in spatial scales. Here we show that there exists order in cloud size distribution of low-level clouds, and that it follows a power-law distribution with exponent gamma close to 2. gamma is insensitive to yearly variations in environmental conditions, but has regional variations and land-ocean contrasts. More importantly, we demonstrate this self-organizing behavior of clouds emerges naturally from a complex network model with simple, physical organizing principles: random clumping and merging. We also demonstrate symmetry between clear and cloudy skies in terms of macroscopic organization because of similar fundamental underlying organizing principles. The order in the apparently complex cloud-clear field thus has its root in random local interactions. Studying cloud organization with complex network models is an attractive new approach that has wide applications in climate science. We also propose a concept of cloud statistic mechanics approach. This approach is fully complementary to deterministic models, and the two approaches provide a powerful framework to meet the challenge of representing clouds in our climate models when working in tandem.

Yuan, Tianle↗

Quantifying the Impact of Vertical Resolution on the Representation of Marine Boundary Layer Physics for Global-Scale Models

Abstract While GCM horizontal resolution has received the majority of scale improvements in recent years, ample evidence suggests that a model’s vertical resolution exerts a strong control on its ability to accurately simulate the physics of the marine boundary layer. Here we show that, regardless of parameter tuning, the ability of a single-column model (SCM) to simulate the subtropical marine boundary layer improves when its vertical resolution is improved. We introduce a novel objective tuning technique to optimize the parameters of an SCM against profiles of temperature and moisture and their turbulent fluxes, horizontal winds, cloud water, and rainwater from large-eddy simulations (LES). We use this method to identify optimal parameters for simulating marine stratocumulus and shallow cumulus. The novel tuning method utilizes an objective performance metric that accounts for the uncertainty in the LES output, including the covariability between model variables. Optimization is performed independently for different vertical grid spacings and value of time step, ranging from coarse scales often used in current global models (120 m, 180 s) to fine scales often used in parameterization development and large-eddy simulations (10 m, 15 s). Uncertainty-weighted disagreement between the SCM and LES decreases by a factor of ∼5 when vertical grid spacing is improved from 120 to 10 m, with time step reductions being of secondary importance. Model performance is shown to converge at a vertical grid spacing of 20 m, with further refinements to 10 m leading to little further improvement. Significance Statement In successive generations of computer models that simulate Earth’s atmosphere, improvements have been mainly accomplished by reducing the horizontal sizes of discretized grid boxes, while the vertical grid spacing has seen comparatively lesser refinements. Here we advocate for additional attention to be paid to the number of vertical layers in these models, especially in the model layers closest to Earth’s surface where climatologically important marine stratocumulus and shallow cumulus clouds reside. Our experiments show that the ability of a one-dimensional model to represent physical processes important to these clouds is strongly dependent on the model’s vertical grid spacing.

Meteorology & Atmospheric Sciences↗

A cloud and radiation model-based algorithm for rainfall retrieval from SSM/I multispectral microwave measurements

A hybrid statistical-physical retrieval scheme is explored which combines a statistical approach with an approach based on the development of cloud-radiation models designed to simulate precipitating atmospheres. The algorithm employs the detailed microphysical information from a cloud model as input to a radiative transfer model which generates a cloud-radiation model database. Statistical procedures are then invoked to objectively generate an initial guess composite profile data set from the database. The retrieval algorithm has been tested for a tropical typhoon case using Special Sensor Microwave/Imager (SSM/I) data and has shown satisfactory results.

Xiang, Xuwu↗

Forecasting Lightning Threat using Cloud-resolving Model Simulations

As numerical forecasts capable of resolving individual convective clouds become more common, it is of interest to see if quantitative forecasts of lightning flash rate density are possible, based on fields computed by the numerical model. Previous observational research has shown robust relationships between observed lightning flash rates and inferred updraft and large precipitation ice fields in the mixed phase regions of storms, and that these relationships might allow simulated fields to serve as proxies for lightning flash rate density. It is shown in this paper that two simple proxy fields do indeed provide reasonable and cost-effective bases for creating time-evolving maps of predicted lightning flash rate density, judging from a series of diverse simulation case study events in North Alabama for which Lightning Mapping Array data provide ground truth. One method is based on the product of upward velocity and the mixing ratio of precipitating ice hydrometeors, modeled as graupel only, in the mixed phase region of storms at the -15\dgc\ level, while the second method is based on the vertically integrated amounts of ice hydrometeors in each model grid column. Each method can be calibrated by comparing domainwide statistics of the peak values of simulated flash rate proxy fields against domainwide peak total lightning flash rate density data from observations. Tests show that the first method is able to capture much of the temporal variability of the lightning threat, while the second method does a better job of depicting the areal coverage of the threat. A blended solution is designed to retain most of the temporal sensitivity of the first method, while adding the improved spatial coverage of the second. Weather Research and Forecast Model simulations of selected North Alabama cases show that this model can distinguish the general character and intensity of most convective events, and that the proposed methods show promise as a means of generating quantitatively realistic fields of lightning threat. However, because models tend to have more difficulty in correctly predicting the instantaneous placement of storms, forecasts of the detailed location of the lightning threat based on single simulations can be in error. Although these model shortcomings presently limit the precision of lightning threat forecasts from individual runs of current generation models, the techniques proposed herein should continue to be applicable as newer and more accurate physically-based model versions, physical parameterizations, initialization techniques and ensembles of cloud-allowing forecasts become available.

McCaul, E. W., Jr.↗

A multishock model for the density variance of anisotropic, highly magnetized, supersonic turbulence

ABSTRACT Shocks form the basis of our understanding for the density and velocity statistics of supersonic turbulent flows, such as those found in the cool interstellar medium (ISM). The variance of the density field, $\sigma ^2_{\rho /\rho _0}$, is of particular interest for molecular clouds (MCs), the birthplaces of stars in the Universe. The density variance may be used to infer underlying physical processes in an MC, and parametrizes the star formation (SF) rate of a cloud. However, models for $\sigma ^2_{\rho /\rho _0}$ all share a common feature – the variance is assumed to be isotropic. This assumption does not hold when a trans-/sub-Alfvénic mean magnetic field, ${B}_0$, is present in the cloud, which observations suggest is relevant for some MCs. We develop an anisotropic model for $\sigma _{\rho /\rho _0}^2$, using contributions from hydrodynamical and fast magnetosonic shocks that propagate orthogonal to each other. Our model predicts an upper bound for $\sigma _{\rho /\rho _0}^2$ in the high Mach number $(\mathcal {M})$ limit as small-scale density fluctuations become suppressed by the strong ${B}_0$. The model reduces to the isotropic $\sigma _{\rho /\rho _0}^2\!-\!\mathcal {M}$ relation in the hydrodynamical limit. To validate our model, we calculate $\sigma _{\rho /\rho _0}^2$ from 12 high-resolution, three-dimensional, supersonic, sub-Alfvénic magnetohydrodynamical (MHD) turbulence simulations and find good agreement with our theory. We discuss how the two MHD shocks may be the bimodally oriented overdensities observed in some MCs and the implications for SF theory in the presence of a sub-Alfvénic ${B}_0$. By creating an anisotropic, supersonic density fluctuation model, this study paves the way for SF theory in the highly anisotropic regime of interstellar turbulence.

Beattie, James R.↗

Radiation properties and emissivity parameterization of high level thin clouds

To parameterize emissivity of clouds at 11 microns, a study has been made in an effort to understand the radiation field of thin clouds. The contributions to the intensity and flux from different sources and through different physical processes are calculated by using the method of successive orders of scattering. The effective emissivity of thin clouds is decomposed into the effective absorption emissivity, effective scattering emissivity, and effective reflection emissivity. The effective absorption emissivity depends on the absorption and emission of the cloud; it is parameterized in terms of optical thickness. The effective scattering emissivity depends on the scattering properties of the cloud; it is parameterized in terms of optical thickness and single scattering albedo. The effective reflection emissivity follows the similarity relation as in the near infrared cases. This is parameterized in terms of the similarity parameter and optical thickness, as well as the temperature difference between the cloud and ground.

Wu, M.-L. C.↗

Using Multi-scale Modeling System to Study the Interactions between Clouds, Precipitation, Aerosols, Radiation and Land Surface

Numerical cloud models, which are based the non-hydrostatic equations of motion, have been extensively applied to cloud-scale and mesoscale processes during the past four decades. Because cloud-scale dynamics are treated explicitly, uncertainties stemming from convection that have to be parameterized in (hydrostatic) large-scale models are obviated, or at least mitigated, in cloud models. Global models will use the non-hydrostatic framework when their horizontal resolution becomes about 10 kilometers, the theoretical limit for the hydrostatic approximation. This juncture will be reached one to two decades from now. Over the past generation, voluminous datasets on atmospheric convection have been accumulated from radar, instrumented aircraft, satellites, and rawinsonde measurements in field campaigns, enabling the detailed evaluation of models. Improved numerical methods have resulted in more accurate and efficient dynamical cores in models. Improvements have been made in the parameterizations of microphysical processes, radiation, boundary-layer effects, and turbulence; however, microphysical parameterizations remain a major source of uncertainty in all classes of atmospheric models. In recent years, exponentially increasing computer power has extended cloud-resolving-model integrations from hours to months, the number of computational grid points from less than a thousand to close to ten million. Three-dimensional models are now more prevalent. Much attention is devoted to precipitating cloud systems where the crucial 1-kilometer scales are resolved in horizontal domains as large as 10,000 kilometers in two-dimensions, and 1,000 x 1,000 square kilometers in three-dimensions. Cloud models now provide statistical information useful for developing more realistic physically based parameterizations for climate models and numerical weather prediction models. It is also expected that NWP and mesoscale model can be run in grid size similar to cloud resolving model through nesting technique. A review of developments, improvements and applications of cloud models (GCE and WRF) at Goddard will be presented in this talk. In particular, a new approach to using multi-scale modeling system to study the interactions between clouds, precipitation, aerosols and land will be presented.

Tao, Wei-Kuo↗

Using Multi-scale Modeling System to Study the Interactions between Clouds, Precipitation, Aerosols, Radiation and Land Surface

Numerical cloud models, which are based the non-hydrostatic equations of motion, have been extensively applied to cloud-scale and mesoscale processes during the past four decades. Because cloud-scale dynamics are treated explicitly, uncertainties stemming from convection that have to be parameterized in (hydrostatic) large-scale models are obviated, or at least mitigated, in cloud models. Global models will use the non-hydrostatic framework when their horizontal resolution becomes about 10 kilometers, the theoretical limit for the hydrostatic approximation. This juncture will be reached one to two decades from now. Over the past generation, voluminous datasets on atmospheric convection have been accumulated from radar, instrumented aircraft, satellites, and rawinsonde measurements in field campaigns, enabling the detailed evaluation of models. Improved numerical methods have resulted in more accurate and efficient dynamical cores in models. Improvements have been made in the parameterizations of microphysical processes, radiation, boundary-layer effects, and turbulence; however, microphysical parameterizations remain a major source of uncertainty in all classes of atmospheric models. In recent years, exponentially increasing computer power has extended cloud-resolving-model integrations from hours to months, the number of computational grid points from less than a thousand to close to ten million. Three-dimensional models are now more prevalent. Much attention is devoted to precipitating cloud systems where the crucial-lkm scales are resolved in horizontal domains as large as 10,000 kilometers in two-dimensions, and 1,000 x 1,000 square kilometers in three-dimensions. Cloud models now provide statistical information useful for developing more realistic physically based parameterizations for climate models and numerical weather prediction models. It is also expected that NWP and mesoscale model can be run in grid size similar to cloud resolving model through nesting technique. A review of developments, improvements and applications of cloud models (GCE and WRF) at Goddard wlll be is presented in this talk. In particular, a new approach to using multi-scale modeling system to study the interactions between clouds, precipitation, aerosols and land will be presented.

Tao, Wei-Kuo↗

Using Multi-scale Modeling System to Study the Interactions between Clouds, Precipitation, Aerosols, Radiation and Land Surface

Numerical cloud models, which are based the non-hydrostatic equations of motion, have been extensively applied to cloud-scale and mesoscale processes during the past four decades. Because cloud-scale dynamics are treated explicitly, uncertainties stemming from convection that have to be parameterized in (hydrostatic) large-scale models are obviated, or at least mitigated, in cloud models. Global models will use the non-hydrostatic framework when their horizontal resolution becomes about 10 km, the theoretical limit for the hydrostatic approximation. This juncture will be reached one to two decades from now. Over the past generation, voluminous datasets on atmospheric convection have been accumulated from radar, instrumented aircraft, satellites, and rawinsonde measurements in field campaigns, enabling the detailed evaluation of models. Improved numerical methods have resulted in more accurate and efficient dynamical cores in models. Improvements have been made in the parameterizations of microphysical processes, radiation, boundary-layer effects, and turbulence; however, microphysical parameterizations remain a major source of uncertainty in all classes of atmospheric models. In recent years, exponentially increasing computer power has extended cloud-resolving-model integrations from hours to months, the number of computational grid points from less than a thousand to close to ten million. Three-dimensional models are now more prevalent. Much attention is devoted to precipitating cloud systems where the crucial 1-km scales are resolved in horizontal domains as large as 10,000 km in two-dimensions, and 1,000 x 1,000 sq km in three-dimensions. Cloud models now provide statistical information useful for developing more realistic physically based parameterizations for climate models and numerical weather prediction models. It is also expected that NWP and mesoscale model can be run in grid size similar to cloud resolving model through nesting technique. A review of developments, improvements and applications of cloud models (GCE and WRF) at Goddard will be presented in this talk. In particular, a new approach to using multi-scale modeling system to study the interactions between clouds, precipitation, aerosols and land will be presented.

Tao, Wei-Kuo↗

Impact of Turbulence Representation on the Relationship Between Cloud Feedback and Aerosol‐Cloud Interaction in an E3SMv2 Perturbed Parameter Ensemble

Recent studies reveal an anti‐correlation between global cloud feedback (CF) and effective radiative forcing due to aerosol‐cloud interaction (ERFaci) in Earth system models, but the physical mechanisms underlying it remain uncertain. Here we investigate how different turbulence representations contribute to this relationship over the global ocean using an ensemble of Energy Exascale Earth System Model version 2 simulations with perturbed turbulence parameters. The anti‐correlation appears only in the tropical ascent regime. In the Northern Hemisphere midlatitude and high latitude regimes, there is no significant correlation, and in the tropical marine low cloud and Southern Ocean regimes, the correlation is positive. These opposite correlations are primarily driven by opposing CF responses to perturbed parameters. We find that the mean‐state turbulent mixing strength affects both CF and ERFaci, enabling strong correlations in certain regimes. This study highlights the complex linkages between CF and ERFaci through turbulent processes across diverse cloud regimes.

54 ENVIRONMENTAL SCIENCES↗

RACORO Continental Boundary Layer Cloud Investigations: 3. Separation of Parameterization Biases in Single-Column Model CAM5 Simulations of Shallow Cumulus

Climatically important low-level clouds are commonly misrepresented in climate models. The FAst-physics System TEstbed and Research (FASTER) Project has constructed case studies from the Atmospheric Radiation Measurement Climate Research Facility's Southern Great Plain site during the RACORO aircraft campaign to facilitate research on model representation of boundary-layer clouds. This paper focuses on using the single-column Community Atmosphere Model version 5 (SCAM5) simulations of a multi-day continental shallow cumulus case to identify specific parameterization causes of low-cloud biases. Consistent model biases among the simulations driven by a set of alternative forcings suggest that uncertainty in the forcing plays only a relatively minor role. In-depth analysis reveals that the model's shallow cumulus convection scheme tends to significantly under-produce clouds during the times when shallow cumuli exist in the observations, while the deep convective and stratiform cloud schemes significantly over-produce low-level clouds throughout the day. The links between model biases and the underlying assumptions of the shallow cumulus scheme are further diagnosed with the aid of large-eddy simulations and aircraft measurements, and by suppressing the triggering of the deep convection scheme. It is found that the weak boundary layer turbulence simulated is directly responsible for the weak cumulus activity and the simulated boundary layer stratiform clouds. Increased vertical and temporal resolutions are shown to lead to stronger boundary layer turbulence and reduction of low-cloud biases.

parameterization↗

CoURAGE KAZR b1 Processing: Corrections, Calibrations, and Processing Report

The U.S. Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) User Facility supports atmospheric and earth system research through a comprehensive network of fixed and mobile observatories. These facilities provide long-term and intensive campaign-based observations of clouds, aerosols, precipitation, radiation, and meteorological state variables. ARM observations are designed to improve the physical understanding and numerical representation of atmospheric processes in earth system models, with particular emphasis on cloud-radiation interactions and precipitation processes. The Coast-Urban-Rural Atmospheric Gradient Experiment (CoURAGE) deploys one of the ARM Mobile Facilities (AMF) to the Mid-Atlantic region surrounding Baltimore, Maryland, for the period 1 December 2024 through 30 November 2025. This deployment focuses on characterizing atmospheric structure, cloud properties, and precipitation processes across strong land-use and surface heterogeneity gradients associated with urban, rural, and coastal (Chesapeake Bay) environments. The CoURAGE deployment complements the Baltimore Social-Environmental Collaborative (BSEC), a DOE Urban Integrated Field Laboratory (UIFL), by providing high-quality atmospheric observations needed to connect urban surface processes, emissions, and meteorology to cloud and precipitation responses. In addition to the central urban site, ancillary observing sites were deployed to rural Maryland northwest of Baltimore and to an island site in Chesapeake Bay. These measurements further complement a long-term atmospheric observatory operated in Beltsville, Maryland, by Howard University in collaboration with the Maryland Department of the Environment. Together, these assets form a four-node regional atmospheric observatory network representing Baltimore and its three primary surrounding environments—urban, rural, and coastal/bay. This coordinated observational strategy enables investigation of spatial gradients in boundary-layer structure, cloud occurrence, precipitation evolution, and aerosol-cloud interactions across complex surface regimes. Within this network, vertically pointing cloud radars play a critical role by providing continuous, high-resolution measurements of cloud and precipitation vertical structure.

54 ENVIRONMENTAL SCIENCES↗

A spinar model of Cygnus A

It is proposed that a gravitationally bound supermassive object is located in the 'core' of each of the Cyg A radio clouds and that these objects are responsible for the prolonged ejection of magnetic plasma which produces the synchrotron radio emission. The objects are considered to be analogous to the massive rotators, or spinars, which have been used to model the central radio and optical activity of quasars. The formation of Cyg A is described in terms of the breakup of such an object in the central optical galaxy, and the physical properties of both the compact and the giant clouds are discussed, including the effects of intergalactic drag. It is shown that the spinars are capable of effectively replenishing the extended radio clouds with relativistic electrons, even if they are embedded within the compact clouds. The spinars are modeled quantitatively, and their possible detection is considered. The requirements of the proposed model are compared with those of several other models for bright double radio galaxies.

Flasar, F. M.↗

Cloud based tool for analysis of chemical kinetic mechanisms

The size and complexity of chemical kinetic models used to simulate combustion of hydrocarbon fuels continue to increase with our improved understanding of the underlying physical mechanisms, along with the improved ability to utilize such models as the result of increased computational power and efficiency. As mechanisms grow beyond thousands of species and tens of thousands of reactions, it becomes increasingly difficult to manually check for errors and physical inconsistencies. In this work, we present several automated methods to check for such issues in the specification of chemical reaction models. First, we demonstrate how discontinuities in thermodynamic data can cause simulation difficulties manifested as long wall-clock times and convergence failures. To correct this type of problem, we describe an automated method for refitting thermodynamic parameters. We also outline several methods to check the timescales of reaction rate coefficients to ensure physical consistency. All the methods are made available through a web-based tool (https://combustiontools.llnl.gov) intended to aid mechanism developers and users to improve the accuracy and performance of fuel models used by the combustion community.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Microparticle cloud imaging and tracking for data-driven plasma science

Oceans of image and particle track data encountered in plasma interactions with microparticle clouds motivate development and applications of machine-learning (ML) algorithms. A local-constant-velocity tracker, a Kohonen neural network or self-organizing map, the feature tracking kit, and U-Net are described and compared with each other for microparticle cloud datasets generated from exploding wires, dusty plasmas, and atmospheric plasmas. Particle density and the signal-to-noise ratio have been identified as two important factors that affect the tracking accuracy. Fast Fourier transform is used to reveal how U-Net, a deep convolutional neural network developed for non-plasma applications, achieves the improvements for noisy scenes. Viscous effects are revealed in the ballistic motions of the particles from the exploding wires and atmospheric plasmas. Subdiffusion of microparticles satisfying Δr 2 ∝t k (k=0.84±0.02) is obtained from the dusty plasma datasets. Microparticle cloud imaging and tracking, when enhanced with data and ML models, present new possibilities for plasma physics.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Prediction of the diffuse far infrared flux from the galactic plane

A basic model and simple numerical relations useful for future far infrared studies of the galaxy are presented. Making use of recent CO and other galactic surveys, the diffuse far infrared flux distribution from the galactic plane as a function of galactic longitude alternate theta for 4 deg or = alternate theta or = 90 and the far infrared emissivity as a function of galactocentric distance is predicted. Future measurements of the galactic far infrared flux would yield valuable information on the physical properties and distribution of dust and molecular clouds in the galaxy, particularly the inner region.

Fazio, G. G.↗