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At least 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Vertical dependence of horizontal variation of cloud microphysics: observations from the ACE-ENA field campaign and implications for warm-rain simulation in climate models

Abstract. In the current global climate models (GCMs), the nonlinearity effect of subgrid cloud variations on the parameterization of warm-rain process, e.g., the autoconversion rate, is often treated by multiplying the resolved-scale warm-rain process rates by a so-called enhancement factor (EF). In this study, we investigate the subgrid-scale horizontal variations and covariation of cloud water content (qc) and cloud droplet number concentration (Nc) in marine boundary layer (MBL) clouds based on the in situ measurements from a recent field campaign and study the implications for the autoconversion rate EF in GCMs. Based on a few carefully selected cases from the field campaign, we found that in contrast to the enhancing effect of qc and Nc variations that tends to make EF > 1, the strong positive correlation between qc and Nc results in a suppressing effect that tends to make EF < 1. This effect is especially strong at cloud top, where the qc and Nc correlation can be as high as 0.95. We also found that the physically complete EF that accounts for the covariation of qc and Nc is significantly smaller than its counterpart that accounts only for the subgrid variation of qc, especially at cloud top. Although this study is based on limited cases, it suggests that the subgrid variations of Nc and its correlation with qc both need to be considered for an accurate simulation of the autoconversion process in GCMs.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1991

Numerical simulation of a subtropical squall line over the Taiwan Strait

The present 2D nonhydrostatic, time-dependent numerical cloud model, which is applied to the structure and development of a subtropical squall line observed during the Taiwan Area Mesoscale Experiment, includes (1) a parameterized ice-phase microphysical scheme, (2) longwave and shortwave radiative-transfer processes, and (3) heat and moisture fluxes from the ocean surface. A comparative study is made of the vertical transport of horizontal momentum and of the latent heat release by the simulated subtropical squall system, as well as the heat release of squall systems occurring in other geographic locations.

Tao, Wei-Kuo↗
DOE OSTI2016

Fast All-Sky Radiation Model for Solar Applications (FARMS): A Brief Overview of Mechanisms, Performance, and Applications: Preprint

Solar radiation can be computed using radiative transfer models, such as the Rapid Radiation Transfer Model (RRTM) and its general circulation model applications, and used for various energy applications. Due to the complexity of computing radiation fields in aerosol and cloudy atmospheres, simulating solar radiation can be extremely time-consuming, but many approximations--e.g., the two-stream approach and the delta-M truncation scheme--can be utilized. To provide a new fast option for computing solar radiation, we developed the Fast All-sky Radiation Model for Solar applications (FARMS) by parameterizing the simulated diffuse horizontal irradiance and direct normal irradiance for cloudy conditions from the RRTM runs using a 16-stream discrete ordinates radiative transfer method. The solar irradiance at the surface was simulated by combining the cloud irradiance parameterizations with a fast clear-sky model, REST2. To understand the accuracy and efficiency of the newly developed fast model, we analyzed FARMS runs using cloud optical and microphysical properties retrieved using GOES data from 2009-2012. The global horizontal irradiance for cloudy conditions was simulated using FARMS and RRTM for global circulation modeling with a two-stream approximation and compared to measurements taken from the U.S. Department of Energy's Atmospheric Radiation Measurement Climate Research Facility Southern Great Plains site. Our results indicate that the accuracy of FARMS is comparable to or better than the two-stream approach; however, FARMS is approximately 400 times more efficient because it does not explicitly solve the radiative transfer equation for each individual cloud condition. Radiative transfer model runs are computationally expensive, but this model is promising for broad applications in solar resource assessment and forecasting. It is currently being used in the National Solar Radiation Database, which is publicly available from the National Renewable Energy Laboratory at http://nsrdb.nrel.gov.

14 SOLAR ENERGY↗
DOE OSTI2018

The path to CAM6: coupled simulations with CAM5.4 and CAM5.5

This paper documents coupled simulations of two developmental versions of the Community Atmosphere Model (CAM) towards CAM6. The configuration called CAM5.4 introduces new microphysics, aerosol, and ice nucleation changes, among others to CAM. The CAM5.5 configuration represents a more radical departure, as it uses an assumed probability density function (PDF)-based unified cloud parameterization to replace the turbulence, shallow convection, and warm cloud macrophysics in CAM. This assumed PDF method has been widely used in the last decade in atmosphere-only climate simulations but has never been documented in coupled mode. Here, we compare the simulated coupled climates of CAM5.4 and CAM5.5 and compare them to the control coupled simulation produced by CAM5.3. We find that CAM5.5 has lower cloud forcing biases when compared to the control simulations. Improvements are also seen in the simulated amplitude of the Niño-3.4 index, an improved representation of the diurnal cycle of precipitation, subtropical surface wind stresses, and double Intertropical Convergence Zone biases. Degradations are seen in Amazon precipitation as well as slightly colder sea surface temperatures and thinner Arctic sea ice. Simulation of the 20th century results in a credible simulation that ends slightly colder than the control coupled simulation. The authors find this is due to aerosol indirect effects that are slightly stronger in the new version of the model and propose a solution to ameliorate this. Altogether, in these early coupled simulations, CAM5.5 produces a credible climate that is appropriate for science applications and is ready for integration into the National Center for Atmospheric Research's (NCAR's) next-generation climate model.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

Effects of Lateral Entrainment Mixing With Entrained Aerosols on Cloud Microphysics

The effects of entrained environment air and aerosols on cloud properties remain a critical yet understudied topic. This study first introduces a new entraining cloud parcel model considering entrained aerosols. With entrained aerosols represented by a newly introduced parameter, the impacts of entrainment rate and entrained aerosols on cloud microphysical properties are investigated. The results here show that the relationships between entrainment rate and cloud microphysical properties are highly nonlinear when the entrained aerosols vary. With the lateral entrainment mixing, a new phenomenon is revealed that the height of maximum parcel supersaturation cannot be reached when the entrainment rate is beyond a certain critical value. This new critical entrainment rate is different from the critical entrainment rate defined by Barahona and Nenes (2007, https://doi.org/10.1029/2007JD008473), beyond which clouds cannot form. This finding has important implications for developing parameterization of droplet activation, which is based primarily on the assumption of the existence of maximum supersaturation.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS2014

Intercomparison of Large-Eddy Simulations of Arctic Mixed-Phase Clouds: Importance of Ice Size Distribution Assumptions

Large-eddy simulations of mixed-phase Arctic clouds by 11 different models are analyzed with the goal of improving understanding and model representation of processes controlling the evolution of these clouds. In a case based on observations from the Indirect and Semi-Direct Aerosol Campaign (ISDAC), it is found that ice number concentration, Ni, exerts significant influence on the cloud structure. Increasing Ni leads to a substantial reduction in liquid water path (LWP), in agreement with earlier studies. In contrast to previous intercomparison studies, all models here use the same ice particle properties (i.e., mass-size, mass-fall speed, and mass-capacitance relationships) and a common radiation parameterization. The constrained setup exposes the importance of ice particle size distributions (PSDs) in influencing cloud evolution. A clear separation in LWP and IWP predicted by models with bin and bulk microphysical treatments is documented and attributed primarily to the assumed shape of ice PSD used in bulk schemes. Compared to the bin schemes that explicitly predict the PSD, schemes assuming exponential ice PSD underestimate ice growth by vapor deposition and overestimate mass-weighted fall speed leading to an underprediction of IWP by a factor of two in the considered case. Sensitivity tests indicate LWP and IWP are much closer to the bin model simulations when a modified shape factor which is similar to that predicted by bin model simulation is used in bulk scheme. These results demonstrate the importance of representation of ice PSD in determining the partitioning of liquid and ice and the longevity of mixed-phase clouds.

clouds↗
NASA NTRS1988

A diagnostic study of vertical fluxes in cumulus clouds using model output data

Midlatitude supercell, Florida sea-breeze convection, and tropical squall line simulations were performed using the Colorado State University Regional Atmospheric Modeling System which employed the non-hydrostatic, fully compressible, equations with parameterized microphysics. The time evolution of the vertical velocity variance (VVV) is especially interesting since it is a measure of the strength of convection. This variance exhibits striking similarities when analyzed for the aforementioned cases which may indicate that all convection is fundamentally similar. Vertical mass and moisture covariances appear strongly linked to VVV, so it may be the appropriate variable on which to base a hybrid cumulus parameterization scheme.

Weissbluth, M. J.↗
DOE OSTI2018

Investigating the impacts of Saharan dust on tropical deep convection using spectral bin microphysics

Abstract. To better understand the impacts of dust aerosols on deep convective cloud (DCC) systems reported by previous observational studies, a case study in the tropical eastern Atlantic was investigated using the Weather Research and Forecasting (WRF) model coupled with a spectral bin microphysics (SBM) model. A detailed set of ice nucleation parameterizations linking ice formation with aerosol particles has been implemented in the SBM. Increasing ice nuclei (IN) concentration in the dust cases results in the formation of more numerous small ice particles in the heterogeneous nucleation regime (between −5 and −38 °C) compared to the background ( Clean ) case. Convective updrafts are invigorated by increased latent heat release due to depositional growth and riming of these more numerous particles, which results in increased overshooting and higher convective core top heights. Competition between the more numerous particles for available water vapor during diffusional growth and available smaller crystals and/or drops during collection reduces particle growth rates and shifts precipitation formation to higher altitudes in the heterogeneous nucleation regime. A greater number of large snow particles form in the dust cases, which are transported from the core into the stratiform regime and sediment out quickly. Together with reduced homogeneous ice formation, the stratiform and/or anvil cloud occurrence shifts frequency to warmer temperatures and reduces anvil cloud extents. Total surface precipitation accumulation is reduced proportionally as IN concentration is increased; though the stratiform precipitation accumulation is increased due to greater snow formation and growth, it does not counteract the reduced convective accumulation due to less efficient graupel formation. Radar reflectivity values are increased in the dust cases at temperatures below 0 °C in both the convective and stratiform regimes due to more large snow particles, and reduced in the convective core near the surface due to melt of small ice or graupel particles, consistent with case study observations.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS2019

Quantifying the Relative Impact of Model Microphysics Parameterizations and Scattering Models in Simulating Synthetic Radar and Microwave Radiometer Observations

Output from numerical weather models is often used to simulate observations from remote sensing instruments, for purposes ranging from data assimilation, synthetic retrievals of geophysical quantities, and optimization of observing systems. However, when hydrometeors are present, the level of detail provided by the weather model is generally insufficient to fully constrain the input to the radiative transfer model (RTM), and further assumptions must be made by the RTM user in order to produce synthetic observations. Using a hierarchy of models including cloud-resolving, double-moment, bin microphysical, and ice-habit predicting models, along with scattering properties from the OpenSSP, Atmospheric Radiative Transfer Simulator (ARTS) databases, as well as relatively simple geometries (e.g., cylindrical plates and columns), we demonstrate the spread in synthetic observation output and the extent to which it is reduced when microphysics is more strongly constrained by the model. As an intermediate step, an error budget for the RTM simulations was derived and from that we developed and will describe best practices for observation simulation (e.g., optimal number of hydrometeor size bins, truncation of the particle size distribution, angular resolution of scattering phase function). Some statistical comparisons with observations will also be presented.

Munchak, S. Joseph↗
DOE OSTI2019

Reconciling differences between large-eddy simulations and Doppler-lidar observations of continental shallow cumulus cloud-base vertical velocity

We investigate a significant model-observation difference found between cloud-base vertical velocity for continental shallow cumulus simulated using large-eddy simulations (LES) and observed by Doppler lidar measurements over the U.S. Southern Great Plains ARM Facility. The LES cloud-base vertical velocity is dominated by updrafts that are consistent with a general picture for convective clouds but is inconsistent with Doppler lidar observations that also show the presence of considerable downdrafts. The underestimation of simulated downdrafts is found to be a robust feature, being insensitive to various numerical, physical, or dynamical choices. We find that simulations can more closely reproduce observations only after improving the model physics to use size-resolved microphysics and horizontal longwave radiation, both of which modify the cloud buoyancy and velocity structure near cloud side edges. The results suggest that treatments that capture these structures are needed for the proper simulation and subsequent parameterization development of shallow cumulus vertical transport.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Vortex Structure of Head Bubble in Convective Cloud Starting Plume

Abstract A developing cumulus cloud (Cu) was modeled, and dynamic, thermodynamic, and microphysical properties of an ascending head bubble reproducing the upper part of a developing Cu were investigated. The data for analysis are taken from 10-m-resolution LES of trade wind Cu under BOMEX conditions. The detection of a rising bubble is carried out using wavelet filtering of the velocity fields and microphysical fields, while a low-frequency signal of the filtering is associated with the convective-scale structure of cloud. We substantiate and discuss the representation of the bubble as a vortex ring, and estimate the parameters of this vortex ring. The simplest Hill’s vortex was chosen as a model of a vortex ring inside cloud. Analytical approximations of the radial profiles of the vertical velocity and of conservative quantities (such as total water mixing ratio and liquid water potential temperature inside and outside the bubble) are obtained. The spatial structure of these quantities is investigated using analytical expressions. Analytical models for spatial distributions of liquid water content (LWC) and adiabatic fraction (AF) are also designed and analyzed. The results demonstrate the existence of a cloud core with high values of LWC and AF up to the height of 1800 m. The horizontally averaged value of the adiabatic fraction, calculated analytically using the Hill’s vortex concept, is evaluated as 0.39, which is the typical AF value in the upper parts of such Cu. The vertical profiles of different important quantities characterizing cloud structure are presented. The analysis performed in this study allows us to conclude that a rising vortex ring plays the dominating role in formation of the thermodynamic and microphysical structure of developing Cu. Significance Statement 1) Dynamic and thermodynamic fields of a developing cumulus cloud simulated by high-resolution LES with spectral bin microphysics are separated into convective and turbulent components by means of the wavelet technique. 2) The analysis of convective component of the cloud revealed the existence of a vortex ring at the developing stage of the cloud and evaluate its parameters. 3) The analysis performed in this study allows us to conclude that a rising vortex ring plays the important role in formation of the thermodynamic and microphysical structure of developing Cu. 4) The study provides a novel insight into the cloud–environment interaction. 5) The approximating equations describing the vortex ring can be usefully applied for developing new schemes of convective parameterization.

Meteorology & Atmospheric Sciences↗
NASA NTRS2011

New Challenges of Cloud Remote Sensing from Space

Clouds are recognized as a major source of uncertainties in predicting climate change. Lack of observational constraints on cloud processes has hindered the development of more reliable climate and weather models. With the advanced NASA EOS and A-Train sensors, we now have better knowledge about vertical distribution, water content, and occurrence frequency of global cloudiness. New ice water content (IWC) measurements from CloudSat and MLS have led to several improvements in model physics and parameterization schemes. MISR and GPS high-resolution data start to reveal deep insights on cloud processes and dynamicS in the planetary boundary layer (PBL). However, Earth sciences are still facing urgent needs to measure cloud microphysical properties, interactions between clouds and aerosol/precipitation, and processes that are coupled in 3-D space but not adequately sampled by the A-Train curtains or by the 2-D imageries. Submillimeter- wave, multi-angle imaging, and GPS radio occultation have emerged as promising remote sensing techniques to meet the challenges and enable new sciences for mid-tropospheric and PBL clouds.

Wu, Dong L.↗
DOE OSTI2024

EAGLES Liquid Cloud Testbed Large Eddy Simulation Library (v3)

This library consists of large eddy simulation (LES) model output using the PINACLES codebase coupled to the Hebrew University Fast Spectral Bin Microphysics scheme representing shallow, liquid phase clouds from four global liquid cloud testbed regions. This data is particularly suitable for examining interactions among cloud, aerosol, and turbulence in warm, boundary-layer clouds and for developing, assessing and refining parameterizations for coarser-scale models.

Kaul, Colleen M↗
DOE OSTI2015

Incorporating an advanced aerosol activation parameterization into WRF-CAM5: Model evaluation and parameterization intercomparison

Aerosol activation into cloud droplets is an important process that governs aerosol indirect effects. The advanced treatment of aerosol activation by Fountoukis and Nenes (2005) and its recent updates, collectively called the FN series, have been incorporated into a newly developed regional coupled climate-air quality model based on the Weather Research and Forecasting model with the physics package of the Community Atmosphere Model version 5 (WRF-CAM5) to simulate aerosol-cloud interactions in both resolved and convective clouds. The model is applied to East Asia for two full years of 2005 and 2010. A comprehensive model evaluation is performed for model predictions of meteorological, radiative, and cloud variables, chemical concentrations, and column mass abundances against satellite data and surface observations from air quality monitoring sites across East Asia. The model performs overall well for major meteorological variables including near-surface temperature, specific humidity, wind speed, precipitation, cloud fraction, precipitable water, downward shortwave and longwave radiation, and column mass abundances of CO, SO 2 , NO 2 , HCHO, and O 3 in terms of both magnitudes and spatial distributions. Larger biases exist in the predictions of surface concentrations of CO and NO x at all sites and SO 2 , O 3 , PM 2.5 , and PM 10 concentrations at some sites, aerosol optical depth, cloud condensation nuclei over ocean, cloud droplet number concentration (CDNC), cloud liquid and ice water path, and cloud optical thickness. Compared with the default Abdul-Razzack Ghan (2000) parameterization, simulations with the FN series produce ~107–113% higher CDNC, with half of the difference attributable to the higher aerosol activation fraction by the FN series and the remaining half due to feedbacks in subsequent cloud microphysical processes. With the higher CDNC, the FN series are more skillful in simulating cloud water path, cloud optical thickness, downward shortwave radiation, shortwave cloud forcing, and precipitation. Here, the model evaluation identifies several areas of improvements including emissions and their vertical allocation as well as model formulations such as aerosol formation, cloud droplet nucleation, and ice nucleation.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2016

Comparison of observed and simulated spatial patterns of ice microphysical processes in tropical oceanic mesoscale convective systems

To equitably compare the spatial pattern of ice microphysical processes produced by three microphysical parameterizations with each other, observations, and theory, simulations of tropical oceanic mesoscale convective systems (MCSs) in the Weather Research and Forecasting (WRF) model were forced to develop the same mesoscale circulations as observations by assimilating radial velocity data from a Doppler radar. The same general layering of microphysical processes was found in observations and simulations with deposition anywhere above the 0°C level, aggregation at and above the 0°C level, melting at and below the 0°C level, and riming near the 0°C level. Thus, this study is consistent with the layered ice microphysical pattern portrayed in previous conceptual models and indicated by dual-polarization radar data. Spatial variability of riming in the simulations suggests that riming in the midlevel inflow is related to convective-scale vertical velocity perturbations. Lastly, this study sheds light on limitations of current generally available bulk microphysical parameterizations. In each parameterization, the layers in which aggregation and riming took place were generally too thick and the frequency of riming was generally too high compared to the observations and theory. Additionally, none of the parameterizations produced similar details in every microphysical spatial pattern. Discrepancies in the patterns of microphysical processes between parameterizations likely factor into creating substantial differences in model reflectivity patterns. It is concluded that improved parameterizations of ice-phase microphysics will be essential to obtain reliable, consistent model simulations of tropical oceanic MCSs.

58 GEOSCIENCES↗
NASA NTRS2004

A New Approach to Using a Cloud-Resolving Model to Study the Interactions between Clouds, Precipitation and Aerosols

Recently Grabowski and Khairoutdinov and Randall have proposed and demonstrated, the feasibility of using 2D CRMs (these did not include detailed land processes and used periodic lateral boundary conditions) as 'super-parameterizations or a multi-scale modeling framework MMF' for cloud processes within atmospheric generation circulation models (GCMs). In the MMF, a fine-resolution 2D CRM takes the place of the single-column parameterization that is used in conventional GCMs. An overview of this approach is given by Randall et al.. Since a CRM can explicitly simulate cloud processes at the natural space and time scales of clouddynamical processes, it's possible to compute statistics of cloud fields, including radiances and radar backscatters, directly simulated/produced by the CRM. A seed fund is available at NASA Goddard to build a MMF based on a 2D GCE model and the Goddard fvGCM. A prototype MMF will be developed at end of 2004. The major objectives are to test and evaluate its performance using the GCE model as the super parameterization and compare its results with other super parameterizations, conventional GCMs and observations. Recently, a detailed spectral-bin microphysical scheme was implemented into the Goddard Cumulus Ensemble (WE) model. The GCE model is a CRM and it has been improved and applied to study the precipitation processes in the past two decades. Atmospheric aerosols are also described using number density size-distribution functions. The model is tested by studying the evolution of deep tropical clouds in the west Pacific warm pool region and summertime convection over a mid-latitude continent with different concentrations of CCN: a low 'clean' concentration and a high 'dirty' concentration. The impact of atmospheric aerosol concentration on cloud and precipitation will be investigated. In addition, the similarities and differences between bulk- and spectral-bin microphysics will be presented,

Tao, Wei-Kuo↗
DOE OSTI2019

Real-Time Ice Nuclei Measurements at Oliktok Point Field Campaign Report

The ability of atmospheric particles (aerosols) to serve as heterogeneous ice nuclei (IN) for formation of ice and mixed-phase clouds is one of the least understood microphysical processes resulting in large uncertainties in climate modeling. As stated in the Intergovernmental Panel on Climate Change 2007 report, “The effects of aerosol particles on heterogeneous ice formation are currently insufficiently understood and present another level of challenge for both observations and modeling.” IN can strongly affect cloud radiative properties and precipitation formation. Only a small fraction (often less than 1%) of all aerosol particles act as atmospheric IN. Therefore, it becomes important to identify the active fraction of aerosol particles under a variety of environmental conditions. Proposed IN measurement techniques provide IN concentrations as a function of temperature and humidity conditions, and this unique data set can be analyzed to develop or constrain IN parameterizations that models rely on to calculate primary ice formation. Currently, the processes of ice formation within models are poorly represented, such that climate models often inaccurately partition cloud water between liquid and solid ice. Model studies have shown that uncertainties in cloud phase partitioning induce large uncertainties in the impact of mixedphase clouds on climate change. Arctic clouds composed of ice particles and supercooled liquid droplets at temperatures warmer than -37°C are important for the Arctic’s climate because of their extensive horizontal coverage over long periods, and their impact on the surface radiative balance and thus ice sheet coverage. The complexity of microphysical processes within these clouds and the sensitivity of precipitation to the details of microphysics are not fully understood. In addition, representation of these processes in a variety of cloud models continues to be challenging. Arctic clouds provide unique conditions to understand heterogeneous ice nucleation mechanisms, particularly immersion freezing, and will be the initial focus of the ice nucleation group. The U.S. Department of Energy Atmospheric Radiation Measurement (ARM)-North Slope of Alaska (NSA) site is well equipped with other aerosol measurement (number concentration, size) instruments, and a relationship among long-term IN measurements and aerosol physical and chemical properties under different environmental conditions can be identified and established. The long-term IN measurements will be critical for understanding the ice nucleation process in response to the seasonal variations of aerosol and developing robust ice nucleation parameterizations for cloud and climate models. In the future, IN measurements could also be made at other ARM-NSA sites, specifically, Oliktok Point.

Arctic processes (meteorology), Oliktok Point, aer↗
DOE OSTI2016

Parameterizing microphysical effects on variances and covariances of moisture and heat content using a multivariate probability density function: a study with CLUBB (tag MVCS)

Abstract. Microphysical processes, such as the formation, growth, and evaporation of precipitation, interact with variability and covariances (e.g., fluxes) in moisture and heat content. For instance, evaporation of rain may produce cold pools, which in turn may trigger fresh convection and precipitation. These effects are usually omitted or else crudely parameterized at subgrid scales in weather and climate models.A more formal approach is pursued here, based on predictive, horizontally averaged equations for the variances, covariances, and fluxes of moisture and heat content. These higher-order moment equations contain microphysical source terms. The microphysics terms can be integrated analytically, given a suitably simple warm-rain microphysics scheme and an approximate assumption about the multivariate distribution of cloud-related and precipitation-related variables. Performing the integrations provides exact expressions within an idealized context.A large-eddy simulation (LES) of a shallow precipitating cumulus case is performed here, and it indicates that the microphysical effects on (co)variances and fluxes can be large. In some budgets and altitude ranges, they are dominant terms. The analytic expressions for the integrals are implemented in a single-column, higher-order closure model. Interactive single-column simulations agree qualitatively with the LES. The analytic integrations form a parameterization of microphysical effects in their own right, and they also serve as benchmark solutions that can be compared to non-analytic integration methods.

54 ENVIRONMENTAL SCIENCES↗