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NASA NTRS2001

Parameterizations of Depositional Growth of Cloud Ice in a Bulk Microphysical Scheme

The conversion of cloud ice to snow by depositional growth, designated P(sub SFI), in the Goddard Cumulus Ensemble Model cloud physics parameterization is examined. The original formulation of P(sub SFI) is shown to produce excessive conversion of cloud ice to snow because of an implicit assumption that the relative humidity is 100% with respect to water even though the air may actually be quite less humid. Two possible corrections to this problem are proposed, the first involving application of a relative humidity dependent correction factor to the original formulation of P(sub SFI), and the second involving a new formulation of P(sub SFI) based on the equation for depositional growth of cloud ice.

Braun, Scott A.↗

NASA NTRS2003

The Impact of Aerosols on Cloud and Precipitation Processes: Cloud-Resolving Model Simulations

Cloud microphysics are inevitably affected by the smoke particle (CCN, cloud condensation nuclei) size distributions below the clouds. Therefore, size distributions parameterized as spectral bin microphysics are needed to explicitly study the effects of atmospheric aerosol concentration on cloud development, rainfall production, and rainfall rates for convective clouds. Recently, two detailed spectral-bin microphysical schemes were implemented into the Goddard Cumulus Ensemble (GCE) model. The formulation for the explicit spectral-bin microphysical processes is based on solving stochastic kinetic equations for the size distribution functions of water droplets (i.e., cloud droplets and raindrops), and several types of ice particles [i.e.,pristine ice crystals (columnar and plate-like), snow (dendrites and aggregates), graupel and frozen drops/hail]. Each type is described by a special size distribution function containing many categories (i.e. 33 bins). Atmospheric aerosols are also described using number density size-distribution functions.A spectral-bin microphysical model is very expensive from a from a computational point of view and has only been implemented into the 2D version of the GCE at the present time. The model is tested by studying the evolution of deep tropical clouds in the west Pacific warm pool region using identical thermodynamic conditions but with different concentrations of CCN: a low "clean" concentration and a high "dirty" concentration. Besides the initial differences in aerosol concentration, preliminary results indicate that the low CCN concentration case produces rainfall at the surface sooner than the high CCN case but has less cloud water mass aloft. Because the spectral-bin model explicitly calculates and allows for the examination of both the mass and number concentration of species in each size categor, a detailed analysis of the instantaneous size spectrum can be obtained for the two cases. It is shown that since the low CCN case produces fewer droplets, larger sized develop due to the greater condensational and collectional growth, leading to a broader size spectrum in comparison to the high CCN case.

Tao, Wei-Kuo↗

NASA NTRS2003

The Impact of Aerosols on Cloud and Precipitation Processes: Cloud-Resolving Model Simulations

Cloud microphysics are inevitable affected by the smoke particle (CCN, cloud condensation nuclei) size distributions below the clouds. Therefore, size distribution parameterized as spectral bin microphysics are needed to explicitly study the effect of atmospheric aerosol concentration on cloud development, rainfall production, and rainfall rates convective clouds. Recently, two detailed spectral-bin microphysical schemes were implemented into the Goddard Cumulus Ensembel (GCE) model. The formulation for the explicit spectral-bim microphysical processes is based on solving stochastic kinetic equations for the size distribution functions of water droplets (i.e., cloud droplets and raindrops), and several types of ice particles [i.e., pristine ice crystals (columnar and plate-like), snow (dendrites and aggregates), groupel and frozen drops/hall] Each type is described by a special size distribution function containing many categories (i.e., 33 bins). Atmospheric aerosols are also described using number density size-distribution functions.A spectral-bin microphysical model is very expensive from a computational point of view and has only been implemented into the 2D version of the GCE at the present time. The model is tested by studying the evolution of deep cloud systems in the west Pacific warm pool region and in the mid-latitude using identical thermodynamic conditions but with different concentrations of CCN: a low "clean" concentration and a high "dirty" concentration. Besides the initial differences in aerosol concentration, preliminary results indicate that the low CCN concentration case produces rainfall at the surface sooner than the high CCN case but has less cloud water mass aloft. Because the spectral-bim model explicitly calculates and allows for the examination of both the mass and number concentration of cpecies in each size category, a detailed analysis of the instantaneous size spectrum can be obtained for the two cases. It is shown that since the low CCN case produces fever droplets, larger size develop due to greater condencational and collectional growth, leading to a broader size spectrum in comparison to the high CCN case.

Tao, Wei-Kuo↗

NASA NTRS2004

The Impact of Aerosols on Cloud and Precipitation Processes: Cloud-resolving Model Simulations

Tao, Wei-Kuo↗

NASA NTRS2004

The Impact of Aerosols on Cloud and Precipitation Processes: Cloud-Resolving Model Simulations

Tao, Wei-Kuo↗

NASA NTRS2004

The Impact of Aerosols on Cloud and Precipitation Processes: Cloud-Resolving Model Simulations

Cloud microphysics are inevitably affected by the smoke particle (CCN, cloud condensation nuclei) size distributions below the clouds. Therefore, size distributions parameterized as spectral bin microphysics are needed to explicitly study the effects of atmospheric aerosol concentration on cloud development, rainfall production, and rainfall rates for convective clouds. Recently, two detailed spectral-bin microphysical schemes were implemented into the Goddard Cumulus Ensemble (GCE) model. The formulation for the explicit spectral-bin microphysical processes is based on solving stochastic kinetic equations for the size distribution functions of water droplets (i.e., cloud droplets and raindrops), and several types of ice particles (i.e., pristine ice crystals (columnar and plate-like), snow (dendrites and aggregates), graupel and frozen drops/hail). Each type is described by a special size distribution function containing many categories (i.e. 33 bins). Atmospheric aerosols are also described using number density size-distribution functions. A spectral-bin microphysical model is very expensive from a computational point of view and has only been implemented into the 2D version of the GCE at the present time. The model is tested by studying the evolution of deep cloud systems in the west Pacific warm pool region, in the sub-tropics (Florida) and in the mid-latitude using identical thermodynamic conditions but with different concentrations of CCN: a low 'clean' concentration and a high 'dirty' concentration.

Tao, Wei-Kuo↗

NASA NTRS2005

The Impact of Aerosols on Cloud and Precipitation Processes: Cloud-Resolving Model Simulations

Cloud microphysics are inevitable affected by the smoke particle (CCN, cloud condensation nuclei) size distributions below the clouds, Therefore, size distributions parameterized as spectral bin microphysics are needed to explicitly study the effect of atmospheric aerosol concentration on cloud development, rainfall production, and rainfall rates for convective clouds. Recently, a detailed spectral-bin microphysical scheme was implemented into the the Goddard Cumulus Ensemble (GCE) model. The formulation for the explicit spectral-bim microphysical processes is based on solving stochastic kinetic equations for the size distribution functions of water droplets (i.e., cloud droplets and raindrops), and several types of ice particles [i.e., pristine ice crystals (columnar and plate-like), snow (dendrites and aggregates), graupel and frozen drops/hail]. Each type is described by a special size distribution function containing many categories (i.e., 33 bins). Atmospheric aerosols are also described using number density size-distribution functions.

Tao, Wei-Kuo↗

NASA NTRS2012

The Role of Aerosols on Precipitation Processes: Cloud Resolving Model Simulations

Cloud microphysics is inevitably affected by the smoke particle (CCN, cloud condensation nuclei) size distributions below the clouds. Therefore, size distributions parameterized as spectral bin microphysics are needed to explicitly study the effects of atmospheric aerosol concentration on cloud development, rainfall production, and rainfall rates for convective clouds. Recently, a detailed spectral-bin microphysical scheme was implemented into the Goddard Cumulus Ensemble (GCE) model. The formulation for the explicit spectral bin microphysical processes is based on solving stochastic kinetic equations for the size distribution functions of water droplets (i.e., cloud droplets and raindrops), and several types of ice particles [i.e. pristine ice crystals (columnar and plate-like), snow (dendrites and aggregates), graupel and frozen drops/hail]. Each type is described by a special size distribution function containing many categories (i.e., 33 bins). Atmospheric aerosols are also described using number density size-distribution functions. The model is tested by studying the evolution of deep cloud systems in the west Pacific warm pool region, the sub-tropics (Florida) and midlatitudes using identical thermodynamic conditions but with different concentrations of CCN: a low "clean" concentration and a high "dirty" concentration. Results indicate that the low CCN concentration case produces rainfall at the surface sooner than the high CeN case but has less cloud water mass aloft. Because the spectral-bin model explicitly calculates and allows for the examination of both the mass and number concentration of species in each size category, a detailed analysis of the instantaneous size spectrum can be obtained for these cases. It is shown that since the low (CN case produces fewer droplets, larger sizes develop due to greater condensational and collection growth, leading to a broader size spectrum in comparison to the high CCN case. Sensitivity tests were performed to identify the impact of ice processes, radiation and large-scale influence on cloud-aerosol interactive processes, especially regarding surface rainfall amounts and characteristics (i.e., heavy or convective versus light or stratiform types). In addition, an inert tracer was included to follow the vertical redistribution of aerosols by cloud processes. We will also give a brief review from observational evidence on the role of aerosol on precipitation processes.

Tao, Wei-Kuo↗

NASA NTRS2007

New, Improved Bulk-microphysical Schemes for Studying Precipitation Processes in WRF: Comparisons with Other Schemes - Part 1

Advances in computing power allow atmospheric prediction models to be mn at progressively finer scales of resolution, using increasingly more sophisticated physical parameterizations and numerical methods. The representation of cloud microphysical processes is a key component of these models, over the past decade both research and operational numerical weather prediction models have started using more complex microphysical schemes that were originally developed for high-resolution cloud-resolving models (CRMs). A recent report to the United States Weather Research Program (USWRP) Science Steering Committee specifically calls for the replacement of implicit cumulus parameterization schemes with explicit bulk schemes in numerical weather prediction (NWP) as part of a community effort to improve quantitative precipitation forecasts (QPF). An improved Goddard bulk microphysical parameterization is implemented into a state-of the-art of next generation of Weather Research and Forecasting (WRF) model. High-resolution model simulations are conducted to examine the impact of microphysical schemes on two different weather events (a midlatitude linear convective system and an Atllan"ic hurricane). The results suggest that microphysics has a major impact on the organization and precipitation processes associated with a summer midlatitude convective line system. The 31CE scheme with a cloud ice-snow-hail configuration led to a better agreement with observation in terms of simulated narrow convective line and rainfall intensity. This is because the 3ICE-hail scheme includes dense ice precipitating (hail) particle with very fast fall speed (over 10 m/s). For an Atlantic hurricane case, varying the microphysical schemes had no significant impact on the track forecast but did affect the intensity (important for air-sea interaction)

Tao, W.-K.↗

DOE OSTI2022

Toward Better Understanding of Microphysical Processes and Resulting Precipitation Physics: A Merger Of Observations and Cloud Models (Final Report)

A large database of global disdrometer observations and a diverse set of simulations from the Regional Atmospheric Modeling System (RAMS) were compared using principal component analysis (PCA) in order to better understand warm and ice-based precipitation processes. The analysis demonstrated that six distinct precipitation groups (PGs) with common characteristics were revealed in both the observations and model simulations. These PGs were defined by similar co-variability of rain parameters. However, the model showed large concentrations of drops with a given size compared to the observations. Detailed investigations determined the parameterization of drop breakup was forcing the drops to an equilibrium size with a stronger constraint than was indicated by the observations. A series of sensitivity studies permutating the rain shape parameter in the RAMS 2-moment bulk scheme demonstrated how the assumed shape of the rain distribution influences the microphysics as well as precipitation characteristics. Results were compared to the same simulations performed with a bin microphysics scheme where the rain shape parameter is freely evolving. In the bin scheme, a wide range of shape parameters were found, with horizontal and vertical variability, in addition to being a function of storm lifetime. The results of this study highlight the limitations of a fixed assumed rain distribution in 2-moment microphysics schemes. Model simulations were used to probe the microphysical origins of the six PGs. Rain budgets from a supercell case showed that several of the groups hypothesized to be associated with strong ice microphysical processes had significant contributions from melting hail, but the complexity of ice and warm-rain processes in the supercell precluded definitive determination of microphysical origins for many of the PGs. A case study from the Mid-latitude Continental Convection and Clouds Experiment (MC3E) compared the model PGs to disdrometer observations, and demonstrated the spatial and temporal variability of the PGs which is not possible from disdrometer point measurements alone. The analyses performed during this project demonstrated how statistical analysis can provide a robust method for comparing model simulations and observations, which ultimately resulted identification of limitations in some current microphysics parameterizations as well as insights into precipitation variability which ultimately benefits both observations and modeling efforts.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS2004

Parameterization of the Vertical Variability of Tropical Cirrus Cloud Microphysical and Optical Properties

Cloud Integrating Nephelometers (CIN) were flown on the U. North Dakota Citation aircraft and the NASA WB-57 aircraft for the purpose of measuring in-situ the optical extinction coefficient and the asymmetry parameter (g) at a wavelength of 635 nm of primarily ice particles encountered during the NASA CRYSTAL-FACE study of large cumulus clouds (Cu) and their anvils found in the southern Florida region. The probes performance was largely successful and produced archived data for vertical profiles of extinction, asymmetry parameter, and effective radius (Re), the latter being obtained by combining CIN and CVI (total water; Oregon State U.) measurements. Composites of the CIN and CVI data describing the average microphysical and optical behavior of the Cu and their anvils showed the following: The extinction increases with height as a result of the size of the particles also decreasing with height as shown by the Re measurements; near the top of anvils the size of the primary ice particles is about 10-um radius; and the value of g does not vary significantly with height and has a mean value of about 0.73 consistent with the idea that ambient ice crystals are primarily of complex shape and reflect solar radiation more efficiently than particles of pristine crystal shape. Other observations include: The g measurements were found to be an indicator of the phase of the cloud permitting identification of the clouds with water droplets, rain, and ice; visual ranges as small as several tens of meters were occasionally found in "extinction cores" that coincided with strong updraft cores; and comparison of the cloud probes on the Citation showed significant disagreement.

Gerber, Hermann E.↗

NASA NTRS2013

Saharan Dust Event Impacts on Cloud Formation and Radiation over Western Europe

We investigated the impact of mineral dust particles on clouds, radiation and atmospheric state during a strong Saharan dust event over Europe in May 2008, applying a comprehensive online-coupled regional model framework that explicitly treats particle-microphysics and chemical composition. Sophisticated parameterizations for aerosol activation and ice nucleation, together with two-moment cloud microphysics are used to calculate the interaction of the different particles with clouds depending on their physical and chemical properties. The impact of dust on cloud droplet number concentration was found to be low, with just a slight increase in cloud droplet number concentration for both uncoated and coated dust. For temperatures lower than the level of homogeneous freezing, no significant impact of dust on the number and mass concentration of ice crystals was found, though the concentration of frozen dust particles reached up to 100 l-1 during the ice nucleation events. Mineral dust particles were found to have the largest impact on clouds in a temperature range between freezing level and the level of homogeneous freezing, where they determined the number concentration of ice crystals due to efficient heterogeneous freezing of the dust particles and modified the glaciation of mixed phase clouds. Our simulations show that during the dust events, ice crystals concentrations were increased twofold in this temperature range (compared to if dust interactions are neglected). This had a significant impact on the cloud optical properties, causing a reduction in the incoming short-wave radiation at the surface up to -75Wm-2. Including the direct interaction of dust with radiation caused an additional reduction in the incoming short-wave radiation by 40 to 80Wm-2, and the incoming long-wave radiation at the surface was increased significantly in the order of +10Wm-2. The strong radiative forcings associated with dust caused a reduction in surface temperature in the order of -0.2 to -0.5K for most parts of France, Germany, and Italy during the dust event. The maximum difference in surface temperature was found in the East of France, the Benelux, and Western Germany with up to -1 K. This magnitude of temperature change was sufficient to explain a systematic bias in numerical weather forecasts during the period of the dust event.

Bangert, M.↗

NASA NTRS2001

Retrieval of Polar Stratospheric Cloud Microphysical Properties from Lidar Measurements: Dependence on Particle Shape Assumptions

A retrieval algorithm has been developed for the microphysical analysis of polar stratospheric cloud (PSC) optical data obtained using lidar instrumentation. The parameterization scheme of the PSC microphysical properties allows for coexistence of up to three different particle types with size-dependent shapes. The finite difference time domain (FDTD) method has been used to calculate optical properties of particles with maximum dimensions equal to or less than 2 mu m and with shapes that can be considered more representative of PSCs on the scale of individual crystals than the commonly assumed spheroids. Specifically. these are irregular and hexagonal crystals. Selection of the optical parameters that are input to the inversion algorithm is based on a potential data set such as that gathered by two of the lidars on board the NASA DC-8 during the Stratospheric Aerosol and Gas Experiment 0 p (SAGE) Ozone Loss Validation experiment (SOLVE) campaign in winter 1999/2000: the Airborne Raman Ozone and Temperature Lidar (AROTEL) and the NASA Langley Differential Absorption Lidar (DIAL). The 0 microphysical retrieval algorithm has been applied to study how particle shape assumptions affect the inversion of lidar data measured in leewave PSCs. The model simulations show that under the assumption of spheroidal particle shapes, PSC surface and volume density are systematically smaller than the FDTD-based values by, respectively, approximately 10-30% and approximately 5-23%.

Reichardt, J.↗

NASA NTRS2011

Advancements in the Representation of Cloud-Aerosol Microphysics in the GEOS-5 AGCM

Despite numerous challenges, the physical parameterization of cloud-aerosol interactions in atmospheric GCMs has become a top priority for advancement because of our need to simulate and understand past, current, and future indirect effects of aerosols on clouds. The challenges stem from the involvement of wide range of cloud-scale dynamics and aerosol activation physical processes. Cloud dynamics modulate cloud areal extent and condensate, while aerosol activation depends on aerosol mass load, size distribution, internal mixing state, and nucleating properties, and ultimately determines cloud optical properties via particle sizes. Both macro- and micro-scale processes are obviously important for cloud-radiation interactions. We will present the main features of cloud microphysical properties in the GEOS- 5 Atmospheric GCM (AGCM) as simulated by the McRAS-AC (Microphysics of Clouds with Relaxed Arakawa-Schubert and Aerosol-Cloud interaction) scheme. McRAS-AC uses Fountoukis and Nenes (2005) aerosol activation for liquid clouds, and has an option for either Liu and Penner (2005) or Barahona and Nenes (2008, 2009) aerosol activation for ice clouds. Aerosol loading (on-line or climatological) comes from GOCART, with an assumed log-normal size distribution. Other features of McRAS-AC are level-by-level cloud-scale thermodynamics, and Seifert-Beheng (2001)-type precipitation microphysics, particularly from moist convection. Results from Single-Column Model simulations will be shown to demonstrate how cloud radiative properties, lifetimes, and precipitation are influenced by different parameterization assumptions. Corresponding fields from year-long simulations of the full AGCM will also be presented with geographical distributions of cloud effective particle sizes compared to satellite retrievals. While the primary emphasis will be on current climate, simulation results with perturbed aerosol loadings will also be shown to expose the radiative sensitivity of the microphysical parameterization.

Lee, D.↗

DOE OSTI2021

Quantifying physical parameterization uncertainties associated with land-atmosphere interactions in the WRF model over Amazon

The Weather Research and Forecasting (WRF) model can be used to diagnose regional land-atmosphere (L-A) coupling strength in the absence of sufficient observations but subjected to uncertainties associated with model physical parameterizations. In this study, we propose a framework to quantify and reduce model physical parameterization uncertainties associated with surface fluxes and L-A coupling. An ensemble of WRF simulations with different physical schemes is used to simulate surface fluxes and land-atmosphere coupling strength over the Amazon region. The physical parameterizations investigated include cloud microphysics (MP), land surface processes (LSM), planetary boundary layer (PBL), surface layer (SL), and cumulus (CU). We perform 120 ensemble simulations using the WRF model and different combinations of six MPs, three LSMs, six PBLs and SLs and three CUs. The measurements from the GoAMAZON field campaign and satellite data are used to evaluate model performance. A Multi-way analysis of variance (ANOVA) approach is applied to quantify the relative importance of different physics processes on L-A coupling. The Tukey’s test is used to sort schemes that have no significant differences into one group. The suite of physics that result in the best simulations of the corresponding variables are selected based on the Taylor skill score. Results show that the relative importance of processes and their interaction vary with the variables of interest. For example, CU was the most important process in modulating soil moisture, 2 m-humidity, latent heat, and net radiation. LSM showed dominant effects on 2 m-temperature and also has the largest impact on sensible heat and the lifting condensation level. The best physical parameterization ensembles show much narrower ranges of the variables of interest than the priori ensemble. Results of this study show the roles of different physical processes in modulating L-A interactions, quantify model uncertainties from physical processes, and provide insights for improving the model physics parameterizations.

Wang, Chen↗

DOE OSTI2024

HSW-V v1.0: localized injections of interactive volcanic aerosols and their climate impacts in a simple general circulation model

Abstract. A new set of standalone parameterizations is presented for simulating the injection, evolution, and radiative forcing by stratospheric volcanic aerosols against an idealized Held–Suarez–Williamson (HSW) atmospheric background in the Energy Exascale Earth System Model version 2 (E3SMv2). In this model configuration (HSW with enabled volcanism, HSW-V), sulfur dioxide (SO2) and ash are injected into the atmosphere with a specified profile in the vertical, and they proceed to follow a simple exponential decay. The SO2 decay is modeled as a perfect conversion to a long-living sulfate aerosol which persists in the stratosphere. All three species are implemented as tracers in the model framework and are transported by the dynamical core's advection algorithm. The aerosols contribute simultaneously to local heating of the stratosphere and cooling of the surface by a simple plane-parallel Beer–Lambert law applied on two zonally symmetric radiation broadbands in the longwave and shortwave ranges. It is shown that the implementation parameters can be tuned to produce realistic temperature anomaly signatures of large volcanic events. In particular, results are shown for an ensemble of runs that mimic the volcanic eruption of Mt. Pinatubo in 1991. The design requires no coupling to microphysical subgrid-scale parameterizations and thus approaches the computational affordability of prescribed aerosol forcing strategies. The idealized simulations contain a single isolated volcanic event against a statistically uniform climate, where no background aerosols or other sources of externally forced variability are present. HSW-V represents a simpler-to-understand tool for the development of climate source-to-impact attribution methods.

Hollowed, Joseph P. (ORCID:0000000286581672)↗

DOE OSTI2020

Effects of Cloud Liquid-Phase Microphysical Processes in Mixed-Phase Cumuli Over the Tibetan Plateau

Numerical simulations often overpredict precipitation over the Tibetan Plateau (TP). To examine the factors causing precipitation overprediction, different parameterizations of liquid-phase microphysical processes (accretion, autoconversion, and entrainment-mixing) are implemented into the Morrison microphysics scheme to simulate a TP precipitation event in summer with the Weather Research and Forecasting (WRF) model. The general spatial distribution and temporal trend of precipitation are captured by all simulations, but the precipitation rate is overpredicted. The results from sensitivity experiments suggest that compared to other examined liquid-phase processes, the accretion process is more important in precipitation simulation over the TP region. Further investigation with the Heidke skill scores reveals that accretion parameterization that takes into account the raindrop size produces the most accurate results in terms of the total surface precipitation. This parameterization suppresses spurious accretion and does not produce liquid-phase precipitation until cloud droplets are big enough. Here, it is also confirmed that increasing the model resolution can reduce precipitation overprediction. Results from the case study are confirmed by the use of a one-month simulation.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

How Snow Aggregate Ellipsoid Shape and Orientation Variability Affects Fall Speed and Self-Aggregation Rates

Snow aggregate shapes and orientations have long been known to exhibit substantial variability. Despite this observed variability, most weather and climate prediction models use fixed power-law functions that deterministically map particle size to mass and fall speed. As such, integrated quantities like precipitation and self-aggregation rates currently ignore nonlinear effects resulting from variation in shape and orientation for aggregates of the same size. This study therefore develops an analytic framework that couples an empirically based bivariate distribution of ellipsoid shapes to classical hydrodynamic theory so as to capture an appropriate dispersion of masses, projected areas, and fall speeds for an assumed size distribution. For a fixed aggregate size, shape variations produce approximately ±0.13 m s -1 standard deviation of fall speed which increases the mass flux fall speed dispersion by more than 100% over traditional microphysics models. This increased fall speed dispersion results predominantly from shape-induced mass dispersion whereas orientation and drag dispersion play a lesser role. Shape variations can increase mass- and reflectivity-weighted fall speeds by up to 60% of traditional models whereas self-aggregation rates can increase by a factor of 100 for very small slope parameters. This implies that aggregate shape variations effectively forestall the theorized onset of fall speed distribution narrowing and subsequent quenching of the aggregation process. As a result, it is likely that secondary ice formation is necessary to prevent an ever decreasing slope parameter. The mathematical theory presented in this study is used to develop simple correction factors for snow forecast and climate models.

54 ENVIRONMENTAL SCIENCES↗