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At least 523 records · Page 29

Ensemble Kalman filter for data assimilation coupled with low-resolution computations techniques applied in fluid dynamics

This paper presents an innovative Reduced-order model (ROM) for merging experimental and simulation data using data assimilation (DA) to estimate the "True" state of a fluid dynamics system, leading to more accurate predictions. Our methodology introduces a novel approach by implementing the ensemble Kalman filter (EnKF) within a reduced-dimensional framework, grounded in a robust theoretical foundation and applied to fluid dynamics. To address the substantial computational demands of DA, the proposed ROM employs low-resolution (LR) techniques to drastically reduce computational costs. This innovative approach involves downsampling datasets for DA computations, followed by an advanced reconstruction technique based on low-cost singular value decomposition (lcSVD). The lcSVD method, a key innovation in this paper, has never been applied to DA before and offers a highly efficient way to enhance resolution with minimal computational resources. Our results demonstrate significant reductions in both computation time and RAM usage through these LR techniques without compromising the accuracy of the estimations. For instance, in a turbulent test case, for a data compression rate of 15.9, the LR approach can achieve a speed-up of 13.7 and a RAM compression of 90.9% while maintaining a low relative root mean square error (RRMSE) of 2.6%, compared to 0.8% in the high-resolution (HR) reference. Furthermore, we highlight the effectiveness of the EnKF in estimating and predicting the state of fluid flow systems based on limited observations and given low-fidelity numerical data. This paper highlights the potential of the proposed DA method in fluid dynamics applications, particularly for improving computational efficiency in CFD and related fields. Its ability to balance accuracy with low computational and memory costs makes it especially suitable for large-scale and real-time applications, such as environmental monitoring or engineering design. This method will be incorporated into ModelFLOWs-app.

Data Assimilation↗

Study of alloy disorder in quantum dots through multi-million atom simulations

A tight binding model which includes s, p, d, s orbitals is used to examine the electronic structures of an ensemble of dome-shaped In0.6 Ga0.4 As quantum dots. Given ensembles of identically sized quantum dots, variations in composition and configuration yield a linewidth broadening of less than 0.35 meV, much smaller than the total broadening determined from photoluminescence experiments. It is also found that the computed disorder-induced broadening is very sensitive to the applied boundary conditions, so that care must be taken to ensure proper convergence of the numerical results. Examination of local eigenenergies as functions of position shows similar convergence problems and indicates that an inaccurate resolution of the equilibrium atomic positions due to truncation of the simulation domain may be the source of the slow ground state convergence.

disorder↗

Microphysics, Radiation and Surface Processes in the Goddard Cumulus Ensemble (GCE) Model

The response of cloud systems to their environment is an important link in a chain of processes responsible for monsoons, frontal depression, El Nino Southern Oscillation (ENSO) episodes and other climate variations (e.g., 30-60 day intra-seasonal oscillations). Numerical models of cloud properties provide essential insights into the interactions of clouds with each other, with their surroundings, and with land and ocean surfaces. Significant advances are currently being made in the modeling of rainfall and rain-related cloud processes, ranging in scales from the very small up to the simulation of an extensive population of raining cumulus clouds in a tropical- or midlatitude-storm environment. The Goddard Cumulus Ensemble (GCE) model is a multi-dimensional nonhydrostatic dynamic/microphysical cloud resolving model. It has been used to simulate many different mesoscale convective systems that occurred in various geographic locations. In this paper, recent GCE model improvements (microphysics, radiation and surface processes) will be described as well as their impact on the development of precipitation events from various geographic locations. The performance of these new physical processes will be examined by comparing the model results with observations. In addition, the explicit interactive processes between cloud, radiation and surface processes will be discussed.

Tao, Wei-Kuo↗

The influence of assimilating leaf area index in a land surface model on global water fluxes and storages

Vegetation plays a fundamental role not only in theenergy and carbon cycles but also in the global water balanceby controlling surface evapotranspiration (ET). Thus, accu-rately estimating vegetation-related variables has the poten-tial to improve our understanding and estimation of the dy-namic interactions between the water, energy, and carbon cy-cles. This study aims to assess the extent to which a land sur-face model (LSM) can be optimized through the assimilationof leaf area index (LAI) observations at the global scale. Twoobserving system simulation experiments (OSSEs) are per-formed to evaluate the efficiency of assimilating LAI into anLSM through an ensemble Kalman filter (EnKF) to estimateLAI, ET, canopy-interception evaporation (CIE), canopy wa-ter storage (CWS), surface soil moisture (SSM), and terres-trial water storage (TWS). Results show that the LAI dataassimilation framework not only effectively reduces errorsin LAI model simulations but also improves all the modeledwater flux and storage variables considered in this study (ET,CIE, CWS, SSM, and TWS), even when the forcing pre-cipitation is strongly positively biased (extremely wet con-ditions). However, it tends to worsen some of the modeledwater-related variables (SSM and TWS) when the forcingprecipitation is affected by a dry bias. This is attributed tothe fact that the amount of water in the LSM is conservative,and the LAI assimilation introduces more vegetation, whichrequires more water than what is available within the soil

Terrestrial Vegetation↗

The Impact of Model and Rainfall Forcing Errors on Characterizing Soil Moisture Uncertainty in Land Surface Modeling

The contribution of rainfall forcing errors relative to model (structural and parameter) uncertainty in the prediction of soil moisture is investigated by integrating the NASA Catchment Land Surface Model (CLSM), forced with hydro-meteorological data, in the Oklahoma region. Rainfall-forcing uncertainty is introduced using a stochastic error model that generates ensemble rainfall fields from satellite rainfall products. The ensemble satellite rain fields are propagated through CLSM to produce soil moisture ensembles. Errors in CLSM are modeled with two different approaches: either by perturbing model parameters (representing model parameter uncertainty) or by adding randomly generated noise (representing model structure and parameter uncertainty) to the model prognostic variables. Our findings highlight that the method currently used in the NASA GEOS-5 Land Data Assimilation System to perturb CLSM variables poorly describes the uncertainty in the predicted soil moisture, even when combined with rainfall model perturbations. On the other hand, by adding model parameter perturbations to rainfall forcing perturbations, a better characterization of uncertainty in soil moisture simulations is observed. Specifically, an analysis of the rank histograms shows that the most consistent ensemble of soil moisture is obtained by combining rainfall and model parameter perturbations. When rainfall forcing and model prognostic perturbations are added, the rank histogram shows a U-shape at the domain average scale, which corresponds to a lack of variability in the forecast ensemble. The more accurate estimation of the soil moisture prediction uncertainty obtained by combining rainfall and parameter perturbations is encouraging for the application of this approach in ensemble data assimilation systems.

Maggioni, V.↗

Long‐Term Large‐Scale Atmospheric Forcing Data From Three‐Dimensional Constrained Variational Analysis for the ARM SGP Site

Here, this study presents a long‐term three‐dimensional large‐scale forcing data set (VARANAL3D) derived from the three‐dimensional constrained variational analysis (3DCVA) method at the Atmospheric Radiation Measurement (ARM) program Southern Great Plains (SGP) site from 2004 to 2018. Building on the same input data sets as the conventional continuous forcing data set (VARANAL), VARANAL3D maintains overall consistency in domain‐averaged fields while introducing spatial variability, offering critical insights into the influence of mesoscale synoptic systems on cloud‐related processes. Evaluations are conducted across four cloud and precipitation regimes: Clear‐sky, Shallow‐clouds, Afternoon‐precipitation, and Nocturnal‐precipitation, presenting high consistency of the domain‐mean forcing data sets while emphasizing the role of subdomain forcing variability particularly in precipitating regimes. Single column model (SCM) simulations demonstrate that subdomain VARANAL3D forcing improves cloud and precipitation representation, with the ensemble outperforming domain‐mean forcing in three cloudy and precipitating regimes. Overall, these results highlight VARANAL3D's value for investigating the impacts of spatial variability of large‐scale forcing on atmospheric processes. The VARANAL3D data set provides new opportunities for evaluating model physics, advancing the development of scale‐aware parameterizations and deepening our understanding of cloud and precipitation dynamics.

Environmental sciences↗

Structure of unsteady flows at leading- and trailing-edges: Flow visualization and its interpretation

Unsteady two- and three-dimensional flow structure at leading and trailing edges of bodies can be characterized effectively using recently developed techniques for acquisition and interpretation of flow visualization. The techniques addressed here include: flow image/surface pressure correlations; 3-D reconstruction of flow structure from flow images; and interactive interpretation of flow images with theoretical simulations. These techniques can be employed in conjunction with: visual correlation and ensemble-averaging, both within a given image and between images; recognition of patterns from images; and estimates of velocity eigenfunctions from images.

Rockwell, D.↗

Impacts of Snow Darkening Effects by Light Absorbing Aerosols on Hydroclimate of Eurasia During Boreal Spring and Summer

In this study, we used the NASA GEOS-5 climate model to investigate the impact of snow darkening by light-absorbing aerosols on the hydro-climate if Eurasia during boreal spring and summer. Two sets of 10-member ensemble model integrations with prescribed sea surface temperature were carried out for 10 simulated years (2002-2011); one includes snow-darkening effects (SDE) by light-absorbing aerosols, and one does not (NSDE). Differences between the two experiments in the hydro-climates over Eurasia were evaluated. Results show that SDE warming is most pronounced during the melting season due to strong snow-albedo feedback in the vicinity of the retreating seasonal snowline. SDE spurs a Wet-First-Dry-Later (WFDL) modulation of the surface energy and water balances, characterized by an accelerated (days-to-weeks) snowmelt along with excessive runoff and a warming and wetting of the land during the early part of the melting season. The fast snowmelt is followed by a prolonged (> months) warming and desiccation of the land, stemming from a deficit of late-season snowmelt water and increased surface evaporation. The prolonged warming and drying of the land atmosphere system persist through the summer, sustained by atmospheric conditions favorable for the development of atmospheric blocking, i.e., higher mid-to-upper tropospheric geopotential height, lower relative humidity, reduced cloudiness and enhanced atmospheric subsidence. Overall, SDE by light absorbing aerosols leads to a warmer and drier boreal summer hydro-climate, increasing the frequency of the top 5% and 1% extreme hot days (as defined by NSDE statistics) over western and northern Eurasia by approximately three-fold and tenfold, respectively.

Snow darkening↗

Increases in Future AR Count and Size: Overview of 2 the ARTMIP Tier 2 CMIP5/6 Experiment

The Atmospheric River (AR) Tracking Method Intercomparison Project (ARTMIP) is a community effort to systematically assess how the uncertainties from AR detectors (ARDTs) impact our scientific understanding of ARs. This study describes the ARTMIP Tier 2 experimental design and initial results using the Coupled Model Intercomparison Project (CMIP) Phases 5 and 6 multi-model ensembles. We show that AR statistics from a given ARDT in CMIP5/6 historical simulations compare remarkably well with the MERRA-2 reanalysis. In CMIP5/6 future simulations, most ARDTs project a global increase in AR frequency, counts, and sizes, especially along the western coastlines of the Pacific and Atlantic oceans. We find that the choice of ARDT is the dominant contributor to the uncertainty in projected AR frequency when compared with model choice. These results imply that new projects investigating future changes in ARs should explicitly consider ARDT uncertainty as a core part of the experimental design. Plain Language Summary: Atmospheric rivers (ARs) are a type of weather pattern known to be important for moving water vapor from the warm, moist tropics to the cool, dry polar regions; when they reach midlatitudes in the winter time, they are commonly associated with heavy precipitation. Recent studies that assess the impacts of global climate change on ARs tend to agree that there will be more ARs in a warmer climate, and that ARs will tend to be more extreme. However, it has been increasingly recognized by the AR research community that these results may depend on the method used to identify ARs and the choice of climate model. This study reports results from a controlled experiment, involving an international research community, that aims to show how different AR identification methods and climate models might impact our scientific understanding of ARs in the future. This experiment shows that there will likely be more ARs in the future, and that ARs will generally have a larger spatial footprint. This experiment also shows that uncertainty in these results are large, with the uncertainty from AR identification methods outweighing that of climate models. Future efforts to better understand the physics of ARs may help us reduce this uncertainty.

T A OBrien↗

Enhancing Extreme Precipitation Predictions With Dynamical Downscaling: A Convection‐Permitting Modeling Study in Texas and Oklahoma

Precipitation in the Southern Plains of the United States is relatively well depicted by the Community Earth System Model (CESM). However, despite its ability to capture seasonal mean precipitation anomalies, CESM consistently underestimates extreme pluvial and drought events, rendering it an insufficient tool for extending simulation lead times for exceptional events, such as the abnormally dry May 2011, which helped drive Texas into its worst period of drought in more than a century, and the abnormally wet May 2015, which led to widespread flooding in that state. Ensemble‐based regional climate experiments are completed for the two extreme years using Weather Research and Forecasting model (WRF) and downscaled from CESM. WRF simulations are at convection‐permitting grid resolution for improved physical representation of simulated precipitation over the Southern Great Plains. By integrating convection‐permitting models (CPMs) into each individual member of a CESM ocean data assimilation ensemble, this study demonstrates that high‐resolution dynamical downscaling can improve model skillfulness at capturing these two events and is thus a potentially useful tool for forecasting extremely high and extremely low precipitation events at subseasonal or even seasonal lead times.

54 ENVIRONMENTAL SCIENCES↗

Observing and Projecting the Lasting Fate of the Hunga Eruption on Atmospheric Water Vapor and Hydroxyl Radical

On 15th Jan. 2022 the submarine Hunga volcano erupted injecting approximately 0.5 Tg of sulfur dioxide into the stratosphere, but more significantly added 150-170 Tg of water vapor to the stratospheric background (over a 10% perturbation) in a matter of several hours. The sulfur dioxide rapidly converted to sulfate aerosol and along with water vapor, was transported around the Southern Hemisphere sub-tropics into midlatitudes and beyond with some transport into the Northern Hemisphere. With a much longer lifetime than sulfate aerosol, measurable water vapor mass anomalies have persisted with only small losses over more than 2 years since the eruption and are likely to continue above background for the remainder of the decade. Satellite measurements from limb and nadir viewing observing instruments provide the information needed to reasonably initialize the Hunga eruption in the Goddard Earth Observing System (GEOS) model using the “replay” framework coupled to the Global Modeling Initiative (GMI) stratosphere-troposphere chemical mechanism for the recent past and have continued the simulations into the future with the free running chemistry climate model (CCM). Using a number of model ensemble members together with the satellite observations, we quantify how the Hunga eruption is perturbing atmospheric composition and climate and projecting the influences to come as the enhanced water vapor continues in the stratosphere with only very slow removal mechanisms. Modeling suggests that the water rich eruption enhanced stratospheric hydroxyl radiacal (OH) and is supported by the speed of the observed conversion of SO2 into sulfate aerosol. These simulations also suggest a significant reduction in tropospheric OH over the mid-high latitude Southern Hemisphere, which has implications for numerous species that are impacted by this radical. The Hunga eruption provides a useful test of chemistry climate models and an opportunity for observation-based process understanding and in making new connections.

Hunga volcano↗

Groundwater Recharge Estimated by Land Surface Models: An Evaluation in the Conterminous U.S.

Estimating diffuse recharge of precipitation is fundamental to assessing groundwater sustainability. Diffuse recharge is also the process through which climate and climate change directly affect groundwater. In this study, we evaluated diffuse recharge over the conterminous U.S. simulated by a suite of land surface models (LSMs) that were forced using a common set of meteorological input data. Simulated annual recharge exhibited spatial patterns that were similar among the LSMs, with the highest values in the eastern U.S. and Pacific Northwest. However, the magnitudes of annual recharge varied significantly among the models and were associated with differences in simulated ET, runoff and snow. Evaluation against two independent datasets did not answer the question of whether the ensemble mean performs the best, due to inconsistency between those datasets. The amplitude and timing of seasonal maximum recharge differed among the models, influenced strongly by model physics governing deep soil moisture drainage rates and, in cold regions, snowmelt. Evaluation using in situ soil moisture observations suggested that true recharge peaks 1-3 months later than simulated recharge, indicating systematic biases in simulating deep soil moisture. However, recharge from lateral flows and through preferential flows cannot be inferred from soil moisture data, and the seasonal cycle of simulated groundwater storage actually compared well with in situ groundwater observations. Long-term trends in recharge were not consistently correlated with either precipitation trends or temperature trends. This study highlights the need to employ dynamic flow models in LSMs, among other improvements, to enable more accurate simulation of recharge.

snowmelt↗

Numerical Mixing Suppresses Submesoscale Baroclinic Instabilities Over Sloping Bathymetry

Abstract The impacts of spurious numerical salinity mixing on the larger‐scale flow and tracer fields are characterized using idealized simulations. The idealized model is motivated by realistic simulations of the Texas‐Louisiana shelf and features oscillatory near‐inertial wind forcing. can exceed the physical mixing from the turbulence closure in frontal zones and within the mixed layer. This suggests that simulated mixing processes in frontal zones are driven largely by . Near‐inertial alongshore wind stress amplitude is varied to identify a base case that maximizes the ratio of to in simulations with no prescribed horizontal mixing. We then test the sensitivity of the base case with three tracer advection schemes (MPDATA, U3HC4, and HSIMT) and conduct ensemble runs with perturbed bathymetry. Instability growth is evaluated using the volume‐integrated eddy kinetic energy and available potential energy . While all schemes have similar total mixing, the HSIMT simulations have over double the volume‐integrated and 20% less relative to other schemes, which suppresses the release of and reduces the by roughly 25%. This results in reduced isohaline variability and steeper isopycnals, evidence that enhanced suppresses instability growth. Differences in and between the MPDATA and U3HC4 simulations are marginal. However, the U3HC4 simulations have 25% more . Experiments with variable horizontal viscosity and diffusivity coefficients show that small amounts of prescribed horizontal mixing improve the representation of the ocean state for all advection schemes by reducing the and increasing the .

54 ENVIRONMENTAL SCIENCES↗

Second-order closure PBL model with new third-order moments: Comparison with LES data

This paper contains two parts. In the first part, a new set of diagnostic equations is derived for the third-order moments for a buoyancy-driven flow, by exact inversion of the prognostic equations for the third-order moment equations in the stationary case. The third-order moments exhibit a universal structure: they all are a linear combination of the derivatives of all the second-order moments, bar-w(exp 2), bar-w theta, bar-theta(exp 2), and bar-q(exp 2). Each term of the sum contains a turbulent diffusivity D(sub t), which also exhibits a universal structure of the form D(sub t) = a nu(sub t) + b bar-w theta. Since the sign of the convective flux changes depending on stable or unstable stratification, D(sub t) varies according to the type of stratification. Here nu(sub t) approximately equal to wl (l is a mixing length and w is an rms velocity) represents the 'mechanical' part, while the 'buoyancy' part is represented by the convective flux bar-w theta. The quantities a and b are functions of the variable N(sub tau)(exp 2), where N(exp 2) = g alpha derivative of Theta with respect to z and tau is the turbulence time scale. The new expressions for the third-order moments generalize those of Zeman and Lumley, which were subsequently adopted by Sun and Ogura, Chen and Cotton, and Finger and Schmidt in their treatments of the convective boundary layer. In the second part, the new expressions for the third-order moments are used to solve the ensemble average equations describing a purely convective boundary laye r heated from below at a constant rate. The computed second- and third-order moments are then compared with the corresponding Large Eddy Simulation (LES) results, most of which are obtained by running a new LES code, and part of which are taken from published results. The ensemble average results compare favorably with the LES data.

Canuto, V. M.↗

Evaluation of ACCMIP Outgoing Longwave Radiation from Tropospheric Ozone Using TES Satellite Observations.

We use simultaneous observations of tropospheric ozone and outgoing longwave radiation (OLR) sensitivity to tropospheric ozone from the Tropospheric Emission Spectrometer (TES) to evaluate model tropospheric ozone and its effect on OLR simulated by a suite of chemistry-climate models that participated in the Atmospheric Chemistry and Climate Model Intercomparison Project (ACCMIP). The ensemble mean of ACCMIP models show a persistent but modest tropospheric ozone low bias (5-20 ppb) in the Southern Hemisphere (SH) and modest high bias (5-10 ppb) in the Northern Hemisphere (NH) relative to TES ozone for 2005-2010. These ozone biases have a significant impact on the OLR. Using TES instantaneous radiative kernels (IRK), we show that the ACCMIP ensemble mean tropospheric ozone low bias leads up to 120mW/ sq. m OLR high bias locally but zonally compensating errors reduce the global OLR high bias to 39+/- 41mW/ sq. m relative to TES data. We show that there is a correlation (Sq. R = 0.59) between the magnitude of the ACCMIP OLR bias and the deviation of the ACCMIP preindustrial to present day (1750-2010) ozone radiative forcing (RF) from the ensemble ozone RF mean. However, this correlation is driven primarily by models whose absolute OLR bias from tropospheric ozone exceeds 100mW/ sq. m. Removing these models leads to a mean ozone radiative forcing of 394+/- 42mW/ sq. m. The mean is about the same and the standard deviation is about 30% lower than an ensemble ozone RF of 384 +/- 60mW/ sq. m derived from 14 of the 16 ACCMIP models reported in a companion ACCMIP study. These results point towards a profitable direction of combining satellite observations and chemistry-climate model simulations to reduce uncertainty in ozone radiative forcing.

troposphere↗

Coherent Eigenmodes in Homogeneous MHD Turbulence

The statistical mechanics of Fourier models of ideal, homogeneous, incompressible magnetohydrodynamic (MHD) turbulence is discussed, along with their relevance for dissipative magnetofluids. Although statistical theory predicts that Fourier coefficients of fluid velocity and magnetic field are zero-mean random variables, numerical simulations clearly show that certain coefficients have a non-zero mean value that can be very large compared to the associated standard deviation, i.e., we have coherent structure. We use eigenanalysis of the modal covariance matrices in the probability density function to explain this phenomena in terms of `broken ergodicity', which is defined to occur when dynamical behavior does not match ensemble predictions on very long time-scales. We provide examples from 2-D and 3-D magnetohydrodynamic simulations of homogeneous turbulence, and show new results from long-time simulations of MHD turbulence with and without a mean magnetic field

Shebalin, John V.↗

Sensitivity of Precipitation Processes to Microphysics and Resolution in a Cloud-Resolving Model

The Goddard Cumulus Ensemble (GCE) model is used to examine the impact of various microphysical schemes, and vertical and horizontal resolution on the development, intensity and rainfall associated with mesoscale convective systems, idealized hurricanes and an ensemble of clouds. The model variables include horizontal and vertical velocities, potential temperature, perturbation pressure, turbulent kinetic energy, and mixing ratios of all water phases (vapor, liquid, and ice). The major characteristics of the GCE model are the explicit representation of warm rain and ice microphysical processes, and their complex interactions with solar and infrared radiative transfer processes, and with surface processes. For idealized hurricane, an axisymmetric version of the GCE model was developed and used successfully to simulate the tropical cyclogenesis process using both a Rankin vortex and saturated air within a specified radius as initial conditions. For mesoscale convective systems, the 3-D version of the GCE model was used to simulate squall lines that developed in the western Pacific, eastern Atlantic and central US. For the cloud ensemble, the GCE model was integrated for several days in order to have good sampling of cloud statistics. In this paper, the sensitivities of hurricane intensity to various microphysical processes and model grid resolution will be examined.

Tao, Wei-Kuo↗

Sensitivity of Precipitation Processes of Microphysics and Resolution in a Cloud-Resolving Model

The Goddard Cumulus Ensemble (GCE) model is used to examine the impact of various microphysical schemes, and vertical and horizontal resolution ion the development, intensity and rainfall associated with mesoscale convective systems, idealized hurricanes and an ensemble f clouds. The model variables include horizontal and vertical velocities, potential temperatures, perturbation pressure, turbulent kinetic energy, and mixing ratios of all water phases (vapor, liquid, and ice). The major characteristics of the GCE model are the explicit representation of warm rain and ice microphysical processes, and their complex interactions with solar and infrared radiative transfer processes, and with surface processes. For idealized hurricane, an axisymmetric version of the GCE model was developed and used successfully to simulate the tropical cyclogenesis process using both a Rankin vortex and saturated air within a specified radius as initial conditions. For mesoscale convective systems, the 3-D version of the GCE model was used to simulated squall lines that developed in the western Pacific, eastern Atlantic and central US. For the cloud ensemble, the GCE model was integrated for several days in order to have good sampling of cloud statistics. In this paper, the sensitivities of hurricane intensity to various microphysical processes and model grid resolution will be examined. This will be mainly achieved by performing sensitivity tests using various horizontal (from 1- to 5-km) and vertical resolutions (from 20- to 200-m in the lower troposphere to 200- to 500-m in the middle and upper troposphere). Sensitivity tests using various microphysical schemes (warm rain only, and three ice with either graupel or hail) will also be performed. The thermodynamic and water budget associated with various types of precipitation systems will also be evaluated. The budgets will be calculated for different regions (i.e., convective and stratiform regions).

Tao, Wei-Kuo↗