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NASA NTRS2014

Toward Improved Land Surface Initialization in Support of Regional WRF Forecasts at the Kenya Meteorological Department

Flooding and drought are two key forecasting challenges for the Kenya Meteorological Department (KMD). Atmospheric processes leading to excessive precipitation and/or prolonged drought can be quite sensitive to the state of the land surface, which interacts with the boundary layer of the atmosphere providing a source of heat and moisture. The development and evolution of precipitation systems are affected by heat and moisture fluxes from the land surface within weakly-sheared environments, such as in the tropics and sub-tropics. These heat and moisture fluxes during the day can be strongly influenced by land cover, vegetation, and soil moisture content. Therefore, it is important to represent the land surface state as accurately as possible in numerical weather prediction models. Enhanced regional modeling capabilities have the potential to improve forecast guidance in support of daily operations and high-end events over east Africa. KMD currently runs a configuration of the Weather Research and Forecasting (WRF) model in real time to support its daily forecasting operations, invoking the Nonhydrostatic Mesoscale Model (NMM) dynamical core. They make use of the National Oceanic and Atmospheric Administration / National Weather Service Science and Training Resource Center's Environmental Modeling System (EMS) to manage and produce the WRF-NMM model runs on a 7-km regional grid over eastern Africa. Two organizations at the National Aeronautics and Space Administration Marshall Space Flight Center in Huntsville, AL, SERVIR and the Short-term Prediction Research and Transition (SPoRT) Center, have established a working partnership with KMD for enhancing its regional modeling capabilities. To accomplish this goal, SPoRT and SERVIR will provide experimental land surface initialization datasets and model verification capabilities to KMD. To produce a land-surface initialization more consistent with the resolution of the KMD-WRF runs, the NASA Land Information System (LIS) will be run at a comparable resolution to provide real-time, daily soil initialization data in place of interpolated Global Forecast System soil moisture and temperature data. Additionally, real-time green vegetation fraction data from the Visible Infrared Imaging Radiometer Suite will be incorporated into the KMD-WRF runs, once it becomes publicly available from the National Environmental Satellite Data and Information Service. Finally, model verification capabilities will be transitioned to KMD using the Model Evaluation Tools (MET) package, in order to quantify possible improvements in simulated temperature, moisture and precipitation resulting from the experimental land surface initialization. The transition of these MET tools will enable KMD to monitor model forecast accuracy in near real time. This presentation will highlight preliminary verification results of WRF runs over east Africa using the LIS land surface initialization.

Case. Jonathan↗

NASA NTRS1995

An airborne study of microwave surface sensing and boundary layer heat and moisture fluxes for FIFE

The objectives of this work were to perform imaging radar and scatterometer measurements over the Konza Prairie as a part of the First International land surface climatology project Field Experiments (EIFE) and to develop an mm-wave radiometer and the data acquisition system for this radiometer. We collected imaging radar data with the University of Kansas Side-Looking Airborne Radar (SLAR) operating at 9.375 GHz and scatterometer data with a helicopter-mounted scatterometer at 5.3 and 9.6 GHz. We also developed a 35-GHz null-balancing radiometer and data acquisition system. Although radar images showed good delineation of various features of the FIFE site, the data were not useful for quantitative analysis for extracting soil moisture information because of day-to-day changes in the system transfer characteristics. Our scatterometer results show that both C and X bands are sensitive to soil moisture variations over grass-covered soils. Scattering coefficients near vertical are about 4 dB lower for unburned areas because of the presence of a thatch layer, in comparison with those for burned areas. The results of the research have been documented in reports, oral presentations, and published papers.

Gogineni, S. P.↗

NASA NTRS2014

Toward Improved Land Surface Initialization in Support of Regional WRF Forecasts at the Kenya Meteorological Service (KMS)

Flooding and drought are two key forecasting challenges for the Kenya Meteorological Service (KMS). Atmospheric processes leading to excessive precipitation and/or prolonged drought can be quite sensitive to the state of the land surface, which interacts with the planetary boundary layer (PBL) of the atmosphere providing a source of heat and moisture. The development and evolution of precipitation systems are affected by heat and moisture fluxes from the land surface, particularly within weakly-sheared environments such as in the tropics and sub-tropics. These heat and moisture fluxes during the day can be strongly influenced by land cover, vegetation, and soil moisture content. Therefore, it is important to represent the land surface state as accurately as possible in land surface and numerical weather prediction (NWP) models. Enhanced regional modeling capabilities have the potential to improve forecast guidance in support of daily operations and high-impact weather over eastern Africa. KMS currently runs a configuration of the Weather Research and Forecasting (WRF) NWP model in real time to support its daily forecasting operations, making use of the NOAA/National Weather Service (NWS) Science and Training Resource Center's Environmental Modeling System (EMS) to manage and produce the KMS-WRF runs on a regional grid over eastern Africa. Two organizations at the NASA Marshall Space Flight Center in Huntsville, AL, SERVIR and the Shortterm Prediction Research and Transition (SPoRT) Center, have established a working partnership with KMS for enhancing its regional modeling capabilities through new datasets and tools. To accomplish this goal, SPoRT and SERVIR is providing enhanced, experimental land surface initialization datasets and model verification capabilities to KMS as part of this collaboration. To produce a land-surface initialization more consistent with the resolution of the KMS-WRF runs, the NASA Land Information System (LIS) is run at a comparable resolution to provide real-time, daily soil initialization data in place of data interpolated from the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS) model soil moisture and temperature fields. Additionally, realtime green vegetation fraction (GVF) data from the Visible Infrared Imaging Radiometer Suite (VIIRS) on the Suomi National Polar-orbiting Partnership (Suomi- NPP) satellite will be incorporated into the KMS-WRF runs, once it becomes publicly available from the National Environmental Satellite Data and Information Service (NESDIS). Finally, model verification capabilities will be transitioned to KMS using the Model Evaluation Tools (MET; Brown et al. 2009) package in conjunction with a dynamic scripting package developed by SPoRT (Zavodsky et al. 2014), to help quantify possible improvements in simulated temperature, moisture and precipitation resulting from the experimental land surface initialization. Furthermore, the transition of these MET tools will enable KMS to monitor model forecast accuracy in near real time. This paper presents preliminary efforts to improve land surface model initialization over eastern Africa in support of operations at KMS. The remainder of this extended abstract is organized as follows: The collaborating organizations involved in the project are described in Section 2; background information on LIS and the configuration for eastern Africa is presented in Section 3; the WRF configuration used in this modeling experiment is described in Section 4; sample experimental WRF output with and without LIS initialization data are given in Section 5; a summary is given in Section 6 followed by acknowledgements and references.

Case, Johnathan L.↗

NASA NTRS2010

Mechanisms of Diurnal Precipitation over the United States Great Plains: A Cloud-Resolving Model Simulation

The mechanisms of summertime diurnal precipitation in the US Great Plains were examined with the two-dimensional (2D) Goddard Cumulus Ensemble (GCE) cloud-resolving model (CRM). The model was constrained by the observed large-scale background state and surface flux derived from the Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) Program s Intensive Observing Period (IOP) data at the Southern Great Plains (SGP). The model, when continuously-forced by realistic surface flux and large-scale advection, simulates reasonably well the temporal evolution of the observed rainfall episodes, particularly for the strongly forced precipitation events. However, the model exhibits a deficiency for the weakly forced events driven by diurnal convection. Additional tests were run with the GCE model in order to discriminate between the mechanisms that determine daytime and nighttime convection. In these tests, the model was constrained with the same repeating diurnal variation in the large-scale advection and/or surface flux. The results indicate that it is primarily the surface heat and moisture flux that is responsible for the development of deep convection in the afternoon, whereas the large-scale upward motion and associated moisture advection play an important role in preconditioning nocturnal convection. In the nighttime, high clouds are continuously built up through their interaction and feedback with long-wave radiation, eventually initiating deep convection from the boundary layer. Without these upper-level destabilization processes, the model tends to produce only daytime convection in response to boundary layer heating. This study suggests that the correct simulation of the diurnal variation in precipitation requires that the free-atmospheric destabilization mechanisms resolved in the CRM simulation must be adequately parameterized in current general circulation models (GCMs) many of which are overly sensitive to the parameterized boundary layer heating.

Lee, M.-I.↗

NASA NTRS1997

The Aggregate Description of Semi-Arid Vegetation with Precipitation-Generated Soil Moisture Heterogeneity

Meteorological measurements in the Walnut Gulch catchment in Arizona were used to synthesize a distributed, hourly-average time series of data across a 26.9 by 12.5 km area with a grid resolution of 480 m for a continuous 18-month period which included two seasons of monsoonal rainfall. Coupled surface-atmosphere model runs established the acceptability (for modelling purposes) of assuming uniformity in all meteorological variables other than rainfall. Rainfall was interpolated onto the grid from an array of 82 recording rain gauges. These meteorological data were used as forcing variables for an equivalent array of stand-alone Biosphere-Atmosphere Transfer Scheme (BATS) models to describe the evolution of soil moisture and surface energy fluxes in response to the prevalent, heterogeneous pattern of convective precipitation. The calculated area-average behaviour was compared with that given by a single aggregate BATS simulation forced with area-average meteorological data. Heterogeneous rainfall gives rise to significant but partly compensating differences in the transpiration and the intercepted rainfall components of total evaporation during rain storms. However, the calculated area-average surface energy fluxes given by the two simulations in rain-free conditions with strong heterogeneity in soil moisture were always close to identical, a result which is independent of whether default or site-specific vegetation and soil parameters were used. Because the spatial variability in soil moisture throughout the catchment has the same order of magnitude as the amount of rain failing in a typical convective storm (commonly 10% of the vegetation's root zone saturation) in a semi-arid environment, non-linearitv in the relationship between transpiration and the soil moisture available to the vegetation has limited influence on area-average surface fluxes.

White, Cary B.↗

DOE OSTI2026

Observed Land Surface Influence on Atmospheric Heat and Moisture Profiles During Interstorms

Land-atmospheric (L-A) feedbacks have historically been studied using models whose structure and parameterizations influence outcomes and insights. The representation of L-A feedbacks based on observations alone remains an ongoing challenge for understanding boundary layer development and precipitation. To address this gap, we use ground-based passive remote sensing and in-situ observations to present an analysis of the atmosphere during 103 interstorm soil moisture drydown events spanning nine warm seasons (2016–2024) in the U.S. Southern Great Plains region. By separating events based on local L-A coupling signals and characterizing the profiles of atmospheric heat and moisture to surface energy flux behavior, we investigate the physical mechanisms linking land surface processes to boundary layer development. We find that during interstorm drydowns, the atmospheric column follows a consistent pattern: moisture increases within the boundary layer, peaks near its top, and declines rapidly above, while warming occurs through the depth. Drydowns that shift toward evaporation produce stronger and deeper thermodynamic responses than cases dominated by sensible heating, which are weaker and shallower. Additionally, moisture is accumulated faster within the boundary layer during shorter drydowns, with longer drydowns representing slower, moisture-limited growth. Drydowns with wetter initial soil moisture will sustain stronger moistening within and above the boundary layer, accelerating buoyancy growth and convective potential toward the next storm. These results provide observational evidence linking surface flux evolution to boundary layer thermodynamics and offer a process-level benchmark for evaluating coupled L-A representations in models and demonstrating the influence of soil moisture on short-term weather forecasting skill.

Zhang, M. S. [Massachusetts Inst. of Technology (M↗

NASA NTRS1993

Session on coupled land surface/hydrological/atmospheric models

The current model capabilities in the context of land surface interactions with the atmosphere include only one-dimensional characteristics of vegetation and soil surface heat, moisture, momentum, and selected other trace gas fluxes (e.g., CO2). The influence of spatially coherent fluxes that result from landscape heterogeneity were not included. Valuable representations of several aspects of the landscape pattern currently exist. These include digital elevation data and measures of the leaf area index (i.e., Normalized Difference Vegetation Index (NDVI) from Advanced Very High Resolution Radiometer (AVHRR) data). A major deficiency, however, is the lack of an ability to sample spatially representative shallow and (especially) deep soil moisture. Numerous mesoscale modeling and observed studies demonstrated the sensitivity of planetary boundary layer structure and deep convection to the magnitude of the surface moisture flux.

Pielke, Roger↗

DOE OSTI2025

Improved Representations of Land‐Atmosphere Interactions Over the Continental U.S. Through Dynamic Root Modeling

Previous studies have identified the oversimplified root system representation as a key factor leading to inaccuracies in vegetation‐atmosphere feedbacks. In this study, a dynamic root water uptake scheme in the Noah‐MP land surface model has been coupled to the Weather Research and Forecasting (WRF) model to investigate its impact on the surface hydroclimate variables and land‐atmosphere interactions. To evaluate the impact of the dynamic root, two coupled simulations were conducted, one with the dynamic root water uptake scheme (DynRt) and one with the static root water uptake scheme (StcRt), which is based on the default root representation in Noah‐MP, with slight modifications, primarily in vegetation‐related parameters. Both DynRt and StcRt simulations were conducted with a small ensemble of three members to account for variations in physical parameterizations, initial and boundary forcing and model setup. When compared with reference data sets, the DynRt simulations show improved results than the StcRt simulations, reducing biases in the simulated leaf area index, surface energy fluxes, soil moisture and precipitation. Two different mechanisms through which roots affect land‐atmosphere coupling have been identified. Over the transitional climate zone between the dry and wet climate, the dynamic root scheme affects surface climate and land‐atmosphere coupling mainly through changes in soil moisture through hydraulic redistribution by plant root system. Over the energy‐limited mesic zone, the dynamic root affects regional land‐atmosphere coupling mainly through changes in carbon allocation. This work highlights the importance of dynamic root representation in improving vegetation‐atmosphere simulations by enhancing predictions of water, energy, and carbon fluxes.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS2021

The Great Plains Irrigation Experiment (GRAINEX)

Extensive expansion in irrigated agriculture has taken place over the last half century. Due to increased irrigation and resultant land-use–land-cover change, the central United States has seen a decrease in temperature and changes in precipitation during the second half of the twentieth century. To investigate the impacts of widespread commencement of irrigation at the beginning of the growing season and continued irrigation throughout the summer on local and regional weather, the Great Plains Irrigation Experiment (GRAINEX) was conducted in the spring and summer of 2018 in southeastern Nebraska. GRAINEX consisted of two 15-day intensive observation periods. Observational platforms from multiple agencies and universities were deployed to investigate the role of irrigation in surface moisture content, heat fluxes, diurnal boundary layer evolution, and local precipitation. This article provides an overview of the data collected and an analysis of the role of irrigation in land–atmosphere interactions on time scales from the seasonal to the diurnal. The analysis shows that a clear irrigation signal was apparent during the peak growing season in mid-July. This paper shows the strong impact of irrigation on surface fluxes, near-surface temperature and humidity, and boundary layer growth and decay.

Atmosphere-land interaction↗

NASA NTRS1989

An airborne study of microwave surface sensing and boundary layer heat and moisture fluxes for Project FIFE

This is the third semiannual status report describing the University of Wyoming participation in project FIFE, under NASA Grant NAG 5-913. The reader is referred to the first and second semiannual status reports for this grant for descriptions of University of Wyoming participation in the field operations of 1987. This report is concerned with continuing analysis of field data. Most of the instrument calibration effort has concentrated on interpretation and use of the high-rate water vapor measurements with the University of Wyoming Lyman-alpha device. In addition, fluxes of water vapor have been corrected for sensible heat flux, and all the high-rate (10 Hz for the Wyoming King Air) data have been filtered before the flux calculations. For each flight the Lyman-alpha data (10 Hz) are calibrated against the Cambridge dew-point hygrometer (1 Hz), by using data from the take-off sounding. The reasoning here is that the takeoff sounding (held to 500 fpm rate-of-climb specifically for the calibration) covers the widest range of humidity conditions of any sequence of the flight. For each takeoff sounding, then, the Lyman-alpha data, averaged to 1 Hz, are matched to the dewpoint data with a second-order least-squares fit. The resulting quadratic equation is then used in all subsequent analyses to convert the Lyman-alpha voltages to vapor desities for that specific flight. An alternative method is to use the range of fluctuation for the same two devices from horizontal passes to establish the calibration function. However, since the Lyman-alpha data are used only in the flux calculations, i.e., only the fluctuation and not the absolute values are used, this method is thought unnecessary.

Kelly, Robert D.↗

NASA NTRS2013

Impacts of Soil-aquifer Heat and Water Fluxes on Simulated Global Climate

Climate models have traditionally only represented heat and water fluxes within relatively shallow soil layers, but there is increasing interest in the possible role of heat and water exchanges with the deeper subsurface. Here, we integrate an idealized 50m deep aquifer into the land surface module of the GISS ModelE general circulation model to test the influence of aquifer-soil moisture and heat exchanges on climate variables. We evaluate the impact on the modeled climate of aquifer-soil heat and water fluxes separately, as well as in combination. The addition of the aquifer to ModelE has limited impact on annual-mean climate, with little change in global mean land temperature, precipitation, or evaporation. The seasonal amplitude of deep soil temperature is strongly damped by the soil-aquifer heat flux. This not only improves the model representation of permafrost area but propagates to the surface, resulting in an increase in the seasonal amplitude of surface air temperature of >1K in the Arctic. The soil-aquifer water and heat fluxes both slightly decrease interannual variability in soil moisture and in landsurface temperature, and decrease the soil moisture memory of the land surface on seasonal to annual timescales. The results of this experiment suggest that deepening the modeled land surface, compared to modeling only a shallower soil column with a no-flux bottom boundary condition, has limited impact on mean climate but does affect seasonality and interannual persistence.

soil↗

DOE OSTI2021

Ocean Surface Flux Algorithm Effects on Earth System Model Energy and Water Cycles

Earth system models parameterize ocean surface fluxes of heat, moisture, and momentum with empirical bulk flux algorithms, which introduce biases and uncertainties into simulations. We investigate the atmosphere and ocean model sensitivity to algorithm choice in the Energy Exascale Earth System Model (E3SM). Flux differences between algorithms are larger in atmosphere simulations (where wind speeds can vary) than ocean simulations (where wind speeds are fixed by forcing data). Surface flux changes lead to global scale changes in the energy and water cycles, notably including ocean heat uptake and global mean precipitation rates. Compared to the control algorithm, both COARE and University of Arizona (UA) algorithms reduce global mean precipitation and top of atmosphere radiative biases. Further, UA may slightly reduce biases in ocean meridional heat transport. We speculate that changes seen here, especially in the ocean, could be even larger in coupled simulations.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1988

Use of LANDSAT images of vegetation cover to estimate effective hydraulic properties of soils

The estimation of the spatially variable surface moisture and heat fluxes of natural, semivegetated landscapes is difficult due to the highly random nature of the vegetation (e.g., plant species, density, and stress) and the soil (e.g., moisture content, and soil hydraulic conductivity). The solution to that problem lies, in part, in the use of satellite remotely sensed data, and in the preparation of those data in terms of the physical properties of the plant and soil. The work was focused on the development and testing of a stochastic geometric canopy-soil reflectance model, which can be applied to the physically-based interpretation of LANDSAT images. The model conceptualizes the landscape as a stochastic surface with bulk plant and soil reflective properties. The model is particularly suited for regional scale investigations where the quantification of the bulk landscape properties, such as fractional vegetation cover, is important on a pixel by pixel basis. A summary of the theoretical analysis and the preliminary testing of the model with actual aerial radiometric data is provided.

Eagleson, Peter S.↗

DOE OSTI2022

An Evaluation of Tropical Cyclone Rainfall Structures in the HighResMIP Simulations against Satellite Observations

This study evaluates tropical cyclone (TC) rainfall structures in the CMIP6 HighResMIP global climate model (GCM) simulations against satellite rainfall retrievals. Here, we specifically focus on TCs within the deep tropics (25°S–25°N). Analysis of TC rain rate composites indicates that in comparison to the satellite observations at the same intensity, many HighResMIP simulations tend to overproduce rain rates around TCs, in terms of both maximum rain rate magnitude and area-averaged rain rates. In addition, as model horizontal resolution increases, the magnitude of the peak rain rate appears to increase. However, the area-averaged rain rates decrease with increasing horizontal resolution, partly due to the TC eyewall being located closer to the TC center, thus occupying a smaller area and contributing less to the area-averaged rain rates. The effect of ocean coupling is to lower the TC rain rates, bringing them closer to the satellite observations, due to reduced horizontal moisture flux convergence and surface latent heat flux beneath TCs. Examination of horizontal rain rate distributions indicates that vertical wind shear–induced rainfall asymmetries in HighResMIP-simulated TCs are qualitatively consistent with the observations. In addition, a positive relationship is observed between the area-averaged inner-core rainfall and TC intensification likelihoods across the HighResMIP simulations, as GCM simulations producing stronger TCs more frequently have the greater rainfall close to the center, in agreement with previous theoretical and GCM simulation results.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Toward Hybrid Physics-Machine Learning to improve Land Surface Model predictions

A critical challenge for Land Surface Models (LSMs) is to simulate processes at the surface and the subsurface and their feedbacks to the atmosphere. Even using the same climate forcings, different LSMs predict different surface fluxes and soil moisture conditions due to differences in the formulations of individual processes, parameterizations, and representation of spatial heterogeneity. Ultimately, these differences contribute to the LSM prediction errors and uncertainty. This research seeks to address this challenge by coupling physics-based modeling with state-of-the-art machine learning (ML) techniques to describe complex physical and biogeochemical processes and narrow the gap between model predictions and observations.

58 GEOSCIENCES↗

DOE OSTI2024

Global 1 km land surface parameters for kilometer-scale Earth system modeling

Earth system models (ESMs) are progressively advancing towards the kilometer scale (“k-scale”). However, the surface parameters for land surface models (LSMs) within ESMs running at the k-scale are typically derived from coarse-resolution and outdated datasets. This study aims to develop a new set of global land surface parameters with a resolution of 1 km for multiple years from 2001 to 2020, utilizing the latest and most accurate available datasets. Specifically, the datasets consist of parameters related to land use and land cover, vegetation, soil, and topography. Differences between the newly developed 1 km land surface parameters and conventional parameters emphasize their potential for higher accuracy due to the incorporation of the most advanced and latest data sources. To demonstrate the capability of these new parameters, we conducted 1 km resolution simulations using the E3SM Land Model version 2 (ELM2) over the contiguous United States. Our results demonstrate that land surface parameters contribute to significant spatial heterogeneity in ELM2 simulations of soil moisture, latent heat, emitted longwave radiation, and absorbed shortwave radiation. On average, about 31 % to 54 % of spatial information is lost by upscaling the 1 km ELM2 simulations to a 12 km resolution. Using eXplainable Machine Learning (XML) methods, the influential factors driving the spatial variability and spatial information loss of ELM2 simulations were identified, highlighting the substantial impact of the spatial variability and information loss of various land surface parameters, as well as the mean climate conditions. The comparison against four benchmark datasets indicates that ELM generally performs well in simulating soil moisture and surface energy fluxes. The new land surface parameters are tailored to meet the emerging needs of k-scale LSM and ESM modeling with significant implications for advancing our understanding of water, carbon, and energy cycles under global change. The 1 km land surface parameters are publicly available at https://doi.org/10.5281/zenodo.10815170 (Li et al., 2024).

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1977

The multiple nesting of mesoscale, submesoscale, and nonhydrostatic microscale numerical models for a case of mesocyclogenesis and severe storm development

A fundamental approach involving the integration of two different numerical models over several different spatial mesh lengths is considered. Model I is a hydrostatic, incompressible, and adiabatic primitive (Newtonian) equation set with no boundary layer forcing, i.e., surface heat or moisture flux or external frictional stresses. Typically this model is initialized from conventional radiosonde data and run for a 12-15 hour forecast period over a 40 km horizontal mesh. If phase I of the tornado developmental processes develops, the data is 'frozen' in space and time, interpolated to a 20 km mesh, and integrated for a shorter time period (3 hours) with a shorter time step. Model II is a nonhydrostatic compressible 18-level complement of model I. If model II indicates the development of the intensification of phase V dynamical processes, the data is frozen and interpolated to finer mesh lengths until the tornadic circulation system is approximated

Kaplan, M. L.↗

NASA NTRS1981

A variable vertical resolution weather model with an explicitly resolved planetary boundary layer

A version of the fourth order weather model incorporating surface wind stress data from SEASAT A scatterometer observations is presented. The Monin-Obukhov similarity theory is used to relate winds at the top of the surface layer to surface wind stress. A reasonable approximation of surface fluxes of heat, moisture, and momentum are obtainable using this method. A Richardson number adjustment scheme based on the ideas of Chang is used to allow for turbulence effects.

Helfand, H. M.↗