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At least 1,189 records · Page 66

Energy balance in the DSSAT-CSM-CROPGRO model

One potential way to improve crop growth models is for the models to predict energy balance and evapotranspiration (ET) from first principles, thus serving as a check on “engineered” ET methodology. In this paper, we present new implementations and the results of an energy balance model (EBL) developed by Jagtap and Jones (1989) and then implemented in DSSAT’s CROPGRO (CG-EBL) model by Pickering et al. (1995) as a linked energy balance-photosynthesis model that has not been field-tested until now. The energy balance code computes evapotranspiration and other energy balance components, as well as a canopy air temperature, based on three sources (sunlit leaves, shaded leaves, soil surface). Model performance was evaluated with measured biomass and energy fluxes from two sites in Nebraska, namely, the US-Ne2 irrigated maize-soybean rotation field and the US-Ne3 rainfed maize-soybean rotation field, which are part of the Ameriflux eddy covariance network (htt ps://ameriflux.lbl.gov/sites). After implementing new aerodynamic resistances and the stomatal conductance model of the Ball–Berry–Leuning, crop growth, evapotranspiration and soil temperature were simulated well by the EBL model. The EBL improved ET predictions slightly over the often-used FAO56 method [Penman–Monteith (Allen et al., 1998)] for 4 of the 5 years evaluated for both irrigated and rainfed conditions. Further, a significant improvement was achieved using EBL for the simulation of soil temperature at the various depths compared to STEMP, the original subroutine in DSSAT for simulating soil temperature. Compared to the other available DSSAT methods, the EBL explicitly simulates the impacts of crop morphology, physiology and management on the crop’s environment and energy and mass exchange, which in turn directly affect the water use and irrigation requirements, phenology, photosynthesis, growth, sterility, and yield of the crop.

54 ENVIRONMENTAL SCIENCES↗

Solar position confounds the relationship between ecosystem function and vegetation indices derived from solar and photosynthetically active radiation fluxes

Vegetation indices derived from solar and photosynthetically active radiation (PAR) sensors (i.e. radiation derived) have been under-utilized in inferring ecosystem function, despite measurement capability at hundreds of sites. This under-utilization may be attributed to reported mismatches among the seasonality of radiation- and satellite-derived vegetation indices and canopy photosynthesis; herein referred to as measurement biases. In this work, biases in radiation derived reflectance and vegetation indices were assessed using a decadal record of satellite and ground based spectroradiometer data, ecosystem phenology and CO 2 fluxes, and radiation derived vegetation indices (i.e. the Normalized Difference Vegetation Index [NDVI], the two band Enhanced Vegetation Index [EVI2]) from a high latitude tundra site (i.e. Imnaviat). At Imnaviat, we found poor correspondence between the three types of reflectance and vegetation indices, especially during the latter part of the growing season. Radiation derived vegetation indices resulted in incorrect estimates of phenological timing of up to a month and poor relationships with canopy photosynthesis (i.e. Gross Ecosystem Exchange (GEE)). These mismatches were attributed to solar position (i.e. solar zenith and azimuth angle) and a method, based on the diel visible and near-infrared albedo variation, was developed to improve the performance of the vegetation indices. The ability of radiation derived vegetation indices to infer GEE and phenological dates drastically improved once radiation derived vegetation indices were corrected for solar position associated biases at Imnaviat. Moreover, radiation derived vegetation indices became better aligned with MODerate resolution Imaging Spectroradiometer (MODIS) satellite estimates after solar position associated biases were corrected at Imnaviat and at 25 Fluxnet sites (~90 site years) across North America. Corrections developed here provide a way forward in understanding daily ecosystem function or filling large gaps in eddy covariance data at a significant number of Fluxnet sites.

54 ENVIRONMENTAL SCIENCES↗

Quantifying carbon budget, crop yields and their responses to environmental variability using the ecosys model for U.S. Midwestern agroecosystems

As one of the major agricultural production areas in the world, the United States (U.S.) Midwest plays a vital role in the global food supply and agricultural ecosystem services. Although significant efforts have been made in modeling the carbon cycle dynamics over this area, large uncertainty still exists in the previous simulations in terms of reproducing individual components of the carbon cycle and their responses to environmental variability. Here we evaluated the performance of an advanced agroecosystem model, ecosys, in simulating carbon budgets over the U.S. Midwest, considering both the magnitude of carbon flux/yield and its response to environmental (climate and soil) variability. We conducted model simulations and evaluations at 7 cropland eddy-covariance sites as well as over 293 counties of Illinois, Indiana, and Iowa in the U.S. Midwest. The site-level simulations showed that ecosys captured both the magnitude and seasonal patterns of carbon fluxes (i.e., net ecosystem carbon exchange (NEE), ecosystem gross primary production (GPP), and ecosystem respiration (Reco)), leaf area index (LAI), and dynamic plant carbon allocation processes, with R2 equal to 0.92, 0.87, 0.87, and 0.78 for GPP, NEE, Reco, and LAI, respectively across all the sites compared with the observations. For regional scale simulations, ecosys reproduced the spatial distribution and interannual variability of corn and soybean yields with the constraints of observed yields and a new remotely sensed GPP product, with R2 of multi-year averaged simulated and observed yield equal 0.83 and 0.80 for corn and soybean, respectively. The simulated responses of carbon cycle dynamics to environmental variability were consistent with that from the empirical observations at both site and regional scales. Furthermore, our results demonstrated the applicability of ecosys in simulating the carbon cycle and soil carbon dynamics of the U.S. Midwestern agroecosystems under different climate and soil conditions.

54 ENVIRONMENTAL SCIENCES↗

Aboveground and belowground contributions to ecosystem respiration in a temperate deciduous forest

In this study, we developed a three-way carbon dioxide (CO 2 ) flux-partitioning algorithm that separates net ecosystem exchange (NEE) into aboveground plant respiration (R above ), belowground root and soil respiration (R below ), and gross primary production (GPP). We applied this algorithm to a coupled dataset of continuous chamber-measured soil respiration and eddy covariance (EC)-measured NEE of CO 2 in an oak-hickory (Quercus-Carya) deciduous broadleaf forest from 2006 to 2015. We found that on annual time scale, R below dominated over R above with the former accounting for 66.9–86.4% and the latter 13.6–33.1%, of the total ecosystem respiration (R eco ). The ratio of R below to R above varied seasonally, ranging from 1.77 to 7.25 in growing season, and 1.02 to 4.57 in non-growing season. The temperature sensitivity (E 0 ) of R below was significantly higher than that of R above , and E 0 of R eco responded differently to air and soil temperature. Over the whole study period, annual mean R above , R below , and GPP were 243, 806, and 1170 g C m –2 , respectively, with annual R eco accounting for 89.6% of GPP, of which 68.8% was lost as R below and 20.8% lost as R above , and leaving only 10% of the carbon fixation in ecosystems. Furthermore, these estimates, however, did not consider potential light inhibition of leaf respiration. If we accept the presence of light inhibition, then the daytime three-way partitioning method would underestimate annual R above by 20.4% whereas the nighttime method would overestimate R above by 23.9% and GPP by 4.7%, compared with estimates accounting for light inhibition in leaves.

54 ENVIRONMENTAL SCIENCES↗

Drought-induced decoupling between carbon uptake and tree growth impacts forest carbon turnover time

The ability of forests to withstand, and recover from, acute drought stress is a critical uncertainty regarding the impacts of climate change on the terrestrial carbon (C) cycle, but it is unclear how drought responses scale from individual trees to whole forests. Here, we assembled a dataset of tree-ring chronologies co-located within the footprint of eddy covariance towers across North America and Europe, with the aim of quantifying the sensitivity of tree radial growth versus gross primary productivity (GPP) during and following drought. We found that drought induced a large decoupling across C cycle processes, whereby GPP was relatively resistant to water stress despite large reductions in tree-ring widths. This decoupling also occurred in the year following drought (i.e., a ‘drought legacy effect’), and was similar in magnitude in response to both summer and winter droughts. By modeling whole-forest C turnover time, we show that a radial growth-GPP decoupling has important ramifications for the forest C cycle, especially if the C not used to support radial growth is instead allocated towards pools with short residence times. Furthermore, our results demonstrate that quantifications of drought impacts that rely solely on C uptake are missing this fundamental pathway through which drought alters the forest C cycle and the resulting feedbacks to the climate system.

54 ENVIRONMENTAL SCIENCES↗

Dryness controls temperature-optimized gross primary productivity across vegetation types

Temperature response of gross primary productivity (GPP) is a well-known property of ecosystem, but GPP at the optimum temperature (GPP_T opt ) has not been fully discussed. Our understanding of how GPP_T opt responds to warming and water availability is highly limited. Here, in this study, we analyzed data at 326 globally distributed eddy covariance sites (79°N-37°S), to identify controlling factors of GPP_T opt . Although GPP_T opt was significantly influenced by soil moisture, global solar radiation, mean annual temperature, and vapor pressure deficit in a non-linear pattern (R 2 = 0.47), the direction and magnitude of these climate variables’ effects on GPP_T opt depend on the dryness index (DI), a ratio of potential evapotranspiration to precipitation. The spatial pattern showed that soil moisture did not affect GPP_T opt across energy-limited sites with DI < 1 while dominated GPP_T opt across water-limited sites with DI >1. The temporal pattern showed that GPP_T opt was lowered by warming or low precipitation in water-limited sites while energy-limited sites tended to maintain a stable GPP_T opt regardless of changes in air temperature. Vegetation types in humid climates tended to have higher GPP_T opt and were more likely to benefit from a warmer climate since it was not restricted by water conditions. This study highlights that the response of GPP_T opt to global warming depends on the dryness conditions, which explains the nonlinear control of water and temperature over GPP_T opt . Our finding is essential to realistic prediction of terrestrial carbon uptake under future climate and vegetation conditions.

54 ENVIRONMENTAL SCIENCES↗

Fine resolution remote sensing spectra improves estimates of gross primary production of croplands

Gross primary production (GPP) is a fundamental measure of the terrestrial carbon cycle critical to our understanding of ecosystem function under the changing climate and land use. Remote sensing enables access to continuous spatial coverage, but remains challenged in heterogeneous croplands. Coarse resolution products, like MOD17A (500 m), may aggregate fragmented land cover types commonly found in heavily managed landscapes and misrepresent their respective contribution to carbon production. Consequently, this study demonstrates the capability of fine-resolution imagery (20-30 m) and available red-edge vegetation indices to characterize GPP across seven Midwest cropping systems. Four sites were established on a 22-year-old USDA Conservation Reserve Program (CRP); and the other three on land conventionally farmed with corn-soybean-wheat rotation (AGR). We compare in situ GPP estimates from eddy-covariance towers with ten satellite models: eight variants of the vegetation photosynthesis models (VPM), of which five include a red-edge vegetation index, as well as conventional products Landsat CONUS GPP (30 m) and MOD17A2H V6 (500 m). Daily and cumulative fine-resolution imagery integrated within VPM generally agreed with tower-based GPP in heterogeneous landscapes more than those from MODIS 500 m VPM or conventional GPP products from MOD17AH V6 or Landsat 8 CONUS. Here, replacing EVI2 with red-edge indices NDRE2, NDRE1, and MTCI in Sentinel 2 VPMs notably improved explanation of variance and estimation of cumulative GPP. While existing methods using MODIS- and Landsat-derived GPP are important baselines for regional and global studies, future research may benefit from the higher spatial, temporal, and radiometric resolution.

54 ENVIRONMENTAL SCIENCES↗

Quantifying the drivers of ecosystem fluxes and water potential across the soil-plant-atmosphere continuum in an arid woodland

Dryland ecosystems occupy a vast swath of the terrestrial land surface and exert a sizeable impact on the cycling of carbon and water globally. These biomes are characterized by tightly coupled carbon and water cycles that respond rapidly to transient pulses in water availability. However, there exist many mechanistic uncertainties regarding the environmental drivers of, and linkages between, plant and ecosystem processes. Thus, drylands are often poorly represented in many vegetation and land surface models. An enhanced understanding of dryland ecosystem function is limited by the lack of long-term, co-located, and frequent measurements of plant and ecosystem processes. At a piñon-juniper woodland in southeastern Utah, USA, we collected a continuous dataset of meteorological conditions, soil water potential from surface to bedrock, tree water potential, and ecosystem carbon and water fluxes from eddy covariance. We found that predawn and midday tree water potential and daily ecosystem fluxes were highly sensitive to fluctuations in soil water availability, particularly in shallower layers, and that daytime variability in atmospheric drivers only loosely controlled these processes. The strong connections between shallow soil water potential, tree water potential, and ecosystem fluxes occurred because of the dominant role of precipitation pulses in driving vegetation activity, as even small pulses of moisture stimulated shallow soil water potential, tree water potential, and evapotranspiration for between 1 and 2 weeks. Carbon fluxes (net ecosystem exchange and gross primary productivity) were sensitive to precipitation pulses for longer, up to 3 weeks. Our results highlight that improved monitoring and sensing of shallow soil moisture can greatly enhance our understanding of dryland ecosystem function. Here, a better mechanistic understanding of the impacts of precipitation pulses is also needed to improve vegetation modeling of dryland ecosystems.

54 ENVIRONMENTAL SCIENCES↗

Remote sensing of seasonal variation of LAI and fAPAR in a deciduous broadleaf forest

Climate change is affecting the phenology of terrestrial ecosystems. In deciduous forests, phenology in leaf area index (LAI) is the primary driver of seasonal variation in the fraction of absorbed photosynthetically active radiation (fAPAR), which drives photosynthesis. Remote sensing has been widely used to estimate LAI and fAPAR. However, while many studies have examined both empirical and model-based relationships among LAI, fAPAR, and spectral vegetation indices (SVI) from remote sensing, few studies have systematically and empirically examined how relationships among these variables change over the growing season. In this study, we examine how and why seasonal-scale covariation differs among time series of remotely sensed SVIs and both LAI and fAPAR based on current understanding and theory. To do this we use newly available remote sensing data sets in combination with time series of in-situ measurements and a canopy radiative transfer model to analyze how seasonal variation in canopy and environmental conditions affect relationships among remotely sensed SVIs, LAI, and fAPAR at a temperate deciduous forest site in central Massachusetts. Our results show that accounting for seasonal variation in canopy shadowing, which is driven by variation in solar zenith angle, improved remote sensing-based estimates of LAI, fAPAR, and daily total APAR. Specifically, we show that the phenology of SVIs is strongly influenced by seasonal variation in near infrared (NIR) reflectance arising from systematic variation in the canopy shadow fraction that is independent of changes in LAI or fAPAR. Therefore, results of this work provide a refined basis for understanding how remote sensing can be used to monitor and model the phenology of LAI, fAPAR, APAR, and gross primary productivity in temperate deciduous forests.

54 ENVIRONMENTAL SCIENCES↗

Simulation of evapotranspiration and yield of maize: An inter-comparison among 41 maize models

Accurate simulation of crop water use (evapotranspiration, ET) can help crop growth models to assess the likely effects of climate change on future crop productivity, as well as being an aid for irrigation scheduling for today's growers. To determine how well maize (Zea mays L.) growth models can simulate ET, an initial inter-comparison study was conducted in 2019 under the umbrella of AgMIP (Agricultural Model Inter-Comparison and Improvement Project). Herein, we present results of a second inter-comparison study of 41 maize models that was conducted using more comprehensive datasets from two additional sites - Mead, Nebraska, USA and Bushland, Texas, USA. There were 20 treatment-years with varying irrigation levels over multiple seasons at both sites. ET was measured using eddy covariance at Mead and using large weighing lysimeters at Bushland. A wide range in ET rates was simulated among the models, yet several generally were able to simulate ET rates adequately. The ensemble median values were generally close to the observations, but a few of the models sometimes performed better than the median. Many of the models that did well at simulating ET for the Mead site did poorly for drier, windy days at the Bushland site, suggesting they need to improve how they handle humidity and wind. Additional variability came from the approaches used to simulate soil water evaporation. Fortunately, several models were identified that did well at simulating soil water evaporation, canopy transpiration, biomass accumulation, and grain yield. These models were older and have been widely used, which suggests that a larger number of users have tested these models over a wider range of conditions leading to their improvement. These revelations of the better approaches are leading to model improvements and more accurate simulations of ET.

60 APPLIED LIFE SCIENCES↗

Spectral distortions to momentum and scalar exchanges by non-turbulent motion and patchy landscape variability

Modifications to the spectra of turbulent velocity and scalars and co-spectra of vertical fluxes of momentum and scalars due to patchy landscape heterogeneity and non-stationarity are explored for a Mediterranean ecosystem. About 9 months of high frequency measurements of the three velocity components, water vapor concentration, carbon dioxide concentration, and air temperature were analyzed for different seasons (spring/summer) and prevalent wind directions (southeast/northwest). The two wind directions sampled a contrast of clumped and patchy landscape comprised of olive trees (southeast) and wall bounded flow disturbed by the presence of few upwind trees (northwest). The measured spectra and co-spectra were also compared to theoretical scaling forms from stationary, planar homogeneous flow, in the absence of subsidence as derived from the Kansas experiment. To assess the role of low frequency non-turbulent motion on the spectral and co-spectral content, a 5-min Fourier cutoff was introduced and the analysis was limited to near-neutral conditions where the boundary layer depth is shallow compared to its unstable counterpart. It was shown that the velocity statistics were not appreciably impacted by the low-frequency motion causing non-stationarity. Moreover, the turbulent scalar fluxes were also shown not to be significantly impacted by such low frequency motion. The scalar variances were impacted, especially the water vapor variance and its concomitant spectral shape. When the non-turbulent motion was filtered, the scalar spectra at low wavenumbers followed expectations from the so-called attached eddy hypothesis (i.e. exhibited a $k^{-1}_x$ scaling with $k_x$ defining the longitudinal wavenumber) applicable for near-neutral conditions. For momentum co-spectra, the canonical shapes from the Kansas experiment appear to describe well the measurements here and in both dominant directions and seasons with some adjustment to the integral time scales based on wind direction. For the scalar co-spectra, deviations from the Kansas experiment were prevalent. The most noticeable and surprising deviations were their slow decay with increased sampling frequency at inertial subrange scales. This slow decay was shown not to contribute appreciably to the overall scalar fluxes. At those fine scales, predictions from local isotropy were expected to hold. The scalar co-spectral deviations from local isotropy were then discussed using a simplified co-spectral budget model where scalar–scalar co-spectra naturally emerged and the interplay between landscape heterogeneity and a scale-dependent pressure-scalar de-correlation time was postulated. It is also envisaged that the findings here offer a preliminary template for analyzing eddy-covariance data in situations that deviate from ideal conditions, especially regarding low-frequency modulations of scalar spectra and vertical scalar flux co-spectra.

Canopy turbulence↗

Explainable machine learning to quantify the value of proximal remote sensing in latent energy flux estimation

Proximal remote sensing has the potential to provide critical information on vegetation biophysical factors that can predict land-atmosphere exchange of water and energy. Latent energy (LE) flux is traditionally estimated using process-based models which rely on vegetation parameters that change during the growing season. Data-driven models have the potential to address these issues by offering flexible predictor selection and more efficient utilization of the information in predictor sets. These models require careful choice of predictors to avoid redundancy and allow robust cross-validation. In this study we present a systematic and comprehensive evaluation of machine learning (ML) models to assess the capability of meteorological and proximal sensing data for predicting LE at a half-hourly temporal resolution across multiple growing seasons for an agricultural system. The results presented here demonstrate that a model using four environmental predictors in combination with two proximal sensing variables can capture 88 % of the variability in LE. ML models using only three predictors (one meteorological and two proximal remote sensing) captured 81 % of LE variability, offering the best trade-off between performance and complexity. An ML model utilizing only two predictors, one proximal remote sensing variable and downwelling radiation, captured 77 % of LE variability. These results demonstrate the power of proximal remote sensing and meteorological observations to estimate land-atmosphere water vapor exchange, providing a solution where more direct methods such as eddy covariance are not available and for evaluations of agronomic management and genotypic variations.

60 APPLIED LIFE SCIENCES↗

Post-Combat-Injury Opioid Prescription and Alcohol Use Disorder in the Military

Previous studies have identified combat exposure and combat traumatic experience as problematic drinking risk factors. Increasing evidence suggests that opioid use increases the risk of alcohol use disorder. This study investigated the association between opioid prescription use after injury and (1) alcohol use disorder and (2) severity of alcohol use disorder among deployed military servicemembers. Deidentified health records data of 9,029 deployed servicemembers from a retrospective cohort study were analyzed. Data were randomly selected from the Department of Defense Trauma Registry and included servicemembers with combat injuries during deployment in Iraq or Afghanistan (2002–2016). Pharmacy records and International Classification of Diseases, Ninth and Tenth Revision diagnosis codes were used. Three groups were identified (no opioid prescription use, nonpersistent opioid prescription use, and persistent opioid prescription use) and were compared on the basis of alcohol use disorder risk using Cox proportional hazard models. Data analyses were performed in 2021. Of the 9,029 servicemembers with combat injury, 2,262 developed alcohol use disorder (1,322 developed severe alcohol use disorder). Compared with no opioid prescription use, increased alcohol use disorder risk was associated with persistent opioid prescription use, with a hazard ratio of 1.13 (95% CI=1.02, 1.26). After covariate adjustment, increased risk remained statistically significant (hazards ratio=1.24; 95% CI=1.10, 1.39). There was no significant difference in alcohol use disorder risk between no opioid prescription use and nonpersistent opioid prescription use. The risk of severe alcohol use disorder did not vary by opioid use among servicemembers with alcohol use disorder diagnosis. Furthermore, the findings of the study suggest that the incidence of alcohol use disorder was higher among injured servicemembers with persistent opioid prescription use than among those without opioid use. If replicated in prospective studies, the findings highlight the need for clinicians to consider the current and history of alcohol use of patients in initiating treatment involving opioids.

60 APPLIED LIFE SCIENCES↗

An uncertainty quantification method relevant to material test reactors

Within material test reactor calculations, energy dependent flux and reaction rate uncertainties are typically not quantified when performing as-run analyses to determine the neutron field experienced by the experiment. When high fidelity Monte-Carlo codes are used in such analyses, straight forward methods to calculate output uncertainties are not available, instead expert opinion is used to postulate computational uncertainties. New methods to propagate uncertainties through these high fidelity simulations are available when sufficient computational power is available. A tool is developed here for sampling any part of an MCNP input from random distributions to determine output uncertainties based on those inputs. Another tool is developed to sample nuclear data cross-section in ACE format using multi-group nuclear data covariances. The Total Monte-Carlo Method and Gesellschaft für Anlagen-und Reaktorsicherheit method (GRS) are implemented and compared to one another as well as MCNP sensitivity and uncertainty calculations. The methods were applied to the Godiva critical sphere k-eigenvalue, the UAM pin-cell benchmark energy dependent flux and reaction rates, and the Advanced Test Reactor energy dependent flux within an experimental location. Furthermore, the two methods agree well, with GRS allowing for an order of magnitude speedup for reaction rate uncertainty calculations and several orders of magnitude for eigenvalue uncertainty calculations.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

A new 181 Ta neutron resolved resonance region evaluation

A new 181 Ta neutron resolved resonance region evaluation has been performed from the thermal energy range up to approximately 2.5 keV. The R-matrix SAMMY code was used with the Reich–Moore approximation to evaluate resonance parameters from several experimental data sets. A Monte Carlo approach was used for resonance spin assignments and generating 59 small fictitious resonance levels which were shown to improve the cumulative level, Porter-Thomas, and Wigner distributions as compared to theoretical predictions. Covariance information was also generated for the entire resolved resonance region. Finally, the positive impact of the new evaluation was validated through benchmark calculations which were sensitive to the 181 Ta cross section and showed improvement in the reactivity bias for several benchmark cases.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Relativistic levels of mesic atoms

We revisit the derivation of the covariant two-body scalar-fermion equation with a Coulomb interaction, presented in a previous paper. We show that it can be given the formal aspect of a Dirac equation, but for the fact that the eigenvalue is also contained in one of the coefficients and thus it is not linearly included. The discussion of the boundary value problem is therefore different, although some properties of the Dirac equation can be recovered in an approximation bringing back to the concept of reduced mass. We discuss a mixed analytic-numerical method of solution which allows to obtain very accurate results and we calculate the lowest levels, states and QED corrections for pionic and kaonic atoms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Current-conserving relativistic linear response for collisional plasmas

Here we investigate the response of a relativistic plasma to electromagnetic fields in the framework of the Boltzmann equation incorporating a collision term in the relaxation rate approximation selected in a form assuring current conservation. We obtain an explicit solution for the linearized perturbation of the Fermi–Dirac equilibrium distribution in terms of the average relaxation rate k. We study the resulting covariant, gauge invariant, and current conserving form of the polarization tensor in the ultrarelativistic and non-relativistic limits. We evaluate the susceptibility in the ultrarelativistic limit and explore their dependence on k. Finally, we study the dispersion relations for the longitudinal and transverse poles of the propagator. We show that for k >2ω p , where ω p is the plasma frequency, the plasma wave modes are overdamped. In the opposite case, k$\ll$ ω p , the propagating plasma modes are weakly damped.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗