On complex-valued 2D eikonals. Part four: continuation past a caustic

Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude -- in other words, are complex-valued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a non-zero imaginary part, comes on stage. In the present paper we explore such a principle in dimension $2.$ We investigate a partial differential system that governs the real and the imaginary parts of complex-valued two-dimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, ill-posedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasi-reversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions.Comment: 48 pages, 15 figure

Preliminary evidence of nocturnal transpiration and stomatal conductance in potato and their interaction with drought and yield.

In spite of the growing evidence showing the occurrence of nocturnal water exchange measured by stomatal conductance (SCnight, related to stomatal openness) and transpiration (TRnight, related to water losses) in different crops, how those processes are affected by soil water availability is still unknown. A preliminary study was conducted to test the occurrence of TRnight and SCnight (assessed in two moments) in two potato cultivars (UNICA and Sarnav), subjected to a long-term water restriction. The overall TRnight average were 9.0 ± 0.6 and 5.4 ± 2.6% for total daily transpiration under well-watered and water restricted conditions, respectively. Cultivar differences in early night and predawn SCnight and TRnight were more evident under non-restricted water conditions. Tuber yield was linearly correlated with TRnight and predawn and early-night SCnight. These preliminary findings emphasize the need for future research to corroborate the detection of TRnight and SCnight under water restrictions in potato, addressing the functional implications of these traits and the usefulness of SCnight for screening large sets of genotypes for potato breeding

Is Discrimination of 13C in potato leaflets and tubers an appropriate trait to describe genotype responses to restrictive and well-watered conditions?

The article objectives were: (i) to compare physiological performance and biomass accumulation among genotypes along the growing period and (ii) to test the reliability of Δleaflet and Δtuber to determine key mechanisms responsible of the varietal differences in biomass accumulation and tuber yield under different water availability levels

Key ecosystem services and ecological intensification of agriculture in the tropical high-Andean Puna as affected by land-use and climate changes.

High-altitude ecosystems shelter important reserves of biodiversity, water provision and soil organic carbon (SOC) stocks. Climate change, agricultural encroachment, overgrazing, and mining activities are endangering ecosystems sustainability, particularly in the high-Andean Puna. Increasing food demands in a region with limited agricultural land calls for agricultural intensification. Ecological intensification of agriculture is a framework for increasing agricultural productivity by fostering supporting and regulating ecosystem services (ES) while reducing negative environmental impacts. In this review we examine how agriculture use and disturb the provision of key ES in this ecoregion – food, wool and fiber provision, soil fertility, nutrient cycling, soil carbon sequestration, water provision and regulation, genetic resources, pest and disease control, pollination regulation and microclimate regulation. We also propose a set of technologies, practices and policies to preserve (or restore) the provision of these key ES: long fallowing, soil amendments, conservation tillage, rotational grazing, grassland ecological restoration, conservation of agrobiodiversity, modern irrigation and water harvesting, plant breeding, climate change mitigation schemes and payment for ecosystem services, and adapted traditional technologies

Land use effects on soil fertility and nutrient cycling in the Peruvian High-Andean puna grasslands.

Expansion of crop production into high-altitude native grasslands is occurring in the Peruvian High-Andes due to climate change and agricultural intensification, with little understanding of the consequences to the ecosystem. The objective of this study was to determine the impact of land-use changes on soil fertility and nutrient cycling to guide future land management. Comparisons were made between native grasslands and two alternative farming systems; the first was a system in which native grassland was replaced by an annual cash crop, maca (Lepidium meyenii Walp.), followed by a long fallow and the second was a long-standing perennial cultivated pasture. There was greater N mineralization, extractable Bray-1 P, and K+ in recently converted maca soils relative to adjacent native grassland soils most likely caused by tillage-induced nutrient mineralization and by incorporation of manure and native grassland residue during the first year of land preparation prior to planting maca. Soil fertility, as determined by an ex situ pot trial to measure plant yield, was correlated with plant cover, soil total organic carbon, and water stable aggregates (WSA) under fallow following maca, suggesting that soil erosion might be an issue in the long term. In cultivated pastures, we found an increase in long-term fertility driven primarily by the build-up of soil N and improved soil structure in the perennial grass-legume mixture compared with adjacent native grassland. Responsible management of the Puna agroecosystem requires restoration of plant cover after annual cropping of maca or the establishment of mixed perennial cultivated pasture

Soil organic carbon stocks and fractionation under different land uses in the Peruvian high-Andean Puna.

Soil organic C (SOC), the largest terrestrial C pool, has been shown to be sensitive to agricultural disturbance. In the Andean Puna, agricultural expansion is thought to be jeopardizing the large SOC stock sequestered in its soils. In this paper, we assessed the impact changes in land use have on C sequestration, fractionation and δ13C natural abundance featuring two case studies in the Peruvian Central Puna. We found SOC stocks greater than average temperate grassland soils (between 123 ± 4 and 136 ± 4 Mg C ha− 1 in the 0–30 cm soil profile); however, they did not differ between land uses. In the first case study, depletion of δ13C in mineral associated C in long fallow (> 7 yr) and maca (Lepidium meyenii Walpers) croplands was hypothesized to be caused by disruption of soil aggregates due to tillage and initial greater SOC concentration in maca plots. In addition, SOC loss was found in long fallow plots, and was correlated to steep and low plant cover plots; evidencing that soil degradation does not occur during cropping activity but due to erosion after land abandonment. In the second case study, a long-established perennial ryegrass (Lolium perenne L.) - white clover (Trifolium repens L.) association (> 40 yr) was studied. We did not find an increase in SOC stock under the cultivated pasture. However, we found a better soil aggregation and an increase of 17.3 and 12.2 g C kg− 1WSA in particulate organic C, and SOC within small macroaggregates in cultivated pasture soils. Soil δ13C natural abundance was about 1‰ depleted in all measured C fractions of cultivated pastures when compared to native grasslands, suggesting that pasture-fixed C forms labile and recalcitrant SOC fractions

Defining biological thresholds associated to plant water status for monitoring water restriction effects: Stomatal conductance and photosynthesis recovery as key indicators in potato.

The definition of irrigation schedules depends on the understanding of the response of key plant traits to different water restriction characteristics with the aim to avoid physiological impairment. In this study, different timings (at tuber initiation and bulking) and intensities (four soil moisture levels) of water restriction were tested in the potato crop. The temporal patterns of mid-morning or maximum, light saturated stomatal conductance (gs_max), recovery of net photosynthesis (Arecovery), stem water potential ( stem), carbon isotope discrimination in tubers ( tuber), plant water concentration (PWC), photochemical reflectance index (PRI) and crop water stress index (CWSI) were analyzed. Early-severe water restriction caused a drastic yield reduction, with low recovery of physiological responses (gs_max, tuber, stem, CWSI, Arecovery) after 15 days of post-restriction irrigation and even a continued reduction of some of them (PWC, PRI). It also caused a prolonged gs_max reduction below 0.05 mol H2O m-2 s-1 (˜ 5 µmol CO2 m-2 s-1 of net photosynthesis) suggesting that this value defines a physiological severity threshold in potato, under which a metabolic impairment occurs. CWSI and PRI showed a close linear (R2 = 0.76) and no linear (natural logarithm function, R2 = 0.67) relationship with gs_max respectively. In cloudless dry environments, irrigation schedules in potato should aim to avoiding CWSI values higher than 0.4, especially until before of maximum canopy cover establishment. A close relationship between Arecovery at maximum stress moment and yield reduction was found. The strong relationship between the measured traits (except PWC and stem) and final yield at maximum stress moment found in the present study warrants further research on drought phenotyping immediately before post-restriction irrigation or when the defined severity threshold in potato is reached