Lie algebroid structures on a class of affine bundles

We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibred over the real numbers. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the theory of Lagrangian systems on Lie algebroids. An extensive discussion is given of a way one can think of forms acting on sections of the affine bundle. It is further shown that the affine Lie algebroid structure gives rise to a coboundary operator on such forms. The concept of admissible curves and dynamical systems whose integral curves are admissible, brings an associated affine bundle into the picture, on which one can define in a natural way a prolongation of the original affine Lie algebroid structure.Comment: 28 page

Impact of deoxygenation and warming on global marine species in the 21st century

Ocean temperature and dissolved oxygen shape marine habitats in an interplay with species' physiological characteristics. Therefore, the observed and projected warming and deoxygenation of the world's oceans in the 21st century may strongly affect species' habitats. Here, we implement an extended version of the Aerobic Growth Index (AGI), which quantifies whether a viable population of a species can be sustained in a particular location. We assess the impact of projected deoxygenation and warming on the contemporary habitat of 47 representative marine species covering the epipelagic, mesopelagic, and demersal realms. AGI is calculated for these species for the historical period and into the 21st century using bias-corrected environmental data from six comprehensive Earth system models. While habitat viability decreases nearly everywhere with global warming, the impact of this decrease is strongly species dependent. Most species lose less than 5 % of their contemporary habitat volume at 2 ∘C of global warming relative to preindustrial levels, although some individual species are projected to incur losses 2–3 times greater than that. We find that the in-habitat spatiotemporal variability of O2 and temperature (and hence AGI) provides a quantifiable measure of a species' vulnerability to change. In the event of potential large habitat losses (over 5 %), species vulnerability is the most important indicator. Vulnerability is more critical than changes in habitat viability, temperature, or pO2 levels. Loss of contemporary habitat is for most epipelagic species driven by the warming of ocean water and is therefore elevated with increased levels of global warming. In the mesopelagic and demersal realms, habitat loss is also affected by pO2 decrease for some species. Our analysis is constrained by the uncertainties involved in species-specific critical thresholds, which we quantify; by data limitations on 3D species distributions; and by high uncertainty in model O2 projections in equatorial regions. A focus on these topics in future research will strengthen our confidence in assessing climate-change-driven losses of contemporary habitats across the global oceans.</p

Southern Ocean phytoplankton under climate change: a shifting balance of bottom-up and top-down control

Phytoplankton form the base of the marine food web by transforming CO2 into organic carbon via photosynthesis. Despite the importance of phytoplankton for marine ecosystems and global carbon cycling, projections of phytoplankton biomass in response to climate change differ strongly across Earth system models, illustrating uncertainty in our understanding of the underlying processes. Differences are especially large in the Southern Ocean, a region that is notoriously difficult to represent in models. Here, we argue that total (depth-integrated) phytoplankton biomass in the Southern Ocean is projected to largely remain unchanged under climate change by the Coupled Model Intercomparison Project Phase 6 (CMIP6) multi-model ensemble because of a shifting balance of bottom-up and top-down processes driven by a shoaling mixed-layer depth. A shallower mixed layer is projected on average to improve growth conditions, consequently weaken bottom-up control, and confine phytoplankton closer to the surface. An increase in the phytoplankton concentration promotes zooplankton grazing efficiency, thus intensifying top-down control. However, large differences across the model ensemble exist, with some models simulating a decrease in surface phytoplankton concentrations. To reduce uncertainties in projections of surface phytoplankton concentrations, we employ an emergent constraint approach using the observed sensitivity of surface chlorophyll concentration, taken as an observable proxy for phytoplankton, to seasonal changes in the mixed-layer depth as an indicator for future changes in surface phytoplankton concentrations. The emergent constraint reduces uncertainties in surface phytoplankton concentration projections by around one-third and increases confidence that surface phytoplankton concentrations will indeed rise due to shoaling mixed layers under global warming, thus favouring intensified top-down control. Overall, our results suggest that while changes in bottom-up conditions stimulate enhanced growth, intensified top-down control opposes an increase in phytoplankton and becomes increasingly important for the phytoplankton response to climate change in the Southern Ocean.</p

The inverse problem for Lagrangian systems with certain non-conservative forces

We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free conditions for the existence of a suitable non-singular multiplier matrix, which will lead to an equivalent representation of a given system of second-order equations as one of these Lagrangian systems with non-conservative forces.Comment: 28 page

Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators

Let $t\mapsto A(t)$ for $t\in T$ be a $C^M$-mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here $C^M$ stands for C^\om (real analytic), a quasianalytic or non-quasianalytic Denjoy-Carleman class, $C^\infty$, or a H\"older continuity class C^{0,\al}. The parameter domain $T$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We prove and review results on $C^M$-dependence on $t$ of the eigenvalues and eigenvectors of $A(t)$.Comment: 8 page

Hotspots and drivers of compound marine heatwaves and low net primary production extremes

Extreme events can severely impact marine organisms and ecosystems. Of particular concern are multivariate compound events, namely when conditions are simultaneously extreme for multiple ocean ecosystem stressors. In 2013–2015 for example, an extensive marine heatwave (MHW), known as the Blob, co-occurred locally with extremely low net primary productivity (NPPX) and negatively impacted marine life in the northeast Pacific. Yet, little is known about the characteristics and drivers of such multivariate compound MHW–NPPX events. Using five different satellite-derived net primary productivity (NPP) estimates and large-ensemble-simulation output of two widely used and comprehensive Earth system models, the Geophysical Fluid Dynamics Laboratory (GFDL) ESM2M-LE and Community Earth System Model version 2 (CESM2-LE), we assess the present-day distribution of compound MHW–NPPX events and investigate their potential drivers on the global scale. The satellite-based estimates and both models reveal hotspots of frequent compound events in the center of the equatorial Pacific and in the subtropical Indian Ocean, where their occurrence is at least 3 times higher (more than 10 d yr−1) than if MHWs (temperature above the seasonally varying 90th-percentile threshold) and NPPX events (NPP below the seasonally varying 10th-percentile threshold) were to occur independently. However, the models show disparities in the northern high latitudes, where compound events are rare in the satellite-based estimates and GFDL ESM2M-LE (less than 3 d yr−1) but relatively frequent in CESM2-LE. In the Southern Ocean south of 60∘ S, low agreement between the observation-based estimates makes it difficult to determine which of the two models better simulates MHW–NPPX events. The frequency patterns can be explained by the drivers of compound events, which vary among the two models and phytoplankton types. In the low latitudes, MHWs are associated with enhanced nutrient limitation on phytoplankton growth, which results in frequent compound MHW–NPPX events in both models. In the high latitudes, NPPX events in GFDL ESM2M-LE are driven by enhanced light limitation, which rarely co-occurs with MHWs, resulting in rare compound events. In contrast, in CESM2-LE, NPPX events in the high latitudes are driven by reduced nutrient supply that often co-occurs with MHWs, moderates phytoplankton growth, and causes biomass to decrease. Compound MHW–NPPX events are associated with a relative shift towards larger phytoplankton in most regions, except in the eastern equatorial Pacific in both models, as well as in the northern high latitudes and between 35 and 50∘ S in CESM2-LE, where the models suggest a shift towards smaller phytoplankton, with potential repercussions on marine ecosystems. Overall, our analysis reveals that the likelihood of compound MHW–NPPX events is contingent on model representation of the factors limiting phytoplankton production. This identifies an important need for improved process understanding in Earth system models used for predicting and projecting compound MHW–NPPX events and their impacts.</p

Oxygen and indicators of stress for marine life in multi-model global warming projections

Decadal-to-century scale trends for a range of marine environmental variables in the upper mesopelagic layer (UML, 100–600 m) are investigated using results from seven Earth System Models forced by a high greenhouse gas emission scenario. The models as a class represent the observation-based distribution of oxygen (O2) and carbon dioxide (CO2), albeit major mismatches between observation-based and simulated values remain for individual models. By year 2100 all models project an increase in SST between 2 °C and 3 °C, and a decrease in the pH and in the saturation state of water with respect to calcium carbonate minerals in the UML. A decrease in the total ocean inventory of dissolved oxygen by 2% to 4% is projected by the range of models. Projected O2 changes in the UML show a complex pattern with both increasing and decreasing trends reflecting the subtle balance of different competing factors such as circulation, production, remineralization, and temperature changes. Projected changes in the total volume of hypoxic and suboxic waters remain relatively small in all models. A widespread increase of CO2 in the UML is projected. The median of the CO2 distribution between 100 and 600m shifts from 0.1–0.2 mol m−3 in year 1990 to 0.2–0.4 mol m−3 in year 2100, primarily as a result of the invasion of anthropogenic carbon from the atmosphere. The co-occurrence of changes in a range of environmental variables indicates the need to further investigate their synergistic impacts on marine ecosystems and Earth System feedbacks

Dispersionless limit of the noncommutative potential KP hierarchy and solutions of the pseudodual chiral model in 2+1 dimensions

The usual dispersionless limit of the KP hierarchy does not work in the case where the dependent variable has values in a noncommutative (e.g. matrix) algebra. Passing over to the potential KP hierarchy, there is a corresponding scaling limit in the noncommutative case, which turns out to be the hierarchy of a `pseudodual chiral model' in 2+1 dimensions (`pseudodual' to a hierarchy extending Ward's (modified) integrable chiral model). Applying the scaling procedure to a method generating exact solutions of a matrix (potential) KP hierarchy from solutions of a matrix linear heat hierarchy, leads to a corresponding method that generates exact solutions of the matrix dispersionless potential KP hierarchy, i.e. the pseudodual chiral model hierarchy. We use this result to construct classes of exact solutions of the su(m) pseudodual chiral model in 2+1 dimensions, including various multiple lump configurations.Comment: 37 pages, 10 figures, 2nd version: some extensions (Fig 3, Appendix A, additional references), 3rd version: some minor changes, additional reference