We consider the numerical calculation of several matrix eigenvalue problems which require some manipulation before the standard algorithms may be used. This includes finding the stationary values of a quadratic form subject to linear constraints and determining the eigenvalues of a matrix which is modified by a matrix of rank one. We also consider several inverse eigenvalue problems. This includes the problem of determining the coefficients for the Gauss–Radau and Gauss–Lobatto quadrature rules. In addition, we study several eigenvalue problems which arise in least squares.

Some modified matrix eigenvalue problems.

Climate extremes threaten human health, economic stability, and the well-being of natural and built environments (e.g., 2003 European heat wave). As the world continues to warm, climate hazards are...

https://www.annualreviews.org/doi/pdf/10.1146/annurev-earth-071719-055228

Climate Extremes and Compound Hazards in a Warming World

We review the characteristics of sea level variability at the coast focussing on how it differs from the variability in the nearby deep ocean. Sea level variability occurs on all timescales, with processes at higher frequencies tending to have a larger magnitude at the coast due to resonance and other dynamics. In the case of some processes, such as the tides, the presence of the coast and the shallow waters of the shelves results in the processes being considerably more complex than offshore. However, ‘coastal variability’ should not always be considered as ‘short spatial scale variability’ but can be the result of signals transmitted along the coast from 1000s km away. Fortunately, thanks to tide gauges being necessarily located at the coast, many aspects of coastal sea level variability can be claimed to be better understood than those in the deep ocean. Nevertheless, certain aspects of coastal variability remain under-researched, including how changes in some processes (e.g., wave setup, river runoff) may have contributed to the historical mean sea level records obtained from tide gauges which are now used routinely in large-scale climate research.

/pdf/forcing-factors-affecting-sea-level-changes-at-the-coast-ujfq8aa0sx.pdf

Forcing Factors Affecting Sea Level Changes at the Coast

There have been significant changes in sea level over the past two million years, and a complete understanding of natural cycles of change as well as anthropogenic effects is imperative for future global development. This book reviews the history of research into these sea-level changes and summarises the methods and analytical approaches used to interpret evidence for sea-level changes. It provides an overview of changing climates during the Quaternary, examines processes responsible for global variability of sea-level records, and presents detailed reviews of sea-level changes for the Pleistocene and Holocene. The book concludes by discussing current trends in sea levels and likely future sea-level changes. This is an important and authoritative resource for academic researchers and graduate and advanced undergraduate students working in tectonics, stratigraphy, geomorphology, physical geography, environmental science and other aspects of Quaternary studies.

Quaternary Sea-Level Changes: A Global Perspective

Sea-level rise and coastal change: the past as a guide to the future

Significance River discharge exerts an important influence on coastal ocean circulation but has been overlooked as a driver of historical coastal sea-level change and future coastal flood risk. We explore the relation between observed river discharge and sea level on the United States Atlantic and Gulf coasts over interannual and longer periods. We formulate a theory that predicts the observed correspondence between river discharge and sea level, demonstrating a causal relation between the two variables. Our results highlight a significant but overlooked driver of coastal sea level, indicating the need for (1) improved resolution in remote sensing and modeling of the coastal zone and (2) inclusion of realistic river runoff variability in climate models. Identifying physical processes responsible for historical coastal sea-level changes is important for anticipating future impacts. Recent studies sought to understand the drivers of interannual to multidecadal sea-level changes on the United States Atlantic and Gulf coasts. Ocean dynamics, terrestrial water storage, vertical land motion, and melting of land ice were highlighted as important mechanisms of sea-level change along this densely populated coast on these time scales. While known to exert an important control on coastal ocean circulation, variable river discharge has been absent from recent discussions of drivers of sea-level change. We update calculations from the 1970s, comparing annual river-discharge and coastal sea-level data along the Gulf of Maine, Mid-Atlantic Bight, South Atlantic Bight, and Gulf of Mexico during 1910–2017. We show that river-discharge and sea-level changes are significantly correlated (p<0.01), such that sea level rises between 0.01 and 0.08 cm for a 1 km3 annual river-discharge increase, depending on region. We formulate a theory that describes the relation between river-discharge and halosteric sea-level changes (i.e., changes in sea level related to salinity) as a function of river discharge, Earth’s rotation, and density stratification. This theory correctly predicts the order of observed increment sea-level change per unit river-discharge anomaly, suggesting a causal relation. Our results have implications for remote sensing, climate modeling, interpreting Common Era proxy sea-level reconstructions, and projecting coastal flood risk.

River-discharge effects on United States Atlantic and Gulf coast sea-level changes

[1] Sea level reconstructions spanning the late Holocene (the past 2000 years) provide a preindustrial context for understanding the patterns and causes of contemporary and future change. The Intergovernmental Panel on Climate Change (IPCC) assumed that global sea level change during the past two millennia (prior to the middle of the nineteenth century) was close to zero [Bindoff et al., 2007], but understanding of late Holocene sea level variability is limited. Glaciers and ice sheets changed significantly in size during this period, and therefore sea level likely oscillated on the order of several decimeters. In addition, ocean dynamics, solid Earth movements, steric (density) changes, and gravitational effects contributed to complex regional patterns of sea level change.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2011EO350001

Two millennia of sea level data: The key to predicting change

Characterizing the spatio-temporal variability of relative sea level (RSL) and estimating local, regional, and global RSL trends requires statistical analysis of RSL data. Formal statistical treatments, needed to account for the spatially and temporally sparse distribution of data and for geochronological and elevational uncertainties, have advanced considerably over the last decade. Time-series models have adopted more flexible and physically-informed specifications with more rigorous quantification of uncertainties. Spatio-temporal models have evolved from simple regional averaging to frameworks that more richly represent the correlation structure of RSL across space and time. More complex statistical approaches enable rigorous quantification of spatial and temporal variability, the combination of geographically disparate data, and the separation of the RSL field into various components associated with different driving processes. We review the range of statistical modeling and analysis choices used in the literature, reformulating them for ease of comparison in a common hierarchical statistical framework. The hierarchical framework separates each model into different levels, clearly partitioning measurement and inferential uncertainty from process variability. Placing models in a hierarchical framework enables us to highlight both the similarities and differences among modeling and analysis choices. We illustrate the implications of some modeling and analysis choices currently used in the literature by comparing the results of their application to common datasets within a hierarchical framework. In light of the complex patterns of spatial and temporal variability exhibited by RSL, we recommend non-parametric approaches for modeling temporal and spatio-temporal RSL.

Andrew Kemp

Papers

River-discharge effects on United States Atlantic and Gulf coast sea-level changes

Two millennia of sea level data: The key to predicting change

Data for: Statistical modeling of rates and trends in Holocene relative sea level

Salt Marsh Sediments as Recorders of Holocene Relative Sea-Level Change

A Bayesian Hierarchical Model for Reconstructing Sea Level from the Common Era: From Raw Data to Rates of Change