Multimode solutions of first-order elliptic quasilinear systems obtained from Riemann invariants

Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links between two techniques, that of the symmetry reduction method and of the generalized method of characteristics. A variant of the conditional symmetry method for constructing this type of solution is proposed. A specific feature of that approach is an algebraic-geometric point of view, which allows the introduction of specific first-order side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method for solving elliptic homogeneous systems of PDEs. A further generalization of the Riemann invariants method to the case of inhomogeneous systems, based on the introduction of specific rotation matrices, enables us to weaken the integrability condition. It allows us to establish a connection between the structure of the set of integral elements and the possibility of constructing specific classes of simple mode solutions. These theoretical considerations are illustrated by the examples of an ideal plastic flow in its elliptic region and a system describing a nonlinear interaction of waves and particles. Several new classes of solutions are obtained in explicit form, including the general integral for the latter system of equations

Source apportionment of atmospheric trace gases and particulate matter: comparison of log-ratio and traditional approaches

In this paper we compare multivariate methods using both traditional approaches, which ignore issues of closure and provide relatively simple methods to deal with censored or missing data, and log-ratio methods to determine the sources of trace constituents in the atmosphere. The data set examined was collected from April to July 2008 at a sampling site near Woods Hole, Massachusetts, along the northeastern United States Atlantic coastline. The data set consists of trace gas mixing ratios (O3, SO2, NOx, elemental mercury [Hgo], and reactive gaseous mercury [RGM]), and concentrations of trace elements in fine (<2.5 μm) particulate matter (Al, As, Ba, Ca, Cd, Ce, Co, Cs, Fe, Ga, Hg, K, La, Mg, Mn, Na, P, Pb, Rb, Sb, Sr, Th, Ti, V, Y, and Zn) with varying percentages of censored values for each species. The data were separated into two subcompositions: s1, which is comprised by RGM and particulate Hg (HgP), which are both highly censored; and s2 which includes all of the trace elements associated with particulate matter except Hg, and the trace gases O3, SO2, NOx, and Hgo. Principal factor analysis (PFA) was successful in determining the primary sources for constituents in s2 using both traditional and log-ratio approaches. Using the traditional approach, regression between factor scores and RGM and particulate Hg concentrations suggested that none of the sources identified during PFA led to positive contributions of either reactive mercury compound. This finding is counter to most conventional thinking and is likely specious, resulting from removal of censored data (up to >80% of the entire dataset) during the analysis. Log-ratio approaches to find relationships between constituents comprising s2 with RGM and HgP (i.e., s1) focused on log-ratio correlation and regression analyses of alr-transformed data, using Al as the divisor. Regression models accounted for large fractions of the variance in concentrations of the two reactive mercury species and generally agreed with conceptualizations about the formation and behavior of these species. An analysis of independence between the subcompositions demonstrated that the behavior of the two constituents comprising s1 (i.e., RGM and HgP) is dependent on changes in s2. Our findings suggest that although problems related to closure are largely unknown or ignored in the atmospheric sciences, much insight can be gleaned from the application of log-ratio methods to atmospheric chemistry data

An updated radiocarbon-based ice margin chronology for the last deglaciation of the North American Ice Sheet Complex

The North American Ice Sheet Complex (NAISC; consisting of the Laurentide, Cordilleran and Innuitian ice sheets) was the largest ice mass to repeatedly grow and decay in the Northern Hemisphere during the Quaternary. Understanding its pattern of retreat following the Last Glacial Maximum is critical for studying many facets of the Late Quaternary, including ice sheet behaviour, the evolution of Holocene landscapes, sea level, atmospheric circulation, and the peopling of the Americas. Currently, the most up-to-date and authoritative margin chronology for the entire ice sheet complex is featured in two publications (Geological Survey of Canada Open File 1574 [Dyke et al., 2003]; ‘Quaternary Glaciations – Extent and Chronology, Part II’ [Dyke, 2004]). These often-cited datasets track ice margin recession in 36 time slices spanning 18 ka to 1 ka (all ages in uncalibrated radiocarbon years) using a combination of geomorphology, stratigraphy and radiocarbon dating. However, by virtue of being over 15 years old, the ice margin chronology requires updating to reflect new work and important revisions. This paper updates the aforementioned 36 ice margin maps to reflect new data from regional studies. We also update the original radiocarbon dataset from the 2003/2004 papers with 1541 new ages to reflect work up to and including 2018. A major revision is made to the 18 ka ice margin, where Banks and Eglinton islands (once considered to be glacial refugia) are now shown to be fully glaciated. Our updated 18 ka ice sheet increased in areal extent from 17.81 to 18.37 million km2, which is an increase of 3.1% in spatial coverage of the NAISC at that time. Elsewhere, we also summarize, region-by-region, significant changes to the deglaciation sequence. This paper integrates new information provided by regional experts and radiocarbon data into the deglaciation sequence while maintaining consistency with the original ice margin positions of Dyke et al. (2003) and Dyke (2004) where new information is lacking; this is a pragmatic solution to satisfy the needs of a Quaternary research community that requires up-to-date knowledge of the pattern of ice margin recession of what was once the world’s largest ice mass. The 36 updated isochrones are available in PDF and shapefile format, together with a spreadsheet of the expanded radiocarbon dataset (n = 5195 ages) and estimates of uncertainty for each interval