Complementary vertices and adjacency testing in polytopes

Our main theoretical result is that, if a simple polytope has a pair of complementary vertices (i.e., two vertices with no facets in common), then it has at least two such pairs, which can be chosen to be disjoint. Using this result, we improve adjacency testing for vertices in both simple and non-simple polytopes: given a polytope in the standard form {x \in R^n | Ax = b and x \geq 0} and a list of its V vertices, we describe an O(n) test to identify whether any two given vertices are adjacent. For simple polytopes this test is perfect; for non-simple polytopes it may be indeterminate, and instead acts as a filter to identify non-adjacent pairs. Our test requires an O(n^2 V + n V^2) precomputation, which is acceptable in settings such as all-pairs adjacency testing. These results improve upon the more general O(nV) combinatorial and O(n^3) algebraic adjacency tests from the literature.Comment: 14 pages, 5 figures. v1: published in COCOON 2012. v2: full journal version, which strengthens and extends the results in Section 2 (see p1 of the paper for details

ESA's Ice Sheets CCI: validation and inter-comparison of surface elevation changes derived from laser and radar altimetry over Jakobshavn Isbræ, Greenland – Round Robin results

In order to increase the understanding of the changing climate, the European Space Agency has launched the Climate Change Initiative (ESA CCI), a program which joins scientists and space agencies into 13 projects either affecting or affected by the concurrent changes. This work is part of the Ice Sheets CCI and four parameters are to be determined for the Greenland Ice Sheet (GrIS), each resulting in a dataset made available to the public: Surface Elevation Changes (SEC), surface velocities, grounding line locations, and calving front locations. All CCI projects have completed a so-called Round Robin exercise in which the scientific community was asked to provide their best estimate of the sought parameters as well as a feedback sheet describing their work. By inter-comparing and validating the results, obtained from research institutions world-wide, it is possible to develop the most optimal method for determining each parameter. This work describes the SEC Round Robin and the subsequent conclusions leading to the creation of a method for determining GrIS SEC values. The participants used either Envisat radar or ICESat laser altimetry over Jakobshavn Isbræ drainage basin, and the submissions led to inter-comparisons of radar vs. altimetry as well as cross-over vs. repeat-track analyses. Due to the high accuracy of the former and the high spatial resolution of the latter, a method, which combines the two techniques will provide the most accurate SEC estimates. The data supporting the final GrIS analysis stem from the radar altimeters on-board Envisat, ERS-1 and ERS-2. The accuracy of laser data exceeds that of radar altimetry; the Round Robin analysis has, however, proven the latter equally capable of dealing with surface topography thereby making such data applicable in SEC analyses extending all the way from the interior ice sheet to margin regions. This shows good potential for a~future inclusion of ESA CryoSat-2 and Sentinel-3 radar data in the analysis, and thus for obtaining reliable SEC estimates throughout the entire GrIS

The impact of glacier geometry on meltwater plume structure and submarine melt in Greenland fjords

Meltwater from the Greenland Ice Sheet often drains subglacially into fjords, driving upwelling plumes at glacier termini. Ocean models and observations of submarine termini suggest that plumes enhance melt and undercutting, leading to calving and potential glacier destabilization. Here we systematically evaluate how simulated plume structure and submarine melt during summer months depends on realistic ranges of subglacial discharge, glacier depth, and ocean stratification from 12 Greenland fjords. Our results show that grounding line depth is a strong control on plume-induced submarine melt: deep glaciers produce warm, salty subsurface plumes that undercut termini, and shallow glaciers produce cold, fresh surface-trapped plumes that can overcut termini. Due to sustained upwelling velocities, plumes in cold, shallow fjords can induce equivalent depth-averaged melt rates compared to warm, deep fjords. These results detail a direct ocean-ice feedback that can affect the Greenland Ice Sheet

Discrete integrable systems and Poisson algebras from cluster maps

We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised them in terms of the skew-symmetric matrix that defines the quiver. The associated nonlinear recurrences are equivalent to birational maps, and we explain how these maps can be endowed with an invariant Poisson bracket and/or presymplectic structure. Upon applying the algebraic entropy test, we are led to a series of conjectures which imply that the entropy of the cluster maps can be determined from their tropical analogues, which leads to a sharp classification result. Only four special families of these maps should have zero entropy. These families are examined in detail, with many explicit examples given, and we show how they lead to discrete dynamics that is integrable in the Liouville-Arnold sense.Comment: 49 pages, 3 figures. Reduced to satisfy journal page restrictions. Sections 2.4, 4.5, 6.3, 7 and 8 removed. All other results remain, with minor editin

The impact of glacier geometry on meltwater plume structure and submarine melt in Greenland fjords

Meltwater from the Greenland Ice Sheet often drains subglacially into fjords, driving upwelling plumes at glacier termini. Ocean models and observations of submarine termini suggest that plumes enhance melt and undercutting, leading to calving and potential glacier destabilization. Here we systematically evaluate how simulated plume structure and submarine melt during summer months depends on realistic ranges of subglacial discharge, glacier depth, and ocean stratification from 12 Greenland fjords. Our results show that grounding line depth is a strong control on plume-induced submarine melt: deep glaciers produce warm, salty subsurface plumes that undercut termini, and shallow glaciers produce cold, fresh surface-trapped plumes that can overcut termini. Due to sustained upwelling velocities, plumes in cold, shallow fjords can induce equivalent depth-averaged melt rates compared to warm, deep fjords. These results detail a direct ocean-ice feedback that can affect the Greenland Ice Sheet

Experimental protocol for sea level projections from ISMIP6 stand-alone ice sheet models

Projection of the contribution of ice sheets to sea level change as part of the Coupled Model Intercomparison Project Phase 6 (CMIP6) takes the form of simulations from coupled ice sheet–climate models and stand-alone ice sheet models, overseen by the Ice Sheet Model Intercomparison Project for CMIP6 (ISMIP6). This paper describes the experimental setup for process-based sea level change projections to be performed with stand-alone Greenland and Antarctic ice sheet models in the context of ISMIP6. The ISMIP6 protocol relies on a suite of polar atmospheric and oceanic CMIP-based forcing for ice sheet models, in order to explore the uncertainty in projected sea level change due to future emissions scenarios, CMIP models, ice sheet models, and parameterizations for ice–ocean interactions. We describe here the approach taken for defining the suite of ISMIP6 stand-alone ice sheet simulations, document the experimental framework and implementation, and present an overview of the ISMIP6 forcing to be used by participating ice sheet modeling groups

ESA ice sheet CCI: derivation of the optimal method for surface elevation change detection of the Greenland ice sheet - round robin results

For more than two decades, radar altimetry missions have provided continuous elevation estimates of the Greenland ice sheet (GrIS). Here, we propose a method for using such data to estimate ice-sheet-wide surface elevation changes (SECs). The final data set will be based on observations acquired from the European Space Agency’s Environmental Satellite (ENVISAT), European Remote Sensing (ERS)-1 and -2, CryoSat-2, and, in the longer term, Sentinel-3 satellites. In order to find the best-performing method, an intercomparison exercise has been carried out in which the scientific community was asked to provide their best SEC estimates as well as feedback sheets describing the applied method. Due to the hitherto few radar-based SEC analyses as well as the higher accuracy of laser data, the participants were asked to use either ENVISAT radar or ICESat (Ice, Cloud, and land Elevation Satellite) laser altimetry over the Jakobshavn Isbræ drainage basin. The submissions were validated against airborne laser-scanner data, and intercomparisons were carried out to analyse the potential of the applied methods and to find whether the two altimeters were capable of resolving the same signal. The analyses found great potential of the applied repeat-track and cross-over techniques, and, for the first time over Greenland, that repeat-track analyses from radar altimetry agreed well with laser data. Since topography-related errors can be neglected in cross-over analyses, it is expected that the most accurate, ice-sheet-wide SEC estimates are obtained by combining the cross-over and repeat-track techniques. It is thus possible to exploit the high accuracy of the former and the large spatial data coverage of the latter. Based on CryoSat’s different operation modes, and the increased spatial and temporal data coverage, this shows good potential for a future inclusion of CryoSat-2 and Sentinel-3 data to continuously obtain accurate SEC estimates both in the interior and margin ice sheet

Future Sea Level Change Under Coupled Model Intercomparison Project Phase 5 and Phase 6 Scenarios From the Greenland and Antarctic Ice Sheets

Projections of the sea level contribution from the Greenland and Antarctic ice sheets (GrIS and AIS) rely on atmospheric and oceanic drivers obtained from climate models. The Earth System Models participating in the Coupled Model Intercomparison Project phase 6 (CMIP6) generally project greater future warming compared with the previous Coupled Model Intercomparison Project phase 5 (CMIP5) effort. Here we use four CMIP6 models and a selection of CMIP5 models to force multiple ice sheet models as part of the Ice Sheet Model Intercomparison Project for CMIP6 (ISMIP6). We find that the projected sea level contribution at 2100 from the ice sheet model ensemble under the CMIP6 scenarios falls within the CMIP5 range for the Antarctic ice sheet but is significantly increased for Greenland. Warmer atmosphere in CMIP6 models results in higher Greenland mass loss due to surface melt. For Antarctica, CMIP6 forcing is similar to CMIP5 and mass gain from increased snowfall counteracts increased loss due to ocean warming