Existence and non uniqueness of constant scalar curvature toric Sasaki metrics

We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using existence result of [25], we show that a co-oriented compact toric contact 5-manifold whose moment cone has 4 facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on $S^2\times S^3$ admitting two non isometric and non transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.Comment: 22 pages, no figure

Toric geometry of convex quadrilaterals

We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric K\"ahler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including K\"ahler-Einstein ones, and show that for a toric orbi-surface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of K\"ahler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.Comment: 36 pages; v2: small changes (typos and sign convention); v3: few typos corrected, adjustments in the trapezoid case; to appear in Journal of Symplectic Geometr

Interaction between two spherical bubbles rising in a viscous liquid

The three-dimensional flow around two spherical bubbles moving in a viscous fluid is studied numerically by solving the full Navier-Stokes equations. The study considers the interaction between two bubbles for moderate Reynolds numbers (50 ≤ Re ≤ 500, Re being based on the bubble diameter) and for positions described by the separation S (2.5 ≤ S ≤ 10, S being the distance between the bubble centres normalized by the bubble radius) and the angle θ (0o ≤ θ ≤ 90o ) formed between the line of centre and the direction perpendicular to the direction of the motion. We provide a general description of the interaction extending the results obtained for two bubbles moving side by side (θ = 0o ) by Legendre, Magnaudet & Mougin 2003 (J. Fluid Mech., 497,133-166) and for two bubbles moving in line (θ = 90o ) by Yuan & Prosperetti 1994 (J. Fluid Mech., 278, 325-349). Simple models based on physical arguments are given for the drag and lift forces experienced by each bubble. The interaction is the combination of three effects: a potential effect, a viscous correction (Moore correction) and a significant wake effect observed on both the drag and the transverse force of the second bubble when located in the wake of the first one

Integrating functional diversity, food web processes, and biogeochemical carbon fluxes into a conceptual approach for modeling the upper ocean in a high-CO2 world

Marine food webs influence climate by channeling carbon below the permanent pycnocline, where it can be sequestered. Because most of the organic matter exported from the euphotic zone is remineralized within the "upper ocean" (i.e., the water column above the depth of sequestration), the resulting CO2 would potentially return to the atmosphere on decadal timescales. Thus ocean-climate models must consider the cycling of carbon within and from the upper ocean down to the depth of sequestration, instead of only to the base of the euphotic zone. Climate-related changes in the upper ocean will influence the diversity and functioning of plankton functional types. In order to predict the interactions between the changing climate and the ocean's biology, relevant models must take into account the roles of functional biodiversity and pelagic ecosystem functioning in determining the biogeochemical fluxes of carbon. We propose the development of a class of models that consider the interactions, in the upper ocean, of functional types of plankton organisms (e.g., phytoplankton, heterotrophic bacteria, microzooplankton, large zooplankton, and microphagous macrozooplankton), food web processes that affect organic matter (e.g., synthesis, transformation, and remineralization), and biogeochemical carbon fluxes (e.g., photosynthesis, calcification, respiration, and deep transfer). Herein we develop a framework for this class of models, and we use it to make preliminary predictions for the upper ocean in a high-CO2 world, without and with iron fertilization. Finally, we suggest a general approach for implementing our proposed class of models