Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds

We investigate a potential relationship between mirror symmetry for Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and Landau-Ginzburg models. More precisely, we show that if a Calabi-Yau admits a so-called Tyurin degeneration to a union of two Fano varieties, then one should be able to construct a mirror to that Calabi-Yau by gluing together the Landau-Ginzburg models of those two Fano varieties. We provide evidence for this correspondence in a number of different settings, including Batyrev-Borisov mirror symmetry for K3 surfaces and Calabi-Yau threefolds, Dolgachev-Nikulin mirror symmetry for K3 surfaces, and an explicit family of threefolds that are not realized as complete intersections in toric varieties.Comment: v2: Section 5 has been completely rewritten to accommodate results removed from Section 5 of arxiv:1501.04019. v3: Final version, to appear in String-Math 2015, forthcoming volume in the Proceedings of Symposia in Pure Mathematics serie

Eddy Generation and Jet Formation via Dense Water Outflows across the Antarctic Continental Slope

Along various stretches of the Antarctic margins, dense Antarctic Bottom Water (AABW) escapes its formation sites and descends the continental slope. This export necessarily raises the isopycnals associated with lighter density classes over the continental slope, resulting in density surfaces that connect the near-freezing waters of the continental shelf to the much warmer circumpolar deep water (CDW) at middepth offshore. In this article, an eddy-resolving process model is used to explore the possibility that AABW export enhances shoreward heat transport by creating a pathway for CDW to access the continental shelf without doing any work against buoyancy forces. In the absence of a net alongshore pressure gradient, the shoreward CDW transport is effected entirely by mesoscale and submesoscale eddy transfer. Eddies are generated partly by instabilities at the pycnocline, sourcing their energy from the alongshore wind stress, but primarily by instabilities at the CDW–AABW interface, sourcing their energy from buoyancy loss on the continental shelf. This combination of processes induces a vertical convergence of eddy kinetic energy and alongshore momentum into the middepth CDW layer, sustaining a local maximum in the eddy kinetic energy over the slope and balancing the Coriolis force associated with the shoreward CDW transport. The resulting slope turbulence self-organizes into a series of alternating along-slope jets with strongly asymmetrical contributions to the slope energy and momentum budgets. Cross-shore variations in the potential vorticity gradient cause the jets to drift continuously offshore, suggesting that fronts observed in regions of AABW down-slope flow may in fact be transient features

Connecting Antarctic Cross-Slope Exchange with Southern Ocean Overturning

Previous idealized investigations of Southern Ocean overturning have omitted its connection with the Antarctic continental shelves, leaving the influence of shelf processes on Antarctic Bottom Water (AABW) export unconsidered. In particular, the contribution of mesoscale eddies to setting the stratification and overturning circulation in the Antarctic Circumpolar Current (ACC) is well established, yet their role in cross-shelf exchange of water masses remains unclear. This study proposes a residual-mean theory that elucidates the connection between Antarctic cross-shelf exchange and overturning in the ACC, and the contribution of mesoscale eddies to the export of AABW. The authors motivate and verify this theory using an eddy-resolving process model of a sector of the Southern Ocean. The strength and pattern of the simulated overturning circulation strongly resemble those of the real ocean and are closely captured by the residual-mean theory. Over the continental slope baroclinic instability is suppressed, and so transport by mesoscale eddies is reduced. This suppression of the eddy fluxes also gives rise to the steep “V”-shaped isopycnals that characterize the Antarctic Slope Front in AABW-forming regions of the continental shelf. Furthermore, to produce water on the continental shelf that is dense enough to sink to the deep ocean, the deep overturning cell must be at least comparable in strength to wind-driven mean overturning on the continental slope. This results in a strong sensitivity of the deep overturning strength to changes in the polar easterly winds

Sensitivity of the ocean's deep overturning circulation to easterly Antarctic winds

We investigate the sensitivity of the deep meridional overturning circulation (MOC) in the Southern Ocean to changes in the easterly wind stress over the Antarctic continental slope. The deep MOC is driven by export of dense Antarctic Bottom Water from the Antarctic continental shelf, and exerts a strong influence on climate by ventilating the deep ocean with oxygen and storing carbon dioxide. The possibility that inter-climatic modifications of the deep overturning may have been driven by changes in the atmospheric circulation has motivated the recent interest in evaluating the sensitivity of the Southern Ocean MOC to modifications of the mid-latitude westerlies. Using a high-resolution eddy-resolving model of a sector of the Southern Ocean, we show that the deep cell of the MOC is highly sensitive to the strength of the polar easterlies, and relatively insensitive to the strength of the mid-latitude westerlies. Our results highlight that determining the deep ocean ventilation in past and future climates demands an accurate evaluation of the concurrent surface wind stress throughout the Southern Ocean

Families of lattice polarized K3 surfaces with monodromy

We extend the notion of lattice polarization for K3 surfaces to families over a (not necessarily simply connected) base, in a way that gives control over the action of monodromy on the algebraic cycles, and discuss the uses of this new theory in the study of families of K3 surfaces admitting fibrewise symplectic automorphisms. We then give an application of these ideas to the study of Calabi-Yau threefolds admitting fibrations by lattice polarized K3 surfaces

Multidecadal warming of Antarctic waters

Decadal trends in the properties of seawater adjacent to Antarctica are poorly known, and the mechanisms responsible for such changes are uncertain. Antarctic ice sheet mass loss is largely driven by ice shelf basal melt, which is influenced by ocean-ice interactions and has been correlated with Antarctic Continental Shelf Bottom Water (ASBW) temperature. We document the spatial distribution of long-term large-scale trends in temperature, salinity, and core depth over the Antarctic continental shelf and slope. Warming at the seabed in the Bellingshausen and Amundsen seas is linked to increased heat content and to a shoaling of the mid-depth temperature maximum over the continental slope, allowing warmer, saltier water greater access to the shelf in recent years. Regions of ASBW warming are those exhibiting increased ice shelf melt

Calabi-Yau Threefolds Fibred by Mirror Quartic K3 Surfaces

We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic K3 surfaces. This is then used to give a complete explicit description of all Calabi-Yau threefolds fibred by mirror quartic K3 surfaces. We conclude by studying the properties of such Calabi-Yau threefolds, including their Hodge numbers and deformation theory.Comment: v2: Significant changes at the request of the referee. Section 3 has been rearranged to accommodate a revised proof of Proposition 3.5 (formerly 3.2). Section 5 has been removed completely, it will instead appear as part of Section 5 in arxiv:1601.0811

Bottom Boundary Potential Vorticity Injection from an Oscillating Flow: A PV Pump

Oceanic boundary currents over the continental slope exhibit variability with a range of time scales. Numerical studies of steady, along-slope currents over a sloping bathymetry have shown that cross-slope Ekman transport can advect buoyancy surfaces in a bottom boundary layer (BBL) so as to produce vertically sheared geostrophic flows that bring the total flow to rest: a process known as buoyancy shutdown of Ekman transport or Ekman arrest. This study considers the generation and evolution of near-bottom flows due to a barotropic, oscillating, and laterally sheared flow over a slope. The sensitivity of the boundary circulation to changes in oscillation frequency ω, background flow amplitude, bottom slope, and background stratification is explored. When ω/f ≪ 1, where f is the Coriolis frequency, oscillations allow the system to escape from the steady buoyancy shutdown scenario. The BBL is responsible for generating a secondary overturning circulation that produces vertical velocities that, combined with the potential vorticity (PV) anomalies of the imposed barotropic flow, give rise to a time-mean, rectified, vertical eddy PV flux into the ocean interior: a “PV pump.” In these idealized simulations, the PV anomalies in the BBL make a secondary contribution to the time-averaged PV flux. Numerical results show the domain-averaged eddy PV flux increases nonlinearly with ω with a peak near the inertial frequency, followed by a sharp decay for ω/f > 1. Different physical mechanisms are discussed that could give rise to the temporal variability of boundary currents

Ocean Circulation

The ocean moderates the Earth's climate due to its vast capacity to store and transport heat; the influence of the large-scale ocean circulation on changes in climate is considered in this chapter. The ocean experiences both buoyancy forcing (through heating/cooling and evaporation/precipitation) and wind forcing. Almost all ocean forcing occurs at the surface, but these changes are communicated throughout the entire depth of the ocean through the meridional overturning circulation (MOC). In a few localized regions, water become sufficiently dense to penetrate thousands of meters deep, where it spreads, providing a continuous source of deep dense water to the entire ocean. Dense water returns to the surface and thus closes the MOC, either through density modification due to diapycnal mixing or by upwelling along sloping isopycnals across the Southern Ocean. Determination of the relative contributions of these two processes in the MOC remains an active area of research. Observations obtained primarily from isotopic compositions in ocean sediments provide substantial evidence that the structure of the MOC has changed significantly in the past. Indeed, large and abrupt changes to the Earth's climate during the past 120,000 years can be linked to either a reorganization or a complete collapse of the MOC. Two of the more dramatic instances of abrupt change include Dansgaard-Oeschger events, abrupt warmings that could exceed 10°C over a period as short as a few decades, and Heinrich events, which are associated with massive freshwater fluxes due to rapid iceberg discharges into the North Atlantic. Numerical models of varying complexity that have captured these abrupt transitions all underscore that the MOC is a highly nonlinear system with feedback loops, multiple equilibria, and hysteresis effects. Prediction of future abrupt shifts in the MOC or “tipping points” remains uncertain. However, the inferred behavior of the MOC during glacial climates suggests that significant modifications to the present circulation are possible and that any change is likely to have a large effect on the Earth's climate