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Moon W.,Yale University | Moon W.,Institute of Theoretical Geophysics | Wettlaufer J.S.,Yale University | Wettlaufer J.S.,University of Oxford
Journal of Mathematical Physics

We develop a perturbation theory for a class of first order nonlinear non-autonomous stochastic ordinary differential equations that arise in climate physics. The perturbative procedure produces moments in terms of integral delay equations, whose order by order decay is characterized in a Floquet-like sense. Both additive and multiplicative sources of noise are discussed and the question of how the nature of the noise influences the results is addressed theoretically and numerically. By invoking the Martingale property, we rationalize the transformation of the underlying Stratonovich form of the model to an Itô form, independent of whether the noise is additive or multiplicative. The generality of the analysis is demonstrated by developing it both for a Brownian particle moving in a periodically forced quartic potential, which acts as a simple model of stochastic resonance, as well as for our more complex climate physics model. The validity of the approach is shown by comparison with numerical solutions. The particular climate dynamics problem upon which we focus involves a low-order model for the evolution of Arctic sea ice under the influence of increasing greenhouse gas forcing δF0. The deterministic model, developed by Eisenman and Wettlaufer ["Nonlinear threshold behavior during the loss of Arctic sea ice," Proc. Natl. Acad. Sci. U.S.A.106(1), 28-32 (2009)] exhibits several transitions as δF0 increases and the stochastic analysis is used to understand the manner in which noise influences these transitions and the stability of the system. © 2013 AIP Publishing LLC. Source

Worster M.G.,Institute of Theoretical Geophysics | Jones D.W.R.,University of Oxford
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Significant changes in the state of the Arctic ice cover are occurring. As the summertime extent of sea ice diminishes, the Arctic is increasingly characterized by first-year rather than multi-year ice. It is during the early stages of ice growth that most brine is injected into the oceans, contributing to the buoyancy flux that mediates the thermo-haline circulation. Current operational sea-ice components of climate models often treat brine rejection between sea ice and the ocean similarly to a thermodynamic segregation process, assigning a fixed salinity to the sea ice, typical of multi-year ice. However, brine rejection is a dynamical, buoyancy-driven process and the salinity of sea ice varies significantly during the first growth season. As a result, current operational models may over predict the early brine fluxes from newly formed sea ice, which may have consequences for coupled simulations of the polar oceans. Improvements both in computational power and our understanding of the processes involved have led to the emergence of a new class of sea-ice models that treat brine rejection dynamically and should enhance predictions of the buoyancy forcing of the oceans. © 2015 The Author(s) Published by the Royal Society. All rights reserved. Source

Pegler S.S.,Institute of Theoretical Geophysics | Worster M.G.,Institute of Theoretical Geophysics
Journal of Fluid Mechanics

We present an experimental and theoretical study of a thin, viscous fluid layer that flows radially under gravity from a point source into a denser inviscid fluid layer of uniform depth above a rigid horizontal surface. Near the source, the viscous layer lies in full contact with the surface, forming a vertical-shear-dominated viscous gravity current. At a certain distance from the source, the layer detaches from the surface to form a floating current whose dynamics are controlled by the viscous stresses due to longitudinal extension. We describe the dynamics of the grounded and floating components using distinct thin-layer theories. Separating the grounded and floating regions is the freely moving line of detachment, or grounding line, whose evolution we model by balancing the horizontal forces between the two regions. Using numerical and asymptotic analysis, we calculate the evolution of the system from a self-similar form at early times towards a steady state at late times. We use our solutions to illustrate how three-dimensional stresses within marine ice sheets, such as that of West Antarctica, can lead to stabilization of the grounding line. To assess the validity of the assumptions underlying our model, we compare its predictions with data from a series of laboratory experiments. © 2013 Cambridge University Press. Source

Boait F.C.,Bullard Laboratories | White N.J.,Bullard Laboratories | Bickle M.J.,Bullard Laboratories | Chadwick R.A.,British Geological Survey | And 3 more authors.
Journal of Geophysical Research: Solid Earth

Time-lapse, three-dimensional (3D) seismic surveys have imaged an accumulation of injected CO2 adjacent to the Sleipner field in the North Sea basin. The changing pattern of reflectivity suggests that CO 2 accumulates within a series of interbedded sandstones and mudstones beneath a thick caprock of mudstone. Nine reflective horizons within the reservoir have been mapped on six surveys acquired between 1999 and 2008. These horizons have roughly elliptical planforms with eccentricities ranging between two and four. In the top half of the reservoir, horizon areas grow linearly with time. In the bottom half, horizon areas initially grow linearly for about eight years and then progressively shrink. The central portions of deeper reflective horizons dim with time. Amplitude analysis of horizons above, within, and below the reservoir show that this dimming is not solely caused by acoustic attenuation. Instead, it is partly attributable to CO2 migration and/or CO2 dissemination, which reduce the impedance contrast between sandstone and mudstone layers. Growth characteristics and permeability constraints suggest that each horizon grows by lateral spreading of a gravity current. This model is corroborated by the temporal pattern of horizon velocity pushdown beneath the reservoir. Horizon shrinkage may occur if the distal edge of a CO2-filled layer penetrates the overlying mudstone, if the buoyant plume draws CO2 upward, or if the effective permeability of deeper mudstone layers increases once interstitial brine has been expelled. Topographic control is evident at later times and produces elliptical planforms, especially toward the top of the reservoir. Our results show that quantitative mapping and analysis of time-lapse seismic surveys yield fluid dynamical insights which are testable, shedding light on the general problem of CO 2 sequestration. Copyright 2012 by the American Geophysical Union. Source

Pegler S.S.,Institute of Theoretical Geophysics | Worster M.G.,Institute of Theoretical Geophysics
Journal of Fluid Mechanics

We present a theoretical and experimental study of a viscous fluid layer spreading over a deep layer of denser, inviscid fluid. Specifically, we study an axisymmetric flow produced by a vertical line source. Close to the source, the flow is controlled viscously, with a balance between radial compressive stresses and hoop stresses. Further out, the flow is driven by gradients in the buoyancy force and is resisted by viscous extensional and hoop stresses. An understanding of these different fluid-mechanical relationships is developed by asymptotic analyses for early times and for the near and far fields at late times. Confirmation of the late-time, far-field behaviour is obtained from a series of laboratory experiments in which golden syrup was injected into denser solutions of potassium carbonate. We use our mathematical solutions to discuss a physical mechanism by which horizontal viscous stresses in a spreading ice shelf, such as those in West Antarctica, can buttress the grounded ice sheet that supplies it. © 2012 Cambridge University Press. Source

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