April 11 - 15, 2011
Keywords of the presentation: CO2 sequestration, fossil energy, multiscale methods, heterogeneity
Currently, mankind extracts most of the fuel for the global economy from underground resources, including oil, gas, and uranium deposits. The byproducts of consuming this fuel enter the atmosphere or remain on the surface. After years of waste build-up, this practice is no longer tenable. Clean technologies that sharply reduce CO2 emissions and other pollutants from power plants are essential bridge technologies on the path towards a sustainable energy future. A critical step will be the ability to cycle fuel byproducts back to their original home: the Earth's subsurface. Applications of this concept include storing CO2 in deep geologic formations and securing radioactive materials in appropriately engineered repositories. It is difficult to design and manage geologic sequestration efforts. Predictive computational simulation may be the only means to account for the lack of complete characterization of the subsurface environment, the multiple scales of the various interacting processes, the large areal extent of reservoirs, and the need for long time predictions. This talk will discuss background issues, and recent work by the author on multiscale methods for computing flow fields in porous media with extreme natural heterogeneities. The methods are suitable for parallel computation through the use of nonoverlapping domain decomposition mortar methods with a restricted set of degrees of freedom on the interfaces. We devise an effective but purely local multiscale method that incorporates information from homogenization theory. We also use this decomposition method approach to devise effective preconditioners that incorporate exact coarse-scale information to iteratively solve the full fine-scale problem.
We study a convection-diffusion type transport equation that is fully coupled to the Navier-Stokes/Darcy flow via velocity field and concentration. This problem is related to the groundwater contamination through rivers. On the interface, we accept balance of forces, continuity of the flux and the Beavers-Joseph-Saffman condition. Existence of a weak solution is shown by a method based on Galerkin approach in time. Furthermore, for the special case where the coupling is only one way via the velocity, we provide numerical analysis and simulations with methods based on continuous and discontinuous Galerkin methods.
Keywords of the presentation: tropical cyclone, direct impact assessment
Society is entering a new era of catastrophes in which natural hazards are causing more damage than in the past. Recent years have seen a steep rise in economic and insured losses from weather and climate related hazards, largely due to a significant increase in exposure, but the likely scenario of the hazards themselves becoming more damaging in the future will only add to increased societal vulnerability. Recent failures in current risk management strategies have highlighted the need for a step change in our understanding of weather and climate risk.
Earth system models are poised to bring about this change by providing new and independent risk assessments with respect to traditional practice based in historical observations. This talk will address the opportunities and challenges of incorporating dynamical models into climate risk assessment with a focus on tropical cyclone risk. Results will be presented from collaborations between NCAR and the reinsurance and energy industries to understand tropical cyclone risk and resulting impacts and losses.
We are at the dawn of this new era of weather and climate risk assessment and these initial developments hold profound possibilities for the future. I will address the mathematical and computational challenges in achieving this step change in our understanding of climate risk.
Keywords of the presentation: tsunamis, finite volume methods, adaptive mesh refinement, debris flows
Mathematical and computational modeling plays an important role in
many aspects of risk mitigation for tsunamis and other hazardous geophysical flows (flooding, landslides and debris flows). Modeling these phenomena accurately and efficiently requires specialized numerical methods and software, as they present unique computational challenges. For instance, with tsunami modeling, the vastly different spatial
scales between propagation over the ocean and the study of a small
region of the coast makes the use of adaptive mesh refinement
crucial. Studying inundation requires wetting/drying algorithms
that can handle the depth going to zero at the shoreline. Well-balanced
methods must be used to accurately capture waves on the open ocean,
where their amplitude is very small relative to the fluid depth. Modeling landslides and debris flows requires the development of suitable mathematical models that can account for the complicated internal stresses of a flowing mixture of solid particles and fluid. The spectrum of flows, ranging from landslides and debris flows to tsunamis, are often modeled with depth-averaged equations of which the shallow water equations are the simplest example. Depth-averaged models for landslides requires additional equations to account for the solid volume fraction and pore-fluid pressure. However, these equations present similar mathematical difficulties.
I will discuss some of these models, challenges and algorithms. I will also introduce the GeoClaw software,
a specialized version of Clawpack that is aimed at solving these real-world
geophysical flow problems over topography. I will show results
from some recent tsunamis and potential future events, and discuss
some of the ways modeling can be used to assess hazards.
Read More...
Whether modeling is performed on a local, regional or global scale, for scientific or engineering purposes, uncertainty is inherently present due to lack of data and lack of understanding of the underlying phenomena and processes taking place.
In this tutorial, we will discuss tools available for modeling uncertainty of complex Earth systems as well as the impact it has on practical geo-engineering decision problems.
The general circulation of the ocean plays a major role in the global climate system, and the local circulation in near-shore regions is also of substantial scientific and societal interest. This talk will give an introduction to some of the physical and computational issues that arise when ocean circulation is modeled numerically. These include motivations for ocean modeling; length, time, and mixing scales; the choice of vertical coordinate; governing equations; multiple time scales and time-stepping; spatial grids and discretization schemes; and some current work and upcoming issues.
Read More...
Keywords of the presentation: sea level, ice-ocean interaction
Global sea level has fluctuated significantly in the past, over glacial and inter-glacial time scales. Such variations arise from the slow buildup and even more rapid collapse of major ice sheets, including the present day Greenland and Antarctic Sheets. In this presentation mechanisms of ice sheet collapse are reviewed and implications for society over the next century and beyond are explored. Both ongoing observational and computational efforts aimed at addressing sea level change are surveyed. Areas for scientific advance are discussed.
Joint work with Paul A. Ullrich (University of Michigan).
The future generation of atmospheric models used for weather and climate predictions will likely rely on both high-order accuracy and Adaptive Mesh Refinement (AMR) techniques in order to properly capture the atmospheric features of interest. We present our ongoing research on developing a set of conservative and highly accurate numerical methods for simulating the atmospheric fluid flow (the so-called dynamical core). In particular, we have developed a fourth-order finite-volume scheme for a nonhydrostatic dynamical core on a cubed-sphere grid that makes use of an implicit-explicit Runge-Kutta-Rosenbrock time integrator and Riemann solvers. The poster surveys the algorithmic steps, presents results from idealized dynamical core test cases and outlines the inclusion of AMR into the model design.
Keywords of the presentation: variable-resolution climate modeling, nonhydrostatic dynamical core, cubed-sphere grid, high-order finite-volume technique, model intercomparisons
Joint work with Paul A.Ullrich and Kevin A. Reed (University of Michigan).
Numerical predictions of high-impact local weather events and the vastly growing demand for regional-local climate predictions are grand challenge problems and one of the main drivers for
novel atmospheric models that are ready for high-performance computing architectures. Future-generation atmospheric General Circulation Models (GCMs) and their fluid dynamics components (the so-called dynamical cores) will likely rely on both high-order accuracy and variable-resolution techniques in order to seamlessly capture the multi-scale flow regimes.
The talk surveys the design of conservative and highly accurate numerical methods for an Adaptive-Mesh-Refinement (AMR) dynamical core of a GCM. In particular, the talk discusses a fourth-order finite-volume scheme for a nonhydrostatic dynamical core on a cubed-sphere grid that makes use of an implicit-explicit Runge-Kutta-Rosenbrock time integrator and Riemann solvers. The talk overviews the algorithmic steps, presents results from idealized dynamical core test cases, outlines the inclusion of AMR into the model design and discusses strategies how to test and intercompare GCMs.
Keywords of the presentation: climate computing
Climate models are used to understand the complex interactions that result in climate change as well as provide projections of future climate change and its impacts on society. I will give a broad overview of current climate and Earth system models, including the diversity of algorithms represented, the complexity of the models and their application to both science and policy problems. Future demands on climate model applications as well as a changing computing landscape will require rethinking the design and implementation of our models. I will describe some of the bigger challenges and potential paths forward.
Joint work with
Matthias Ruth^{1}, Ning Zeng^{1},
Safa Motesharrei^{1}, and Jorge Rivas^{2}.
Earth System Models (ESM) designed to study climate change should include a fully coupled Human model, with submodels such as Population, Energy, Agriculture and Fisheries, Water, as well as environmental sources and sinks. Specifically, fully coupled means that subcomponents of a model (e.g., the atmosphere and the ocean) are coupled in a two-way fashion: for example, the atmosphere can change the ocean and in turn is affected by the feedback of this change. The importance of having positive, negative, and delayed feedbacks is obvious: the phenomenon of El Niño-Southern Oscillation (ENSO) is the result of such feedbacks, and thus, one-way coupled ocean-atmosphere models (used until about 1990) were not able to reproduce ENSO.
ESMs are currently much more comprehensive than they used to be (e.g., the Community ESM, CESM), and include fully coupled land-ocean-atmosphere components. Vegetation models, which used to influence the climate of the model but without feedbacks (one-way coupling), are now also driven by the climate (two-way coupling). However, there is an important component of the Earth System (ES) that is not included in the ESMs: the Human System, which in reality is not only strongly coupled but actually dominates many components of the ES. For example, the human appropriated portion of the Natural Primary Productivity (HANPP) is estimated to be at least 25%, and about 60% is affected by human activities. Nevertheless, population remains a taboo subject in the discussions and policies on climate change. In this talk we discuss how to develop a fully coupled ESM-Human model in order to be able to study the role of population in climate change, and the new mathematical problems that may arise.
^{1}University of Maryland, ^{2}University of Minnesota
Keywords of the presentation: Inverse Problems, Stochastic, Imaging, Geophysics, Hydrology
The subsurface is where most of the available freshwater is stored; in the United States, groundwater is the primary source of water for over 50 percent of Americans, and roughly 95 percent for those in rural areas. Cleaning up the surface from industrial and nuclear wastes is quite challenging. A major impediment in studying processes in the subsurface and in managing resources is that it is difficult to achieve accurate and reliable imaging, i.e., identification of properties, of geologic formations. Some of the difficulties one has to overcome are heterogeneity of subsurface environments that manifests itself in complex ways and at all spatial scales, field measurements that are not only expensive to get but are affected by disturbances and factors that are hard to manage or model, and the non-uniqueness of the mapping from observables to the underlying formation properties. In this talk, we will discuss stochastic methods to explore the range of solutions or “images” that are consistent with measurements and to quantify the uncertainty in predictions. We will also discuss computational challenges posed by the need to process large data sets and to resolve variability at small scales.
An important emerging scientific issue in many practical problems ranging from climate and weather prediction to biological science involves the real time filtering and prediction through partial observations of noisy turbulent signals for complex dynamical systems with many degrees of freedom as well as the statistical accuracy of various strategies to cope with the “curse of dimensions”. The speaker and his collaborators, Harlim (North Carolina State University), Gershgorin (CIMS Post doc), and Grote (University of Basel) have developed a systematic applied mathematics perspective on all of these issues. One part of these ideas blends classical stability analysis for PDE's and their finite difference approximations, suitable versions of Kalman filtering, and stochastic models from turbulence theory to deal with the large model errors in realistic systems. Many new mathematical phenomena occur. Another aspect involves the development of test suites of statistically exactly solvable models and new NEKF algorithms for filtering and prediction for slow-fast system, moist convection, and turbulent tracers. Here a stringent suite of test models for filtering and stochastic parameter estimation is developed based on NEKF algorithms in order to systematically correct both multiplicative and additive bias in an imperfect model. As briefly described in the talk, there are both significantly increased filtering and predictive skill through the NEKF stochastic parameter estimation algorithms provided that these are guided by mathematical theory. The recent paper by Majda et al (Discrete and Cont. Dyn. Systems, 2010, Vol. 2, 441-486) as well as a forthcoming introductory graduate text by Majda and Harlim (Cambridge U. Press) provide an overview of this research.
Geophysical flows are a rich source of novel problems for applied mathematics and the contemporary theory of partial differential equations. The reason for this is that many physically important geophysical flows involve complex nonlinear interaction over multi-scales in both time and space so developing simplified reduced models which are simpler yet capture key physical phenomena is of central importance. In mid-latitudes, the fact that the rotational Coriolis terms are bounded away from zero leads to a strict temporal frequency scale separation between slow potential vorticity dynamics and fast gravity waves; this physical fact leads to new theorems justifying the quasi-geostrophic midlatitude dynamics even with general unbalanced initial data for both rapidly rotating shallow water equations and completely stratified flows.
At the equator, the tangential projection of the Coriolis force from rotation vanishes identically so that there is no longer a time scale separation between potential vortical flows and gravity waves. This has profound consequences physically that allow the tropics to behave as a waveguide with extremely warm surface temperatures. The resulting behavior profoundly influences longer term mid-latitude weather prediction and climate change through hurricanes, monsoons, El Nino, and global teleconnections with the mid-latitude atmosphere. How this happens through detailed physical mechanisms is one of the most important contemporary problems in the atmosphere-ocean science community with a central role played by nonlinear interactive heating involving the interaction of clouds, moisture, and convection. The variable coefficient degeneracy of the Coriolis term at the equator alluded to earlier leads to both important new physical effects as well as fascinating new mathematical phenomena and PDE’s. In this equatorial context, new multi-scale reduced dynamical PDE models are even relatively recent in origin.
After a brief discussion of the observational record as background, this lecturer surveys the remarkable new hyperbolic systems that have emerged recently in applications including their physical properties, applied mathematical and rigorous mathematical theory. These last topics include novel relaxation limits for climate models with active moisture and new singular limits for hyperbolic PDE’s with variable coefficients. All of the references in this lecture can be found at
.http://www.math.nyu.edu/faculty/majda/.
We propose a simple and computationally inexpensive model for the
description of the sea bed displacement during an underwater
earthquake, based on the finite fault solution for the slip
distribution under some assumptions on the dynamics of the rupturing
process. Once the bottom motion is reconstructed, we study waves
induced on the free surface of the ocean using three different models
approximating the Euler equations of the water wave theory. The
developments of the present study are illustrated on the July 17, 2006
Java event.
Keywords of the presentation: eruption, multiphase, CFD, turbulence, volcano
The violent nature of explosive volcanic eruptions makes understanding their behavior both imperative and extremely challenging. These dangerous natural phenomena threaten society in a variety of ways ranging from destruction of local communities to disrupting global air traffic to influencing global climate change. Our ability to mitigate the risks posed by volcanoes is hampered by our limited understanding of their controlling physics. The opacity and violence of eruptions makes them difficult and dangerous to measure directly. We therefore depend strongly on numerical models to study and understand eruptive dynamics. These models range in sophistication and computational expense depending on their purpose. Real-time hazard assessment requires parameterized models that can run quickly on a desktop computer during eruptions. On the other hand, 3D time-dependent computational fluid dynamics models run on massively parallel supercomputers are necessary to capture and study the turbulent, multiphase flow that controls eruption behavior. This talk will discuss some of the computational issues and challenges currently faced by the volcanology community.
Geostrophic turbulence on a surface of a rotating sphere (so called beta-plane turbulence) has been simulated using the newly developed beta-sQG+1 numerical model. This model incorporates higher order terms beyond the standard quasi-geostrophy. The domain occupied by the fluid has a channel geometry with 512 by 256 grid points, periodic boundary conditions in x-direction and rigid boundaries in y-direction. To better understand wave-vortices dynamics both cases, with and without random forcing, are investigated. Both simulations start from identical random initial conditions and exhibit different dynamical properties.
In the freely evolving case, adding a wave term that competes with inertia on larger scales produces high meridional asymmetry in eddies spatial and time scales. This novel asymmetry is added to the standard for the beta-plane turbulence zonal asymmetry. The model in the forced regime exhibits not only anisotropy in eddies deformation radius, but also in their orientation. The warm anomalies are elongated in the north-western direction, while the cold anomalies are elongated in the north-eastern direction. As a result there is a meridional meandering in the formed zonal jets.
Keywords of the presentation: coupled PDE models, multiphase multicomponent flow and transport, phase transitions, methane hydrates, coalbed methane recovery, porescale modeling, hybrid modeling
In the talk we describe two applications important for global climate
and energy studies: methane hydrates and coalbed methane. Methane
hydrates also known as "ice that burns" are present in large amounts
along continental slopes and in permafrost regions and, therefore, are
a possible source of energy and at the same time a potential
environmental hazard. Their evolution critically depends on how the
hydrate formation and dissociation affects the porescale properties.
This so far has been only modeled with ad-hoc phenomenological
approaches on top of the continuum models which account for multiple
flowing phases, energy conservation, and phase change with or without
latent heat. A similar situation arises in coalbed methane recovery
where the traditional models of multicomponent adsorption appear
inadequate to capture the dynamics of coupled porescale processes
involving matrix swelling, competitive adsorption between carbon
dioxide and methane, and adsorption hysteresis.
Both applications have had comprehensive computational realizations
based on coupled nonlinear PDE systems whose analysis has not yet been
carried out. More importantly, both call for broadening the scope of
modeling tools from traditional continuum PDE-based models to include
a variety of discrete models which help to understand processes at
porescale and to formulate constitutive relationships useful at
continuum scale. In the talk we outline challenges of traditional
continuum models, present some porescale results, and introduce some promising hybrid modeling approaches.
Keywords of the presentation: cloud, turbulence, rain, preferential concentration, collision
Clouds contain fine water droplets in moist air, that may grow to form rain drops. A precise and quantitative understanding of the growth of droplets remains elusive, despite the fact that the essential physical mechanisms responsible for the process are known. In particular, the accepted scenario of rain formation has long rested on the notion of droplets of different sizes falling at different terminal velocities in a quiescent fluid, therefore colliding and coalescing. It has been realized that such approaches lead to wrong predictions of the time it takes for rain to form, in particular in warm clouds.
The turbulent motion of air in clouds is likely to play an important role in the formation of rain drops. I will review in this talk the physical effects induced by turbulence, acting on particles that are much heavier than the surrounding fluids, such as water droplets in air. The main ones are preferential concentration -- the very uneven distribution of droplets, as well as the formation of "caustics". These effects lead to a strong enhancement of collision rates, which can play an important role in enhancing the rate of drop formation in clouds.
I will also discuss some of the current experimental efforts to document the role of turbulence in cloud microphysics.
Keywords of the presentation: stochastic parametrization, wave current interactions, multiscale ocean dynamics
The challenge of producing practical models that accurately capture the interaction of waves and currents
is primarily a problem of the enormous spatio-temporal scales: on the Continental Shelf, we require capturing
dynamics that span seconds to seasons, and spatial structures spanning meters to basins. Using filtering
and asymptotics we have been able to develop a comprehensive model for waves and currents, that capture
the conservative portion of the dynamics.
An important source of momentum exchanges between waves and currents are breaking events. These
whitecapping events are short-lived yet not ignorable even on the very largest of spatio-temporal scales.
We have developed a stochastic parametrization that models breaking events as sources of uncertainty in
the Lagrangian paths of fluid parcels. Application of the projection to the Eulerian frame along with filtering
yields their contribution to the dynamics of waves and currents.
Keywords of the presentation: global atmospheric chemistry, multi-scale analysis, chemical kinetics
Understanding the global-scale dynamics of the chemical composition of our atmosphere is essential for addressing a wide range of environmental issues from air quality to climate change. Understanding this phenomenon enables us to evaluate and devise appropriate environmental policies, such as the Kyoto Protocol on global greenhouse gases emissions. Numerical modeling of global atmospheric chemical dynamics presents an enormous challenge associated with simulating hundreds of chemical species with time scales varying from milliseconds to years. In my talk, I will present an overview of the state of the art in global atmospheric chemistry modeling and will point out some of the mathematical challenges that need attention in this field.
A large, unpredicted, water level increase appeared along a
substantial section of the western
Louisiana and northern Texas (LATEX) coasts 12-24 hrs in
advance of the landfall of Hurricane
Ike (2008), with water levels in some areas reaching 3m above
mean sea level. During this time
the cyclonic wind field was largely shore parallel throughout
the region. A similar early water
level rise was reported for both the 1900 and the 1915
Galveston Hurricanes. The Ike forerunner
anomaly occurred over a much larger area and prior to the
primary coastal surge which was
driven by onshore directed winds to the right of the storm
track. We diagnose the forerunner
surge as being generated by Ekman setup on the wide and shallow
LATEX shelf by simulating the hindcast with
Coriolis turned on and off as well as with various frictional
formulations. The longer
forerunner time scale additionally served to increase water
levels significantly in narrow entranced
coastal bays.
The forerunner surge generated a freely propagating continental
shelf wave with greater than
1.4m peak elevation that travelled coherently along the coast
to Southern Texas, and was 300km
in advance of the storm track at the time of landfall. This
was, at some locations, the largest
water level increase seen throughout the storm, and appears to
be the largest freely-propagating
shelf wave ever reported. Ekman setup-driven forerunners will
be most significant on wide,
forecasting in these cases.
Reference:
Kennedy, A.B., U. Gravois, B.C. Zachry, J.J. Westerink, M.E.
Hope, J.C. Dietrich, M.D. Powell, A.T. Cox, R.A. Luettich, R.G.
Dean, "Origin of the Hurricane Ike Forerunner Surge,"
Geophysical Research Letters, In Press, 2011.
Keywords of the presentation: Storm Surge, wind waves, hurricane, flooding, inundation, Shallow Water Equations, Finite Elements, unstructured grids, coupled wave-circulation modeling, high performance computing, scalability
Coastal Louisiana and Texas are characterized by tremendous complexity and variability in their geography, topography, bathymetry, continental shelf, estuarine systems, and surface roughness. Hurricane Ike significantly impacted both coastal Texas and Louisiana producing a storm surge of more than 5.3m in eastern Texas and more than 2.2 m in eastern Louisiana (more than 500 km away from the storm landfall location). Particularly important was that more than 2m of hurricane forerunner developed prior to the storm coming onto the continental shelf (more than 15 hours prior to landfall), while coastal winds were shore parallel or coming off of the land. The forerunner flooded much of western Louisiana and eastern Texas, and filled Galveston Bay in its entirety, reaching into the heart of Houston. The forerunner then propagated down the Louisiana-Texas (LATEX) shelf as a free wave, passing Corpus Christi with an amplitude of more than 1m. The forerunner is the largest ever recorded.
The rapid evolution of data collection systems allows the physical system to be accurately defined, and the rapid evolution of unstructured grid computational models allows these characteristics and the resulting waves and flows to be numerically resolved. The SWAN+ADCIRC unstructured grid modeling system has been developed to simulate fully coupled hurricane winds, wind-waves, storm surge, tides and river flow in this complex region. This is accomplished by defining a domain and computational resolution appropriate for the relevant processes, specifying realistic boundary conditions, and implementing accurate, robust, and highly parallel unstructured grid algorithms for both the wind waves and the long wave current/storm surge/tide model. Basin to channel scale domains and high resolution grids which resolve features down to 30 meters and contain up to 3.3 million nodes have been developed. This modeling system is run on up to 4,096 processors and requires as little as 18 minutes of wall clock time per day of simulation.
Hindcasts of the storm indicate an excellent match of measured wave and surge records. Numerical experiments indicate that the unprecedented forerunner was generated by very fast shore parallel currents driven by the early shore parallel winds that allow for a Coriolis driven set up to be pushed up against the coast. Achieving fast enough currents on the mid and outer shelf is vital for the driving mechanism to work. This in turn requires low frictional resistance which is consistent with the smooth and muddy LATEX shelf.
In this work we discuss separation of time scales for rotating and stratified flow in the limit of fast rotation and weak stratification.
Keywords of the presentation: artic ocean, projection operator, vortices, Taylor Proudman flows, separation of time scales
Earth's high latitudes stand to be among the first regions affected by
climate change issues due to changes induced by melting ice in the
Arctic and Antarctic. Motivated by gaining fundamental understanding
of ocean dynamics at high latitudes my collaborators and I have
derived new equations, based on the method of multiple scales
presented in Embid and Majda (1996,1998), that address the scale
separation between slow- and fast-time scale dynamics in the limit of
fast rotation while retaining order one affects due to stratification.
The new slow equations and their conservation laws describe the {sl
dynamics of Taylor-Proudman flows}. We also present numerical
results that support the theory and that show the spontaneous creation
of Taylor-Proudman columns. We also show recent measurements from the NSF Beaufort Gyre exploration program that show strong, deep columnar vortices with speeds as much as 30cm/s and which span the depth of the weakly stratified water column. The projection operators derived as a part
of these results have mathematical and numerical implications for the
development of time-stepping algorithms in next-generation exascale climate models.