Adaptive Multiscale Methods for the Atmosphere and Ocean
Wednesday 22nd August 2012 to Friday 24th August 2012
13:00 to 13:30  Welcome and introduction to programme and workshop  
13:30 to 14:30 
M Piggott (Imperial College London) Modelling geophysical fluid dynamics with anisotropic adaptive mesh methods
Many geophysical fluid dynamics problems include large variations in spatial scales that are important to resolve in a numerical simulation. An important example being the global ocean where dynamics at spatial scales of thousands of kms has strong twoway coupling with processes occurring at the km, and subkm, scale, e.g. boundary layers, eddies and buoyancy driven flows interacting with bathymetry. Adaptive and unstructured mesh methods represent a possible means to simulate these multiscale systems efficiently. In addition, smaller scale processes often have high aspect ratios and hence anisotropic mesh methods should be considered. In this talk our work applying anisotropic adaptive methods to geophysical fluid dynamics problems will be reviewed. A series of recent applications will also be presented.

INI 1  
14:30 to 15:00 
J Steppeler (Deutscher Wetterdienst (DWD)) Variable resolution and uniform second and third Order approximation order
Approximations on polygonal grids, such as cube sphere or icosahedral require a slightly irregular grid, as for high resolution no uniform polygonal cover by cells is possible. For variable resolution sudden refinement is considered not possible by some authors and a gradual change of resolution is preferred,such as with the new MPAS model of NCAR. It will be shown that such problems can be traced back to a decrease of approximation order to 1 in points where the resolution is irregular. Examples will be given to show that a jump of resolution is possible without problems when care is taken that in such points an approximation order 2 or 3 is maintained.

INI 1  
15:00 to 15:30  Afternoon Tea  
15:30 to 16:30 
Development of the Nonhydrostatic Unified Model of the Atmosphere (NUMA): a unified model for both localarea modeling and global modeling
In this talk I will give an overview of the Nonhydrostatic Unified Model of the Atmosphere (NUMA). NUMA solves the fully compressible nonhydrostatic equations with the goal to have it unified across various fronts including: applications (localarea and global modeling); numerics (using both continuous and discontinuous Galerkin methods); timeintegration (explicit and implicitexplicit methods); iterative solvers and preconditioners (using a suite of these); grid generation (using both conforming and nonconforming grids); and finally, parallelization (using both CPU and GPU based parallelization). In this talk, we will describe what we mean by each of these components and will report on the status of each of these components. The work that we shall describe will set the stage for the work we wish to carry out during the stay of my group at the Newton Institute.

INI 1  
16:30 to 17:00 
M Taylor (Sandia National Laboratories) Variable resolution experiments using CAM's spectral finite element dynamical core
Much recent work in the Community Earth System Model (CESM) and its atmosphere component (CAM) has been devoted to developing higher resolution configurations, motivating new dynamical cores which can use unstructured quasiisotropic grids. These dynamical cores also allow for variable resolution configurations in CAM, but making use of variable resolution (adaptive or statically refined grids) is difficult due to the strong resolution sensitivity of CAM's many subgrid physical parameterizations.
Here we will describe our work with statically refined grids in CAM, using CAM's spectral finite element dynamical core. This work is supporting the development of a model/observation "test bed". Testbeds combine models, observations and uncertainty quantification methodologies in order to evaluate existing models, quickly develop and test new parameterizations and constrain parameters with observations. It is hoped that variable resolution can provide a 10100 times more efficient way to calibrate and evaluate highresolution configurations of CAM. Our initial focus is on central U.S. precipitation, using a global 14km grid and a variable resolution grid with 14km resolution over the central U.S., transitioning to 110km over most of the globe. For both configurations, we will present computational performance and compare precipitation related diagnostics.

INI 1  
17:00 to 18:00  Welcome Drinks Reception 
09:00 to 10:00 
J Behrens (Universität Hamburg) Reviewing a roadmap for adaptive atmospheric modeling
In my 2006 monograph on Adaptive Atmospheric Modeling the last section titled "Roadmap for the next five years" outlines several aspects of numerical methods for multiscale phenomena in the atmosphere. Now, more than five years later, it is time to review a few of the issues raised in 2006. As one would expect, the grand solution to adaptive multiscale modeling has not been found so far, and some solutions have generated new questions. But a lot has been achieved since 2006 and it is interesting to see, which directions have been taken. I will cover the aspects of consistent numerical methods, refinement criteria and strategies, applications and efficiency of adaptive methods in atmosphere and ocean applications.

INI 1  
10:00 to 10:30 
A highly adaptive three dimensional hybrid vortex method for inviscid flows
Motivated by outstanding problems surrounding vortex stretching, a new numerical method to solve the inviscid Euler equations for a threedimensional, incompressible fluid is presented. Special emphasis on spatial adaptivity is given to resolve as broad a range of scales as possible in a completely selfsimilar fashion. We present a hybrid vortex method whereby we discretise the vorticity in Lagrangian filaments and perform an inversion to compute velocity on an adapted finitevolume grid. This allows for a twofold adaptivity strategy. First, although naturally spatially adaptive by definition, the vorticity filaments undergo `renoding'. We redistribute nodes along the filament to concentrate their density in regions of high curvature. Secondly the Eulerian mesh is adapted to follow high strain by increasing resolution based on local filament dimensions. These features allow vortex stretching and folding to be resolved in a completely automatic and selfsimilar way. The method is validated via well known vortex rings and newly discovered helical vortex equilibria are also used to test the method.

INI 1  
10:30 to 11:00  Morning Coffee  
11:00 to 11:30 
Adaptive mesh method in the Met Office variational data assimilation system
A frequent problem in forecasting fog or icy roads in a numerical weather prediction system is attributed to the misinterpretation of the boundary layer structure in the assimilation procedure. Case studies showed that much of the misinterpretation of temperature inversions and stratocumulus layers in the assimilation is due to inappropriate background error covariances. This paper looks at the application of adaptive mesh methods in the Met Office variational assimilation system to modify the background error correlations in the boundary layer when temperature inversions or stratocumulus layers are present in the background state.

INI 1  
11:30 to 12:30 
Adaptive Numerical Simulation of Idealized Cyclones
The processes causing a cyclone's formation, its intensification, its motion and finally its terminating all proceed at multiple and interacting temporal and spatial scales. Therefore the forecasting of a cyclone's dynamic can make great benefits of adaptive techniques such as local mesh refinement. Mesh adaptation strategies are often based on problemdependent indicators that need to be determined and which have different properties with respect to the associated computational effort and the quality of the resulting meshes. For example, the computation of goaloriented error indicators requires sensitivity information provided by the solution of an additional linear problem which leads to a remarkable overhead in computation time and additional storage requirements. In that context, we address the question whether the complexity of different criteria for refinement is justifiable. To this end, we investigate idealized cyclone scenarios and systematically analyze the efficiency of adaptive numerical methods employing a selection of different indicators based on physical criteria, heuristic criteria, a posteriori error estimators, and goaloriented approaches. We compare these approaches with regard to the related computational costs, storage requirements, implementation complexity, and the accuracy of the resulting solution.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:00 
A Müller (Naval Postgraduate School) Are adaptive simulations more accurate than uniform simulations?
Adaptive mesh refinement generally serves to increase computational efficiency without compromising the accuracy of the numerical solution significantly. However it is an open question in which regions the spatial resolution can actually be coarsened without affecting the accuracy of the result significantly. Another open question is the following: does an adaptive computation simulate large scale features of the flow more accurately than a uniform simulation when both use the same CPU time? These questions are investigated in the case of a 2D dry warm air bubble with the help of a recently developed adaptive discontinuous Galerkin model.
A method is introduced which allows one to compare the accuracy between different choices of refinement regions even in a case when the exact solution is not known. Essentially this is done by comparing features of the solution that are strongly sensitive to spatial resolution. The additional error by using adaptivity is smaller than 1% of the total numerical error if the average number of elements used for the adaptive simulation is about 50% smaller than the number used for the simulation with the uniform fineresolution grid. Correspondingly the adaptive simulation is almost two times faster than the uniform simulation. Furthermore the adaptive simulation is more accurate than a uniform simulation when both use the same CPU time.

INI 1  
14:00 to 15:00  Breakout and discussion: the Newton Institute Programme, where should it lead, what will we achieve?  INI 2  
15:00 to 15:30  Afternoon Tea  
15:30 to 16:30 
D Holm (Imperial College London) Parameterizing interaction of disparate scales: selective decay by Casimir dissipation in fluids
The problem of parameterizing the interactions of disparate scales in fluid flows is addressed by considering a property of twodimensional incompressible turbulence. The property we consider is selective decay, in which a Casimir of the ideal formulation (enstrophy in 2D flows) decays in time, while the energy stays essentially constant. This paper introduces a mechanism that produces selective decay by enforcing Casimir dissipation in fluid dynamics. This mechanism turns out to be related in certain cases to the numerical method of anticipated vorticity discussed in \cite{SaBa1981,SaBa1985}. Several examples are given and a general theory of selective decay is developed that uses the LiePoisson structure of the ideal theory. A scaleselection operator allows the resulting modifications of the fluid motion equations to be interpreted in several examples as parameterizing the nonlinear, dynamical interactions between disparate scales. The type of modified fluid equation systems derived here may be useful in turbulent geophysical flows where it is computationally prohibitive to rely on the slower, indirect effects of a realistic viscosity, such as in largescale, coherent, oceanic flows interacting with much smaller eddies.

INI 1  
16:30 to 17:00 
Adaptive multiscale discontinuous Galerkin methods for multiphase morphodynamics
We present a strongly coupled, eigendecomposition problem for an extension of the Saint–Venant shallow water equations in two dimensions strongly coupled to a completely generalized Exner form of the sediment discharge equation. This formulation is used to implement an adaptive discontinuous Galerkin (DG) finite element method, using a Roe Flux for the advective components and the unified form for the dissipative components. We discuss important mathematical and numerical nuances that arise due to the emergence of nonconservative product formalisms in the presence of sharp gradients, and present some large scale candidate application models with examples

INI 1  
19:30 to 22:00  Conference Dinner at Corpus Christi College 
09:00 to 10:00 
A Framework for the Development of Computable Error Bounds for Finite Element Approximations
We present an overview of our recent work on the development of fully computable upper bounds for the discretisation error measured in the natural (energy) norm for a variety of problems including linear elasticity, convectiondiffusionreaction and Stokes flow in three space dimensions. The upper bounds are genuine upper bounds in the sense that the actual numerical value of the estimated error exceeds the actual numerical value of the true error regardless of the coarseness of the mesh or the nature of the data for the problem, and are applicable to a variety of discretisation schemes including conforming, nonconforming and discontinuous Galerkin finite element schemes. All constants appearing in the bounds are fully specified. Numerical examples show the estimators are reliable and accurate even in the case of complicated three dimensional problems, and are suitable for driving adaptive finite element solution algorithms.

INI 1  
10:00 to 10:30 
Adapting to life: Ocean ecosystem modelling using an unstructured and adaptive mesh ocean model
Primary production in the world ocean is significantly controlled by meso and submesocale process. Thus existing general circulation models applied at the basin and global scale are limited by two opposing requirements: to have high enough spatial resolution to resolve fully the processes involved (down to order 1km) and the need to realistically simulate the basin scale. No model can currently satisfy both of these constraints. Adaptive unstructured mesh techniques offer a fundamental advantage over standard fixed structured mesh models by automatically generating very high resolution at locations only where and when it is required. Mesh adaptivity automatically resolves finescale physical or biological features as they develop, optimising computational cost by reducing resolution where it is not required.
Here, we describe FluidityICOM, a nonhydrostatic, finiteelement, unstructured mesh ocean model, into which we have embedded a sixcomponent ecosystem model, that has been validated at a number of ocean locations. We show the different meshes that arise from using different metrics to create the adaptive mesh and from the underlying physical and biological processes that occur at each station. We then apply the model to a threedimensional restratification problem and examine the effect of mesh resolution on simulated biological productivity on both fixed and adaptive meshes.

INI 1  
10:30 to 11:00  Morning Coffee  
11:00 to 11:30 
Y Li (Chinese Academy of Sciences) A new approach to implement sigma coordinate in a numerical model
This study shows a new way to implement terrainfollowing σcoordinate in a numerical model, which does not lead to the wellknown “pressure gradient force (PGF)” problem. First, the causes of the PGF problem are analyzed with existing methods that are categorized into two different types based on the causes. Then, the new method that bypasses the PGF problem all together is proposed. By comparing these three methods and analyzing the expression of the scalar gradient in a curvilinear coordinate system, this study finds out that only when using the covariant scalar equations of σcoordinate will the PGF computational form have one term in each momentum component equation, thereby avoiding the PGF problem completely.
A convenient way of implementing the covariant scalar equations of σcoordinate in a numerical atmospheric model is illustrated, which is to set corresponding parameters in the scalar equations of the Cartesian coordinate. Finally, two idealized experiments manifest that the PGF calculated with the new method is more accurate than using the classic one. Specifically, the relative error of PGF in the new method is reduced by orders of magnitude compared with the result obtained by the classic method; and the pattern of PGF in the new method is more consistent with the analytical PGF pattern than using the classic method. This new method can be used for oceanic models as well, and needs to be tested in both the atmospheric and oceanic models.

INI 1  
11:30 to 12:30 
Discontinuous Galerkin Methods for Adaptive Atmospheric Flow
In this talk we present higher order discontinuous Galerkin methods for convection
dominated problems (using a limiter based stabilization [3]) or diusion
dominated problems (see [4]). A comparison of these methods with COSMO, a
well established dynamical core for weather forecast, for standard test cases for atmospheric
ow [5] is presented. This talk also highlights software techniques as
well as recent development of the software package Dune [1] and the discretization
module DuneFem [2]. In particular we comment on implemented techniques to
allows for local grid adaptivity even in parallel environments. In this case dynamic
loadbalancing is applied to maintain scalability of the simulation code.
References
[1] P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klofkorn, R. Kornhuber,
M. Ohlberger, and O. Sander. A generic grid interface for parallel and adaptive
scientic computing. II: Implementation and tests in Dune. Computing,
82(23):121{138, 2008. [2] A. Dedner, R. Klofkorn, M. Nolte, and M. Ohlberger. A Generic Interface for Parallel and Adaptive Scientic Computing: Abstraction Principles and the Dune Fem Module. Computing, 90(3{4):165{196, 2010. [3] A. Dedner and R. Klofkorn. A Generic Stabilization Approach for Higher Order Discontinuous Galerkin Methods for Convection Dominated Problems. J. Sci. Comput., 47(3):365{388, 2011. [4] S. Brdar, A. Dedner, and R. Klofkorn. Compact and stable Discontinuous Galerkin methods for convectiondiusion problems. Preprint no. 2/2010, Mathematisches Institut, Universitat Freiburg, 2010. accepted for publication in SIAM J. Sci. Comput. [5] S. Brdar, M. Baldauf, A. Dedner, and R. Klofkorn. Comparison of dynamical cores for NWP models. Theor. Comput. Fluid Dyn., 2012. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:30 
Monge ampere based moving mesh methods with applications to numerical weather prediction
Moving mesh methods can be very effective for problems with many scales such as those which arise in numerical weather prediction and data assimilation. However traditional moving mesh methods can have problems with implementation and mesh tangling which have made them less effective than other adaptive methods for problems in meteorology. In this talk I will describe a moving mesh method based on ideas in the theory of optimal transport which derives a mesh by solving a MongeAmpere equation. This can then be coupled to a CFD solver to provide an effective method for solving multiscale incompressible flows. I will describe the method and apply it to several meteorological problems.
Joint work with mike Cullen , chiara piccolo , Emily Walsh and Phil Browne

INI 1  
14:30 to 15:00 
Parallelization and Software Concepts for Tsunami Simulation on Dynamically Adaptive Triangular Grids
We present a memory and cacheefficient approach for simulations on recursively refined dynamically adaptive triangular grids. Grid cells are stored and processed in an order defined by the Sierpinski curve; the resulting locality properties are exploited for optimised serial implementation and parallelisation. The approach is particularly designed for Finite Volume and discontinuous Galerkin solvers, with Tsunami simulation as the main target application. In the talk, we will discuss approaches for parallelisation in shared and distributed memory. We will present a classical partitioningbased strategy, as well as a novel sharedmemory approach based on the dynamical scheduling of many small subpartitions. Here, the intention is to allow for strongly varying computational load per element (as required for inundation modelling, or for local timestepping methods).
In addition, we would like to discuss some ideas on how to provide bathymetry data and (timedependent) displacements for a simulation with dynamically adaptive refinement. We'll present some first results of adaptive simulations using the augmented Riemann solvers provided with GeoClaw.

INI 1  
15:00 to 15:30  Afternoon Tea  
15:30 to 16:00 
Unified Multiscale Operational Weather Forecasting at the Canadian Meteorological Centre
The genesis of the unified Global Environmental Multiscale model and forecasting system that was gradually deployed since 1997 will be presented. It was originally designed for both forecasting and data assimilation at uniform resolution global scale and variable resolution continental scale. It could also be run for mesoscale forecasting over smaller areas by increasing the stretching of the grid. The formulation of the model allowed it to be run either in hydrostatic or nonhydrostatic mode. Tangent linear and adjoint models were developed for variational data assimilation, and data assimilation was performed first using 3D Var and later 4D Var. More recently a global ensemble system was developed based on the GEM model for both ensemble forecasting and data assimilation. The modelling system has been generalised in different directions to become a complete environmental prediction system: emergency response, volcanic ashes, air quality, stratospheric ozone, wave modelling, coupling to rivers and oceans etc.
The development of a nested version of the model is having a profound impact on the forecasting system. After thorough testing, it was used successfully for operational forecasting at 1 km during the Vancouver Olympics in 2010 and it will become operational at 2.5 km over several windows in Canada. A large uniform resolution nested model has been shown to be equivalent to the variable resolution version for continental forecast and was implemented operationally. The nested version has allowed the development of a regional ensemble system. A replacement of the uniform resolution global grid by the composite YinYang grid based on two uniform resolution limitedarea grids is under study. Preliminary testing is showing this version to be equivalent in accuracy, but free of the “pole problem” affecting the grid point latitudelongitude models and much more suitable for future supercomputer architectures.

INI 1  
16:00 to 16:30 
Finite element exterior calculus framework for geophysical fluid dynamics
Finite element exterior calculus provides an extension of the mimetic differencing approach that underlies the Cgrid schemes used in the ICON and MPAS projects. This talk will lay out an extension of this approach, in particular of the approach of Ringler, Thuburn, Skamarock and Klemp (2010), to a finite element framework. The required properties of (a) stationary geostrophic linear modes on the fplane, (b) local conservation of mass, and (c) conservative, consistent advection of potential vorticity are retaining in this framework, whilst allowing higherorder accuracy and increased flexibility which can be used to alter the balance of vorticity and mass degrees of freedom to minimise the potential for spurious modes.

INI 1 