Philip Browne

Fast three dimensional r-adaptive mesh redistribution

This paper describes a fast and reliable method for redistributing a computational mesh in three dimensions which can generate a complex three dimensional mesh without any problems due to mesh tangling. The method relies on a three dimensional implementation of the parabolic Monge-Ampere (PMA) technique, for finding an optimally transported mesh. The method for implementing PMA is described in detail and applied to both static and dynamic mesh redistribution problems, studying both the convergence and the computational cost of the algorithm. The algorithm is applied to a series of problems of increasing complexity. In particular very regular meshes are generated to resolve real meteorological features (derived from a weather forecasting model covering the UK area) in grids with over 2x10^7 degrees of freedom. The PMA method computes these grids in times commensurate with those required for operational weather forecasting.

A simple method for integrating a complex model into an ensemble data assimilation system using MPI

This paper details a strategy for modifying the source code of a complex model so that the model may be used in a data assimilation context, and gives the standards for implementing a data assimilation code to use such a model. The strategy relies on keeping the model separate from any data assimilation code, and coupling the two through the use of Message Passing Interface (MPI) functionality. This strategy limits the changes necessary to the model and as such is rapid to program, at the expense of ultimate performance. The implementation technique is applied in different models with state dimension up to .2.7?i??108 The overheads added by using this implementation strategy in a coupled ocean-atmosphere climate model are shown to be an order of magnitude smaller than the addition of correlated stochastic random errors necessary for some nonlinear data assimilation techniques. We introduce an MPI coupling of a model and an ensemble data assimilation system.We show the full changes to the Lorenz 1963 model that implement the MPI coupling.The MPI coupling method is shown to exhibit perfect scaling on large-scale examples.

Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Ampère type equation

Abstract An equation of Monge–Ampere type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport generates meshes that keep the same connectivity as the original mesh, making them suitable for r-adaptive simulations, in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers. The semi-implicit solution of the Monge–Ampere type equation involves a new linearisation of the Hessian term, and exponential maps are used to map from old to new meshes on the sphere. The determinant of the Hessian is evaluated as the change in volume between old and new mesh cells, rather than using numerical approximations to the gradients. OT meshes are generated to compare with centroidal Voronoi tessellations on the sphere and are found to have advantages and disadvantages; OT equidistribution is more accurate, the number of iterations to convergence is independent of the mesh size, face skewness is reduced and the connectivity does not change. However anisotropy is higher and the OT meshes are non-orthogonal. It is shown that optimal transport on the sphere leads to meshes that do not tangle. However, tangling can be introduced by numerical errors in calculating the gradient of the mesh potential. Methods for alleviating this problem are explored. Finally, OT meshes are generated using observed precipitation as a monitor function, in order to demonstrate the potential power of the technique.

A systematic method of parameterisation estimation using data assimilation

In numerical weather prediction, parameterisations are used to simulate missing physics in the model. These can be due to a lack of scientific understanding or a lack of computing power available to address all the known physical processes. Parameterisations are sources of large uncertainty in a model as parameter values used in these parameterisations cannot be measured directly and hence are often not well known; and the parameterisations themselves are also approximations of the processes present in the true atmosphere. Whilst there are many efficient and effective methods for combined state/parameter estimation in data assimilation (DA), such as state augmentation, these are not effective at estimating the structure of parameterisations. A new method of parameterisation estimation is proposed that uses sequential DA methods to estimate errors in the numerical models at each space-time point for each model equation. These errors are then fitted to pre-determined functional forms of missing physics or parameterisations that are based upon prior information. We applied the method to a one-dimensional advection model with additive model error, and it is shown that the method can accurately estimate parameterisations, with consistent error estimates. Furthermore, it is shown how the method depends on the quality of the DA results. The results indicate that this new method is a powerful tool in systematic model improvement.

Data Assimilation in the Solar Wind: Challenges and First Results†

Abstract Data assimilation (DA) is used extensively in numerical weather prediction (NWP) to improve forecast skill. Indeed, improvements in forecast skill in NWP models over the past 30 years have directly coincided with improvements in DA schemes. At present, due to data availability and technical challenges, DA is underused in space weather applications, particularly for solar wind prediction. This paper investigates the potential of advanced DA methods currently used in operational NWP centers to improve solar wind prediction. To develop the technical capability, as well as quantify the potential benefit, twin experiments are conducted to assess the performance of the Local Ensemble Transform Kalman Filter (LETKF) in the solar wind model ENLIL. Boundary conditions are provided by the Wang‐Sheeley‐Arge coronal model and synthetic observations of density, temperature, and momentum generated every 4.5 h at 0.6 AU. While in situ spacecraft observations are unlikely to be routinely available at 0.6 AU, these techniques can be applied to remote sensing of the solar wind, such as with Heliospheric Imagers or interplanetary scintillation. The LETKF can be seen to improve the state at the observation location and advect that improvement toward the Earth, leading to an improvement in forecast skill in near‐Earth space for both the observed and unobserved variables. However, sharp gradients caused by the analysis of a single observation in space resulted in artificial wavelike structures being advected toward Earth. This paper is the first attempt to apply DA to solar wind prediction and provides the first in‐depth analysis of the challenges and potential solutions.

A comparison of the equivalent weights particle filter and the local ensemble transform Kalman filter in application to the barotropic vorticity equation

Data assimilation methods that work in high-dimensional systems are crucial to many areas of the geosciences: meteorology, oceanography, climate science and so on. The equivalent weights particle filter (EWPF) has been designed for, and recently shown to scale to, problems that are of use to these communities. This article performs a systematic comparison of the EWPF with the established and widely used local ensemble transform Kalman filter (LETKF). Both methods are applied to the barotropic vorticity equation for different networks of observations. In all cases, it was found that the LETKF produced lower root mean–squared errors than the EWPF. The performance of the EWPF is shown to depend strongly on the form of nudging used, and a nudging term based on the local ensemble transform Kalman smoother is shown to improve the performance of the filter. This indicates that the EWPF must be considered as a truly two-stage filter and not only by its final step which avoids weight collapse.