M. J. P. Cullen

Unified Modeling and Prediction of Weather and Climate: A 25-Year Journey

In recent years there has been a growing appreciation of the potential advantages of using a seamless approach to weather and climate prediction. However, what exactly should this mean in practice? To help address this question, we document some of the experiences already gathered over 25 years of developing and using the Met Office Unified Model (MetUM) for both weather and climate prediction. Overall, taking a unified approach has given enormous benefits, both scientific and in terms of efficiency, but we also detail some of the challenges it has presented and the approaches taken to overcome them.

An Overview of Numerical Methods for the Next Generation U.K. NWP and Climate Model

ABSTRACT The U.K. Meteorological Office now uses a single model for atmospheric simulation and forecasting from all scales from mesoscale to climate. The constraints which numerical methods for such a model have to satisfy are described. A new version of the model is being developed with the aims of improving its accuracy by better treatment of the ‘balanced’ part of the flow, and increasing its applicability by including non-hydrostatic effects. Unusual features of this version are the use of the Charney-Phillips grid in the vertical, to improve the geostrophic adjustment properties, and the method of constructing the semi-implicit algorithm for solving the fully compressible equations. Idealized tests of these two aspects of the scheme are presented, showing that the Charney-Phillips grid reduces spurious gravity wave generation without compromising the treatment of the atmospheric boundary layer, and that the semi-implicit integration scheme can give stable solutions without the need for added temporal...

Monge-Ampére based moving mesh methods for numerical weather prediction, with applications to the Eady problem

We derive a moving mesh method based upon ideas from optimal transport theory which is suited to solving PDE problems in meteorology. In particular we show how the Parabolic Monge-Ampere method for constructing a moving mesh in two-dimensions can be coupled successfully to a pressure correction method for the solution of incompressible flows with significant convection and subject to Coriolis forces. This method can be used to resolve evolving small scale features in the flow. In this paper the method is then applied to the computation of the solution to the Eady problem which is observed to develop large gradients in a finite time. The moving mesh method is shown to work and be stable, and to give significantly better resolution of the evolving singularity than a fixed, uniform mesh.

Modelling atmospheric flows

This article demonstrates how numerical methods for atmospheric models can be validated by showing that they give the theoretically predicted rate of convergence to relevant asymptotic limit solutions. This procedure is necessary because the exact solution of the Navier–Stokes equations cannot be resolved by production models. The limit solutions chosen are those most important for weather and climate prediction. While the best numerical algorithms for this purpose largely reflect current practice, some important limit solutions cannot be captured by existing methods. The use of Lagrangian rather than Eulerian averaging may be required in these cases.

Processes Controlling Tropical Tropopause Temperature and Stratospheric Water Vapor in Climate Models

A warm bias in tropical tropopause temperature is found in the Met Office Unified Model (MetUM), in common with most models from phase 5 of CMIP (CMIP5). Key dynamical, microphysical, and radiative processes influencing the tropical tropopause temperature and lower-stratospheric water vapor concentrations in climate models are investigated using the MetUM. A series of sensitivity experiments are run to separate the effects of vertical advection, ice optical and microphysical properties, convection, cirrus clouds, and atmospheric composition on simulated tropopause temperature and lower-stratospheric water vapor concentrations in the tropics. The numerical accuracy of the vertical advection, determined in the MetUM by the choice of interpolation and conservation schemes used, is found to be particularly important. Microphysical and radiative processes are found to influence stratospheric water vapor both through modifying the tropical tropopause temperature and through modifying upper-tropospheric water vapor concentrations, allowing more water vapor to be advected into the stratosphere. The representation of any of the processes discussed can act to significantly reduce biases in tropical tropopause temperature and stratospheric water vapor in a physical way, thereby improving climate simulations.

The use of quadratic finite element methods and irregular grids in the solution of hyperbolic problems

Abstract The Galerkin method for first order hyperbolics using most higher order finite elements on any mesh, or using any type of element on an irregular mesh, is known to give a low order of accuracy. This is because the exact solutions become distorted, though the propagation speeds are handled to a higher order of accuracy. The distortion comes about because the method makes no smoothness assumptions in its formulation. Computations with these methods can show rapid growth of small scale noise. The algorithms can be improved by making smoothness assumptions. Many other techniques for obtaining higher order accuracy with such elements ignore the structure of the error and give worse results than standard Galerkin methods. This paper presents analysis and computational examples to support these statements.

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.

Implicit finite difference methods for modelling discontinuous atmospheric flows

Abstract An important model for describing discontinuous atmospheric flows is obtained by making the geostrophic momentum approximation. The solutions vary smoothly along trajectories but may be discontinuous in an Eulerian sense. An implicit finite difference method is presented for modelling such flows. It is demonstrated that it is able to approximate the correct solution in two test problems.

Current progress and prospects in numerical techniques for weather prediction models

Abstract Progress in numerical weather prediction since about 1977 is reviewed. The development of larger and faster computers now allows a global forecast to be made in three minutes per day using a 150 km grid in the horizontal, 15 levels in the vertical, and one of the very efficient integration schemes now available. The forecasts produced are useful guidance to forecasters to about four days on average, but there is a large difference in performance from case to case. In this review recent developments in understanding the solutions to the governing equations are discussed, some of the efficient integration schemes are described in detail, and an example of the current standard of forecasts is displayed.

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.