The composite-tendency RAW filter in semi-implicit integrations

The time discretization in weather and climate models introduces truncation errors that limit the accuracy of the simulations. Recent work has yielded a method for reducing the amplitude errors in leapfrog integrations from first-order to fifth-order. This improvement is achieved by replacing the Robert--Asselin filter with the RAW filter and using a linear combination of the unfiltered and filtered states to compute the tendency term. The purpose of the present paper is to apply the composite-tendency RAW-filtered leapfrog scheme to semi-implicit integrations. A theoretical analysis shows that the stability and accuracy are unaffected by the introduction of the implicitly treated mode. The scheme is tested in semi-implicit numerical integrations in both a simple nonlinear stiff system and a medium-complexity atmospheric general circulation model, and yields substantial improvements in both cases. We conclude that the composite-tendency RAW-filtered leapfrog scheme is suitable for use in semi-implicit integrations.

The composite-tendency Robert-Asselin-Williams (RAW) filter in semi-implicit integrations

Timediscretization in weatherandclimate modelsintroduces truncation errors that limit the accuracy of the simulations. Recent work has yielded a method for reducing the amplitude errors in leap-frog integrations from first-order to fifth-order.This improvement is achieved by replacing the Robert–Asselin filter with the Robert–Asselin–Williams (RAW) filter and using a linear combination of unfiltered and filtered states to compute the tendency term. The purpose of the present article is to apply the composite-tendency RAW-filtered leapfrog scheme to semi-implicit integrations. A theoretical analysis shows that the stability and accuracy are unaffected by the introduction of the implicitly treated mode. The scheme is tested in semi-implicit numerical integrations in both a simple nonlinear stiff system and a medium-complexity atmospheric general circulation model and yields substantial improvements in both cases. We conclude that the composite-tendency RAW-filtered leap-frog scheme is suitable for use in semi-implicit integrations.

Runge-Kutta IMEX schemes for the Horizontally Explicit/Vertically Implicit (HEVI) solution of wave equations

Many operational weather forecasting centres use semi-implicit time-stepping schemes because of their good efficiency. However, as computers become ever more parallel, horizontally explicit solutions of the equations of atmospheric motion might become an attractive alternative due to the additional inter-processor communication of implicit methods. Implicit and explicit (IMEX) time-stepping schemes have long been combined in models of the atmosphere using semi-implicit, split-explicit or HEVI splitting. However, most studies of the accuracy and stability of IMEX schemes have been limited to the parabolic case of advection–diffusion equations. We demonstrate how a number of Runge–Kutta IMEX schemes can be used to solve hyperbolic wave equations either semi-implicitly or HEVI. A new form of HEVI splitting is proposed, UfPreb, which dramatically improves accuracy and stability of simulations of gravity waves in stratified flow. As a consequence it is found that there are HEVI schemes that do not lose accuracy in comparison to semi-implicit ones. The stability limits of a number of variations of trapezoidal implicit and some Runge–Kutta IMEX schemes are found and the schemes are tested on two vertical slice cases using the compressible Boussinesq equations split into various combinations of implicit and explicit terms. Some of the Runge–Kutta schemes are found to be beneficial over trapezoidal, especially since they damp high frequencies without dropping to first-order accuracy. We test schemes that are not formally accurate for stiff systems but in stiff limits (nearly incompressible) and find that they can perform well. The scheme ARK2(2,3,2) performs the best in the tests.

Numerical analyses of Runge–Kutta implicit–explicit schemes for horizontally explicit, vertically implicit solutions of atmospheric models

With the prospect of exascale computing, computational methods requiring only local data become especially attractive. Consequently, the typical domain decomposition of atmospheric models means horizontally-explicit vertically-implicit (HEVI) time-stepping schemes warrant further attention. In this analysis, Runge-Kutta implicit-explicit schemes from the literature are analysed for their stability and accuracy using a von Neumann stability analysis of two linear systems. Attention is paid to the numerical phase to indicate the behaviour of phase and group velocities. Where the analysis is tractable, analytically derived expressions are considered. For more complicated cases, amplification factors have been numerically generated and the associated amplitudes and phase diagnosed. Analysis of a system describing acoustic waves has necessitated attributing the three resultant eigenvalues to the three physical modes of the system. To do so, a series of algorithms has been devised to track the eigenvalues across the frequency space. The result enables analysis of whether the schemes exactly preserve the non-divergent mode; and whether there is evidence of spurious reversal in the direction of group velocities or asymmetry in the damping for the pair of acoustic modes. Frequency ranges that span next-generation high-resolution weather models to coarse-resolution climate models are considered; and a comparison is made of errors accumulated from multiple stability-constrained shorter time-steps from the HEVI scheme with a single integration from a fully implicit scheme over the same time interval. Two schemes, “Trap2(2,3,2)” and “UJ3(1,3,2)”, both already used in atmospheric models, are identified as offering consistently good stability and representation of phase across all the analyses. Furthermore, according to a simple measure of computational cost, “Trap2(2,3,2)” is the least expensive.

Comparison of terrain following and cut cell grids using a non-hydrostatic model

Terrain following coordinates are widely used in operational models but the cut cell method has been proposed as an alternative that can more accurately represent atmospheric dynamics over steep orography. Because the type of grid is usually chosen during model implementation, it becomes necessary to use different models to compare the accuracy of different grids. In contrast, here a C-grid finite volume model enables a like-for-like comparison of terrain following and cut cell grids. A series of standard two-dimensional tests using idealised terrain are performed: tracer advection in a prescribed horizontal velocity field, a test starting from resting initial conditions, and orographically induced gravity waves described by nonhydrostatic dynamics. In addition, three new tests are formulated: a more challenging resting atmosphere case, and two new advection tests having a velocity field that is everywhere tangential to the terrain following coordinate surfaces. These new tests present a challenge on cut cell grids. The results of the advection tests demonstrate that accuracy depends primarily upon alignment of the flow with the grid rather than grid orthogonality. A resting atmosphere is well-maintained on all grids. In the gravity waves test, results on all grids are in good agreement with existing results from the literature, although terrain following velocity fields lead to errors on cut cell grids. Due to semi-implicit timestepping and an upwind-biased, explicit advection scheme, there are no timestep restrictions associated with small cut cells. We do not find the significant advantages of cut cells or smoothed coordinates that other authors find.

A comparison of two methods for developing the linearization of a shallow-water model

We develop the linearization of a semi-implicit semi-Lagrangian model of the one-dimensional shallow-water equations using two different methods. The usual tangent linear model, formed by linearizing the discrete nonlinear model, is compared with a model formed by first linearizing the continuous nonlinear equations and then discretizing. Both models are shown to perform equally well for finite perturbations. However, the asymptotic behaviour of the two models differs as the perturbation size is reduced. This leads to difficulties in showing that the models are correctly coded using the standard tests. To overcome this difficulty we propose a new method for testing linear models, which we demonstrate both theoretically and numerically. © Crown copyright, 2003. Royal Meteorological Society

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

An equation of Monge-Ampère 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-Ampère 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 tesselations 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 semi-blind reception scheme based on ICA for multiuser MIMO-STBC system

When the orthogonal space-time block code (STBC), or the Alamouti code, is applied on a multiple-input multiple-output (MIMO) communications system, the optimum reception can be achieved by a simple signal decoupling at the receiver. The performance, however, deteriorates significantly in presence of co-channel interference (CCI) from other users. In this paper, such CCI problem is overcome by applying the independent component analysis (ICA), a blind source separation algorithm. This is based on the fact that, if the transmission data from every transmit antenna are mutually independent, they can be effectively separated at the receiver with the principle of the blind source separation. Then equivalently, the CCI is suppressed. Although they are not required by the ICA algorithm itself, a small number of training data are necessary to eliminate the phase and order ambiguities at the ICA outputs, leading to a semi-blind approach. Numerical simulation is also shown to verify the proposed ICA approach in the multiuser MIMO system.

Evaluation of a photosynthesis-based biogenic isoprene emission scheme in JULES and simulation of isoprene emissions under present-day climate conditions

We have incorporated a semi-mechanistic isoprene emission module into the JULES land-surface scheme, as a first step towards a modelling tool that can be applied for studies of vegetation – atmospheric chemistry interactions, including chemistry-climate feedbacks. Here, we evaluate the coupled model against local above-canopy isoprene emission flux measurements from six flux tower sites as well as satellite-derived estimates of isoprene emission over tropical South America and east and south Asia. The model simulates diurnal variability well: correlation coefficients are significant (at the 95 % level) for all flux tower sites. The model reproduces day-to-day variability with significant correlations (at the 95 % confidence level) at four of the six flux tower sites. At the UMBS site, a complete set of seasonal observations is available for two years (2000 and 2002). The model reproduces the seasonal pattern of emission during 2002, but does less well in the year 2000. The model overestimates observed emissions at all sites, which is partially because it does not include isoprene loss through the canopy. Comparison with the satellite-derived isoprene-emission estimates suggests that the model simulates the main spatial patterns, seasonal and inter-annual variability over tropical regions. The model yields a global annual isoprene emission of 535 ± 9 TgC yr−1 during the 1990s, 78 % of which from forested areas.

Palaeovegetation in China during the late Quaternary: Biome reconstructions based on a global scheme of plant functional types

Two previous reconstructions of palaeovegetation across the whole of China were performed using a simple classification of plant functional types (PFTs). Now a more explicit, global PFT classification scheme has been developed, and a substantial number of additional pollen records have become available. Here we apply the global scheme of PFTs to a comprehensive set of pollen records available from China to test the applicability of the global scheme of PFTs in China, and to obtain a well-founded reconstruction of changing palaeovegetation patterns. A total of 806 pollen surface samples, 188 mid-Holocene (MH, 6000 14C yr BP) and 50 last glacial maximum (LGM, 18,000 14C yr BP) pollen records were used to reconstruct vegetation patterns in China, based on a new global classification system of PFTs and a standard numerical technique for biome assignment (biomization). The biome reconstruction based on pollen surface samples showed convincing agreement with present potential natural vegetation. Coherent patterns of change in biome distribution between MH, LGM and present are observed. In the MH, cold and cool-temperate evergreen needleleaf forests and mixed forests, temperate deciduous broadleaf forest, and warm-temperate evergreen broadleaf and mixed forest in eastern China were shifted northward by 200–500 km. Cold-deciduous forest in northeastern China was replaced by cold evergreen needleleaf forest while in central northern China, cold-deciduous forest was present at some sites now occupied by temperate grassland and desert. The forest–grassland boundary was 200–300 km west of its present position. Temperate xerophytic shrubland, temperate grassland and desert covered a large area on the Tibetan Plateau, but the area of tundra was reduced. Treeline was 300–500 m higher than present in Tibet. These changes imply generally warmer winters, longer growing seasons and more precipitation during the MH. Westward shifts of the forest–shrubland–grassland and grassland–desert boundaries imply greater moisture availability in the MH, consistent with a stronger summer monsoon. During the LGM, in contrast, cold-deciduous forest, cool-temperate evergreen needleleaf forest, cool mixed forests, warm-temperate evergreen broadleaf and mixed forest in eastern China were displaced to the south by 300–1000 km, while temperate deciduous broadleaf forest, pure warm-temperate evergreen forest, tropical semi-evergreen and evergreen broadleaf forests were restricted or absent from the mainland of southern China, implying colder winters than present. Strong shifts of temperate xerophytic shrubland, temperate grassland and desert to the south and east in northern and western China and on the Tibetan Plateau imply drier conditions than present.

Implicit equal-weights particle filter

Filter degeneracy is the main obstacle for the implementation of particle filter in non-linear high-dimensional models. A new scheme, the implicit equal-weights particle filter (IEWPF), is introduced. In this scheme samples are drawn implicitly from proposal densities with a different covariance for each particle, such that all particle weights are equal by construction. We test and explore the properties of the new scheme using a 1,000-dimensional simple linear model, and the 1,000-dimensional non-linear Lorenz96 model, and compare the performance of the scheme to a Local Ensemble Kalman Filter. The experiments show that the new scheme can easily be implemented in high-dimensional systems and is never degenerate, with good convergence properties in both systems.