On the numerical integration of a class of singular perturbation problems

A three-point difference scheme recently proposed in Ref. 1 for the numerical solution of a class of linear, singularly perturbed, two-point boundary-value problems is investigated. The scheme is derived from a first-order approximation to the original problem with a small deviating argument. It is shown here that, in the limit, as the deviating argument tends to zero, the difference scheme converges to a one-sided approximation to the original singularly perturbed equation in conservation form. The limiting scheme is shown to be stable on any uniform grid. Therefore, no advantage arises from using the deviating argument, and the most accurate and efficient results are obtained with the deviation at its zero limit.

Comment on “A modified leapfrog scheme for shallow water equations” by Wen-Yih Sun and Oliver M.T. Sun

A recent paper published in this journal considers the numerical integration of the shallow-water equations using the leapfrog time-stepping scheme [Sun Wen-Yih, Sun Oliver MT. A modified leapfrog scheme for shallow water equations. Comput Fluids 2011;52:69–72]. The authors of that paper propose using the time-averaged height in the numerical calculation of the pressure-gradient force, instead of the instantaneous height at the middle time step. The authors show that this modification doubles the maximum Courant number (and hence the maximum time step) at which the integrations are stable, doubling the computational efficiency. Unfortunately, the pressure-averaging technique proposed by the authors is not original. It was devised and published by Shuman [5] and has been widely used in the atmosphere and ocean modelling community for over 40 years.

A reprocessing for climate of sea surface temperature from the Along-Track Scanning Radiometers: a new retrieval scheme

We present a new coefficient-based retrieval scheme for estimation of sea surface temperature (SST) from the Along Track Scanning Radiometer (ATSR) instruments. The new coefficients are banded by total column water vapour (TCWV), obtained from numerical weather prediction analyses. TCWV banding reduces simulated regional retrieval biases to < 0.1 K compared to biases ~ 0.2 K for global coefficients. Further, detailed treatment of the instrumental viewing geometry reduces simulated view-angle related biases from ~ 0.1 K down to < 0.005 K for dual-view retrievals using channels at 11 and 12 μm. A novel analysis of trade-offs related to the assumed noise level when defining coefficients is undertaken, and we conclude that adding a small nominal level of noise (0.01 K) is optimal for our purposes. When applied to ATSR observations, some inter-algorithm biases appear as TCWV-related differences in SSTs estimated from different channel combinations. The final step in coefficient determination is to adjust the offset coefficient in each TCWV band to match results from a reference algorithm. This reference uses the dual-view observations of 3.7 and 11 μm. The adjustment is independent of in situ measurements, preserving independence of the retrievals. The choice of reference is partly motivated by uncertainty in the calibration of the 12 μm of Advanced ATSR. Lastly, we model the sensitivities of the new retrievals to changes to TCWV and changes in true SST, confirming that dual-view SSTs are most appropriate for climatological applications

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.

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.

Impacts of increasing the aerosol complexity in the Met Office global numerical weather prediction model.

The inclusion of the direct and indirect radiative effects of aerosols in high-resolution global numerical weather prediction (NWP) models is being increasingly recognised as important for the improved accuracy of short-range weather forecasts. In this study the impacts of increasing the aerosol complexity in the global NWP configuration of the Met Office Unified Model (MetUM) are investigated. A hierarchy of aerosol representations are evaluated including three-dimensional monthly mean speciated aerosol climatologies, fully prognostic aerosols modelled using the CLASSIC aerosol scheme and finally, initialised aerosols using assimilated aerosol fields from the GEMS project. The prognostic aerosol schemes are better able to predict the temporal and spatial variation of atmospheric aerosol optical depth, which is particularly important in cases of large sporadic aerosol events such as large dust storms or forest fires. Including the direct effect of aerosols improves model biases in outgoing long-wave radiation over West Africa due to a better representation of dust. However, uncertainties in dust optical properties propagate to its direct effect and the subsequent model response. Inclusion of the indirect aerosol effects improves surface radiation biases at the North Slope of Alaska ARM site due to lower cloud amounts in high-latitude clean-air regions. This leads to improved temperature and height forecasts in this region. Impacts on the global mean model precipitation and large-scale circulation fields were found to be generally small in the short-range forecasts. However, the indirect aerosol effect leads to a strengthening of the low-level monsoon flow over the Arabian Sea and Bay of Bengal and an increase in precipitation over Southeast Asia. Regional impacts on the African Easterly Jet (AEJ) are also presented with the large dust loading in the aerosol climatology enhancing of the heat low over West Africa and weakening the AEJ. This study highlights the importance of including a more realistic treatment of aerosol–cloud interactions in global NWP models and the potential for improved global environmental prediction systems through the incorporation of more complex aerosol schemes.

Two fast radiative transfer methods to improve the temporal sampling of clouds in numerical weather prediction and climate models

The high computational cost of calculating the radiative heating rates in numerical weather prediction (NWP) and climate models requires that calculations are made infrequently, leading to poor sampling of the fast-changing cloud field and a poor representation of the feedback that would occur. This paper presents two related schemes for improving the temporal sampling of the cloud field. Firstly, the ‘split time-stepping’ scheme takes advantage of the independent nature of the monochromatic calculations of the ‘correlated-k’ method to split the calculation into gaseous absorption terms that are highly dependent on changes in cloud (the optically thin terms) and those that are not (optically thick). The small number of optically thin terms can then be calculated more often to capture changes in the grey absorption and scattering associated with cloud droplets and ice crystals. Secondly, the ‘incremental time-stepping’ scheme uses a simple radiative transfer calculation using only one or two monochromatic calculations representing the optically thin part of the atmospheric spectrum. These are found to be sufficient to represent the heating rate increments caused by changes in the cloud field, which can then be added to the last full calculation of the radiation code. We test these schemes in an operational forecast model configuration and find a significant improvement is achieved, for a small computational cost, over the current scheme employed at the Met Office. The ‘incremental time-stepping’ scheme is recommended for operational use, along with a new scheme to correct the surface fluxes for the change in solar zenith angle between radiation calculations.

Understanding advances in the simulation of the Madden-Julian Oscillation in a numerical weather prediction model.

The Madden-Julian Oscillation (MJO) is the dominant mode of intraseasonal variability in the Trop- ics. It can be characterised as a planetary-scale coupling between the atmospheric circulation and organised deep convection that propagates east through the equatorial Indo-Pacific region. The MJO interacts with weather and climate systems on a near-global scale and is a crucial source of predictability for weather forecasts on medium to seasonal timescales. Despite its global signifi- cance, accurately representing the MJO in numerical weather prediction (NWP) and climate models remains a challenge. This thesis focuses on the representation of the MJO in the Integrated Forecasting System (IFS) at the European Centre for Medium-Range Weather Forecasting (ECMWF), a state-of-the-art NWP model. Recent modifications to the model physics in Cycle 32r3 (Cy32r3) of the IFS led to ad- vances in the simulation of the MJO; for the first time the observed amplitude of the MJO was maintained throughout the integration period. A set of hindcast experiments, which differ only in their formulation of convection, have been performed between May 2008 and April 2009 to asses the sensitivity of MJO simulation in the IFS to the Cy32r3 convective parameterization. Unique to this thesis is the attribution of the advances in MJO simulation in Cy32r3 to the mod- ified convective parameterization, specifically, the relative-humidity-dependent formulation for or- ganised deep entrainment. Increasing the sensitivity of the deep convection scheme to environmen- tal moisture is shown to modify the relationship between precipitation and moisture in the model. Through dry-air entrainment, convective plumes ascending in low-humidity environments terminate lower in the atmosphere. As a result, there is an increase in the occurrence of cumulus congestus, which acts to moisten the mid-troposphere. Due to the modified precipitation-moisture relationship more moisture is able to build up which effectively preconditions the tropical atmosphere for the transition to deep convection. Results from this thesis suggest that a tropospheric moisture control on convection is key to simulating the interaction between the physics and large-scale circulation associated with the MJO.