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.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

Optimal convergence estimates for the trace of the polynomial L2-projection operator on a simplex

In this paper we study convergence of the L2-projection onto the space of polynomials up to degree p on a simplex in Rd, d >= 2. Optimal error estimates are established in the case of Sobolev regularity and illustrated on several numerical examples. The proof is based on the collapsed coordinate transform and the expansion into various polynomial bases involving Jacobi polynomials and their antiderivatives. The results of the present paper generalize corresponding estimates for cubes in Rd from [P. Houston, C. Schwab, E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), no. 6, 2133-2163].

The surface urban energy and water balance scheme (SUEWS): evaluation in Los Angeles and Vancouver

An urban energy and water balance model is presented which uses a small number of commonly measured meteorological variables and information about the surface cover. Rates of evaporation-interception for a single layer with multiple surface types (paved, buildings, coniferous trees and/or shrubs, deciduous trees and/or shrubs, irrigated grass, non-irrigated grass and water) are calculated. Below each surface type, except water, there is a single soil layer. At each time step the moisture state of each surface is calculated. Horizontal water movements at the surface and in the soil are incorporated. Particular attention is given to the surface conductance used to model evaporation and its parameters. The model is tested against direct flux measurements carried out over a number of years in Vancouver, Canada and Los Angeles, USA. At all measurement sites the model is able to simulate the net all-wave radiation and turbulent sensible and latent heat well (RMSE = 25–47 W m−2, 30–64 and 20–56 W m−2, respectively). The model reproduces the diurnal cycle of the turbulent fluxes but typically underestimates latent heat flux and overestimates sensible heat flux in the day time. The model tracks measured surface wetness and simulates the variations in soil moisture content. It is able to respond correctly to short-term events as well as annual changes. The largest uncertainty relates to the determination of surface conductance. The model has the potential be used for multiple applications; for example, to predict effects of regulation on urban water use, landscaping and planning scenarios, or to assess climate mitigation strategies.

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.

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

Optimal estimation of sea surface temperature from split-window observations

Optimal estimation (OE) improves sea surface temperature (SST) estimated from satellite infrared imagery in the “split-window”, in comparison to SST retrieved using the usual multi-channel (MCSST) or non-linear (NLSST) estimators. This is demonstrated using three months of observations of the Advanced Very High Resolution Radiometer (AVHRR) on the first Meteorological Operational satellite (Metop-A), matched in time and space to drifter SSTs collected on the global telecommunications system. There are 32,175 matches. The prior for the OE is forecast atmospheric fields from the Météo-France global numerical weather prediction system (ARPEGE), the forward model is RTTOV8.7, and a reduced state vector comprising SST and total column water vapour (TCWV) is used. Operational NLSST coefficients give mean and standard deviation (SD) of the difference between satellite and drifter SSTs of 0.00 and 0.72 K. The “best possible” NLSST and MCSST coefficients, empirically regressed on the data themselves, give zero mean difference and SDs of 0.66 K and 0.73 K respectively. Significant contributions to the global SD arise from regional systematic errors (biases) of several tenths of kelvin in the NLSST. With no bias corrections to either prior fields or forward model, the SSTs retrieved by OE minus drifter SSTs have mean and SD of − 0.16 and 0.49 K respectively. The reduction in SD below the “best possible” regression results shows that OE deals with structural limitations of the NLSST and MCSST algorithms. Using simple empirical bias corrections to improve the OE, retrieved minus drifter SSTs are obtained with mean and SD of − 0.06 and 0.44 K respectively. Regional biases are greatly reduced, such that the absolute bias is less than 0.1 K in 61% of 10°-latitude by 30°-longitude cells. OE also allows a statistic of the agreement between modelled and measured brightness temperatures to be calculated. We show that this measure is more efficient than the current system of confidence levels at identifying reliable retrievals, and that the best 75% of satellite SSTs by this measure have negligible bias and retrieval error of order 0.25 K.

Representativity error for temperature and humidity using the Met Office high-resolution model

The observation-error covariance matrix used in data assimilation contains contributions from instrument errors, representativity errors and errors introduced by the approximated observation operator. Forward model errors arise when the observation operator does not correctly model the observations or when observations can resolve spatial scales that the model cannot. Previous work to estimate the observation-error covariance matrix for particular observing instruments has shown that it contains signifcant correlations. In particular, correlations for humidity data are more significant than those for temperature. However it is not known what proportion of these correlations can be attributed to the representativity errors. In this article we apply an existing method for calculating representativity error, previously applied to an idealised system, to NWP data. We calculate horizontal errors of representativity for temperature and humidity using data from the Met Office high-resolution UK variable resolution model. Our results show that errors of representativity are correlated and more significant for specific humidity than temperature. We also find that representativity error varies with height. This suggests that the assimilation scheme may be improved if these errors are explicitly included in a data assimilation scheme. This article is published with the permission of the Controller of HMSO and the Queen's Printer for Scotland.

Impact of the modal aerosol scheme GLOMAP-mode on aerosol forcing in the Hadley Centre Global Environmental Model

The Hadley Centre Global Environmental Model (HadGEM) includes two aerosol schemes: the Coupled Large-scale Aerosol Simulator for Studies in Climate (CLASSIC), and the new Global Model of Aerosol Processes (GLOMAP-mode). GLOMAP-mode is a modal aerosol microphysics scheme that simulates not only aerosol mass but also aerosol number, represents internally-mixed particles, and includes aerosol microphysical processes such as nucleation. In this study, both schemes provide hindcast simulations of natural and anthropogenic aerosol species for the period 2000–2006. HadGEM simulations of the aerosol optical depth using GLOMAP-mode compare better than CLASSIC against a data-assimilated aerosol re-analysis and aerosol ground-based observations. Because of differences in wet deposition rates, GLOMAP-mode sulphate aerosol residence time is two days longer than CLASSIC sulphate aerosols, whereas black carbon residence time is much shorter. As a result, CLASSIC underestimates aerosol optical depths in continental regions of the Northern Hemisphere and likely overestimates absorption in remote regions. Aerosol direct and first indirect radiative forcings are computed from simulations of aerosols with emissions for the year 1850 and 2000. In 1850, GLOMAP-mode predicts lower aerosol optical depths and higher cloud droplet number concentrations than CLASSIC. Consequently, simulated clouds are much less susceptible to natural and anthropogenic aerosol changes when the microphysical scheme is used. In particular, the response of cloud condensation nuclei to an increase in dimethyl sulphide emissions becomes a factor of four smaller. The combined effect of different 1850 baselines, residence times, and abilities to affect cloud droplet number, leads to substantial differences in the aerosol forcings simulated by the two schemes. GLOMAP-mode finds a presentday direct aerosol forcing of −0.49Wm−2 on a global average, 72% stronger than the corresponding forcing from CLASSIC. This difference is compensated by changes in first indirect aerosol forcing: the forcing of −1.17Wm−2 obtained with GLOMAP-mode is 20% weaker than with CLASSIC. Results suggest that mass-based schemes such as CLASSIC lack the necessary sophistication to provide realistic input to aerosol-cloud interaction schemes. Furthermore, the importance of the 1850 baseline highlights how model skill in predicting present-day aerosol does not guarantee reliable forcing estimates. Those findings suggest that the more complex representation of aerosol processes in microphysical schemes improves the fidelity of simulated aerosol forcings.

Local-scale urban meteorological parameterization scheme (LUMPS): longwave radiation parameterization and seasonality-related developments

Recent developments to the Local-scale Urban Meteorological Parameterization Scheme (LUMPS), a simple model able to simulate the urban energy balance, are presented. The major development is the coupling of LUMPS to the Net All-Wave Radiation Parameterization (NARP). Other enhancements include that the model now accounts for the changing availability of water at the surface, seasonal variations of active vegetation, and the anthropogenic heat flux, while maintaining the need for only commonly available meteorological observations and basic surface characteristics. The incoming component of the longwave radiation (L↓) in NARP is improved through a simple relation derived using cloud cover observations from a ceilometer collected in central London, England. The new L↓ formulation is evaluated with two independent multiyear datasets (Łódź, Poland, and Baltimore, Maryland) and compared with alternatives that include the original NARP and a simpler one using the National Climatic Data Center cloud observation database as input. The performance for the surface energy balance fluxes is assessed using a 2-yr dataset (Łódź). Results have an overall RMSE < 34 W m−2 for all surface energy balance fluxes over the 2-yr period when

Spectral asymmetry of the massless Dirac operator on a 3-torus

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.

Responding to the EU Emissions Trading Scheme: an empirical analysis of corporate carbon disclosure strategy

The purpose of this paper is to explore how companies that hold carbon trading accounts under European Union Emissions Trading Scheme (EU ETS) respond to the climate change by using disclosures on carbon emissions as a means to generate legitimacy compared to others. The study is based on disclosures made in annual reports and stand-alone sustainability reports of UK listed companies from 2001- 2012. The study uses content analysis to capture both the quality and volume of the carbon disclosures. The results show that there is a significant increase in both the quality and volume of the carbon disclosures after the launch of EU ETS. Companies with carbon trading accounts provide greater detailed disclosures as compared to the others without an account. We also find that company size is positively correlated with the disclosures while the association with the industry produces an inconclusive result.