Integrating Hamiltonian systems defined on the Lie groups SO(4) and SO(1,3)

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E³, the spheres S³ and the hyperboloids H³ with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions is illustrated.

Singularities of optimal control problems on some 6-D lie groups

This paper considers the motion planning problem for oriented vehicles travelling at unit speed in a 3-D space. A Lie group formulation arises naturally and the vehicles are modeled as kinematic control systems with drift defined on the orthonormal frame bundles of particular Riemannian manifolds, specifically, the 3-D space forms Euclidean space E-3, the sphere S-3, and the hyperboloid H'. The corresponding frame bundles are equal to the Euclidean group of motions SE(3), the rotation group SO(4), and the Lorentz group SO (1, 3). The maximum principle of optimal control shifts the emphasis for these systems to the associated Hamiltonian formalism. For an integrable case, the extremal curves are explicitly expressed in terms of elliptic functions. In this paper, a study at the singularities of the extremal curves are given, which correspond to critical points of these elliptic functions. The extremal curves are characterized as the intersections of invariant surfaces and are illustrated graphically at the singular points. It. is then shown that the projections, of the extremals onto the base space, called elastica, at these singular points, are curves of constant curvature and torsion, which in turn implies that the oriented vehicles trace helices.

On the structure of Langmuir turbulence

The Stokes drift induced by surface waves distorts turbulence in the wind-driven mixed layer of the ocean, leading to the development of streamwise vortices, or Langmuir circulations, on a wide range of scales. We investigate the structure of the resulting Langmuir turbulence, and contrast it with the structure of shear turbulence, using rapid distortion theory (RDT) and kinematic simulation of turbulence. Firstly, these linear models show clearly why elongated streamwise vortices are produced in Langmuir turbulence, when Stokes drift tilts and stretches vertical vorticity into horizontal vorticity, whereas elongated streaky structures in streamwise velocity fluctuations (u) are produced in shear turbulence, because there is a cancellation in the streamwise vorticity equation and instead it is vertical vorticity that is amplified. Secondly, we develop scaling arguments, illustrated by analysing data from LES, that indicate that Langmuir turbulence is generated when the deformation of the turbulence by mean shear is much weaker than the deformation by the Stokes drift. These scalings motivate a quantitative RDT model of Langmuir turbulence that accounts for deformation of turbulence by Stokes drift and blocking by the air–sea interface that is shown to yield profiles of the velocity variances in good agreement with LES. The physical picture that emerges, at least in the LES, is as follows. Early in the life cycle of a Langmuir eddy initial turbulent disturbances of vertical vorticity are amplified algebraically by the Stokes drift into elongated streamwise vortices, the Langmuir eddies. The turbulence is thus in a near two-component state, with suppressed and . Near the surface, over a depth of order the integral length scale of the turbulence, the vertical velocity (w) is brought to zero by blocking of the air–sea interface. Since the turbulence is nearly two-component, this vertical energy is transferred into the spanwise fluctuations, considerably enhancing at the interface. After a time of order half the eddy decorrelation time the nonlinear processes, such as distortion by the strain field of the surrounding eddies, arrest the deformation and the Langmuir eddy decays. Presumably, Langmuir turbulence then consists of a statistically steady state of such Langmuir eddies. The analysis then provides a dynamical connection between the flow structures in LES of Langmuir turbulence and the dominant balance between Stokes production and dissipation in the turbulent kinetic energy budget, found by previous authors.

Transport in the low-latitude tropopause zone diagnosed using particle trajectories

Recent literature has described a “transition zone” between the average top of deep convection in the Tropics and the stratosphere. Here transport across this zone is investigated using an offline trajectory model. Particles were advected by the resolved winds from the European Centre for Medium-Range Weather Forecasts reanalyses. For each boreal winter clusters of particles were released in the upper troposphere over the four main regions of tropical deep convection (Indonesia, central Pacific, South America, and Africa). Most particles remain in the troposphere, descending on average for every cluster. The horizontal components of 5-day trajectories are strongly influenced by the El Niño–Southern Oscillation (ENSO), but the Lagrangian average descent does not have a clear ENSO signature. Tropopause crossing locations are first identified by recording events when trajectories from the same release regions cross the World Meteorological Organization lapse rate tropopause. Most crossing events occur 5–15 days after release, and 30-day trajectories are sufficiently long to estimate crossing number densities. In a further two experiments slight excursions across the lapse rate tropopause are differentiated from the drift deeper into the stratosphere by defining the “tropopause zone” as a layer bounded by the average potential temperature of the lapse rate tropopause and the profile temperature minimum. Transport upward across this zone is studied using forward trajectories released from the lower bound and back trajectories arriving at the upper bound. Histograms of particle potential temperature (θ) show marked differences between the transition zone, where there is a slow spread in θ values about a peak that shifts slowly upward, and the troposphere below 350 K. There forward trajectories experience slow radiative cooling interspersed with bursts of convective heating resulting in a well-mixed distribution. In contrast θ histograms for back trajectories arriving in the stratosphere have two distinct peaks just above 300 and 350 K, indicating the sharp change from rapid convective heating in the well-mixed troposphere to slow ascent in the transition zone. Although trajectories slowly cross the tropopause zone throughout the Tropics, all three experiments show that most trajectories reaching the stratosphere from the lower troposphere within 30 days do so over the west Pacific warm pool. This preferred location moves about 30°–50° farther east in an El Niño year (1982/83) and about 30° farther west in a La Niña year (1988/89). These results could have important implications for upper-troposphere–lower-stratosphere pollution and chemistry studies.

The climate response to CO2 of the Hadley Centre coupled AOGCM with and without flux adjustment

Coupled atmosphere‐ocean general circulation models have a tendency to drift away from a realistic climatology. The modelled climate response to an increase of CO2 concentration may be incorrect if the simulation of the current climate has significant errors, so in many models, including ours, the drift is counteracted by applying prescribed fluxes of heat and fresh water at the ocean‐atmosphere interface in addition to the calculated surface exchanges. Since the additional fluxes do not have a physical basis, the use of this technique of “flux adjustment” itself introduces some uncertainty in the simulated response to increased CO2. We find that the global‐average temperature response of our model to CO2 increasing at 1% per year is about 30% less without flux adjustment than with flux adjustment. The geographical patterns of the response are similar, indicating that flux adjustment is not causing any gross distortion. The reduced size of the response is due to more effective vertical transport of heat into the ocean, and a somewhat smaller climate sensitivity. Although the response in both cases lies within the generally accepted range for the climate sensitivity, systematic uncertainties of this size are clearly undesirable, and the best strategy for future development is to improve the climate model in order to reduce the need for flux adjustment.

Self-consistent field theory for diblock copolymers grafted to a sphere

An efficient numerical self-consistent field theory (SCFT) algorithm is developed for treating structured polymers on spherical surfaces. The method solves the diffusion equations of SCFT with a pseudospectral approach that combines a spherical-harmonics expansion for the angular coordinates with a modified real-space Crank–Nicolson method for the radial direction. The self-consistent field equations are solved with Anderson-mixing iterations using dynamical parameters and an alignment procedure to prevent angular drift of the solution. A demonstration of the algorithm is provided for thin films of diblock copolymer grafted to the surface of a spherical core, in which the sequence of equilibrium morphologies is predicted as a function of diblock composition. The study reveals an array of interesting behaviors as the block copolymer pattern is forced to adapt to the finite surface area of the sphere.

International evidence on the predictability of returns to securitized real estate assets: econometric models versus neural networks

The performance of various statistical models and commonly used financial indicators for forecasting securitised real estate returns are examined for five European countries: the UK, Belgium, the Netherlands, France and Italy. Within a VAR framework, it is demonstrated that the gilt-equity yield ratio is in most cases a better predictor of securitized returns than the term structure or the dividend yield. In particular, investors should consider in their real estate return models the predictability of the gilt-equity yield ratio in Belgium, the Netherlands and France, and the term structure of interest rates in France. Predictions obtained from the VAR and univariate time-series models are compared with the predictions of an artificial neural network model. It is found that, whilst no single model is universally superior across all series, accuracy measures and horizons considered, the neural network model is generally able to offer the most accurate predictions for 1-month horizons. For quarterly and half-yearly forecasts, the random walk with a drift is the most successful for the UK, Belgian and Dutch returns and the neural network for French and Italian returns. Although this study underscores market context and forecast horizon as parameters relevant to the choice of the forecast model, it strongly indicates that analysts should exploit the potential of neural networks and assess more fully their forecast performance against more traditional models.

Clear-sky biases in satellite infrared estimates of upper tropospheric humidity and its trends

We use microwave retrievals of upper tropospheric humidity (UTH) to estimate the impact of clear-sky-only sampling by infrared instruments on the distribution, variability and trends in UTH. Our method isolates the impact of the clear-sky-only sampling, without convolving errors from other sources. On daily time scales IR-sampled UTH contains large data gaps in convectively active areas, with only about 20-30 % of the tropics (30 S­ 30 N) being sampled. This results in a dry bias of about -9 %RH in the area-weighted tropical daily UTH time series. On monthly scales, maximum clear-sky bias (CSB) is up to -30 %RH over convectively active areas. The magnitude of CSB shows significant correlations with UTH itself (-0.5) and also with the variability in UTH (-0.6). We also show that IR-sampled UTH time series have higher interannual variability and smaller trends compared to microwave sampling. We argue that a significant part of the smaller trend results from the contrasting influence of diurnal drift in the satellite measurements on the wet and dry regions of the tropics.

Ensemble prediction of transitions of the North Atlantic eddy-driven jet

The consistency of ensemble forecasts from three global medium-range prediction systems with the observed transition behaviour of a three-cluster model of the North Atlantic eddy-driven jet is examined. The three clusters consist of a mid jet cluster taken to represent an undisturbed jet and south and north jet clusters representing southward and northward shifts of the jet. The ensemble forecasts span a period of three extended winters (October–February) from October 2007–February 2010. The mean probabilities of transitions between the clusters calculated from the ensemble forecasts are compared with those calculated from a 23-extended-winter climatology taken from the European Centre for Medium-Range Weather Forecasts 40-Year Re-analysis (ERA40) dataset. No evidence of a drift with increasing lead time of the ensemble forecast transition probabilities towards values inconsistent with the 23-extended-winter climatology is found. The ensemble forecasts of transition probabilities are found to have positive Brier Skill at 15 day lead times. It is found that for the three-extended-winter forecast set, probabilistic forecasts initialized in the north jet cluster are generally less skilful than those initialized in the other clusters. This is consistent with the shorter persistence time-scale of the north jet cluster observed in the ERA40 23-extended-winter climatology. Copyright © 2011 Royal Meteorological Society

Was UV spectral solar irradiance lower during the recent low sunspot minimum?

A detailed analysis is presented of solar UV spectral irradiance for the period between May 2003 and August 2005, when data are available from both the Solar Ultraviolet pectral Irradiance Monitor (SUSIM) instrument (on board the pper Atmosphere Research Satellite (UARS) spacecraft) and the Solar Stellar Irradiance Comparison Experiment (SOLSTICE) instrument (on board the Solar Radiation and Climate Experiment (SORCE) satellite). The ultimate aim is to develop a data composite that can be used to accurately determine any differences between the “exceptional” solar minimum at the end of solar cycle 23 and the previous minimum at the end of solar cycle 22 without having to rely on proxy data to set the long‐term change. SUSIM data are studied because they are the only data available in the “SOLSTICE gap” between the end of available UARS SOLSTICE data and the start of the SORCE data. At any one wavelength the two data sets are considered too dissimilar to be combined into a meaningful composite if any one of three correlations does not exceed a threshold of 0.8. This criterion removes all wavelengths except those in a small range between 156 nm and 208 nm, the longer wavelengths of which influence ozone production and heating in the lower stratosphere. Eight different methods are employed to intercalibrate the two data sequences. All methods give smaller changes between the minima than are seen when the data are not adjusted; however, correcting the SUSIM data to allow for an exponentially decaying offset drift gives a composite that is largely consistent with the unadjusted data from the SOLSTICE instruments on both UARS and SORCE and in which the recent minimum is consistently lower in the wave band studied.

Assimilation impacts on Arctic Ocean circulation, heat and freshwater

We investigate the Arctic basin circulation, freshwater content (FWC) and heat budget by using a high-resolution global coupled ice–ocean model implemented with a state-of-the-art data assimilation scheme. We demonstrate that, despite a very sparse dataset, by assimilating hydrographic data in and near the Arctic basin, the initial warm bias and drift in the control run is successfully corrected, reproducing a much more realistic vertical and horizontal structure to the cyclonic boundary current carrying the Atlantic Water (AW) along the Siberian shelves in the reanalysis run. The Beaufort Gyre structure and FWC and variability are also more accurately reproduced. Small but important changes in the strait exchange flows are found which lead to more balanced budgets in the reanalysis run. Assimilation fluxes dominate the basin budgets over the first 10 years (P1: 1987–1996) of the reanalysis for both heat and FWC, after which the drifting Arctic upper water properties have been restored to realistic values. For the later period (P2: 1997–2004), the Arctic heat budget is almost balanced without assimilation contributions, while the freshwater budget shows reduced assimilation contributions compensating largely for surface salinity damping, which was extremely strong in this run. A downward trend in freshwater export at the Canadian Straits and Fram Strait is found in period P2, associated with Beaufort Gyre recharge. A detailed comparison with observations and previous model studies at the individual Arctic straits is also included.

Understanding distributions of chess performances

This paper presents evidence for several features of the population of chess players, and the distribution of their performances measured in terms of Elo ratings and by computer analysis of moves. Evidence that ratings have remained stable since the inception of the Elo system in the 1970’s is given in several forms: by showing that the population of strong players fits a simple logistic-curve model without inflation, by plotting players’ average error against the FIDE category of tournaments over time, and by skill parameters from a model that employs computer analysis keeping a nearly constant relation to Elo rating across that time. The distribution of the model’s Intrinsic Performance Ratings can hence be used to compare populations that have limited interaction, such as between players in a national chess federation and FIDE, and ascertain relative drift in their respective rating systems.

Validation of the GRAPE single view aerosol retrieval for ATSR-2 and insights into the long term global AOD trend over the ocean

The Global Retrieval of ATSR Cloud Parameters and Evaluation (GRAPE) project has produced a global data-set of cloud and aerosol properties from the Along Track Scanning Radiometer-2 (ATSR-2) instrument, covering the time period 1995�2001. This paper presents the validation of aerosol optical depths (AODs) over the ocean from this product against AERONET sun-photometer measurements, as well as a comparison to the Advanced Very High Resolution Radiometer (AVHRR) optical depth product produced by the Global Aerosol Climatology Project (GACP). The GRAPE AOD over ocean is found to be in good agreement with AERONET measurements, with a Pearson's correlation coefficient of 0.79 and a best-fit slope of 1.0±0.1, but with a positive bias of 0.08±0.04. Although the GRAPE and GACP datasets show reasonable agreement, there are significant differences. These discrepancies are explored, and suggest that the downward trend in AOD reported by GACP may arise from changes in sampling due to the orbital drift of the AVHRR instruments.

Geostatistical radar-raingauge combination with nonparametric correlograms: methodological considerations and application in Switzerland

Modelling spatial covariance is an essential part of all geostatistical methods. Traditionally, parametric semivariogram models are fit from available data. More recently, it has been suggested to use nonparametric correlograms obtained from spatially complete data fields. Here, both estimation techniques are compared. Nonparametric correlograms are shown to have a substantial negative bias. Nonetheless, when combined with the sample variance of the spatial field under consideration, they yield an estimate of the semivariogram that is unbiased for small lag distances. This justifies the use of this estimation technique in geostatistical applications. Various formulations of geostatistical combination (Kriging) methods are used here for the construction of hourly precipitation grids for Switzerland based on data from a sparse realtime network of raingauges and from a spatially complete radar composite. Two variants of Ordinary Kriging (OK) are used to interpolate the sparse gauge observations. In both OK variants, the radar data are only used to determine the semivariogram model. One variant relies on a traditional parametric semivariogram estimate, whereas the other variant uses the nonparametric correlogram. The variants are tested for three cases and the impact of the semivariogram model on the Kriging prediction is illustrated. For the three test cases, the method using nonparametric correlograms performs equally well or better than the traditional method, and at the same time offers great practical advantages. Furthermore, two variants of Kriging with external drift (KED) are tested, both of which use the radar data to estimate nonparametric correlograms, and as the external drift variable. The first KED variant has been used previously for geostatistical radar-raingauge merging in Catalonia (Spain). The second variant is newly proposed here and is an extension of the first. Both variants are evaluated for the three test cases as well as an extended evaluation period. It is found that both methods yield merged fields of better quality than the original radar field or fields obtained by OK of gauge data. The newly suggested KED formulation is shown to be beneficial, in particular in mountainous regions where the quality of the Swiss radar composite is comparatively low. An analysis of the Kriging variances shows that none of the methods tested here provides a satisfactory uncertainty estimate. A suitable variable transformation is expected to improve this.

The basic ingredients of the North Atlantic Storm Track. Part II: Sea surface temperatures

The impact of North Atlantic SST patterns on the storm track is investigated using a hierarchy of GCM simulations using idealized (aquaplanet) and “semirealistic” boundary conditions in the atmospheric component (HadAM3) of the third climate configuration of the Met Office Unified Model (HadCM3). This framework enables the mechanisms determining the tropospheric response to North Atlantic SST patterns to be examined, both in isolation and in combination with continental-scale landmasses and orography. In isolation, a “Gulf Stream” SST pattern acts to strengthen the downstream storm track while a “North Atlantic Drift” SST pattern weakens it. These changes are consistent with changes in the extratropical SST gradient and near-surface baroclinicity, and each storm-track response is associated with a consistent change in the tropospheric jet structure. Locally enhanced near-surface horizontal wind convergence is found over the warm side of strengthened SST gradients associated with ascending air and increased precipitation, consistent with previous studies. When the combined SST pattern is introduced into the semirealistic framework (including the “North American” continent and the “Rocky Mountains”), the results suggest that the topographically generated southwest–northeast tilt in the North Atlantic storm track is enhanced. In particular, the Gulf Stream shifts the storm track south in the western Atlantic whereas the strong high-latitude SST gradient in the northeastern Atlantic enhances the storm track there.