Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .

Hankel and Toeplitz transforms on H 1: continuity, compactness and Fredholm properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H 1 and give a new proof of the old result stating that the Hankel operator H a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H a to be compact on H 1. The Fredholm properties of Toeplitz operators on H 1 are studied for symbols in a Banach algebra similar to C + H ∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H 1 and H 2.

Importance of density-compensated temperature change for deep North Atlantic Ocean heat uptake

The efficiency with which the oceans take up heat has a significant influence on the rate of global warming. Warming of the ocean above 700 m over the past few decades has been well documented. However, most of the ocean lies below 700 m. Here we analyse observations of heat uptake into the deep North Atlantic. We find that the extratropical North Atlantic as a whole warmed by 1.45±0.5×1022 J between 1955 and 2005, but Lower North Atlantic Deep Water cooled, most likely as an adjustment from an early twentieth-century warm period. In contrast, the heat content of Upper North Atlantic Deep Water exhibited strong decadal variability. We demonstrate and quantify the importance of density-compensated temperature anomalies for long-term heat uptake into the deep North Atlantic. These anomalies form in the subpolar gyre and propagate equatorwards. High salinity in the subpolar gyre is a key requirement for this mechanism. In the past 50 years, suitable conditions have occurred only twice: first during the 1960s and again during the past decade. We conclude that heat uptake through density-compensated temperature anomalies will contribute to deep ocean heat uptake in the near term. In the longer term, the importance of this mechanism will be determined by competition between the multiple processes that influence subpolar gyre salinity in a changing climate.

Atmospheric point discharge current measurements using a temperature-compensated logarithmic current amplifier

Measurements of atmospheric corona currents have been made for over 100 years to indicate the atmospheric electric field. Corona currents vary substantially, in polarity and in magnitude. The instrument described here uses a sharp point sensor connected to a temperature compensated bi-polar logarithmic current amplifier. Calibrations over a range of currents from ±10 fA to ±3 μA and across ±20 ◦C show it has an excellent logarithmic response over six orders of magnitude from 1 pA to 1 μA in both polarities for the range of atmospheric temperatures likely to be encountered in the southern UK. Comparison with atmospheric electric field measurements during disturbed weather confirms that bipolar electric fields induce corona currents of corresponding sign, with magnitudes ∼0.5 μA.

An improved algorithm for generating global window brightness temperatures from multiple satellite infrared imagery

An improved algorithm for the generation of gridded window brightness temperatures is presented. The primary data source is the International Satellite Cloud Climatology Project, level B3 data, covering the period from July 1983 to the present. The algorithm rakes window brightness, temperatures from multiple satellites, both geostationary and polar orbiting, which have already been navigated and normalized radiometrically to the National Oceanic and Atmospheric Administration's Advanced Very High Resolution Radiometer, and generates 3-hourly global images on a 0.5 degrees by 0.5 degrees latitude-longitude grid. The gridding uses a hierarchical scheme based on spherical kernel estimators. As part of the gridding procedure, the geostationary data are corrected for limb effects using a simple empirical correction to the radiances, from which the corrected temperatures are computed. This is in addition to the application of satellite zenith angle weighting to downweight limb pixels in preference to nearer-nadir pixels. The polar orbiter data are windowed on the target time with temporal weighting to account for the noncontemporaneous nature of the data. Large regions of missing data are interpolated from adjacent processed images using a form of motion compensated interpolation based on the estimation of motion vectors using an hierarchical block matching scheme. Examples are shown of the various stages in the process. Also shown are examples of the usefulness of this type of data in GCM validation.

A study of the sensitivity of ocean overturning circulation and climate to freshwater input in different regions of the North Atlantic

The Atlantic meridional overturning circulation (AMOC) is an important component of the climate system. Models indicate that the AMOC can be perturbed by freshwater forcing in the North Atlantic. Using an ocean-atmosphere general circulation model, we investigate the dependence of such a perturbation of the AMOC, and the consequent climate change, on the region of freshwater forcing. A wide range of changes in AMOC strength is found after 100 years of freshwater forcing. The largest changes in AMOC strength occur when the regions of deepwater formation in the model are forced directly, although reductions in deepwater formation in one area may be compensated by enhanced formation elsewhere. North Atlantic average surface air temperatures correlate linearly with the AMOC decline, but warming may occur in localised regions, notably over Greenland and where deepwater formation is enhanced. This brings into question the representativeness of temperature changes inferred from Greenland ice-core records.

Intercomparison of radiative forcing calculations of stratospheric water vapour and contrails

Seven groups have participated in an intercomparison study of calculations of radiative forcing (RF) due to stratospheric water vapour (SWV) and contrails. A combination of detailed radiative transfer schemes and codes for global-scale calculations have been used, as well as a combination of idealized simulations and more realistic global-scale changes in stratospheric water vapour and contrails. Detailed line-by-line codes agree within about 15 % for longwave (LW) and shortwave (SW) RF, except in one case where the difference is 30 %. Since the LW and SW RF due to contrails and SWV changes are of opposite sign, the differences between the models seen in the individual LW and SW components can be either compensated or strengthened in the net RF, and thus in relative terms uncertainties are much larger for the net RF. Some of the models used for global-scale simulations of changes in SWV and contrails differ substantially in RF from the more detailed radiative transfer schemes. For the global-scale calculations we use a method of weighting the results to calculate a best estimate based on their performance compared to the more detailed radiative transfer schemes in the idealized simulations.

Anthropogenic warming of the oceans: Observations and model results

Observations show the oceans have warmed over the past 40 yr. with appreciable regional variation and more warming at the surface than at depth. Comparing the observations with results from two coupled ocean-atmosphere climate models [the Parallel Climate Model version 1 (PCM) and the Hadley Centre Coupled Climate Model version 3 (HadCM3)] that include anthropogenic forcing shows remarkable agreement between the observed and model-estimated warming. In this comparison the models were sampled at the same locations as gridded yearly observed data. In the top 100 m of the water column the warming is well separated from natural variability, including both variability arising from internal instabilities of the coupled ocean-atmosphere climate system and that arising from volcanism and solar fluctuations. Between 125 and 200 m the agreement is not significant, but then increases again below this level, and remains significant down to 600 m. Analysis of PCM's heat budget indicates that the warming is driven by an increase in net surface heat flux that reaches 0.7 W m(-2) by the 1990s; the downward longwave flux increases bv 3.7 W m(-2). which is not fully compensated by an increase in the upward longwave flux of 2.2 W m(-2). Latent and net solar heat fluxes each decrease by about 0.6 W m(-2). The changes in the individual longwave components are distinguishable from the preindustrial mean by the 1920s, but due to cancellation of components. changes in the net surface heat flux do not become well separated from zero until the 1960s. Changes in advection can also play an important role in local ocean warming due to anthropogenic forcing, depending, on the location. The observed sampling of ocean temperature is highly variable in space and time. but sufficient to detect the anthropogenic warming signal in all basins, at least in the surface layers, bv the 1980s.

The importance of the diurnal and annual cycle of air traffic for contrail radiative forcing

Air traffic condensation trails, or contrails, are believed to have a net atmospheric warming effect(1), although one that is currently small compared to that induced by other sources of human emissions. However, the comparably large growth rate of air traffic requires an improved understanding of the resulting impact of aircraft radiative forcing on climate(2). Contrails have an effect on the Earth's energy balance similar to that of high thin ice clouds(3). Their trapping of outgoing longwave radiation emitted by the Earth and atmosphere (positive radiative forcing) is partly compensated by their reflection of incoming solar radiation (negative radiative forcing). On average, the longwave effect dominates and the net contrail radiative forcing is believed to be positive(1,2,4). Over daily and annual timescales, varying levels of air traffic, meteorological conditions, and solar insolation influence the net forcing effect of contrails. Here we determine the factors most important for contrail climate forcing using a sophisticated radiative transfer model(5,6) for a site in southeast England, located in the entrance to the North Atlantic flight corridor. We find that night-time flights during winter (December to February) are responsible for most of the contrail radiative forcing. Night flights account for only 25 per cent of daily air traffic, but contribute 60 to 80 per cent of the contrail forcing. Further, winter flights account for only 22 per cent of annual air traffic, but contribute half of the annual mean forcing. These results suggest that flight rescheduling could help to minimize the climate impact of aviation.

The quasi-nondispersive regimes of long extratropical baroclinic rossby waves over (slowly varying) topography

Actual energy paths of long, extratropical baroclinic Rossby waves in the ocean are difficult to describe simply because they depend on the meridional-wavenumber-to-zonal-wavenumber ratio tau, a quantity that is difficult to estimate both observationally and theoretically. This paper shows, however, that this dependence is actually weak over any interval in which the zonal phase speed varies approximately linearly with tau, in which case the propagation becomes quasi-nondispersive (QND) and describable at leading order in terms of environmental conditions (i.e., topography and stratification) alone. As an example, the purely topographic case is shown to possess three main kinds of QND ray paths. The first is a topographic regime in which the rays follow approximately the contours f/h(alpha c) = a constant (alpha(c) is a near constant fixed by the strength of the stratification, f is the Coriolis parameter, and h is the ocean depth). The second and third are, respectively, "fast" and "slow" westward regimes little affected by topography and associated with the first and second bottom-pressure-compensated normal modes studied in previous work by Tailleux and McWilliams. Idealized examples show that actual rays can often be reproduced with reasonable accuracy by replacing the actual dispersion relation by its QND approximation. The topographic regime provides an upper bound ( in general a large overestimate) of the maximum latitudinal excursions of actual rays. The method presented in this paper is interesting for enabling an optimal classification of purely azimuthally dispersive wave systems into simpler idealized QND wave regimes, which helps to rationalize previous empirical findings that the ray paths of long Rossby waves in the presence of mean flow and topography often seem to be independent of the wavenumber orientation. Two important side results are to establish that the baroclinic string function regime of Tyler and K se is only valid over a tiny range of the topographic parameter and that long baroclinic Rossby waves propagating over topography do not obey any two-dimensional potential vorticity conservation principle. Given the importance of the latter principle in geophysical fluid dynamics, the lack of it in this case makes the concept of the QND regimes all the more important, for they are probably the only alternative to provide a simple and economical description of general purely azimuthally dispersive wave systems.

The propagation mechanism of a vortex on the β plane

The propagation velocity and propagation mechanism for vortices on a β plane are determined for a reduced-gravity model by integrating the momentum equations over the β plane. Isolated vortices, vortices in a background current, and initial vortex propagation from rest are studied. The propagation mechanism for isolated anticyclones as well as cyclones, which has been lacking up to now, is presented. It is shown that, to first order, the vortex moves to generate a Coriolis force on the mass anomaly of the vortex to compensate for the force on the vortex due to the variation of the Coriolis parameter. Only the mass anomaly of the vortex is of importance, because the Coriolis force due to the motion of the bulk of the layer moving with the vortex is almost fully compensated by the Coriolis force on the motion of the exterior flow. Because the mass anomaly of a cyclone is negative the force and acceleration have opposite sign. The role of dipolar structures in steadily moving vortices is discussed, and it is shown that their overall structure is fixed by the steady westward motion of the mass anomaly. Furthermore, it is shown that reduced-gravity vortices are not advected with a background flow. The reason for this behavior is that the background flow changes the ambient vorticity gradient such that the vortex obtains an extra self-propagation term that exactly cancels the advection by the background flow. Last, it is shown that a vortex initially at rest will accelerate equatorward first, after which a westward motion is generated. This result is independent of the sign of the vortex.

Effects of triazole and strobilurin fungicide programmes, with and without late-season nitrogen fertiliser, on the baking quality of Malacca winter wheat

Field experiments were conducted over 3 years to study the effect of applying triazole and strobilurin fungicides on the bread-making quality of Malacca winter wheat. Averaged over all years the application of a fungicide programme increased yields, particularly when strobilurin fungicides were applied. Reductions in protein concentration, sulphur concentration, Hageberg failing number and loaf volumes also occurred as the amount of fungicide applied increased. However, there were no deleterious effects of fungicide application on sodium dodecyl sulphate (SDS) sedimentation volumes, N:S ratios or dough theology. Effects of fungicide application on bread-making quality were not product specific. Therefore, it appears that new mechanisms to explain strobilurin effects on bread-making quality do not need to be invoked. Where reductions in protein concentration did occur they could be compensated for by a late-season application of nitrogen either as granular ammonium nitrate at flag leaf emergence or foliar urea at anthesis. These applications, however, sometimes increased the N:S ratio of the extracted flour and failed to improve loaf volume. Multiple regression analysis revealed that main effects of year, flour protein concentration and N:S ratio could explain 93% of the variance in loaf volume caused by season, fungicide and nitrogen treatments. However, an equally good fit was achieved by just including sulphur concentration with year. (C) 2004 Elsevier Ltd. All rights reserved.

The effects of adding picoxystrobin, azoxystrobin and nitrogen to a triazole programme on disease control, flag leaf senescence, yield and grain quality of winter wheat

The effect of adding strobilurins to a triazole (epoxiconazole) fungicide programme on the quality of a range of wheat cultivars was assessed in field experiments in three successive years. Strobilurin was applied at just flag leaf emergence (azoxystrobin) or at the start of stem extension (azoxystrobin or picoxystrobin) and again at flag leaf emergence or at flag leaf emergence and again at ear emergence (azoxystrobin). All strobilurin treatments reduced disease levels, delayed senescence of the flag leaf and consistently increased yields, thousand grain weight and specific weight. Reductions in Hagberg falling number were observed, even by fungicide applications at the start of stem extension, but effects were small compared to the variation among cultivars. Application of fungicide (triazole or strobilurin) before ear emergence increased the amount of blackpoint, but this was partly countered by applying azoxystrobin at ear emergence. The effect of fungicide on protein concentration differed over seasons and cultivar. Where they occurred. small reductions in protein concentration could be compensated for by extra application of nitrogen as foliar urea at anthesis. Foliar urea (40 kg N ha(-1)) applied at anthesis also improved Hagberg failing number and reduced blackpoint in one of the growing seasons. In one season, the effect of foliar urea at anthesis was compared with applications of granular fertiliser at flag leaf emergence. The granular treatment produced grain with more concentrated protein, while the later, foliar application produced higher specific weights. (C) 2003 Elsevier Science Ltd. All rights reserved.