Stochastic climate theory and modeling

Stochastic methods are a crucial area in contemporary climate research and are increasingly being used in comprehensive weather and climate prediction models as well as reduced order climate models. Stochastic methods are used as subgrid-scale parameterizations (SSPs) as well as for model error representation, uncertainty quantification, data assimilation, and ensemble prediction. The need to use stochastic approaches in weather and climate models arises because we still cannot resolve all necessary processes and scales in comprehensive numerical weather and climate prediction models. In many practical applications one is mainly interested in the largest and potentially predictable scales and not necessarily in the small and fast scales. For instance, reduced order models can simulate and predict large-scale modes. Statistical mechanics and dynamical systems theory suggest that in reduced order models the impact of unresolved degrees of freedom can be represented by suitable combinations of deterministic and stochastic components and non-Markovian (memory) terms. Stochastic approaches in numerical weather and climate prediction models also lead to the reduction of model biases. Hence, there is a clear need for systematic stochastic approaches in weather and climate modeling. In this review, we present evidence for stochastic effects in laboratory experiments. Then we provide an overview of stochastic climate theory from an applied mathematics perspective. We also survey the current use of stochastic methods in comprehensive weather and climate prediction models and show that stochastic parameterizations have the potential to remedy many of the current biases in these comprehensive models.

The calculation of accurate normal coordinates I: general theory, and application to methane

The calculation of accurate and reliable vibrational potential functions and normal co-ordinates is discussed, for such simple polyatomic molecules as it may be possible. Such calculations should be corrected for the effects of anharmonicity and of resonance interactions between the vibrational states, and should be fitted to all the available information on all isotopic species: particularly the vibrational frequencies, Coriolis zeta constants and centrifugal distortion constants. The difficulties of making these corrections, and of making use of the observed data are reviewed. A programme for the Ferranti Mercury Computer is described by means of which harmonic vibration frequencies and normal co-ordinate vectors, zeta factors and centrifugal distortion constants can be calculated, from a given force field and from given G-matrix elements, etc. The programme has been used on up to 5 × 5 secular equations for which a single calculation and output of results takes approximately l min; it can readily be extended to larger determinants. The best methods of using such a programme and the possibility of reversing the direction of calculation are discussed. The methods are applied to calculating the best possible vibrational potential function for the methane molecule, making use of all the observed data.

Managing threatened species – the ecological toolbox, evolutionary theory and the declining-population paradigm

1. The management of threatened species is an important practical way in which conservationists can intervene in the extinction process and reduce the loss of biodiversity. Understanding the causes of population declines (past, present and future) is pivotal to designing effective practical management. This is the declining-population paradigm identified by Caughley. 2. There are three broad classes of ecological tool used by conservationists to guide management decisions for threatened species: statistical models of habitat use, demographic models and behaviour-based models. Each of these is described here, illustrated with a case study and evaluated critically in terms of its practical application. 3. These tools are fundamentally different. Statistical models of habitat use and demographic models both use descriptions of patterns in abundance and demography, in relation to a range of factors, to inform management decisions. In contrast, behaviourbased models describe the evolutionary processes underlying these patterns, and derive such patterns from the strategies employed by individuals when competing for resources under a specific set of environmental conditions. 4. Statistical models of habitat use and demographic models have been used successfully to make management recommendations for declining populations. To do this, assumptions are made about population growth or vital rates that will apply when environmental conditions are restored, based on either past data collected under favourable environmental conditions or estimates of these parameters when the agent of decline is removed. As a result, they can only be used to make reliable quantitative predictions about future environments when a comparable environment has been experienced by the population of interest in the past. 5. Many future changes in the environment driven by management will not have been experienced by a population in the past. Under these circumstances, vital rates and their relationship with population density will change in the future in a way that is not predictable from past patterns. Reliable quantitative predictions about population-level responses then need to be based on an explicit consideration of the evolutionary processes operating at the individual level. 6. Synthesis and applications. It is argued that evolutionary theory underpins Caughley’s declining-population paradigm, and that it needs to become much more widely used within mainstream conservation biology. This will help conservationists examine critically the reliability of the tools they have traditionally used to aid management decision-making. It will also give them access to alternative tools, particularly when predictions are required for changes in the environment that have not been experienced by a population in the past.

Managing threatened species: the ecological toolbox, evolutionary theory and declining-population paradigm

1. The management of threatened species is an important practical way in which conservationists can intervene in the extinction process and reduce the loss of biodiversity. Understanding the causes of population declines (past, present and future) is pivotal to designing effective practical management. This is the declining-population paradigm identified by Caughley. 2. There are three broad classes of ecological tool used by conservationists to guide management decisions for threatened species: statistical models of habitat use, demographic models and behaviour-based models. Each of these is described here, illustrated with a case study and evaluated critically in terms of its practical application. 3. These tools are fundamentally different. Statistical models of habitat use and demographic models both use descriptions of patterns in abundance and demography, in relation to a range of factors, to inform management decisions. In contrast, behaviour-based models describe the evolutionary processes underlying these patterns, and derive such patterns from the strategies employed by individuals when competing for resources under a specific set of environmental conditions. 4. Statistical models of habitat use and demographic models have been used successfully to make management recommendations for declining populations. To do this, assumptions are made about population growth or vital rates that will apply when environmental conditions are restored, based on either past data collected under favourable environmental conditions or estimates of these parameters when the agent of decline is removed. As a result, they can only be used to make reliable quantitative predictions about future environments when a comparable environment has been experienced by the population of interest in the past. 5. Many future changes in the environment driven by management will not have been experienced by a population in the past. Under these circumstances, vital rates and their relationship with population density will change in the future in a way that is not predictable from past patterns. Reliable quantitative predictions about population-level responses then need to be based on an explicit consideration of the evolutionary processes operating at the individual level. 6. Synthesis and applications. It is argued that evolutionary theory underpins Caughley's declining-population paradigm, and that it needs to become much more widely used within mainstream conservation biology. This will help conservationists examine critically the reliability of the tools they have traditionally used to aid management decision-making. It will also give them access to alternative tools, particularly when predictions are required for changes in the environment that have not been experienced by a population in the past.

Operator Theory and Its Applications: In Memory of V. B. Lidskii (1924-2008)

This book is a collection of articles devoted to the theory of linear operators in Hilbert spaces and its applications. The subjects covered range from the abstract theory of Toeplitz operators to the analysis of very specific differential operators arising in quantum mechanics, electromagnetism, and the theory of elasticity; the stability of numerical methods is also discussed. Many of the articles deal with spectral problems for not necessarily selfadjoint operators. Some of the articles are surveys outlining the current state of the subject and presenting open problems.

Evolutionary developmental biology: impact on systematic theory and practice, and the contribution of systematics

Evolutionary developmental genetics brings together systematists, morphologists and developmental geneticists; it will therefore impact on each of these component disciplines. The goals and methods of phylogenetic analysis are reviewed here, and the contribution of evolutionary developmental genetics to morphological systematics, in terms of character conceptualisation and primary homology assessment, is discussed. Evolutionary developmental genetics, like its component disciplines phylogenetic systematics and comparative morphology, is concerned with homology concepts. Phylogenetic concepts of homology and their limitations are considered here, and the need for independent homology statements at different levels of biological organisation is evaluated. The role of systematics in evolutionary developmental genetics is outlined. Phylogenetic systematics and comparative morphology will suggest effective sampling strategies to developmental geneticists. Phylogenetic systematics provides hypotheses of character evolution (including parallel evolution and convergence), stimulating investigations into the evolutionary gains and losses of morphologies. Comparative morphology identifies those structures that are not easily amenable to typological categorisation, and that may be of particular interest in terms of developmental genetics. The concepts of latent homology and genetic recall may also prove useful in the evolutionary interpretation of developmental genetic data.

Markov models of aging: theory and practice

We review and structure some of the mathematical and statistical models that have been developed over the past half century to grapple with theoretical and experimental questions about the stochastic development of aging over the life course. We suggest that the mathematical models are in large part addressing the problem of partitioning the randomness in aging: How does aging vary between individuals, and within an individual over the lifecourse? How much of the variation is inherently related to some qualities of the individual, and how much is entirely random? How much of the randomness is cumulative, and how much is merely short-term flutter? We propose that recent lines of statistical inquiry in survival analysis could usefully grapple with these questions, all the more so if they were more explicitly linked to the relevant mathematical and biological models of aging. To this end, we describe points of contact among the various lines of mathematical and statistical research. We suggest some directions for future work, including the exploration of information-theoretic measures for evaluating components of stochastic models as the basis for analyzing experiments and anchoring theoretical discussions of aging.

Theory and observations of ice particle evolution in cirrus using Doppler radar: evidence for aggregation

Vertically pointing Doppler radar has been used to study the evolution of ice particles as they sediment through a cirrus cloud. The measured Doppler fall speeds, together with radar-derived estimates for the altitude of cloud top, are used to estimate a characteristic fall time tc for the `average' ice particle. The change in radar reflectivity Z is studied as a function of tc, and is found to increase exponentially with fall time. We use the idea of dynamically scaling particle size distributions to show that this behaviour implies exponential growth of the average particle size, and argue that this exponential growth is a signature of ice crystal aggregation.

An Assessment of the Recent Past and Future Climate Change, Glacier Retreat, and Runoff in the Caucasus Region Using Dynamical and Statistical Downscaling and HBV-ETH Hydrological Model

The paper discusses the observed and projected warming in the Caucasus region and its implications for glacier melt and runoff. A strong positive trend in summer air temperatures of 0.05 degrees C a(-1) is observed in the high-altitude areas providing for a strong glacier melt and continuous decline in glacier mass balance. A warming of 4-7 degrees C and 3-5 degrees C is projected for the summer months in 2071-2100 under the A2 and B2 emission scenarios respectively, suggesting that enhanced glacier melt can be expected. The expected changes in winter precipitation will not compensate for the summer melt and glacier retreat is likely to continue. However, a projected small increase in both winter and summer precipitation combined with the enhanced glacier melt will result in increased summer runoff in the currently glaciated region of the Caucasus (independent of whether the region is glaciated at the end of the twenty-first century) by more than 50% compared with the baseline period.