A stress relaxation model for the viscoelastic solids based on the steady-state creep equation

Creep and stress relaxation are inherent mechanical behaviors of viscoelastic materials. It is considered that both are different performances of one identical physical phenomenon. The relationship between the decay stress and time during stress relaxation has been derived from the power law equation of the steady-state creep. The model was used to analyse the stress relaxation curves of various different viscoelastic materials (such as pure polycrystalline ice, polymers, foods, bones, metal, animal tissues, etc.). The calculated results using the theoretical model agree with the experimental data very well. Here we show that the new mathematical formula is not only simple but its parameters have the clear physical meanings. It is suitable to materials with a very broad scope and has a strong predictive ability.

Response theory for equilibrium and non-equilibrium statistical mechanics: causality and generalized Kramers-Kronig relations

We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.

Closed loop design for a simple nonlinear aircraft model using eigenstructure assignment

Aircraft systems are highly nonlinear and time varying. High-performance aircraft at high angles of incidence experience undesired coupling of the lateral and longitudinal variables, resulting in departure from normal controlled flight. The aim of this work is to construct a robust closed-loop control that optimally extends the stable and decoupled flight envelope. For the study of these systems nonlinear analysis methods are needed. Previously, bifurcation techniques have been used mainly to analyze open-loop nonlinear aircraft models and investigate control effects on dynamic behavior. In this work linear feedback control designs calculated by eigenstructure assignment methods are investigated for a simple aircraft model at a fixed flight condition. Bifurcation analysis in conjunction with linear control design methods is shown to aid control law design for the nonlinear system.

Breakdown of hydrostatic balance at convective scales in the forecast errors in the Met Office Unified Model

For data assimilation in numerical weather prediction, the initial forecast-error covariance matrix Pf is required. For variational assimilation it is particularly important to prescribe an accurate initial matrix Pf, since Pf is either static (in the 3D-Var case) or constant at the beginning of each assimilation window (in the 4D-Var case). At large scales the atmospheric flow is well approximated by hydrostatic balance and this balance is strongly enforced in the initial matrix Pf used in operational variational assimilation systems such as that of the Met Office. However, at convective scales this balance does not necessarily hold any more. Here we examine the extent to which hydrostatic balance is valid in the vertical forecast-error covariances for high-resolution models in order to determine whether there is a need to relax this balance constraint in convective-scale data assimilation. We use the Met Office Global and Regional Ensemble Prediction System (MOGREPS) and a 1.5 km resolution version of the Unified Model for a case study characterized by the presence of convective activity. An ensemble of high-resolution forecasts valid up to three hours after the onset of convection is produced. We show that at 1.5 km resolution hydrostatic balance does not hold for forecast errors in regions of convection. This indicates that in the presence of convection hydrostatic balance should not be enforced in the covariance matrix used for variational data assimilation at this scale. The results show the need to investigate covariance models that may be better suited for convective-scale data assimilation. Finally, we give a measure of the balance present in the forecast perturbations as a function of the horizontal scale (from 3–90 km) using a set of diagnostics. Copyright © 2012 Royal Meteorological Society and British Crown Copyright, the Met Office

Climate variability-observations, reconstructions, and model simulations for the Atlantic-European and Alpine region from 1500-2100 AD

A detailed analysis is undertaken of the Atlantic-European climate using data from 500-year-long proxy-based climate reconstructions, a long climate simulation with perpetual 1990 forcing, as well as two global and one regional climate change scenarios. The observed and simulated interannual variability and teleconnectivity are compared and interpreted in order to improve the understanding of natural climate variability on interannual to decadal time scales for the late Holocene. The focus is set on the Atlantic-European and Alpine regions during the winter and summer seasons, using temperature, precipitation, and 500 hPa geopotential height fields. The climate reconstruction shows pronounced interdecadal variations that appear to “lock” the atmospheric circulation in quasi-steady long-term patterns over multi-decadal periods controlling at least part of the temperature and precipitation variability. Different circulation patterns are persistent over several decades for the period 1500 to 1900. The 500-year-long simulation with perpetual 1990 forcing shows some substantial differences, with a more unsteady teleconnectivity behaviour. Two global scenario simulations indicate a transition towards more stable teleconnectivity for the next 100 years. Time series of reconstructed and simulated temperature and precipitation over the Alpine region show comparatively small changes in interannual variability within the time frame considered, with the exception of the summer season, where a substantial increase in interannual variability is simulated by regional climate models.

Correction of atmospheric delay effects in radar interferometry using a nested mesoscale atmospheric model

The temporal variability of the atmosphere through which radio waves pass in the technique of differential radar interferometry can seriously limit the accuracy with which the method can measure surface motion. A forward, nested mesoscale model of the atmosphere can be used to simulate the variable water content along the radar path and the resultant phase delays. Using this approach we demonstrate how to correct an interferogram of Mount Etna in Sicily associated with an eruption in 2004-5. The regional mesoscale model (Unified Model) used to simulate the atmosphere at higher resolutions consists of four nested domains increasing in resolution (12, 4, 1, 0.3 km), sitting within the analysis version of a global numerical model that is used to initiate the simulation. Using the high resolution 3D model output we compute the surface pressure, temperature and the water vapour, liquid and solid water contents, enabling the dominant hydrostatic and wet delays to be calculated at specific times corresponding to the acquisition of the radar data. We can also simulate the second-order delay effects due to liquid water and ice.

Estimating root–soil contact from 3D X-ray microtomographs

Adequate contact with the soil is essential for water and nutrient adsorption by plant roots, but the determination of root–soil contact is a challenging task because it is difficult to visualize roots in situ and quantify their interactions with the soil at the scale of micrometres. A method to determine root–soil contact using X-ray microtomography was developed. Contact areas were determined from 3D volumetric images using segmentation and iso-surface determination tools. The accuracy of the method was tested with physical model systems of contact between two objects (phantoms). Volumes, surface areas and contact areas calculated from the measured phantoms were compared with those estimated from image analysis. The volume was accurate to within 0.3%, the surface area to within 2–4%, and the contact area to within 2.5%. Maize and lupin roots were grown in soil (<2 mm) and vermiculite at matric potentials of −0.03 and −1.6 MPa and in aggregate fractions of 4–2, 2–1, 1–0.5 and < 0.5 mm at a matric potential of −0.03 MPa. The contact of the roots with their growth medium was determined from 3D volumetric images. Macroporosity (>70 µm) of the soil sieved to different aggregate fractions was calculated from binarized data. Root-soil contact was greater in soil than in vermiculite and increased with decreasing aggregate or particle size. The differences in root–soil contact could not be explained solely by the decrease in porosity with decreasing aggregate size but may also result from changes in particle and aggregate packing around the root.

A ‘Boscastle-type’ quasi-stationary convective system over the UK Southwest Peninsula

An investigation is presented of a quasi-stationary convective system (QSCS) which occurred over the UK Southwest Peninsula on 21 July 2010. This system was remarkably similar in its location and structure to one which caused devastating flash flooding in the coastal village of Boscastle, Cornwall on 16 August 2004. However, in the 2010 case rainfall accumulations were around four times smaller and no flooding was recorded. The more extreme nature of the Boscastle case is shown to be related to three factors: (1) higher rain rates, associated with a warmer and moister tropospheric column and deeper convective clouds; (2) a more stationary system, due to slower evolution of the large-scale flow; and (3) distribution of the heaviest precipitation over fewer river catchments. Overall, however, the synoptic setting of the two events was broadly similar, suggesting that such conditions favour the development of QSCSs over the Southwest Peninsula. A numerical simulation of the July 2010 event was performed using a 1.5-km grid length configuration of the Met Office Unified Model. This reveals that convection was repeatedly initiated through lifting of low-level air parcels along a quasi-stationary coastal convergence line. Sensitivity tests are used to show that this convergence line was a sea breeze front which temporarily stalled along the coastline due to the retarding influence of an offshore-directed background wind component. Several deficiencies are noted in the 1.5-km model’s representation of the storm system, including delayed convective initiation; however, significant improvements are observed when the grid length is reduced to 500 m. These result in part from an improved representation of the convergence line, which enhances the associated low-level ascent allowing air parcels to more readily reach their level of free convection. The implications of this finding for forecasting convective precipitation are discussed.

Some numerical experiments on the effect of the variation of static stability in two-layer quasi-geostrophic models

A method to solve a quasi-geostrophic two-layer model including the variation of static stability is presented. The divergent part of the wind is incorporated by means of an iterative procedure. The procedure is rather fast and the time of computation is only 60–70% longer than for the usual two-layer model. The method of solution is justified by the conservation of the difference between the gross static stability and the kinetic energy. To eliminate the side-boundary conditions the experiments have been performed on a zonal channel model. The investigation falls mainly into three parts: The first part (section 5) contains a discussion of the significance of some physically inconsistent approximations. It is shown that physical inconsistencies are rather serious and for these inconsistent models which were studied the total kinetic energy increased faster than the gross static stability. In the next part (section 6) we are studying the effect of a Jacobian difference operator which conserves the total kinetic energy. The use of this operator in two-layer models will give a slight improvement but probably does not have any practical use in short periodic forecasts. It is also shown that the energy-conservative operator will change the wave-speed in an erroneous way if the wave-number or the grid-length is large in the meridional direction. In the final part (section 7) we investigate the behaviour of baroclinic waves for some different initial states and for two energy-consistent models, one with constant and one with variable static stability. According to the linear theory the waves adjust rather rapidly in such a way that the temperature wave will lag behind the pressure wave independent of the initial configuration. Thus, both models give rise to a baroclinic development even if the initial state is quasi-barotropic. The effect of the variation of static stability is very small, qualitative differences in the development are only observed during the first 12 hours. For an amplifying wave we will get a stabilization over the troughs and an instabilization over the ridges.

Constraints on wave drag parameterization schemes for simulating the quasi-biennial oscillation. Part I: gravity wave forcing

Parameterization schemes for the drag due to atmospheric gravity waves are discussed and compared in the context of a simple one-dimensional model of the quasi-biennial oscillation (QBO). A number of fundamental issues are examined in detail, with the goal of providing a better understanding of the mechanism by which gravity wave drag can produce an equatorial zonal wind oscillation. The gravity wave–driven QBOs are compared with those obtained from a parameterization of equatorial planetary waves. In all gravity wave cases, it is seen that the inclusion of vertical diffusion is crucial for the descent of the shear zones and the development of the QBO. An important difference between the schemes for the two types of waves is that in the case of equatorial planetary waves, vertical diffusion is needed only at the lowest levels, while for the gravity wave drag schemes it must be included at all levels. The question of whether there is downward propagation of influence in the simulated QBOs is addressed. In the gravity wave drag schemes, the evolution of the wind at a given level depends on the wind above, as well as on the wind below. This is in contrast to the parameterization for the equatorial planetary waves in which there is downward propagation of phase only. The stability of a zero-wind initial state is examined, and it is determined that a small perturbation to such a state will amplify with time to the extent that a zonal wind oscillation is permitted.

Constraints on wave drag parameterization schemes for simulating the quasi-biennial oscillation. Part II: combined effects of gravity waves and equatorial planetary waves

This study examines the effect of combining equatorial planetary wave drag and gravity wave drag in a one-dimensional zonal mean model of the quasi-biennial oscillation (QBO). Several different combinations of planetary wave and gravity wave drag schemes are considered in the investigations, with the aim being to assess which aspects of the different schemes affect the nature of the modeled QBO. Results show that it is possible to generate a realistic-looking QBO with various combinations of drag from the two types of waves, but there are some constraints on the wave input spectra and amplitudes. For example, if the phase speeds of the gravity waves in the input spectrum are large relative to those of the equatorial planetary waves, critical level absorption of the equatorial planetary waves may occur. The resulting mean-wind oscillation, in that case, is driven almost exclusively by the gravity wave drag, with only a small contribution from the planetary waves at low levels. With an appropriate choice of wave input parameters, it is possible to obtain a QBO with a realistic period and to which both types of waves contribute. This is the regime in which the terrestrial QBO appears to reside. There may also be constraints on the initial strength of the wind shear, and these are similar to the constraints that apply when gravity wave drag is used without any planetary wave drag. In recent years, it has been observed that, in order to simulate the QBO accurately, general circulation models require parameterized gravity wave drag, in addition to the drag from resolved planetary-scale waves, and that even if the planetary wave amplitudes are incorrect, the gravity wave drag can be adjusted to compensate. This study provides a basis for knowing that such a compensation is possible.

The ModFOLD4 server for the quality assessment of 3D protein models

Once you have generated a 3D model of a protein, how do you know whether it bears any resemblance to the actual structure? To determine the usefulness of 3D models of proteins, they must be assessed in terms of their quality by methods that predict their similarity to the native structure. The ModFOLD4 server is the latest version of our leading independent server for the estimation of both the global and local (per-residue) quality of 3D protein models. The server produces both machine readable and graphical output, providing users with intuitive visual reports on the quality of predicted protein tertiary structures. The ModFOLD4 server is freely available to all at: http://www.reading.ac.uk/bioinf/ModFOLD/.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 1. pseudomomentum-based theory

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics. In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms. The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent. This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii). In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.

Climate determinism revisited: multiple equilibria in a complex climate model

Multiple equilibria in a coupled ocean–atmosphere–sea ice general circulation model (GCM) of an aquaplanet with many degrees of freedom are studied. Three different stable states are found for exactly the same set of parameters and external forcings: a cold state in which a polar sea ice cap extends into the midlatitudes; a warm state, which is ice free; and a completely sea ice–covered “snowball” state. Although low-order energy balance models of the climate are known to exhibit intransitivity (i.e., more than one climate state for a given set of governing equations), the results reported here are the first to demonstrate that this is a property of a complex coupled climate model with a consistent set of equations representing the 3D dynamics of the ocean and atmosphere. The coupled model notably includes atmospheric synoptic systems, large-scale circulation of the ocean, a fully active hydrological cycle, sea ice, and a seasonal cycle. There are no flux adjustments, with the system being solely forced by incoming solar radiation at the top of the atmosphere. It is demonstrated that the multiple equilibria owe their existence to the presence of meridional structure in ocean heat transport: namely, a large heat transport out of the tropics and a relatively weak high-latitude transport. The associated large midlatitude convergence of ocean heat transport leads to a preferred latitude at which the sea ice edge can rest. The mechanism operates in two very different ocean circulation regimes, suggesting that the stabilization of the large ice cap could be a robust feature of the climate system. Finally, the role of ocean heat convergence in permitting multiple equilibria is further explored in simpler models: an atmospheric GCM coupled to a slab mixed layer ocean and an energy balance model