Conditioning and preconditioning of the variational data assimilation problem

Numerical weather prediction (NWP) centres use numerical models of the atmospheric flow to forecast future weather states from an estimate of the current state. Variational data assimilation (VAR) is used commonly to determine an optimal state estimate that miminizes the errors between observations of the dynamical system and model predictions of the flow. The rate of convergence of the VAR scheme and the sensitivity of the solution to errors in the data are dependent on the condition number of the Hessian of the variational least-squares objective function. The traditional formulation of VAR is ill-conditioned and hence leads to slow convergence and an inaccurate solution. In practice, operational NWP centres precondition the system via a control variable transform to reduce the condition number of the Hessian. In this paper we investigate the conditioning of VAR for a single, periodic, spatially-distributed state variable. We present theoretical bounds on the condition number of the original and preconditioned Hessians and hence demonstrate the improvement produced by the preconditioning. We also investigate theoretically the effect of observation position and error variance on the preconditioned system and show that the problem becomes more ill-conditioned with increasingly dense and accurate observations. Finally, we confirm the theoretical results in an operational setting by giving experimental results from the Met Office variational system.

Kinetic energy analysis of the response of the Atlantic meridional overturning circulation to CO2-forced climate change

Atmosphere–ocean general circulation models (AOGCMs) predict a weakening of the Atlantic meridional overturning circulation (AMOC) in response to anthropogenic forcing of climate, but there is a large model uncertainty in the magnitude of the predicted change. The weakening of the AMOC is generally understood to be the result of increased buoyancy input to the north Atlantic in a warmer climate, leading to reduced convection and deep water formation. Consistent with this idea, model analyses have shown empirical relationships between the AMOC and the meridional density gradient, but this link is not direct because the large-scale ocean circulation is essentially geostrophic, making currents and pressure gradients orthogonal. Analysis of the budget of kinetic energy (KE) instead of momentum has the advantage of excluding the dominant geostrophic balance. Diagnosis of the KE balance of the HadCM3 AOGCM and its low-resolution version FAMOUS shows that KE is supplied to the ocean by the wind and dissipated by viscous forces in the global mean of the steady-state control climate, and the circulation does work against the pressure-gradient force, mainly in the Southern Ocean. In the Atlantic Ocean, however, the pressure-gradient force does work on the circulation, especially in the high-latitude regions of deep water formation. During CO2-forced climate change, we demonstrate a very good temporal correlation between the AMOC strength and the rate of KE generation by the pressure-gradient force in 50–70°N of the Atlantic Ocean in each of nine contemporary AOGCMs, supporting a buoyancy-driven interpretation of AMOC changes. To account for this, we describe a conceptual model, which offers an explanation of why AOGCMs with stronger overturning in the control climate tend to have a larger weakening under CO2 increase

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.

The future of the tenanted sector in British agriculture: a review of the proposals for change

The impending decline of the tenanted sector in British agriculture has been forecast for many years. Much debate has surrounded the issues and ensuing legislation has repeatedly attempted to stave-off what some view as the inevitable demise of tenant farmers. Following a flurry of activity after the Northfield Report of 1979 and culminating in the Agricultural Holdings Acts of 1984 and 1986, the debate has recently been fuelled by a strongly pro-market lobby. With the public support of successive Ministers of Agriculture, this lobby has advocated a rejection of the former state intervention in the landlord/tenant relationship in favour of freedom of contract, an option that now appears increasingly likely to reach the statute books. This paper reviews the significant elements of the debate, attempting to explain the principal reasons for the failure of earlier legislation and the primary shortcomings of the current emphasis of consultation. The paper concludes that while there are some significant legislative disincentives to letting land, the freeing-up of contracts in isolation from other, non-contractual issues, will not result in the increase in lettings purportedly desired by the Ministers and their acolytes.

Predictable climate impacts of the decadal changes in the ocean in the 1990s

During the 1990s there was a major change in the state of the world's oceans. In particular, the North Atlantic underwent a rapid warming, with sea surface temperatures (SSTs) in the subpolar gyre region increasing by 1°C in just a few years. Associated with the changes in SST patterns were changes in the surface climate, in particular, a tendency for warm and dry conditions over areas of North America in all seasons, and warm springs and wet summers over areas of Europe. Here, the extent to which a climate prediction system initialized using observations of the ocean state is able to capture the observed changes in seasonal mean surface climate is investigated. Rather than examining predictions of the mid-1990s North Atlantic warming event itself, this study compares hindcasts started before and after the warming, relative to hindcasts that do not assimilate information. It is demonstrated that the hindcasts capture many aspects of the observed changes in seasonal mean surface climate, especially in North, South, and Central America and in Europe. Furthermore, the prediction system retains skill beyond the first year. Finally, it is shown that, in addition to memory of Atlantic SSTs, successfully predicting Pacific SSTs was likely important for the hindcasts to predict surface climate over North America.

An exploration of the equivalent weights particle filter

Particle filters are fully non-linear data assimilation techniques that aim to represent the probability distribution of the model state given the observations (the posterior) by a number of particles. In high-dimensional geophysical applications the number of particles required by the sequential importance resampling (SIR) particle filter in order to capture the high probability region of the posterior, is too large to make them usable. However particle filters can be formulated using proposal densities, which gives greater freedom in how particles are sampled and allows for a much smaller number of particles. Here a particle filter is presented which uses the proposal density to ensure that all particles end up in the high probability region of the posterior probability density function. This gives rise to the possibility of non-linear data assimilation in large dimensional systems. The particle filter formulation is compared to the optimal proposal density particle filter and the implicit particle filter, both of which also utilise a proposal density. We show that when observations are available every time step, both schemes will be degenerate when the number of independent observations is large, unlike the new scheme. The sensitivity of the new scheme to its parameter values is explored theoretically and demonstrated using the Lorenz (1963) model.

Phosphorescence quantum yield enhanced by intermolecular hydrogen bonds in Cu4I4 clusters in the solid state

Organo-copper(I) halide complexes with a Cu4I4 cubane core and cyclic amines as ligands have been synthesized and their crystal structures have been defined. Their solid state photophysical properties have been measured and correlated with the crystal structure and packing. A unique and remarkably high luminescence quantum yield (76%) has been measured for one of the complexes having the cubane clusters arranged in a columnar structure and held together by N–HI hydrogen bonds. This high luminescence quantum yield is correlated with a slow radiationless deactivation rate of the excited state and suggests a rather strong enhancement of the cubane core rigidity bestowed by the hydrogen bond pattern. Some preliminary thin film deposition experiments show that these compounds could be considered to be good candidates for applications in electroluminescent devices because of their bright luminescence, low cost and relatively easy synthesis processes