State estimation using model order reduction for unstable systems

The problem of state estimation occurs in many applications of fluid flow. For example, to produce a reliable weather forecast it is essential to find the best possible estimate of the true state of the atmosphere. To find this best estimate a nonlinear least squares problem has to be solved subject to dynamical system constraints. Usually this is solved iteratively by an approximate Gauss–Newton method where the underlying discrete linear system is in general unstable. In this paper we propose a new method for deriving low order approximations to the problem based on a recently developed model reduction method for unstable systems. To illustrate the theoretical results, numerical experiments are performed using a two-dimensional Eady model – a simple model of baroclinic instability, which is the dominant mechanism for the growth of storms at mid-latitudes. It is a suitable test model to show the benefit that may be obtained by using model reduction techniques to approximate unstable systems within the state estimation problem.

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.

Asymmetric space market adjustment in the London office market

Models of the City of London office market are extended by considering a longer time series of data, covering two cycles, and by explicit modeling of asymmetric rental response to supply and demand model. A long run structural model linking demand for office space, real rental levels and office-based employment is estimated and then rental adjustment processes are modeled using an error correction model framework. Adjustment processes are seen to be asymmetric, dependent both on the direction of the supply and demand shock and on the state of the rental market at the time of the shock. A complete system of equations is estimated: unit shocks produce oscillations but there is a return to a steady equilibrium state in the long run.

Relaxing the Tropics to an ‘observed’ state: analysis using a simple baroclinic model

The technique of relaxation of the tropical atmosphere towards an analysis in a month-season forecast model has previously been successfully exploited in a number of contexts. Here it is shown that when tropical relaxation is used to investigate the possible origin of the observed anomalies in June–July 2007, a simple dynamical model is able to reproduce the observed component of the pattern of anomalies given by an ensemble of ECMWF forecast runs. Following this result, the simple model is used for a range of experiments on time-scales of relaxation, variables and regions relaxed based on a control model run with equatorial heating in a zonal flow. A theory based on scale analysis for the large-scale tropics is used to interpret the results. Typical relationships between scales are determined from the basic equations, and for a specified diabatic heating a chain of deductions for determining the dependent variables is derived. Different critical time-scales are found for tropical relaxation of different dependent variables to be effective. Vorticity has the longest critical time-scale, typically 1.2 days. For temperature and divergence, the time-scales are 10 hours and 3 hours, respectively. However not all the tropical fields, in particular the vertical motion, are reproduced correctly by the model unless divergence is heavily damped. To obtain the correct extra-tropical fields, it is crucial to have the correct rotational flow in the subtropics to initiate the Rossby wave propagation from there. It is sufficient to relax vorticity or temperature on a time-scale comparable or less than their critical time-scales to obtain this. However if the divergent advection of vorticity is important in the Rossby Wave Source then strong relaxation of divergence is required to accurately represent the tropical forcing of Rossby waves.

Ensemble forecasts of a flood-producing storm: comparison of the influence of model-state perturbations and parameter modifications

High-resolution ensemble simulations (Δx = 1 km) are performed with the Met Office Unified Model for the Boscastle (Cornwall, UK) flash-flooding event of 16 August 2004. Forecast uncertainties arising from imperfections in the forecast model are analysed by comparing the simulation results produced by two types of perturbation strategy. Motivated by the meteorology of the event, one type of perturbation alters relevant physics choices or parameter settings in the model's parametrization schemes. The other type of perturbation is designed to account for representativity error in the boundary-layer parametrization. It makes direct changes to the model state and provides a lower bound against which to judge the spread produced by other uncertainties. The Boscastle has genuine skill at scales of approximately 60 km and an ensemble spread which can be estimated to within ∼ 10% with only eight members. Differences between the model-state perturbation and physics modification strategies are discussed, the former being more important for triggering and the latter for subsequent cell development, including the average internal structure of convective cells. Despite such differences, the spread in rainfall evaluated at skilful scales is shown to be only weakly sensitive to the perturbation strategy. This suggests that relatively simple strategies for treating model uncertainty may be sufficient for practical, convective-scale ensemble forecasting.

A time-invariant visco-elastic windkessel model relating blood flow and blood volume

The difference between the rate of change of cerebral blood volume (CBV) and cerebral blood flow (CBF) following stimulation is thought to be due to circumferential stress relaxation in veins (Mandeville, J.B., Marota, J.J.A., Ayata, C., Zaharchuk, G., Moskowitz, M.A., Rosen, B.R., Weisskoff, R.M., 1999. Evidence of a cerebrovascular postarteriole windkessel with delayed compliance. J. Cereb. Blood Flow Metab. 19, 679–689). In this paper we explore the visco-elastic properties of blood vessels, and present a dynamic model relating changes in CBF to changes in CBV. We refer to this model as the visco-elastic windkessel (VW) model. A novel feature of this model is that the parameter characterising the pressure–volume relationship of blood vessels is treated as a state variable dependent on the rate of change of CBV, producing hysteresis in the pressure–volume space during vessel dilation and contraction. The VW model is nonlinear time-invariant, and is able to predict the observed differences between the time series of CBV and that of CBF measurements following changes in neural activity. Like the windkessel model derived by Mandeville, J.B., Marota, J.J.A., Ayata, C., Zaharchuk, G., Moskowitz, M.A., Rosen, B.R., Weisskoff, R.M., 1999. Evidence of a cerebrovascular postarteriole windkessel with delayed compliance. J. Cereb. Blood Flow Metab. 19, 679–689, the VW model is primarily a model of haemodynamic changes in the venous compartment. The VW model is demonstrated to have the following characteristics typical of visco-elastic materials: (1) hysteresis, (2) creep, and (3) stress relaxation, hence it provides a unified model of the visco-elastic properties of the vasculature. The model will not only contribute to the interpretation of the Blood Oxygen Level Dependent (BOLD) signals from functional Magnetic Resonance Imaging (fMRI) experiments, but also find applications in the study and modelling of the brain vasculature and the haemodynamics of circulatory and cardiovascular systems.

Atmospheric blocking in a high resolution climate model: influences of mean state, orography and eddy forcing

An underestimate of atmospheric blocking occurrence is a well-known limitation of many climate models. This article presents an analysis of Northern Hemisphere winter blocking in an atmospheric model with increased horizontal resolution. European blocking frequency increases with model resolution, and this results from an improvement in the atmospheric patterns of variability as well as a simple improvement in the mean state. There is some evidence that the transient eddy momentum forcing of European blocks is increased at high resolution, which could account for this. However, it is also shown that the increase in resolution of the orography is needed to realise the improvement in blocking, consistent with the increase in height of the Rocky Mountains acting to increase the tilt of the Atlantic jet stream and giving higher mean geopotential heights over northern Europe. Blocking frequencies in the Pacific sector are also increased with atmospheric resolution, but in this case the improvement in orography actually leads to a decrease in blocking

Present state of global wetland extent and wetland methane modelling: methodology of a model inter-comparison project (WETCHIMP)

The Wetland and Wetland CH4 Intercomparison of Models Project (WETCHIMP) was created to evaluate our present ability to simulate large-scale wetland characteristics and corresponding methane (CH4) emissions. A multi-model comparison is essential to evaluate the key uncertainties in the mechanisms and parameters leading to methane emissions. Ten modelling groups joined WETCHIMP to run eight global and two regional models with a common experimental protocol using the same climate and atmospheric carbon dioxide (CO2) forcing datasets. We reported the main conclusions from the intercomparison effort in a companion paper (Melton et al., 2013). Here we provide technical details for the six experiments, which included an equilibrium, a transient, and an optimized run plus three sensitivity experiments (temperature, precipitation, and atmospheric CO2 concentration). The diversity of approaches used by the models is summarized through a series of conceptual figures, and is used to evaluate the wide range of wetland extent and CH4 fluxes predicted by the models in the equilibrium run. We discuss relationships among the various approaches and patterns in consistencies of these model predictions. Within this group of models, there are three broad classes of methods used to estimate wetland extent: prescribed based on wetland distribution maps, prognostic relationships between hydrological states based on satellite observations, and explicit hydrological mass balances. A larger variety of approaches was used to estimate the net CH4 fluxes from wetland systems. Even though modelling of wetland extent and CH4 emissions has progressed significantly over recent decades, large uncertainties still exist when estimating CH4 emissions: there is little consensus on model structure or complexity due to knowledge gaps, different aims of the models, and the range of temporal and spatial resolutions of the models.

Nocturnal cooling below a forest canopy: Model and evaluation

Nocturnal cooling of air within a forest canopy and the resulting temperature profile may drive local thermally driven motions, such as drainage flows, which are believed to impact measurements of ecosystem–atmosphere exchange. To model such flows, it is necessary to accurately predict the rate of cooling. Cooling occurs primarily due to radiative heat loss. However, much of the radiative loss occurs at the surface of canopy elements (leaves, branches, and boles of trees), while radiative divergence in the canopy air space is small due to high transmissivity of air. Furthermore, sensible heat exchange between the canopy elements and the air space is slow relative to radiative fluxes. Therefore, canopy elements initially cool much more quickly than the canopy air space after the switch from radiative gain during the day to radiative loss during the night. Thus in modeling air cooling within a canopy, it is not appropriate to neglect the storage change of heat in the canopy elements or even to assume equal rates of cooling of the canopy air and canopy elements. Here a simple parameterization of radiatively driven cooling of air within the canopy is presented, which accounts implicitly for radiative cooling of the canopy volume, heat storage in the canopy elements, and heat transfer between the canopy elements and the air. Simulations using this parameterization are compared to temperature data from the Morgan–Monroe State Forest (IN, USA) FLUXNET site. While the model does not perfectly reproduce the measured rates of cooling, particularly near the top of the canopy, the simulated cooling rates are of the correct order of magnitude.

A comparison of GOCE and drifter-based estimates of the North Atlantic steady-state surface circulation

Over the last decade, due to the Gravity Recovery And Climate Experiment (GRACE) mission and, more recently, the Gravity and steady state Ocean Circulation Explorer (GOCE) mission, our ability to measure the ocean’s mean dynamic topography (MDT) from space has improved dramatically. Here we use GOCE to measure surface current speeds in the North Atlantic and compare our results with a range of independent estimates that use drifter data to improve small scales. We find that, with filtering, GOCE can recover 70% of the Gulf Steam strength relative to the best drifter-based estimates. In the subpolar gyre the boundary currents obtained from GOCE are close to the drifter-based estimates. Crucial to this result is careful filtering which is required to remove small-scale errors, or noise, in the computed surface. We show that our heuristic noise metric, used to determine the degree of filtering, compares well with the quadratic sum of mean sea surface and formal geoid errors obtained from the error variance–covariance matrix associated with the GOCE gravity model. At a resolution of 100 km the North Atlantic mean GOCE MDT error before filtering is 5 cm with almost all of this coming from the GOCE gravity model.

The first real-time worldwide ionospheric predictions network: an advance in support of spaceborne experimentation, on-line model validation, and space weather

We report on the first realtime ionospheric predictions network and its capabilities to ingest a global database and forecast F-layer characteristics and "in situ" electron densities along the track of an orbiting spacecraft. A global network of ionosonde stations reported around-the-clock observations of F-region heights and densities, and an on-line library of models provided forecasting capabilities. Each model was tested against the incoming data; relative accuracies were intercompared to determine the best overall fit to the prevailing conditions; and the best-fit model was used to predict ionospheric conditions on an orbit-to-orbit basis for the 12-hour period following a twice-daily model test and validation procedure. It was found that the best-fit model often provided averaged (i.e., climatologically-based) accuracies better than 5% in predicting the heights and critical frequencies of the F-region peaks in the latitudinal domain of the TSS-1R flight path. There was a sharp contrast however, in model-measurement comparisons involving predictions of actual, unaveraged, along-track densities at the 295 km orbital altitude of TSS-1R In this case, extrema in the first-principle models varied by as much as an order of magnitude in density predictions, and the best-fit models were found to disagree with the "in situ" observations of Ne by as much as 140%. The discrepancies are interpreted as a manifestation of difficulties in accurately and self-consistently modeling the external controls of solar and magnetospheric inputs and the spatial and temporal variabilities in electric fields, thermospheric winds, plasmaspheric fluxes, and chemistry.

The statistical cusp: a flux transfer event model

Traditionally, the cusp has been described in terms of a time-stationary feature of the magnetosphere which allows access of magnetosheath-like plasma to low altitudes. Statistical surveys of data from low-altitude spacecraft have shown the average characteristics and position of the cusp. Recently, however, it has been suggested that the ionospheric footprint of flux transfer events (FTEs) may be identified as variations of the “cusp” on timescales of a few minutes. In this model, the cusp can vary in form between a steady-state feature in one limit and a series of discrete ionospheric FTE signatures in the other limit. If this time-dependent cusp scenario is correct, then the signatures of the transient reconnection events must be able, on average, to reproduce the statistical cusp occurrence previously determined from the satellite observations. In this paper, we predict the precipitation signatures which are associated with transient magnetopause reconnection, following recent observations of the dependence of dayside ionospheric convection on the orientation of the IMF. We then employ a simple model of the longitudinal motion of FTE signatures to show how such events can easily reproduce the local time distribution of cusp occurrence probabilities, as observed by low-altitude satellites. This is true even in the limit where the cusp is a series of discrete events. Furthermore, we investigate the existence of double cusp patches predicted by the simple model and show how these events may be identified in the data.

Software evolution: a graph based model

A model based on graph isomorphisms is used to formalize software evolution. Step by step we narrow the search space by an informed selection of the attributes based on the current state-of-the-art in software engineering and generate a seed solution. We then traverse the resulting space using graph isomorphisms and other set operations over the vertex sets. The new solutions will preserve the desired attributes. The goal of defining an isomorphism based search mechanism is to construct predictors of evolution that can facilitate the automation of ’software factory’ paradigm. The model allows for automation via software tools implementing the concepts.

An evaluation of decadal probability forecasts from state-of-the-art climate models

While state-of-the-art models of Earth's climate system have improved tremendously over the last 20 years, nontrivial structural flaws still hinder their ability to forecast the decadal dynamics of the Earth system realistically. Contrasting the skill of these models not only with each other but also with empirical models can reveal the space and time scales on which simulation models exploit their physical basis effectively and quantify their ability to add information to operational forecasts. The skill of decadal probabilistic hindcasts for annual global-mean and regional-mean temperatures from the EU Ensemble-Based Predictions of Climate Changes and Their Impacts (ENSEMBLES) project is contrasted with several empirical models. Both the ENSEMBLES models and a “dynamic climatology” empirical model show probabilistic skill above that of a static climatology for global-mean temperature. The dynamic climatology model, however, often outperforms the ENSEMBLES models. The fact that empirical models display skill similar to that of today's state-of-the-art simulation models suggests that empirical forecasts can improve decadal forecasts for climate services, just as in weather, medium-range, and seasonal forecasting. It is suggested that the direct comparison of simulation models with empirical models becomes a regular component of large model forecast evaluations. Doing so would clarify the extent to which state-of-the-art simulation models provide information beyond that available from simpler empirical models and clarify current limitations in using simulation forecasting for decision support. Ultimately, the skill of simulation models based on physical principles is expected to surpass that of empirical models in a changing climate; their direct comparison provides information on progress toward that goal, which is not available in model–model intercomparisons.

MetUM-GOML: a near-globally coupled atmosphere–ocean-mixed-layer model

Well-resolved air–sea interactions are simulated in a new ocean mixed-layer, coupled configuration of the Met Office Unified Model (MetUM-GOML), comprising the MetUM coupled to the Multi-Column K Profile Parameterization ocean (MC-KPP). This is the first globally coupled system which provides a vertically resolved, high near-surface resolution ocean at comparable computational cost to running in atmosphere-only mode. As well as being computationally inexpensive, this modelling framework is adaptable– the independent MC-KPP columns can be applied selectively in space and time – and controllable – by using temperature and salinity corrections the model can be constrained to any ocean state. The framework provides a powerful research tool for process-based studies of the impact of air–sea interactions in the global climate system. MetUM simulations have been performed which separate the impact of introducing inter- annual variability in sea surface temperatures (SSTs) from the impact of having atmosphere–ocean feedbacks. The representation of key aspects of tropical and extratropical variability are used to assess the performance of these simulations. Coupling the MetUM to MC-KPP is shown, for example, to reduce tropical precipitation biases, improve the propagation of, and spectral power associated with, the Madden–Julian Oscillation and produce closer-to-observed patterns of springtime blocking activity over the Euro-Atlantic region.