Nonlinear data assimilation in geosciences: an extremely efficient particle filter

Almost all research fields in geosciences use numerical models and observations and combine these using data-assimilation techniques. With ever-increasing resolution and complexity, the numerical models tend to be highly nonlinear and also observations become more complicated and their relation to the models more nonlinear. Standard data-assimilation techniques like (ensemble) Kalman filters and variational methods like 4D-Var rely on linearizations and are likely to fail in one way or another. Nonlinear data-assimilation techniques are available, but are only efficient for small-dimensional problems, hampered by the so-called ‘curse of dimensionality’. Here we present a fully nonlinear particle filter that can be applied to higher dimensional problems by exploiting the freedom of the proposal density inherent in particle filtering. The method is illustrated for the three-dimensional Lorenz model using three particles and the much more complex 40-dimensional Lorenz model using 20 particles. By also applying the method to the 1000-dimensional Lorenz model, again using only 20 particles, we demonstrate the strong scale-invariance of the method, leading to the optimistic conjecture that the method is applicable to realistic geophysical problems. Copyright c 2010 Royal Meteorological Society

Precise calculations of the existence of multiple AMOC equilibria in coupled climate models Part I: equilibrium states

This study examines criteria for the existence of two stable states of the Atlantic Meridional Overturning Circulation (AMOC) using a combination of theory and simulations from a numerical coupled atmosphere–ocean climate model. By formulating a simple collection of state parameters and their relationships, the authors reconstruct the North Atlantic Deep Water (NADW) OFF state behavior under a varying external salt-flux forcing. This part (Part I) of the paper examines the steady-state solution, which gives insight into the mechanisms that sustain the NADW OFF state in this coupled model; Part II deals with the transient behavior predicted by the evolution equation. The nonlinear behavior of the Antarctic Intermediate Water (AAIW) reverse cell is critical to the OFF state. Higher Atlantic salinity leads both to a reduced AAIW reverse cell and to a greater vertical salinity gradient in the South Atlantic. The former tends to reduce Atlantic salt export to the Southern Ocean, while the latter tends to increases it. These competing effects produce a nonlinear response of Atlantic salinity and salt export to salt forcing, and the existence of maxima in these quantities. Thus the authors obtain a natural and accurate analytical saddle-node condition for the maximal surface salt flux for which a NADW OFF state exists. By contrast, the bistability indicator proposed by De Vries and Weber does not generally work in this model. It is applicable only when the effect of the AAIW reverse cell on the Atlantic salt budget is weak.

Superensembles of linear viscoelastic models of polymer melts

The idea of incorporating multiple models of linear rheology into a superensemble, to forge a consensus forecast from the individual model predictions, is investigated. The relative importance of the individual models in the so-called multimodel superensemble (MMSE) was inferred by evaluating their performance on a set of experimental training data, via nonlinear regression. The predictive ability of the MMSE model was tested by comparing its predictions on test data that were similar (in-sample) and dissimilar (out-of-sample) to the training data used in the calibration. For the in-sample forecasts, we found that the MMSE model easily outperformed the best constituent model. The presence of good individual models greatly enhanced the MMSE forecast, while the presence of some bad models in the superensemble also improved the MMSE forecast modestly. While the performance of the MMSE model on the out-of-sample training data was not as spectacular, it demonstrated the robustness of this approach.

Optimal linearization trajectories for tangent linear models

We examine differential equations where nonlinearity is a result of the advection part of the total derivative or the use of quadratic algebraic constraints between state variables (such as the ideal gas law). We show that these types of nonlinearity can be accounted for in the tangent linear model by a suitable choice of the linearization trajectory. Using this optimal linearization trajectory, we show that the tangent linear model can be used to reproduce the exact nonlinear error growth of perturbations for more than 200 days in a quasi-geostrophic model and more than (the equivalent of) 150 days in the Lorenz 96 model. We introduce an iterative method, purely based on tangent linear integrations, that converges to this optimal linearization trajectory. The main conclusion from this article is that this iterative method can be used to account for nonlinearity in estimation problems without using the nonlinear model. We demonstrate this by performing forecast sensitivity experiments in the Lorenz 96 model and show that we are able to estimate analysis increments that improve the two-day forecast using only four backward integrations with the tangent linear model. Copyright © 2011 Royal Meteorological Society

Validation of three new measure-correlate-predict models for the long-term prospection of the wind resource

The estimation of the long-term wind resource at a prospective site based on a relatively short on-site measurement campaign is an indispensable task in the development of a commercial wind farm. The typical industry approach is based on the measure-correlate-predict �MCP� method where a relational model between the site wind velocity data and the data obtained from a suitable reference site is built from concurrent records. In a subsequent step, a long-term prediction for the prospective site is obtained from a combination of the relational model and the historic reference data. In the present paper, a systematic study is presented where three new MCP models, together with two published reference models �a simple linear regression and the variance ratio method�, have been evaluated based on concurrent synthetic wind speed time series for two sites, simulating the prospective and the reference site. The synthetic method has the advantage of generating time series with the desired statistical properties, including Weibull scale and shape factors, required to evaluate the five methods under all plausible conditions. In this work, first a systematic discussion of the statistical fundamentals behind MCP methods is provided and three new models, one based on a nonlinear regression and two �termed kernel methods� derived from the use of conditional probability density functions, are proposed. All models are evaluated by using five metrics under a wide range of values of the correlation coefficient, the Weibull scale, and the Weibull shape factor. Only one of all models, a kernel method based on bivariate Weibull probability functions, is capable of accurately predicting all performance metrics studied.

The role of atmosphere feedbacks during ENSO in the CMIP3 models. Part III: the shortwave flux feedback

Previous studies using coupled general circulation models (GCMs) suggest that the atmosphere model plays a dominant role in the modeled El Nin ̃ o–Southern Oscillation (ENSO), and that intermodel differences in the thermodynamical damping of sea surface temperatures (SSTs) are a dominant contributor to the ENSO amplitude diversity. This study presents a detailed analysis of the shortwave flux feedback (aSW) in 12 Coupled Model Intercomparison Project phase 3 (CMIP3) simulations, motivated by findings that aSW is the primary contributor to model thermodynamical damping errors. A ‘‘feedback decomposition method,’’ developed to elucidate the aSW biases, shows that all models un- derestimate the dynamical atmospheric response to SSTs in the eastern equatorial Pacific, leading to un- derestimated aSW values. Biases in the cloud response to dynamics and the shortwave interception by clouds also contribute to errors in aSW. Changes in the aSW feedback between the coupled and corresponding atmosphere-only simulations are related to changes in the mean dynamics. A large nonlinearity is found in the observed and modeled SW flux feedback, hidden when linearly cal- culating aSW. In the observations, two physical mechanisms are proposed to explain this nonlinearity: 1) a weaker subsidence response to cold SST anomalies than the ascent response to warm SST anomalies and 2) a nonlinear high-level cloud cover response to SST. The shortwave flux feedback nonlinearity tends to be underestimated by the models, linked to an underestimated nonlinearity in the dynamical response to SST. The process-based methodology presented in this study may help to correct model ENSO atmospheric biases, ultimately leading to an improved simulation of ENSO in GCMs.

The troposphere-to-stratosphere transition in kinetic energy spectra and nonlinear spectral fluxes as seen in ECMWF analyses

Global horizontal wavenumber kinetic energy spectra and spectral fluxes of rotational kinetic energy and enstrophy are computed for a range of vertical levels using a T799 ECMWF operational analysis. Above 250 hPa, the kinetic energy spectra exhibit a distinct break between steep and shallow spectral ranges, reminiscent of dual power-law spectra seen in aircraft data and high-resolution general circulation models. The break separates a large-scale ‘‘balanced’’ regime in which rotational flow strongly dominates divergent flow and a mesoscale ‘‘unbalanced’’ regime where divergent energy is comparable to or larger than rotational energy. Between 230 and 100 hPa, the spectral break shifts to larger scales (from n 5 60 to n 5 20, where n is spherical harmonic index) as the balanced component of the flow preferentially decays. The location of the break remains fairly stable throughout the stratosphere. The spectral break in the analysis occurs at somewhat larger scales than the break seen in aircraft data. Nonlinear spectral fluxes defined for the rotational component of the flow maximize between about 300 and 200 hPa. Large-scale turbulence thus centers on the extratropical tropopause region, within which there are two distinct mechanisms of upscale energy transfer: eddy–eddy interactions sourcing the transient energy peak in synoptic scales, and zonal mean–eddy interactions forcing the zonal flow. A well-defined downscale enstrophy flux is clearly evident at these altitudes. In the stratosphere, the transient energy peak moves to planetary scales and zonal mean–eddy interactions become dominant.

Comparison of polar ozone loss rates simulated by one-dimensional and three-dimensional models with Match observations in recent Antarctic and Arctic winters

Simulations of ozone loss rates using a three-dimensional chemical transport model and a box model during recent Antarctic and Arctic winters are compared with experimental loss rates. The study focused on the Antarctic winter 2003, during which the first Antarctic Match campaign was organized, and on Arctic winters 1999/2000, 2002/2003. The maximum ozone loss rates retrieved by the Match technique for the winters and levels studied reached 6 ppbv/sunlit hour and both types of simulations could generally reproduce the observations at 2-sigma error bar level. In some cases, for example, for the Arctic winter 2002/2003 at 475 K level, an excellent agreement within 1-sigma standard deviation level was obtained. An overestimation was also found with the box model simulation at some isentropic levels for the Antarctic winter and the Arctic winter 1999/2000, indicating an overestimation of chlorine activation in the model. Loss rates in the Antarctic show signs of saturation in September, which have to be considered in the comparison. Sensitivity tests were performed with the box model in order to assess the impact of kinetic parameters of the ClO-Cl2O2 catalytic cycle and total bromine content on the ozone loss rate. These tests resulted in a maximum change in ozone loss rates of 1.2 ppbv/sunlit hour, generally in high solar zenith angle conditions. In some cases, a better agreement was achieved with fastest photolysis of Cl2O2 and additional source of total inorganic bromine but at the expense of overestimation of smaller ozone loss rates derived later in the winter.

Interface dynamics in planar neural field models

Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves.

Lateral boundary contributions to wave-activity invariants and nonlinear stability theorems for balanced dynamics

The slow advective-timescale dynamics of the atmosphere and oceans is referred to as balanced dynamics. An extensive body of theory for disturbances to basic flows exists for the quasi-geostrophic (QG) model of balanced dynamics, based on wave-activity invariants and nonlinear stability theorems associated with exact symmetry-based conservation laws. In attempting to extend this theory to the semi-geostrophic (SG) model of balanced dynamics, Kushner & Shepherd discovered lateral boundary contributions to the SG wave-activity invariants which are not present in the QG theory, and which affect the stability theorems. However, because of technical difficulties associated with the SG model, the analysis of Kushner & Shepherd was not fully nonlinear. This paper examines the issue of lateral boundary contributions to wave-activity invariants for balanced dynamics in the context of Salmon's nearly geostrophic model of rotating shallow-water flow. Salmon's model has certain similarities with the SG model, but also has important differences that allow the present analysis to be carried to finite amplitude. In the process, the way in which constraints produce boundary contributions to wave-activity invariants, and additional conditions in the associated stability theorems, is clarified. It is shown that Salmon's model possesses two kinds of stability theorems: an analogue of Ripa's small-amplitude stability theorem for shallow-water flow, and a finite-amplitude analogue of Kushner & Shepherd's SG stability theorem in which the ‘subsonic’ condition of Ripa's theorem is replaced by a condition that the flow be cyclonic along lateral boundaries. As with the SG theorem, this last condition has a simple physical interpretation involving the coastal Kelvin waves that exist in both models. Salmon's model has recently emerged as an important prototype for constrained Hamiltonian balanced models. The extent to which the present analysis applies to this general class of models is discussed.

A spectral view of nonlinear fluxes and stationary-transient interaction in the atmosphere

Nonlinear spectral transfers of kinetic energy and enstrophy, and stationary-transient interaction, are studied using global FGGE data for January 1979. It is found that the spectral transfers arise primarily from a combination, in roughly equal measure, of pure transient and mixed stationary-transient interactions. The pure transient interactions are associated with a transient eddy field which is approximately locally homogeneous and isotropic, and they appear to be consistently understood within the context of two-dimensional homogeneous turbulence. Theory based on spatial wale separation concepts suggests that the mixed interactions may be understood physically, to a first approximation, as a process of shear-induced spectral transfer of transient enstrophy along lines of constant zonal wavenumber. This essentially conservative enstrophy transfer generally involves highly nonlocal stationary-transient energy conversions. The observational analysis demonstrates that the shear-induced transient enstrophy transfer is mainly associated with intermediate-scale (zonal wavenumber m > 3) transients and is primarily to smaller (meridional) scales, so that the transient flow acts as a source of stationary energy. In quantitative terms, this transient-eddy rectification corresponds to a forcing timescale in the stationary energy budget which is of the same order of magnitude as most estimates of the damping timescale in simple stationary-wave models (5 to 15 days). Moreover, the nonlinear interactions involved are highly nonlocal and cover a wide range of transient scales of motion.

Testing the expectations theory of the term structure in threshold models

We test the expectations theory of the term structure of U.S. interest rates in nonlinear systems. These models allow the response of the change in short rates to past values of the spread to depend upon the level of the spread. The nonlinear system is tested against a linear system, and the results of testing the expectations theory in both models are contrasted. We find that the results of tests of the implications of the expectations theory depend on the size and sign of the spread. The long maturity spread predicts future changes of the short rate only when it is high.

Forecasting US output growth with non-linear models in the presence of data uncertainty

We consider the impact of data revisions on the forecast performance of a SETAR regime-switching model of U.S. output growth. The impact of data uncertainty in real-time forecasting will affect a model's forecast performance via the effect on the model parameter estimates as well as via the forecast being conditioned on data measured with error. We find that benchmark revisions do affect the performance of the non-linear model of the growth rate, and that the performance relative to a linear comparator deteriorates in real-time compared to a pseudo out-of-sample forecasting exercise.

The dependence of radiative forcing and feedback on evolving patterns of surface temperature change in climate models

Experiments with CO2 instantaneously quadrupled and then held constant are used to show that the relationship between the global-mean net heat input to the climate system and the global-mean surface-air-temperature change is nonlinear in Coupled Model Intercomparison Project phase 5 (CMIP5) Atmosphere-Ocean General Circulation Models (AOGCMs). The nonlinearity is shown to arise from a change in strength of climate feedbacks driven by an evolving pattern of surface warming. In 23 out of the 27 AOGCMs examined the climate feedback parameter becomes significantly (95% confidence) less negative – i.e. the effective climate sensitivity increases – as time passes. Cloud feedback parameters show the largest changes. In the AOGCM-mean approximately 60% of the change in feedback parameter comes from the topics (30N-30S). An important region involved is the tropical Pacific where the surface warming intensifies in the east after a few decades. The dependence of climate feedbacks on an evolving pattern of surface warming is confirmed using the HadGEM2 and HadCM3 atmosphere GCMs (AGCMs). With monthly evolving sea-surface-temperatures and sea-ice prescribed from its AOGCM counterpart each AGCM reproduces the time-varying feedbacks, but when a fixed pattern of warming is prescribed the radiative response is linear with global temperature change or nearly so. We also demonstrate that the regression and fixed-SST methods for evaluating effective radiative forcing are in principle different, because rapid SST adjustment when CO2 is changed can produce a pattern of surface temperature change with zero global mean but non-zero change in net radiation at the top of the atmosphere (~ -0.5 Wm-2 in HadCM3).