A general method for finding extremal states of Hamiltonian dynamical systems, with applications to perfect fluids

In addition to the Hamiltonian functional itself, non-canonical Hamiltonian dynamical systems generally possess integral invariants known as ‘Casimir functionals’. In the case of the Euler equations for a perfect fluid, the Casimir functionals correspond to the vortex topology, whose invariance derives from the particle-relabelling symmetry of the underlying Lagrangian equations of motion. In a recent paper, Vallis, Carnevale & Young (1989) have presented algorithms for finding steady states of the Euler equations that represent extrema of energy subject to given vortex topology, and are therefore stable. The purpose of this note is to point out a very general method for modifying any Hamiltonian dynamical system into an algorithm that is analogous to those of Vallis etal. in that it will systematically increase or decrease the energy of the system while preserving all of the Casimir invariants. By incorporating momentum into the extremization procedure, the algorithm is able to find steadily-translating as well as steady stable states. The method is applied to a variety of perfect-fluid systems, including Euler flow as well as compressible and incompressible stratified flow.

Dynamical control of the mesosphere by orographic and non-orographic gravity wave drag during the extended northern winters of 2006 and 2009

A version of the Canadian Middle Atmosphere Model (CMAM) that is nudged toward reanalysis data up to 1 hPa is used to examine the impacts of parameterized orographic and non-orographic gravity wave drag (OGWD and NGWD) on the zonal-mean circulation of the mesosphere during the extended northern winters of 2006 and 2009 when there were two large stratospheric sudden warmings. The simulations are compared to Aura Microwave Limb Sounder (MLS) observations of mesospheric temperature, carbon monoxide (CO) and derived zonal winds. The control simulation, which uses both OGWD and NGWD, is shown to be in good agreement with MLS. The impacts of OGWD and NGWD are assessed using simulations in which those sources of wave drag are removed. In the absence of OGWD the mesospheric zonal winds in the months preceding the warmings are too strong, causing increased mesospheric NGWD, which drives excessive downwelling, resulting in overly large lower mesospheric values of CO prior to the warming. NGWD is found to be most important following the warmings when the underlying westerlies are too weak to allow much vertical propagation of the orographic gravity waves to the mesosphere. NGWD is primarily responsible for driving the circulation that results in the descent of CO from the thermosphere following the warmings. Zonal mean mesospheric winds and temperatures in all simulations are shown to be strongly constrained by (i.e. slaved to) the stratosphere. Finally, it is demonstrated that the responses to OGWD and NGWD are non-additive due to their dependence and influence on the background winds and temperatures.

A dynamical systems framework for intermittent data assimilation

We consider the problem of discrete time filtering (intermittent data assimilation) for differential equation models and discuss methods for its numerical approximation. The focus is on methods based on ensemble/particle techniques and on the ensemble Kalman filter technique in particular. We summarize as well as extend recent work on continuous ensemble Kalman filter formulations, which provide a concise dynamical systems formulation of the combined dynamics-assimilation problem. Possible extensions to fully nonlinear ensemble/particle based filters are also outlined using the framework of optimal transportation theory.

A 27 day persistence model of near-Earth solar wind conditions: a long lead-time forecast and a benchmark for dynamical models

Geomagnetic activity has long been known to exhibit approximately 27 day periodicity, resulting from solar wind structures repeating each solar rotation. Thus a very simple near-Earth solar wind forecast is 27 day persistence, wherein the near-Earth solar wind conditions today are assumed to be identical to those 27 days previously. Effective use of such a persistence model as a forecast tool, however, requires the performance and uncertainty to be fully characterized. The first half of this study determines which solar wind parameters can be reliably forecast by persistence and how the forecast skill varies with the solar cycle. The second half of the study shows how persistence can provide a useful benchmark for more sophisticated forecast schemes, namely physics-based numerical models. Point-by-point assessment methods, such as correlation and mean-square error, find persistence skill comparable to numerical models during solar minimum, despite the 27 day lead time of persistence forecasts, versus 2–5 days for numerical schemes. At solar maximum, however, the dynamic nature of the corona means 27 day persistence is no longer a good approximation and skill scores suggest persistence is out-performed by numerical models for almost all solar wind parameters. But point-by-point assessment techniques are not always a reliable indicator of usefulness as a forecast tool. An event-based assessment method, which focusses key solar wind structures, finds persistence to be the most valuable forecast throughout the solar cycle. This reiterates the fact that the means of assessing the “best” forecast model must be specifically tailored to its intended use.

Challenges in communicating and using ensembles in operational flood forecasting.

Following trends in operational weather forecasting, where ensemble prediction systems (EPS) are now increasingly the norm, flood forecasters are beginning to experiment with using similar ensemble methods. Most of the effort to date has focused on the substantial technical challenges of developing coupled rainfall-runoff systems to represent the full cascade of uncertainties involved in predicting future flooding. As a consequence much less attention has been given to the communication and eventual use of EPS flood forecasts. Drawing on interviews and other research with operational flood forecasters from across Europe, this paper highlights a number of challenges to communicating and using ensemble flood forecasts operationally. It is shown that operational flood forecasters understand the skill, operational limitations, and informational value of EPS products in a variety of different and sometimes contradictory ways. Despite the efforts of forecasting agencies to design effective ways to communicate EPS forecasts to non-experts, operational flood forecasters were often skeptical about the ability of forecast recipients to understand or use them appropriately. It is argued that better training and closer contacts between operational flood forecasters and EPS system designers can help ensure the uncertainty represented by EPS forecasts is represented in ways that are most appropriate and meaningful for their intended consumers, but some fundamental political and institutional challenges to using ensembles, such as differing attitudes to false alarms and to responsibility for management of blame in the event of poor or mistaken forecasts are also highlighted. Copyright © 2010 Royal Meteorological Society.

Influence of mass diffusion in sea ice dynamical models

The mixing of floes of different thickness caused by repeated deformation of the ice cover is modeled as diffusion, and the mass balance equation for sea ice accounting for mass diffusion is developed. The effect of deformational diffusion on the ice thickness balance is shown to reach 1% of the divergence effect, which describes ridging and lead formation. This means that with the same accuracy the mass balance equation can be written in terms of mean velocity rather than mean mass-weighted velocity, which one should correctly use for a multicomponent fluid such as sea ice with components identified by floe thickness. Mixing (diffusion) of sea ice also occurs because of turbulent variations in wind and ocean drags that are unresolved in models. Estimates of the importance of turbulent mass diffusion on the dynamic redistribution of ice thickness are determined using empirical data for the turbulent diffusivity. For long-time-scale prediction (≫5 days), where unresolved atmospheric motion may have a length scale on the order of the Arctic basin and the time scale is larger than the synoptic time scale of atmospheric events, turbulent mass diffusion can exceed 10% of the divergence effect. However, for short-time-scale prediction, for example, 5 days, the unresolved scales are on the order of 100 km, and turbulent diffusion is about 0.1% of the divergence effect. Because inertial effects are small in the dynamics of the sea ice pack, diffusive momentum transfer can be disregarded.

Mid-Holocene climates of the Americas: a dynamical response to changed seasonality

Simulations of the climatic response to mid-Holocene (6 ka BP) orbital forcing with two coupled ocean–atmosphere models (FOAM and CSM) show enhancement of monsoonal precipitation in parts of the American Southwest, Central America and northernmost South America during Northern Hemisphere summer. The enhanced onshore flow that brings precipitation into Central America is caused by a northward displacement of the inter-tropical convergence zone, driven by cooling of the equatorial and warming of the northern subtropical and mid-latitude ocean. Ocean feedbacks also enhance precipitation over the American Southwest, although the increase in monsoon precipitation there is largely driven by increases in land-surface temperature. The northward shift in the equatorial precipitation band that causes enhanced precipitation in Central America and the American Southwest has a negative feedback effect on monsoonal precipitation in northern South America. The simulations demonstrate that mid-Holocene aridity in the mid-continent of North America is dynamically linked to the orbitally induced enhancement of the summer monsoon in the American Southwest, with a spatial structure (wet in the Southwest and dry in the mid-continent) similar to that found in strong monsoon years today. Changes in winter precipitation along the west coast of North America, in Central America and along the Gulf Coast, caused by southward-displacement of the westerly storm tracks, indicate that changes in the Northern Hemisphere winter monsoon also play a role in regional climate changes during the mid-Holocene. Although the simulations with FOAM and CSM differ in detail, the general mechanisms and patterns are common to both. The model results thus provide a coherent dynamical explanation for regional patterns of increased or decreased aridity shown by vegetation, lake status and aeolian data from the Americas

Dynamical and observational constraints on tropical Pacific sea surface temperatures at the last glacial maximum

Asynchronously coupled atmosphere and ocean general circulation model simulations are used to examine the consequences of changes in the west/east sea-surface temperature (SST) gradient across the equatorial Pacific at the last glacial maximum (LGM). Simulations forced by the CLIMAP SST for the LGM, where the west/east SST gradient across the Pacific is reduced compared to present, produce a reduction in the strength of the trade winds and a decrease in the west/east slope of the equatorial thermocline that is incompatible with thermocline depths newly inferred from foraminiferal assemblages. Stronger-than-present trade winds, and a more realistic simulation of the thermocline slope, are produced when eastern Pacific SSTs are 2°C cooler than western Pacific SSTs. Our study highlights the importance of spatial heterogeneity in tropical SSTs in determining key features of the glacial climate.

Predictability of the western North Pacific summer climate associated with different ENSO phases by ENSEMBLES multi-model seasonal forecasts

Predictability of the western North Pacific (WNP) summer climate associated with different El Niño–Southern Oscillation (ENSO) phases is investigated in this study based on the 1-month lead retrospective forecasts of five state-of-the-art coupled models from ENSEMBLES. During the period from 1960 to 2005, the models well capture the WNP summer climate anomalies during most of years in different ENSO phases except the La Niña decaying summers. In the El Niño developing, El Niño decaying and La Niña developing summers, the prediction skills are high for the WNP summer monsoon index (WNPMI), with the prediction correlation larger than 0.7. The high prediction skills of the lower-tropospheric circulation during these phases are found mainly over the tropical western Pacific Ocean, South China Sea and subtropical WNP. These good predictions correspond well to their close teleconnection with ENSO and the high prediction skills of tropical SSTs. By contrast, for the La Niña decaying summers, the prediction skills are considerably low with the prediction correlation for the WNPMI near to zero and low prediction skills around the Philippines and subtropical WNP. These poor predictions relate to the weak summer anomalies of the WNPMI during the La Niña decaying years and no significant connections between the WNP lower-tropospheric circulation anomalies and the SSTs over the tropical central and eastern Pacific Ocean in observations. However, the models tend to predict an apparent anomalous cyclone over the WNP during the La Niña decaying years, indicating a linearity of the circulation response over WNP in the models prediction in comparison with that during the El Niño decaying years which differs from observations. In addition, the models show considerable capability in describing the WNP summer anomalies during the ENSO neutral summers. These anomalies are related to the positive feedback between the WNP lower-tropospheric circulation and the local SSTs. The models can capture this positive feedback but with some uncertainties from different ensemble members during the ENSO neutral summers.

Towards a general theory of extremes for observables of chaotic dynamical systems

In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the cho- sen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan–Yorke dimension of the attractor. Preliminary numer- ical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

A spatial view of ensemble spread in convection permitting ensembles

With movement toward kilometer-scale ensembles, new techniques are needed for their characterization. A new methodology is presented for detailed spatial ensemble characterization using the fractions skill score (FSS). To evaluate spatial forecast differences, the average and standard deviation are taken of the FSS calculated over all ensemble member–member pairs at different scales and lead times. These methods were found to give important information about the ensemble behavior allowing the identification of useful spatial scales, spinup times for the model, and upscale growth of errors and forecast differences. The ensemble spread was found to be highly dependent on the spatial scales considered and the threshold applied to the field. High thresholds picked out localized and intense values that gave large temporal variability in ensemble spread: local processes and undersampling dominate for these thresholds. For lower thresholds the ensemble spread increases with time as differences between the ensemble members upscale. Two convective cases were investigated based on the Met Office United Model run at 2.2-km resolution. Different ensemble types were considered: ensembles produced using the Met Office Global and Regional Ensemble Prediction System (MOGREPS) and an ensemble produced using different model physics configurations. Comparison of the MOGREPS and multiphysics ensembles demonstrated the utility of spatial ensemble evaluation techniques for assessing the impact of different perturbation strategies and the need for assessing spread at different, believable, spatial scales.

The Hiccup: a dynamical coupling process during the autumn transition in the Northern Hemisphere – similarities and differences to sudden stratospheric warmings

Sudden stratospheric warmings (SSWs) are the most prominent vertical coupling process in the middle atmosphere, which occur during winter and are caused by the interaction of planetary waves (PWs) with the zonal mean flow. Vertical coupling has also been identified during the equinox transitions, and is similarly associated with PWs. We argue that there is a characteristic aspect of the autumn transition in northern high latitudes, which we call the “hiccup”, and which acts like a “mini SSW”, i.e. like a small minor warming. We study the average characteristics of the hiccup based on a superimposed epoch analysis using a nudged version of the Canadian Middle Atmosphere Model, representing 30 years of historical data. Hiccups can be identified in about half the years studied. The mesospheric zonal wind results are compared to radar observations over Andenes (69N,16E) for the years 2000–2013. A comparison of the average characteristics of hiccups and SSWs shows both similarities and differences between the two vertical coupling processes.

Method for exploratory cluster analysis and visualisation of single-trial ERP ensembles

Background: The validity of ensemble averaging on event-related potential (ERP) data has been questioned, due to its assumption that the ERP is identical across trials. Thus, there is a need for preliminary testing for cluster structure in the data. New method: We propose a complete pipeline for the cluster analysis of ERP data. To increase the signalto-noise (SNR) ratio of the raw single-trials, we used a denoising method based on Empirical Mode Decomposition (EMD). Next, we used a bootstrap-based method to determine the number of clusters, through a measure called the Stability Index (SI). We then used a clustering algorithm based on a Genetic Algorithm (GA)to define initial cluster centroids for subsequent k-means clustering. Finally, we visualised the clustering results through a scheme based on Principal Component Analysis (PCA). Results: After validating the pipeline on simulated data, we tested it on data from two experiments – a P300 speller paradigm on a single subject and a language processing study on 25 subjects. Results revealed evidence for the existence of 6 clusters in one experimental condition from the language processing study. Further, a two-way chi-square test revealed an influence of subject on cluster membership.

The effect of the equivalent-weights particle filter on dynamical balance in a primitive equation model

The disadvantage of the majority of data assimilation schemes is the assumption that the conditional probability density function of the state of the system given the observations [posterior probability density function (PDF)] is distributed either locally or globally as a Gaussian. The advantage, however, is that through various different mechanisms they ensure initial conditions that are predominantly in linear balance and therefore spurious gravity wave generation is suppressed. The equivalent-weights particle filter is a data assimilation scheme that allows for a representation of a potentially multimodal posterior PDF. It does this via proposal densities that lead to extra terms being added to the model equations and means the advantage of the traditional data assimilation schemes, in generating predominantly balanced initial conditions, is no longer guaranteed. This paper looks in detail at the impact the equivalent-weights particle filter has on dynamical balance and gravity wave generation in a primitive equation model. The primary conclusions are that (i) provided the model error covariance matrix imposes geostrophic balance, then each additional term required by the equivalent-weights particle filter is also geostrophically balanced; (ii) the relaxation term required to ensure the particles are in the locality of the observations has little effect on gravity waves and actually induces a reduction in gravity wave energy if sufficiently large; and (iii) the equivalent-weights term, which leads to the particles having equivalent significance in the posterior PDF, produces a change in gravity wave energy comparable to the stochastic model error. Thus, the scheme does not produce significant spurious gravity wave energy and so has potential for application in real high-dimensional geophysical applications.