Ensemble Of Initial Conditions Research Articles

Ensemble Experiments on Numerical Weather Prediction Error and Uncertainty for a North Pacific Forecast Failure

An intense maritime cyclone off the northwest coast of North America during 9–14 February 1999 was remarkable in the repeated inability of numerical weather prediction (NWP) models to forecast its evolution. Each day during that period, the operational and research models of the United States and Canada were initialized with new analysis fields. Each day, the new short-term NWP forecasts predicted the storm to strike the densely populated Lower Mainland (Vancouver) area of southwest British Columbia, Canada, during the next 24–48 h. NWP guidance prompted the local forecast office to issue storm warnings including one or more of the following for Vancouver: heavy snow, heavy rain, and strong winds. Satellite imagery clearly showed the storm off the coast, but the storm did not strike Vancouver until much later and in a decayed state. This synoptic case is studied with an aim to understand the source of the NWP error, and an ensemble of research model runs is made to address three possibilities for failure: 1) initial condition (IC) error, 2) model error for a particularly nonlinear or sensitive event, and 3) sympathetic data denial. To estimate the effect of IC uncertainty, a short-range ensemble system is developed and tested on a limited-area model for a sequence of successive 3-day reforecasts covering the 10-day period surrounding the storm. This IC ensemble shows some correlation between spread and skill and provides one estimate of IC uncertainty. To estimate the effect of model uncertainty, a physics-based ensemble is run for the same period. The effect of data denial is investigated by comparing forecasts made with the same model but from analyses created at different operational centers. Results suggest that if the runs initialized at 0000 UTC 10 February 1999 were used for guidance, model uncertainty was likely responsible for the forecast failure. It had larger-than-average model error but lower-than-average IC error. Subsequent forecast errors were likely dominated by IC uncertainty. An attempt at assessing sympathetic data denial is inconclusive.

Deviations from hydrodynamics on the nanoscale

This study has examined the decay of an instantaneously imposed heat pulse on an equilibrium model of a dense fluid. The spatial extent of the initial pulse is quite small, of the order of 100 cubic nanometres; the amount of energy added to the system is only 5% of the total system kinetic energy. This small pulse decays quite rapidly, within several picoseconds, but the decay proceeds more slowly than predicted by the hydrodynamic equations. During the first picosecond of the decay, the kinetic energy is not equipartitioned, and a rapid process of energy transfer from kinetic energy to potential energy via interparticle interactions takes place. A new transport theory is developed that includes the ‘pre-hydrodynamic’ stage of evolution of non-equilibrium systems. Formally exact expressions for the local density, velocity, and kinetic energy (temperature) fields are developed in terms of Green's functions that depend on dynamic quantities, such as the mean-square displacement, averaged over the ensemble of initial states. No partial differential equations are involved.

Global Optimization by Adiabatic Switching

We apply a recently introduced method for global optimization to determine the ground state energy and configuration for model metallic clusters. The global minimum for a given N–atom cluster is found by following the damped dynamics of the N particle system on an evolving potential energy surface. In this application, the time dependent interatomic potential interpolates adiabatically between the Lennard–Jones (LJ) and the Sutton–Chen (SC) forms. Starting with an ensemble of initial conditions corresponding to the ground state configuration of the Lennard–Jones cluster, the system asymptotically reaches the ground state of the Sutton–Chen cluster. We describe the method and present results for specific cluster size N = 15, when the ground state symmetry of LJN and SCN differ.

Schwinger-Dyson approach to nonequilibrium classical field theory

In this paper we discuss a Schwinger-Dyson (SD) approach for determining the time evolution of the unequal time correlation functions of a nonequilibrium classical field theory, where the classical system is described by an initial density matrix at time $t=0.$ We focus on $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ field theory in $1+1$ space-time dimensions where we can perform exact numerical simulations by sampling an ensemble of initial conditions specified by the initial density matrix. We discuss two approaches. The first, the bare vertex approximation (BVA), is based on ignoring vertex corrections to the SD equations in the auxiliary field formalism relevant for $1/N$ expansions. The second approximation is a related approximation made to the SD equations of the original formulation in terms of $\ensuremath{\varphi}$ alone. We compare these SD approximations as well as a Hartree approximation with exact numerical simulations. We find that both approximations based on the SD equations yield good agreement with exact numerical simulations and cure the late time oscillation problem of the Hartree approximation. We also discuss the relationship between the quantum and classical SD equations.

Replica-symmetry breaking in dynamical glasses

Systems of globally coupled logistic maps (GCLM) can display complex collective behaviour characterized by the formation of synchronous clusters. In the dynamical clustering regime, such systems possess a large number of coexisting attractors and might be viewed as dynamical glasses. Glass properties of GCLM in the thermodynamical limit of large system sizes $N$ are investigated. Replicas, representing orbits that start from various initial conditions, are introduced and distributions of their overlaps are numerically determined. We show that for fixed-field ensembles of initial conditions, as used in previous numerical studies, all attractors of the system become identical in the thermodynamical limit up to variations of order $1/\sqrt{N}$ because the initial value of the coupling field is characterized by vanishing fluctuations, and thus replica symmetry is recovered for $N\to \infty $. In contrast to this, when random-field ensembles of initial conditions are chosen, replica symmetry remains broken in the thermodynamical limit.

Thermalization of quantum fields from time-reversal invariant evolution equations

We study the time evolution of correlation functions in closed quantum systems for nonequilibrium ensembles of initial conditions. For a scalar quantum field theory we show that generic time-reversal invariant evolutions approach equilibrium at large times. The calculation provides a first principles justification of Boltzmann's conjecture that the large-time behavior of isolated macroscopic systems can be described by thermal ensemble averages.

Applications of the Adjoint Technique to Short-Range Ensemble Forecasting of Mesoscale Convective Systems

The feasibility of applying a mesoscale adjoint model to the creation of an ensemble for short-range simulations of mesoscale convective systems (MCSs) is explored. Because past studies show that forecasters routinely improve upon numerical guidance and can identify mesoscale-sized areas of forecast concern with a high level of skill, it is clear that forecasters have insights into the daily weather forecast problems that exceed what can be provided by a numerical weather prediction model. Using an adjoint model, one could develop a system in which a forecaster identifies the area of forecast concern and then designs a set of rapidly produced sensitivity experiments that evaluate the influence of key atmospheric parameters on the model forecast. The output from these sensitivity experiments is then used to create ensemble members for a short-range operational ensemble forecast, which is specifically designed to investigate the forecast concern of the day. This adjoint ensemble approach is tested for the 48-h period beginning 1200 UTC 27 May 1985, in which a long-lived MCS developed underneath a large-scale ridge. A mesoscale adjoint model is used to define the alterations to the model initial conditions necessary to evaluate the influences of key mesoscale structures, which the authors believe have a large influence on later convective development in the model. Results indicate that the adjoint technique is effective in creating the proper directional response in the model simulation. When compared to initial condition and model physics ensembles of this event, the adjoint ensemble produces more variance than the initial condition ensemble and almost as much variance as the model physics ensemble. However, the values of the equitable threat score and the ranked probability score are better for the adjoint ensemble between 6 and 24 h than for either of the other two ensembles. These results suggest that further exploration of ensembles that incorporate the experience and expertise possessed by forecasters is warranted.

Linear Regime Duration: Is 24 Hours a Long Time in Synoptic Weather Forecasting?

Day-to-day variations in the growth of uncertainty in the current state of the atmosphere have led to operational ensemble weather predictions in which an ensemble of different initial conditions, each perturbed from the best estimate of the current state and yet still consistent with the observations, is forecast. Contrasting competing methods for the selection of ensemble members is a subject of active research; the assumption that the ensemble members represent sufficiently small perturbations so as to evolve within the ‘‘linear regime’’ is implicit to several of these methods. This regime, in which the model dynamics are well represented by a linear approximation, is commonly held to extend to 2 or 3 days for operational forecasts. It is shown that this is rarely the case. A new measure, the relative nonlinearity, which quantifies the duration of the linear regime by monitoring the evolution of ‘‘twin’’ pairs of ensemble members, is introduced. Both European and American ensemble prediction systems are examined; in the cases considered for each system (87 and 25, respectively), the duration of the linear regime is often less than a day and never extends to 2 days. The internal consistency of operational ensemble formation schemes is discussed in light of these results. By decreasing the optimization time, a modified singular vector‐based formation scheme is shown to improve consistency while maintaining traditional skill and spread scores in the seven cases considered. The relevance of the linear regime to issues regarding data assimilation, adaptive observations, and model sensitivity is also noted.

Distance-Dependent Filtering of Background Error Covariance Estimates in an Ensemble Kalman Filter

The usefulness of a distance-dependent reduction of background error covariance estimates in an ensemble Kalman filter is demonstrated. Covariances are reduced by performing an elementwise multiplication of the background error covariance matrix with a correlation function with local support. This reduces noisiness and results in an improved background error covariance estimate, which generates a reduced-error ensemble of model initial conditions. The benefits of applying the correlation function can be understood in part from examining the characteristics of simple 2 × 2 covariance matrices generated from random sample vectors with known variances and covariance. These show that noisiness in covariance estimates tends to overwhelm the signal when the ensemble size is small and/or the true covariance between the sample elements is small. Since the true covariance of forecast errors is generally related to the distance between grid points, covariance estimates generally have a higher ratio of noise to signal with increasing distance between grid points. This property is also demonstrated using a two-layer hemispheric primitive equation model and comparing covariance estimates generated by small and large ensembles. Covariances from the large ensemble are assumed to be accurate and are used a reference for measuring errors from covariances estimated from a small ensemble. The benefits of including distance-dependent reduction of covariance estimates are demonstrated with an ensemble Kalman filter data assimilation scheme. The optimal correlation length scale of the filter function depends on ensemble size; larger correlation lengths are preferable for larger ensembles. The effects of inflating background error covariance estimates are examined as a way of stabilizing the filter. It was found that more inflation was necessary for smaller ensembles than for larger ensembles.

Convergence to the critical attractor of dissipative maps: log-periodic oscillations, fractality, and nonextensivity

For a family of logisticlike maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space. We found that the phase-space volume occupied by the ensemble W(t) depicts a power-law decay with log-periodic oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractor's fractal dimension governed by the inflection of the map near its extremal point. Further, we investigate the temporal evolution of W(t) for the circle map whose critical attractor is dense. In this case, we found W(t) to exhibit a rich pattern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of nonextensive Tsallis entropies.

Using Initial Condition and Model Physics Perturbations in Short-Range Ensemble Simulations of Mesoscale Convective Systems

Two separate numerical model ensembles are created by using model configurations with different model physical process parameterization schemes and identical initial conditions, and by using different model initial conditions from a Monte Carlo approach and the identical model configuration. Simulations from these two ensembles are investigated for two 48-h periods during which large, long-lived mesoscale convective systems develop. These two periods are chosen because, in some respects, they span the range of convective forecast problems routinely handled by operational forecasters. Calculations of the root-mean-square error, equitable threat score, and ranked probability score from both ensembles indicate that the model physics ensemble is more skillful than the initial-condition ensemble when the large-scale forcing for upward motion is weak. When the large-scale forcing for upward motion is strong, the initial-condition ensemble is more skillful than the model physics ensemble. This result is consistent with the expectation that model physics play a larger role in model simulations when the large-scale signal is weak and the assumptions used within the model parameterization schemes largely determine the evolution of the simulated weather events. The variance from the two ensembles is created at significantly different rates, with the variance in the physics ensemble being produced two to six times faster during the first 12 h than the variance in the initial-condition ensemble. Therefore, within a very brief time period, the variance from the physics ensemble often greatly exceeds that produced by the initial-condition ensemble. These results suggest that varying the model physics is a potentially powerful method to use in creating an ensemble. In essence, by using different model configurations, the systematic errors of the individual ensemble members are different and, hence, this may allow one to determine a probability density function from this ensemble that is more diffuse than can be accomplished using a single model configuration.

Chaos and universality in the dynamics of inflationary cosmologies

We describe a new statistical pattern in the chaotic dynamics of closed inflationary cosmologies, associated with the partition of the Hamiltonian rotational motion energy and hyperbolic motion energy pieces, in a linear neighborhood of the saddle-center present in the phase space of the models. The hyperbolic energy of orbits visiting a neighborhood of the saddle-center has a random distribution with respect to the ensemble of initial conditions, but the associated histograms define a statistical distribution law of the form $p(x) = C x^{-\gamma}$, for almost the whole range of hyperbolic energies considered. We present numerical evidence that $\gamma$ determines the dimension of the fractal basin boundaries in the ensemble of initial conditions. This distribution is universal in the sense that it does not depend on the parameters of the models and is scale invariant. We discuss possible physical consequences of this universality for the physics of inflation.tribution law of the form $p(x) = C x^{-\gamma}$, for almost the whole range of hyperbolic energies considered. We present numerical evidence that $\gamma$ determines the dimension of the fractal basin boundaries in the ensemble of initial conditions. This distribution is universal in the sense that it does not depend on the parameters of the models and is scale invariant. We discuss possible physical consequences of this universality for the physics of inflation.

Canonical ensemble of initial states leading to chiral fluctuations

In energetic heavy ion collisions, if quark-gluon plasma is formed, its hadronization may lead to observable critical fluctuations, i.e., DCC formation. The strength and observability of these fluctuations depend on the initial state. Here we study the canonical ensemble of initial states of chiral fluctuations in heavy ion collisions and the probability to obtain observable domains of chiral condensates.

Moment equations and closure schemes in chaotic dynamics

Closure schemes for the infinite hierarchy of moment equations describing the time evolution of averages of observables over a non-equilibrium ensemble of initial conditions for the logistic map model are derived with the aid the Frobenius-Perron equation in the long-time regime. For the growth parameter r = 4 the approximations compare well with exact results derived for specific initial probability densities.

Phase mixing in time-independent Hamiltonian systems

Everything you ever wanted to know about what has come to be known as ``chaotic mixing:'' This paper describes the evolution of localised ensembles of initial conditions in 2- and 3-D time-independent potentials which admit both regular and chaotic orbits. The coarse-grained approach towards an invariant, or near-invariant, distribution was probed by tracking (1) phase space moments through order 4 and (2) binned reduced distributions f(Z_a,Z_b,t) for a,b=x,y,z,p_x,p_y,p_z, computed at fixed time intervals. For ``unconfined'' chaotic orbits in 2-D systems not stuck near islands by cantori, the moments evolve exponentially: Quantities like the dispersion in p_x, which start small and eventually asymptote towards a larger value, initially grow exponentially in time at a rate comparable to the largest short time Lyapunov exponent. Quantities like | |, that can start large but eventually asymptote towards zero, decrease exponentially. With respect to a discrete L^p norm, reduced distributions f(t) generated from successive decay exponentially towards a near-invariant f_{niv}, although a plot of Df(t)=||f(t)-f_{niv}|| can exhibit considerable structure. Regular ensembles behave very differently, both moments and Df evolving in a fashion better represented by a power law time dependence. ``Confined'' chaotic orbits, initially stuck near regular islands because of cantori, exhibit an intermediate behaviour. The behaviour of ensembles evolved in 3-D potentials is qualitatively similar, except that, in this case, it is relatively likely to find one direction in configuration space which is ``less chaotic'' than the other two, so that quantities like L_{ab} depend more sensitively on which phase space variables one tracks.

Time evolution of correlation functions in non-equilibrium field theories

We investigate the non-equilibrium properties of an N -component scalar field theory. The time evolution of the equal-time correlation functions for an arbitrary ensemble of initial conditions is described by an exact functional differential equation. In leading order in the 1/ N expansion the system can be understood in terms of infinitely many conserved quantities. They forbid the approach to the canonical thermal distribution. Beyond leading order only energy conservation is apparent generically. Nevertheless, we find a large manifold of stationary distributions both for classical and quantum fields. They are the fixed points of the evolution equation. For small deviations of the correlation functions from a large range of fixed points we observe stable oscillations. These results raise the question of if and in what sense the particular fixed point corresponding to thermal equilibrium dominates the large time behavior of the system.

Sufficient condition for a finite-time singularity in a high-symmetry Euler flow: Analysis and statistics.

A sufficient condition is obtained for the development of a finite-time singularity in a highly symmetric Euler flow, first proposed by Kida [J. Phys. Soc. Jpn. 54, 2132 (1995)] and recently simulated by Boratav and Pelz [Phys. Fluids 6, 2757 (1994)]. It is shown that if the second-order spatial derivative of the pressure (${\mathit{p}}_{\mathit{xx}}$) is positive following a Lagrangian element (on the x axis), then a finite-time singularity must occur. Under some assumptions, this Lagrangian sufficient condition can be reduced to an Eulerian sufficient condition which requires that the fourth-order spatial derivative of the pressure (${\mathit{p}}_{\mathit{xxxx}}$) at the origin be positive for all times leading up to the singularity. Analytical as well as direct numerical evaluation over a large ensemble of initial conditions demonstrate that for fixed total energy, ${\mathit{p}}_{\mathit{xxxx}}$ is predominantly positive with the average value growing with the numbers of modes. \textcopyright{} 1996 The American Physical Society.

A stochastic interrogation method for experimental measurements of global dynamics and basin evolution: Application to a two-well oscillator

An experimental study of local and global bifurcations in a driven two-well magneto-mechanical oscillator is presented. A detailed picture of the local bifurcation structure of the system is obtained using an automated bifurcation data acquisition system. Basins of attractions for the system are obtained using a new experimental technique: an ensemble of initial conditions is generated by switching between stochastic and deterministic excitation. Using this stochastic interrogation method, we observe the evolution of basins of attraction in the nonlinear oscillator as the forcing amplitude is increased, and find evidence for homoclinic bifurcation before the onset of chaos. Since the entire transient is collected for each initial condition, the same data can be used to obtain pictures of the flow of points in phase space. Using Liouville's Theorem, we obtain damping estimates by calculating the contraction of volumes under the action of the Poincare map, and show that they are in good agreement with the results of more conventional damping estimation methods. Finally, the stochastic interrogation data is used to estimate transition probability matrices for finite partitions of the Poincare section. Using these matrices, the evolution of probability densities can be studied.

Classical motion in two-dimensional crystals

The classical motion of an electron of high enough energy in a two-dimensional crystal is diffusive for many potentials with Coulomb singularities. A simple model of the dynamics is developed which predicts the dependence of the diffusion constantD on the particle energyE in the high-energy limit:D(E)∼const·E3/2. This diffusion law is checked for a concrete crystal by numerically integrating the Hamilton equations for an ensemble of initial conditions. Finally this method is compared with other models of the classical dynamics in a crystal, especially the Sinai billiard.

Chaos and quantum thermalization.

We show that a bounded, isolated quantum system of many particles in a specific initial state will approach thermal equilibrium if the energy eigenfunctions which are superposed to form that state obey {\it Berry's conjecture}. Berry's conjecture is expected to hold only if the corresponding classical system is chaotic, and essentially states that the energy eigenfunctions behave as if they were gaussian random variables. We review the existing evidence, and show that previously neglected effects substantially strengthen the case for Berry's conjecture. We study a rarefied hard-sphere gas as an explicit example of a many-body system which is known to be classically chaotic, and show that an energy eigenstate which obeys Berry's conjecture predicts a Maxwell--Boltzmann, Bose--Einstein, or Fermi--Dirac distribution for the momentum of each constituent particle, depending on whether the wave functions are taken to be nonsymmetric, completely symmetric, or completely antisymmetric functions of the positions of the particles. We call this phenomenon {\it eigenstate thermalization}. We show that a generic initial state will approach thermal equilibrium at least as fast as $O(\hbar/\Delta)t^{-1}$, where $\Delta$ is the uncertainty in the total energy of the gas. This result holds for an individual initial state; in contrast to the classical theory, no averaging over an ensemble of initial states is needed. We argue that these results constitute a new foundation for quantum statistical mechanics.