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Recently it has been established, using simple mathematical models of chaos, low-order models, and large numerical models of the atmosphere, that small errors grow in the mean in a superexponential manner. In this paper the mechanisms behind this behaviour are examined with special emphasis on the non-orthogonality of the eigenvectors of the linearized evolution operator and the variability of the local Lyapunov exponents on the attractor. The study reveals a picture that is far more complex and system-dependent than what has been advanced so far in the literature. The general ideas are illustrated by Lorenz's low-order atmospheric model for which a simple phenomenological model of error growth is developed and tested successfully against the simulations.

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... Growth rates larger than predicted by the first Lyapunov exponent have been documented in the literature, but the underlying mechanism is controversial (Trevisan and Legnani 1995;Nicolis et al. 1995). Nonorthogonality of the vectors associated with asymptotic growth rates given by the Lyapunov exponents has been in-voked as a possible mechanism for super-Lyapunov growth. ...

... Based on expressions analogous to (12) and (14), Nicolis et al. (1995) concluded that RMS growth rates larger than those given by the largest Lyapunov are obtained under conditions that they considered too restrictive to be realized. Before we apply it to the periodic orbits of the Lorenz system and arrive at firm conclusions about the envisaged mechanism, we have to extend the analysis to the three-dimensional case. ...

... Ensemble average growth rates associated with the singular vectors are instead characterized by transient behavior, documented for some simple systems which include the Lorenz model (Trevisan 1993;Krishnamurty 1993;Trevisan and Legnani 1995;Nicolis et al. 1995). ...

Some theoretical issues related to the problem of quantifying local predictability of atmospheric flow and the generation of perturbations for ensemble forecasts are investigated in the Lorenz system. A periodic orbit analysis and the study of the properties of the associated tangent linear equations are performed.In this study a set of vectors are found that satisfy Oseledec theorem and reduce to Floquet eigenvectors in the particular case of a periodic orbit. These vectors, called Lyapunov vectors, can be considered the generalization to aperiodic orbits of the normal modes of the instability problem and are not necessarily mutually orthogonal.The relation between singular vectors and Lyapunov vectors is clarified, and transient or asymptotic error growth properties are investigated. The mechanism responsible for super-Lyapunov growth is shown to be related to the nonorthogonality of Lyapunov vectors.The leading Lyapunov vectors, as defined here, as well as the asymptotic final singular vectors, are tangent to the attractor, while the leading initial singular vectors, in general, point away from it. Perturbations that are on the attractor and maximize growth should be considered in meteorological applications such as ensemble forecasting and adaptive observations. These perturbations can be found in the subspace of the leading Lyapunov vectors.

... Despite the high variance of these errors, Lacarra and Talagrand (1988) found a transient growth that affects the short-range average of errors and Trevisan and Legnani (1995) related this behavior to the amplitude of the initial error. However, Nicolis (1992) showed that the first two moments of the distribution (mean and variance) are not enough to characterize the predictability due to a bimodal distribution of errors in the Lorenz 63 system. ...

... By averaging the ensemble member errors, the initial growth of error follows a more complex behavior than a regular exponential growth as demonstrated by Nicolis et al. (1995). They showed that a subexponential initial behavior is followed (in some cases) by a superexponential growth. ...

... Examples and conditions for this behavior can be found inNicolis et al. (1995). ...

Intrinsic predictability is defined as the uncertainty in a forecast due to small errors in the initial conditions. In fact, not only the amplitude but also the structure of these initial errors plays a key role in the evolution of the forecast. Several methodologies have been developed to create an ensemble of forecasts from a feasible set of initial conditions, such as bred vectors or singular vectors. However, these methodologies consider only the fastest growth direction globally, which is represented by the Lyapunov vector.
In this paper, the simple Lorenz 63 model is used to compare bred vectors, random perturbations, and normal modes against analogs. The concept of analogs is based on the ergodicity theory to select compatible states for a given initial condition. These analogs have a complex structure in the phase space of the Lorenz attractor that is compatible with the properties of the nonlinear chaotic system.
It is shown that the initial averaged growth rate of errors of the analogs is similar to the one obtained with bred vectors or normal modes (fastest growth), but they do not share other properties or statistics, such as the spread of these growth rates. An in-depth study of different properties of the analogs and the previous existing perturbation methodologies is carried out to shed light on the consequences of forecasting the choice of the perturbations.

... Since the amplification of this error fluctuates on the inhomogeneous attractor of the system, an average over the attractor is necessary in order to obtain properties that are independent of the initial state. We do not use the classical norm (L 2 norm) but rather the logarithmic norm (Nicolis et al., 1995), ...

... As the error (17) reaches a substantial amplitude, the effect of the nonlinear terms on the dynamics cannot be neglected anymore and the rate of amplification of the logarithm of the error starts to decrease, and for long lead times, saturates due to the finite size of the system's attractor. This evolution is discussed in detail in (Nicolis et al., 1995;Vannitsem and Nicolis, 1994;Vannitsem and Lucarini, 2016). As we are interested in the long-term predictability of the atmosphere in the coupled system, the focus is placed on how the error defined in (17) saturates for a long lead time. ...

The predictability of the atmosphere at short and long time scales, associated with the coupling to the ocean, is explored in a new version of the Modular Arbitrary‐Order Ocean‐Atmosphere Model (MAOOAM). This version features a new ocean basin geometry with periodic boundary conditions in the zonal direction. The analysis presented in this paper considers a low‐order version of the model with 40 dynamical variables. First the increase of surface friction (and the associated heat flux) with the ocean can either induce chaos when the aspect ratio between the meridional and zonal directions of the domain of integration is small, or suppress chaos when it is large. This reflects the potentially counter‐intuitive role that the ocean can play in the coupled dynamics. Second, and perhaps more importantly, the emergence of long‐term predictability within the atmosphere for specific values of the friction coefficient occurs through intermittent excursions in the vicinity of a (long‐period) unstable periodic solution. Once close to this solution the system is predictable for long times, i.e. a few years. The intermittent transition close to this orbit is, however, erratic and probably hard to predict. This new route to long‐term predictability contrasts with the one found in the closed ocean‐basin low‐order version of MAOOAM, in which the chaotic solution is permanently wandering in the vicinity of an unstable periodic orbit for specific values of the friction coefficient. The model solution is thus at any time influenced by the unstable periodic orbit and inherits from its long‐term predictability.

... Since the amplification of this error fluctuates on the inhomogeneous attractor of the system, an average over the attractor is necessary in order to obtain properties that are independent of the initial state. We do not use the classical norm (L 2 norm) but rather the logarithmic norm (Nicolis et al., 1995), ...

... As the error (17) reaches a substantial amplitude, the effect of the nonlinear terms on the dynamics cannot be neglected anymore and the rate of amplification of the logarithm of the error starts to decrease, and for long lead times, saturates due to the finite size of the system's attractor. This evolution is discussed in detail in (Nicolis et al., 1995;Vannitsem and Nicolis, 1994;Vannitsem and Lucarini, 2016). As we are interested in the long-term predictability of the atmosphere in the coupled system, the focus is placed on how the error defined in (17) saturates for a long lead time. ...

The predictability of the atmosphere at short and long time scales, associated with the coupling to the ocean, is explored in a new version of the Modular Arbitrary-Order Ocean-Atmosphere Model (MAOOAM). This version features a new ocean basin geometry with periodic boundary conditions in the zonal direction. The analysis presented in this paper considers a low-order version of the model with 40 dynamical variables. First the increase of surface friction (and the associated heat flux) with the ocean can either induce chaos when the aspect ratio between the meridional and zonal directions of the domain of integration is small, or suppress chaos when it is large. This reflects the potentially counter-intuitive role that the ocean can play in the coupled dynamics. Second, and perhaps more importantly, the emergence of long-term predictability within the atmosphere for specific values of the friction coefficient occurs through intermittent excursions in the vicinity of a (long-period) unstable periodic solution. Once close to this solution the system is predictable for long times, i.e. a few years. The intermittent transition close to this orbit is, however, erratic and probably hard to predict. This new route to long-term predictability contrasts with the one found in the closed ocean-basin low-order version of MAOOAM, in which the chaotic solution is permanently wandering in the vicinity of an unstable periodic orbit for specific values of the friction coefficient. The model solution is thus at any time influenced by the unstable periodic orbit and inherits from its long-term predictability.

... This programme has been followed on many occasions and has led to prominent achievements such as the understanding of the property of sensitivity to initial conditions (e.g. [9][10][11][12]). ...

... 10) after dividing by f 0 , and where H and d are the thickness of the lower atmospheric layer and the thickness of the Ekman surface layer, respectively. Typically, H is of the order of 5000 m and d of the order of 100-1000 m. ...

There is a growing interest in developing stochastic schemes for the description of processes that are poorly represented in atmospheric and climate models, in order to increase their variability and reduce the impact of model errors. The use of such noise could however have adverse effects by modifying in undesired ways a certain number of moments of their probability distributions. In this work, the impact of developing a stochastic scheme (based on stochastic averaging) for the ocean is explored in the context of a low-order coupled (deterministic) ocean-atmosphere system. After briefly analysing its variability, its ability in predicting the oceanic flow generated by the coupled system is investigated. Different phases in the error dynamics are found: for short lead times, an initial overdispersion of the ensemble forecast is present while the ensemble mean follows a dynamics reminiscent of the combined amplification of initial condition and model errors for deterministic systems; for longer lead times, a reliable diffusive ensemble spread is observed. These different phases are also found for ensemble-oriented skill measures like the Brier score and the rank histogram. The implications of these features on building stochastic models are then briefly discussed.

... We hightlight the fact that these last ones may be superior to the classical, infinite-time one. This can be due to the initial divergence that is not infinitesimal [1], to the initial condition or the time [10], or even the eigenvectors of the Jacobian matrix [11]. ...

... Maximum of the finite-time pseudo-Lyapunov exponents (equation(11)) of the dynamical system (9) with a = 0.15, b = −1, c = 1 along the initial condition x0. Thus the finite-time pseudo-Lyapunov exponent may take negative or positive values (decreasing or growing of the divergence), whereas the Lyapunov exponent is negative. ...

Lyapunov exponents measure the sensitivity of a dynamical system to initial conditions [1,2]. In fact an infinitesimal difference of initial conditions may lead to totally different paths. This is the case if the computed Lyapunov exponent is strictly positive. The system is then theoretically chaotic in infinite time, but practically this may occur at finite time considered as “asymptotic” for applications. Mathematically, the numerical value of the Lyapunov exponents is given by the formula:
$$\lambda = \mathop {\lim \;\;\lim }\limits_{t \to + \infty \left\| {\delta x_0 } \right\| \to 0t} \frac{1}{t}in\left({\frac{{\left\| {\delta x\left({t,x_0 } \right)} \right\|}}{{\left\| {\delta x_0 } \right\|}}} \right)$$ (1)
where x
0 is the initial condition (it may be a vector of initial conditions in ℝn), t is the time, ∥̤∥ is a norm for ℝn
, ∂x
0 is an infinitesimal divergence in initial conditions (in ℝn
) and .x (∈ ℝn
) is the path followed by the system starting at x = x
0 at time t =0.
The maximum of this spectrum is often the only one that is computed to detect chaos.

... Compared with the Lorenz model, the variability of the local Lyapunov exponents is reduced in the sophisticated atmospheric and climate models. Nicolis et al. [55] pointed out that a super-exponential growth of errors would arise when the variability is large. Because of the low variability of local Lyapunov exponents in sophisticated atmospheric and climate models, the impact of variability on error saturation would in principle be lower than in the Lorenz system. ...

The relative effects of initial condition and model uncertainties on local predictability are important issues in the atmospheric sciences. This study quantitatively compared the relative effects of these two types of uncertainty on local predictability using the Lorenz model. Local predictability limits were quantitatively estimated using the nonlinear local Lyapunov exponent (NLLE) method. Results show that the relative effects of initial conditions and model uncertainties on local predictability vary with the state. In addition, inverse spatial distributions of local predictability limits are induced by the two types of uncertainty. In the regime transition region, the local predictability limits of modeled states are more sensitive to initial condition uncertainty than to model uncertainty, resulting in lower local predictability limits being induced by initial condition uncertainties. Local predictability limits induced by initial condition uncertainties are 4 time units shorter than those induced by model uncertainties. In the “butterfly wing” regions, the local predictability limits of modeled states are more sensitive to model uncertainty than to initial condition uncertainty, resulting in lower local predictability limits due to model uncertainty. Local predictability limits induced by initial condition uncertainty are larger (0 to 4 time units) than those induced by model uncertainty. These differences in the regions that are sensitive to the two types of uncertainty mean that strategic reductions of uncertainty in sensitive areas may effectively improve forecast skill.

... As an effect of baroclinic instability, cyclonic activity facilitates poleward heat transport, two modes of which are represented by y and z. This appealing low-order model was studied in different contexts (Masoller and Schifino 1992;Pielke and Zeng 1994;Masoller et al. 1995;Nicolis et al. 1995;Roebber 1995;Shil'nikov et al. 1995;Provenzale and Balmforth 1999;Leonardo 1995;Tél and Gruiz 2006;Freire et al. 2008;Bódai et al. 2011aBódai et al. , 2013. The model reads as follows: ...

The authors argue that the concept of snapshot attractors and of their natural probability distributions are the only available tools by means of which mathematically sound statements can be made about averages, variances, etc., for a given time instant in a changing climate. A basic advantage of the snapshot approach, which relies on the use of an ensemble, is that the natural distribution and thus any statistics based on it are independent of the particular ensemble used, provided it is initiated in the past earlier than a convergence time. To illustrate these concepts, a tutorial presentation is given within the framework of a low-order model in which the temperature contrast parameter over a hemisphere decreases linearly in time. Furthermore, the averages and variances obtained from the snapshot attractor approach are demonstrated to strongly differ from the traditional 30-yr temporal averages and variances taken along single realizations. The authors also claim that internal variability can be quantified by the natural distribution since it characterizes the chaotic motion represented by the snapshot attractor. This experience suggests that snapshot-attractor-based calculations might be appropriate to be evaluated in any large-scale climate model, and that the application of 30-yr temporal averages taken along single realizations should be complemented with this more appealing tool for the characterization of climate changes, which seems to be practically feasible with moderate ensemble sizes.

... The model has been studied in many papers (see e.g. [42][43][44]). The time unit corresponds to 5 days, and therefore it was natural for Lorenz to introduce seasonal oscillations by making F sinusoidally time-dependent with a period of T = 73 time units = 365 days about a central value of F 0 [45]. ...

Based on the theory of “snapshot/pullback attractors”, we show that important features of the climate change that we are observing can be understood by imagining many replicas of Earth that are not interacting with each other. Their climate systems evolve in parallel, but not in the same way, although they all obey the same physical laws, in harmony with the chaotic-like nature of the climate dynamics. These parallel climate realizations evolving in time can be considered as members of an ensemble. We argue that the contingency of our Earth’s climate system is characterized by the multiplicity of parallel climate realizations rather than by the variability that we experience in a time series of our observed past. The natural measure of the snapshot attractor enables one to determine averages and other statistical quantifiers of the climate at any instant of time. In this paper, we review the basic idea for climate changes associated with monotonic drifts, and illustrate the large number of possible applications. Examples are given in a low-dimensional model and in numerical climate models of different complexity. We recall that systems undergoing climate change are not ergodic, hence temporal averages are generically not appropriate for the instantaneous characterization of the climate. In particular, teleconnections, i.e. correlated phenomena of remote geographical locations are properly characterized only by correlation coefficients evaluated with respect to the natural measure of a given time instant, and may also change in time. Physics experiments dealing with turbulent-like phenomena in a changing environment are also worth being interpreted in view of the attractor-based ensemble approach. The possibility of the splitting of the snapshot attractor to two branches, near points where the corresponding time-independent system undergoes bifurcation as a function of the changing parameter, is briefly mentioned. This can lead in certain climate-change scenarios to the coexistence of two distinct sub-ensembles representing dramatically different climatic options. The problem of pollutant spreading during climate change is also discussed in the framework of parallel climate realizations.

... Alternatively, one may sample the growth of infinitesimal perturbations along some other well-defined basis; for example, the local orientations of a basis which defines the global Lyapunov exponents. Effective growth rates sampled in this very different basis are also referred to as "local Lyapunov exponents" [38,[46][47][48][49][50]. To avoid increasing the existing confusion, we would refer to the second as finite sample Lyapunov exponents and avoid the moniker local Lyapunov exponent all together. ...

Inasmuch as Lyapunov exponents provide a necessary condition for chaos in a dynamical system, confidence bounds on estimated Lyapunov exponents are of great interest. Estimates derived either from observations or from numerical integrations are limited to trajectories of finite length, and it is the uncertainties in (the distribution of) these finite time Lyapunov exponents which are of interest. Within this context a bootstrap algorithm for quantifying sampling uncertainties is shown to be inappropriate for multiplicative-ergodic statistics of deterministic chaos. This result remains unchanged in the presence of observational noise. As originally proposed, the algorithm is also inappropriate for general nonlinear stochastic processes, a modified version is presented which may prove of value in the case of stochastic dynamics. A new approach towards quantifying the minimum duration of observations required to estimate global Lyapunov exponents is suggested and is explored in a companion paper.

... The equations are nondimensionalized with respect to time by the average damping time of eddies, being about 5 days. This model enjoys popularity in teaching [21,6] as well as theoretically oriented weather and climate research [22][23][24][25]. ...

... Our main illustrative example is based on a low-order model of atmospheric circulation introduced by Lorenz [31] (different from that of the celebrated Lorenz attractor [32]). This appealing model was studied in several contexts [28,[33][34][35][36][37][38][39][40][41][42][43][44]. It represents a coupled dynamics between the averaged wind speed of the Westerlies on one hemisphere, represented by the variable x, and two modes of cyclonic activity, denoted by y and z. ...

In nonautonomous dynamical systems, like in climate dynamics, an ensemble of trajectories initiated in the remote past defines a unique probability distribution, the natural measure of a snapshot attractor, for any instant of time, but this distribution typically changes in time. In cases with an aperiodic driving, temporal averages taken along a single trajectory would differ from the corresponding ensemble averages even in the infinite-time limit: ergodicity does not hold. It is worth considering this difference, which we call the nonergodic mismatch, by taking time windows of finite length for temporal averaging. We point out that the probability distribution of the nonergodic mismatch is qualitatively different in ergodic and nonergodic cases: its average is zero and typically nonzero, respectively. A main conclusion is that the difference of the average from zero, which we call the bias, is a useful measure of nonergodicity, for any window length. In contrast, the standard deviation of the nonergodic mismatch, which characterizes the spread between different realizations, exhibits a power-law decrease with increasing window length in both ergodic and nonergodic cases, and this implies that temporal and ensemble averages differ in dynamical systems with finite window lengths. It is the average modulus of the nonergodic mismatch, which we call the ergodicity deficit, that represents the expected deviation from fulfilling the equality of temporal and ensemble averages. As an important finding, we demonstrate that the ergodicity deficit cannot be reduced arbitrarily in nonergodic systems. We illustrate via a conceptual climate model that the nonergodic framework may be useful in Earth system dynamics, within which we propose the measure of nonergodicity, i.e., the bias, as an order-parameter-like quantifier of climate change.

... th of initial errors in their study with the ECMWF numerical prediction model. On the other hand, a number of authors have presented evidence of superexponential growth of large scale errors. This includes the studies of Schubert and Suarez (1989) with a simple general circulation model and the works of Trevisan (1993) with simple low order models. Nicolis et al. (1995) examined the mechanisms responsible for super-exponential growth in low order models. Frederiksen and Branstator (2001) have found that both large scale barotropic teleconnection pattern FT- NMs and travelling normal modes for zonally varying basic states can grow super-exponentially and sub-exponentially during parts of their life-cycl ...

The structural organization of initially random perturbations or 'errors' evolving in a barotropic tangent linear model with time-dependent basic states taken from observations, is examined for cases of block development, maturation and decay in the Southern Hemisphere atmosphere during April, November and December 1989. We determine statistical results relating the structures of evolved errors to singular vectors (SVs), Lyapunov vectors (LVs) and finite-time normal modes (FTNMs). The statistics of 100 evolved error fields are studied for six day periods or longer and compared with the growth and structures of leading fast growing SVs, LVs and FTNMs. The SVs are studied in the kinetic energy (KE), enstrophy (EN) and streamfunction (SF) norms, while all FTNMs and the first LV are norm independent. The mean of the largest pattern correlations between the 100 error fields and dynamical vectors, taken over the five fastest growing SVs, in any of the three norms, or over the five fastest growing FTNMs, increases with increasing time interval to a value close to 0.6 after six days. Corresponding pattern correlations with the five fastest growing LVs are slightly lower. The leading dynamical vectors (SVs 1, FTNM1 or LV 1) generally, but not always, give the largest pattern correlations with the error fields. It is found that viscosity slightly increases the average correlations between the evolved errors and LV 1 and evolved SVs 1. Mean pattern correlations with fast growing dynamical vectors increase further for time intervals longer than six days. The properties of the dynamical vectors during Southern Hemisphere blocking are briefly outlined. After a few days integration, the structures of the leading evolved SVs in the KE, EN and SF norms, are in general quite similar and also similar to some of the dominant FTNMs that are norm independent. For optimization times of six days or less, the evolved SVs and FTNMs are, in general, different from the dominant LVs on the same day. Nevertheless, amplification factors of the first FTNMs and first LVs are very similar, and also similar to, but slightly larger than, the mean amplification factor of 100 initially random perturbations in the SF norm, while the amplification factors in the SF norm of KE SVs 1 and SF SV 1 are much higher. For longer optimization times, the first SVs and the first FTNM increasingly turn towards the leading LV with convergence achieved within a month.

... Thus, the predictability studies that analyze the sensitivity to initial conditions, reveal the exponential (or close to) growth rate. Regarding to this, one can cite [2], [8], [12], [13], [7] and many others. In this paper we show, that at short time scales, the sensitivity to topography differs from the sensitivity to initial conditions. ...

In this paper, we look for an operator that describes the relationship between small errors in representation of the bottom topography in a barotropic ocean model and the model's solution. The study shows that the model's solution is very sensitive to topography perturbations in regions where the flow is turbulent. On the other hand, the flow exhibits low sensitivity in laminar regions. The quantitative measure of sensitivity is influenced essentially by the error growing time. At short time scales, the sensitivity exhibits the polynomial dependence on the error growing time. And in the long time limit, the dependence becomes exponential.

The use of the data assimilation technique to identify optimal topography is discussed in frames of time-dependent motion governed by nonlinear barotropic ocean model. As- similation of artificially generated data allows to measure the in uence of various error sources and to classify the impact of noise that is present in observational data and model parameters. The choice of length of the assimilation window in 4DVar is discussed. It is shown that using longer window lengths would provide more accurate ocean topography. The topography defined using this technique can be further used in other model runs that start from other initial conditions and are situated in other parts of the model's attractor.

Examination has been made of the skill of ECMWF forecasts of the 500 hPa height field produced daily out to ten days ahead, verifying in the period from 1 December 1980 to 31 May 1994. Over this time accuracy has been improved substantially over the first half of the forecast range. the systematic (seasonal-mean) component of the error has been greatly reduced at all forecast times, but there has been little reduction in the non-systematic (transient) component later in the range.
The simple model proposed by Lorenz for the intrinsic growth of forecast error has been applied to the evolution of differences between consecutive forecasts. the implied growth-rates of small forecast errors have increased significantly since 1981. They do not show much variation with season, and are a little lower in the southern than in the northern hemisphere. the most recent error-doubling times are around 1.5 days for the northern hemisphere and 1.7 days for the southern hemisphere. Error saturation levels are at present similar to or greater than those of the 1981 version of the model, having been significantly lower in intermediate years. the accuracy of recent short- and early medium-range forecasts and realism of the climatology of the forecast model support the view that estimates of intrinsic error-growth parameters from the current forecasting system are more reliable than those obtained earlier. Forecast accuracy later in the medium range may thus not have benefited fully from improvements earlier in the range because of the faster error-growth associated with a more active, though more realistic, forecast model. Overprediction of variance may nevertheless detrimentally affect present levels of skill and estimates of predictability in all seasons other than summer.
The error-growth model currently indicates that it is possible, in principle, to make deterministic mediumrange forecasts for the extratropical 500 hPa height field of the northern hemisphere that are as accurate five days ahead as present forecasts are three days ahead, provided the one-day forecast error can be reduced by the same factor in the future as has actually been achieved in the years since 1981. the level of error currently reached at day seven would then be reached at around day ten. the scope for improvement of forecasts for the southern hemisphere appears to be rather larger. Improvements seem to be possible throughout the spectral range studied, up to total wave-number 40. This is found also for the rotational and divergent wind components at 850 and 200 hPa. For these components, particularly the divergent component, there is a quite pronounced error in the representation of the largest scales.

Every forecast should include an estimate of its likely accuracy, a current measure of predictability. Two distinct types of localized Lyapunov exponents based on infinitesimal uncertainty dynamics are investigated to reflect this predictability. Regions of high predictability within which any initial uncertainty will decrease are proven to exist in two common chaotic systems; potential implications of these regions are considered. The relevance of these results for finite size uncertainties is discussed and illustrated numerically.

A numerical method for detection of unstable periodic orbits on attractors of nonlinear models is proposed. The method requires similar techniques to data assimilation. This fact facilitates its implementation for geophysical models. This method was used to find numerically several low-period orbits for the barotropic ocean model in a square. Some numerical particularities of application of this method are discussed. Knowledge of periodic orbits of the model helps to explain some of these features like bimodality of probability density functions (PDF) of principal parameters. These PDFs have been reconstructed as weighted averages of periodic orbits with weights proportional to the period of the orbit and inversely proportional to the sum of positive Lyapunov exponents. The fraction of time spent in the vicinity of each periodic orbit has been compared with its instability characteristics. The relationship between these values shows the 93% correlation. The attractor dimension of the model has also been approximated as a weighted average of local attractor dimensions in vicinities of periodic orbits.

An initial uncertainty in the state of a chaotic system is expected to grow even under a perfect model; the dynamics of this uncertainty during the early stages of its evolution are investigated. A variety of ‘error growth’ statistics are contrasted, illustrating their relative strengths when applied to chaotic systems, all within a perfect-model scenario. A procedure is introduced which can establish the existence of regions of a strange attractor within which all infinitesimal uncertainties decrease with time. It is proven that such regions exist in the Lorenz attractor, and a number of previous numerical observations are interpreted in the light of this result; similar regions of decreasing uncertainty exist in the Ikeda attractor. It is proven that no such regions exist in either the Rössler system or the Moore-Spiegel system. Numerically, strange attractors in each of these systems are observed to sample regions of state space where the Jacobians have eigenvalues with negative real parts, yet when the Jacobians are not normal matrices this does not guarantee that uncertainties will decrease. Discussions of predictability often focus on the evolution of infinitesimal uncertainties; clearly, as long as an uncertainty remains infinitesimal it cannot pose a limit to predictability. to reflect realistic boundaries, any proposed ‘limit of predictability’ must be defined with respect to the nonlinear behaviour of perfect ensembles. Such limits may vary significantly with the initial state of the system, the accuracy of the observations, and the aim of the forecaster. Perfect-model analogues of operational weather forecasting ensemble schemes with finite initial uncertainties are contrasted both with perfect ensembles and uncertainty statistics based upon the dynamics infinitesimal uncertainties.

We compare the first local Lyapunov vector (LLV) and the leading optimal vectors in a T10/18 level truncated version of the National Centers for Environmental Prediction global spectral model. The leading LLV is a vector toward which all other perturbations turn and hence it is characterized by the fastest possible growth over infinitely long time periods, while the optimal vectors are perturbations that maximize growth for a finite time period, with respect to a chosen norm. Linear tangent model breeding experiments without convection at T10 resolution show that arbitrary random perturbations converge within a transition period of 3 to 4 days to a single LLV. We computed optimal vectors with the Lanczos algorithm, using the total energy norm. For optimization periods shorter than the transition period (about 3 days), the horizontal structure of the leading initial optimal vectors differs substantially from that of the leading LLV, which provides maximum sustainable growth. There are also profound differences between the two types of vectors in their vertical structure. While the 24- h optimal vectors rapidly become similar to the LLV in their vertical structure, changes in their horizontal structure are very slow. As a consequence, their amplification factor drops and stays well below that of the LLV for an extended period after the optimization period ends. This may have an adverse effect when optimal vectors with short optimization periods are used as initial perturbations for medium-range ensemble forecasts. The optimal vectors computed for 3 days or longer are different. In these vectors, the fastest growing initial perturbation has a horizontal structure similar to that of the leading LLV, and its major difference from the LLV, in the vertical structure, tends to disappear by the end of the optimization period. Initially, the optimal vectors are highly unbalanced and the rapid changes in their vertical structure are associated with geostrophic adjustment. The kinetic energy of the initial optimal vectors peaks in the lower troposphere, whereas in the LLV the maximum is around the jet level. During the integration the phase of the streamfunction field of the optimal vectors, with respect to their corresponding temperature field, is rapidly shifted 180°. And, due to drastic changes that also take place in the vertical temperature distribution, the maximum baroclinic shear shifts from the lower troposphere to just below the jet level. Just after initial time, when the geostrophic adjustment dominates, the leading optimal vectors exhibit a growth rate significantly higher than that of the LLV. By the end of the period of optimization, however, the growth rate associated with the leading optimal vectors drops to or below the level of the Lyapunov exponent. The transient super-Lyapunov growth associated with the leading optimal vectors is due to a one-time-only rapid rotation of the optimal vectors toward the leading LLVs. The nature of this rapid rotation depends on the length of the optimization period and the norm chosen. We speculate that the initial optimal vectors computed with commonly used norms may not be realizable

The dynamics of space- and time-averaged observables of a 3-level quasi-geostrophic model is investigated and compared with that of the fine-scale variables, with emphasis on the role of the initial error field on predictability. The invariance of the largest Lyapunov exponent against averaging of up to 10 days is established. Some improvements of predictability of the averaged variables turn out to be possible, at least for certain classes of initial perturbations. Finally, a map of error distribution in space and a classification of weather regimes are derived on the basis of the average properties of the Lyapunov vectors.

Atmospheric Variability: Complicated or Complex?Nonlinear Dynamics and Chaos in the AtmosphereA Case Study: Lorenz's Three-Variable Atmospheric ModelPredictabilityFrom Short-Scale Atmospheric Variability to Global Climate DynamicsDiscussion

ABSTRACTA canonical model describing the instability of plane wave solutions toward inhomogeneous long wave fluctuations emulating mesoscale atmospheric variability is used to analyze the performance of nested limited-area forecasting models. Comparative studies between a reference fine grid “perfect” model and the ones provided, successively, by a globally coarse and a nested, Fine scale, limited-area one are carried out. The statistical properties of the relevant fields are First analyzed, and an optimal size for which the deviations between the “perfect” model and the limited-area one are minimized is identified. Predictability analysis reveals that, although the Lyapunov exponents of the “perfect” model are smaller than those of the globally coarse model, operational error growth proceeds eventually faster in the nested limited-area model than in the globally coarse one.

The dynamical properties of forecasts corrected using model output statistics (MOS) schemes are explored, with emphasis on the respective role of model and initial condition uncertainties. Analytical and numerical investigations of low-order systems displaying chaos indicate that MOS schemes are able to partly correct the impact of both initial and model errors on model forecasting. Nevertheless the amplitude of the correction is much more sensitive to the presence of (state dependent) model errors, and if initial condition errors are much larger than model uncertainties then MOS schemes become less effective. Furthermore, the amplitude of the MOS correction depends strongly on the statistical properties of the phase space velocity difference between the model and reference systems, such as its mean and its covariance with the model predictors in the MOS scheme. Large corrections are expected when the predictors are closely related to the sources of model errors. The practical implications of these results are briefly discussed.

The moment equations associated with the evolution of the probability density are known to form an infinite hierarchy of coupled equations in nonlinear dynamical systems. In the present paper a systematic approach for closing this hierarchy is proposed, based on the ansatz that in the long time limit there exist groups of moments varying on the same time scale. The method is applied to a one-dimensional vector field in the presence of noise, and to two prototypes of chaotic behavior. Excellent agreement with numerical results is obtained. Special emphasis is placed on the role of symmetries, and on the origin of the composite oscillations found for certain types of moments in the chaotic systems.

A systematic formulation of model error is developed accounting for both its dynamical and probabilistic aspects. Some generic features, largely independent of the particular model under consideration, are brought out in connection with power-law behavior in the short time regime. The lifetime of this regime is shown to be related to the spectrum of the Lyapunov exponents and the decay characteristics of the time correlation functions involved. The results are illustrated and tested on representative low-order models of atmospheric dynamics exhibiting oscillations, bistability, and chaotic behavior.

Different strategies for building high-resolution models providing a more detailed description of a limited area of interest as for example, in regional weather forecasts are developed. They are subsequently compared, on the basis of the dynamical behavior generated by the corresponding models. The statistical properties of the relevant fields are analyzed, and predictability experiments are performed on statistical ensembles of close lying trajectories whose mean distance represents the uncertainty in the initial state of the system. The results show that a global, variable-mesh model performs much better than a limited area fine mesh one embedded into a coarser global model.

The dynamical properties of time- and space-averaged observables of a simplified one-dimensional thermal convection model are explored and compared to those of the fine scale variables. It is found that averaging reduces the domain of variability of the system, favors persistence of correlations in time and space, and reduces the ``effective'' dimensionality of the underlying attractor. Furthermore, space and time averages display enhanced predictability, characterized by an error growth rate smaller than the one of the fine scale variables.

The authors report a systematic study on the short and intermediate time predictability properties of a quasigeostrophic T21L3 model in which emphasis is placed on the role of the Lyapunov vectors in the growth patterns of generic initial error fields. It is found that under scale-independent small-amplitude initial perturbations the evolution of the mean error is intimately related to the spectral distribution of the Lyapunov vectors. In the case of perturbations at a particular scale of motion the picture turns out to be more involved, particularly as far as mean error growth over all wavenumbers is concerned, and must appeal to coupling mechanisms between different scales. The role of the norm used for the measure of the mean error growth and the specific predictability properties at different vertical levels of the model are also analyzed.

The dynamics of a low-order coupled wind-driven ocean–atmosphere system is investigated with emphasis on its predictability properties. The low-order coupled deterministic system is composed of a baroclinic atmosphere for which 12 dominant dynamical modes are only retained (Charney and Straus in J Atmos Sci 37:1157–1176, 1980) and a wind-driven, quasi-geostrophic and reduced-gravity shallow ocean whose field is truncated to four dominant modes able to reproduce the large scale oceanic gyres (Pierini in J Phys Oceanogr 41:1585–1604, 2011). The two models are coupled through mechanical forcings only. The analysis of its dynamics reveals first that under aperiodic atmospheric forcings only dominant single gyres (clockwise or counterclockwise) appear, while for periodic atmospheric solutions the double gyres emerge. In the present model domain setting context, this feature is related to the level of truncation of the atmospheric fields, as indicated by a preliminary analysis of the impact of higher wavenumber (“synoptic” scale) modes on the development of oceanic gyres. In the latter case, double gyres appear in the presence of a chaotic atmosphere. Second the dynamical quantities characterizing the short-term predictability (Lyapunov exponents, Lyapunov dimension, Kolmogorov–Sinaï (KS) entropy) displays a complex dependence as a function of the key parameters of the system, namely the coupling strength and the external thermal forcing. In particular, the KS-entropy is increasing as a function of the coupling in most of the experiments, implying an increase of the rate of loss of information about the localization of the system on its attractor. Finally the dynamics of the error is explored and indicates, in particular, a rich variety of short term behaviors of the error in the atmosphere depending on the (relative) amplitude of the initial error affecting the ocean, from polynomial (at
2 + bt
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4) up to exponential-like evolutions. These features are explained and analyzed in the light of the recent findings on error growth (Nicolis et al. in J Atmos Sci 66:766–778, 2009).

The natural instability of the atmosphere is at the origin of the rapid amplification of errors coming from the uncertainty on the initial conditions and from the imperfect representation (the model) of the atmospheric dynamics. In this paper, the short-term dynamics of model error is examined in the context of low-order chaotic dynamical systems. A mathematical model describing the dynamics of this error is first derived where some of the key ingredients of its erratic behavior are incorporated, namely, the variability of the local Lyapunov exponents and of the model error source term along the dominant unstable direction. The analysis of this simplified equation indicates that depending on the nature of the model error sources (here limited to a white noise or an Ornstein-Uhlenbeck process), the mean square error initially follows either a linear or a quadratic evolution, the latter being generic. The numerical analysis of the Lorenz 1984 low-order atmospheric system, in which the model error source is associated with the inaccurate estimate of one of its parameters, supports the main features demonstrated by the simplified mathematical model. However, it also reveals a more involved behavior of the mean square error, which can be traced back to some intrinsic properties of the underlying dynamics not incorporated in the simplified model. The role of the truncation of the small scales of the flow on the dynamics of the larger scales is also studied in two spatially distributed systems. In this context, the mean square error closely follows a quadratic evolution for short times. In the light of these results, the classical view of the linear evolution of the mean square error advanced thus far in the literature should be reassessed.

In a low-order model of the general circulation of the atmosphere we examine the predictability of threshold exceedance events of certain observables. The likelihood of such binary events -- the cornerstone also for the categoric (as opposed to probabilistic) prediction of threshold exceedences -- is established from long time series of one or more observables of the same system. The prediction skill is measured by a summary index of the ROC curve that relates the hit- and false alarm rates. Our results for the examined systems confirm a counterintuitive (and seemingly contrafactual) statement that was previously formulated for more simple autoregressive stochastic processes. Namely, we find that exceedances of higher thresholds are more predictable; or in other words: rare extremes are more predictable than frequent typical events. We find this to hold provided that the bin size for binning time series data is optimized, but not necessarily otherwise. We argue that when there is a sufficient amount of data depending on the precision of observation, the skill of data-driven categoric prediction of threshold exceedences approximates the skill of the model-driven counterpart of the same kind of prediction, assuming strictly no model errors. Therefore, stronger extremes in terms of higher threshold levels are more predictable both in case of data- and model-driven prediction. Furthermore, we show that a quantity commonly regarded as a measure of predictability, the finite-time maximal Lyapunov exponent, does not correspond directly to the ROC-based measure of prediction skill when they are viewed as functions of the prediction lead time and the threshold level. This points to the fact that even if the Lyapunov exponent as an intrinsic property of the system, measuring the instability of trajectories, determines predictability, it does that in a nontrivial manner.

We formulate and study a low-order nonlinear coupled ocean-atmosphere model
with an emphasis on the impact of radiative and heat fluxes and of the
frictional coupling between the two components. This model version extends a
previous 24-variable version by adding a dynamical equation for the passive
advection of temperature in the ocean, together with an energy balance model.
The bifurcation analysis and the numerical integration of the model reveal
the presence of low-frequency variability (LFV) concentrated on and near a
long-periodic, attracting orbit. This orbit combines atmospheric and oceanic
modes, and it arises for large values of the meridional gradient of radiative
input and of frictional coupling. Chaotic behavior develops around this orbit
as it loses its stability; this behavior is still dominated by the LFV on
decadal and multi-decadal time scales that is typical of oceanic processes.
Atmospheric diagnostics also reveals the presence of predominant low- and
high-pressure zones, as well as of a subtropical jet; these features recall
realistic climatological properties of the oceanic atmosphere.
Finally, a predictability analysis is performed. Once the decadal-scale
periodic orbits develop, the coupled system's short-term instabilities --- as
measured by its Lyapunov exponents --- are drastically reduced, indicating the
ocean's stabilizing role on the atmospheric dynamics. On decadal time scales,
the recurrence of the solution in a certain region of the invariant subspace
associated with slow modes displays some extended predictability, as reflected
by the oscillatory behavior of the error for the atmospheric variables at long
lead times.

Weather forecasts produced with numerical weather prediction (NWP) models of the atmosphere possess intrinsic uncertainty. This uncertainty is caused through both errors in the specification of the initial state of the model, as well as errors in the model formulation itself. In the process of NWP, the consideration of both error sources is important, because the nature of atmospheric dynamics is such that it acts to increase errors originating from either error source. In addition to this overall error-growth effect, forecast error possesses considerable day-to-day variability depending, among other things, on the flow regime. The quantitative and reliable assessment (i. e., prediction) of the uncertainty of weather forecasts is important, both for scientific and economic reasons. Scientifically, quantification of atmospheric predictability asks for the rate at which two initially close trajectories diverge (on average) for given atmospheric dynamics. Such estimates place upper bounds on time horizons over which useful forecasts may be expected. Economically, a reliable estimate of the uncertainty of a particular forecast will lead to increased credibility and utility of weather forecasts. The description and discussion of strategies and methods to predict the uncertainty of weather forecasts produced with NWP models are the subject of this article. The limited predictability of atmospheric flows, considered here on time scales of days, as it results essentially from the intrinsic error growth in the atmosphere is briefly discussed. The Liouville equation as the theoretical concept for dealing with the prediction problem of forecast uncertainty is described, as it governs the time-evolution of the probability density function (pdf) of the NWP model state. Related concepts more readily applicable in operational contexts are reviewed. Among these concepts are stochastic-dynamic prediction, the lagged-average forecasting technique, and the Monte Carlo approach. Particular attention is given to the description of methodology and results from currently operational efforts at major forecasting centers directed towards the prediction of forecast uncertainty through multiple (i. e., an ensemble of) NWP model integrations started from different initial states. Such time-evolved ensembles provide partial information about the time-evolved pdf. These ensemble prediction systems at the European Centre for Medium-Range Weather Forecasts, as well as the National Centers for Environmental Prediction are discussed in some detail (e. g., with respect to the selection of the initial states of individual ensemble members). Results presently obtained with ensemble prediction systems are highly promising, although various questions related to, for example, the modeling error source, the validation of products from ensemble prediction systems, as well as to the methodology for the selection of initial states remain to be answered. This review is concluded by mentioning briefly similar efforts at other operational NWP centers, as well as applications of the methodology used in ensemble prediction in related contexts such as stability analysis and data assimilation.

We study a simplified coupled atmosphere-ocean model using the formalism of
covariant Lyapunov vectors (CLVs), which link physically-based directions of
perturbations to growth/decay rates. The model is obtained via a severe
truncation of quasi-geostrophic equations for the two fluids, and includes a
simple yet physically meaningful representation of their
dynamical/thermodynamical coupling. The model has 36 degrees of freedom, and
the parameters are chosen so that a chaotic behaviour is observed. One finds
two positive Lyapunov exponents (LEs), sixteen negative LEs, and eighteen
near-zero LEs. The presence of many near-zero LEs results from the vast
time-scale separation between the characteristic time scales of the two fluids,
and leads to nontrivial error growth properties in the tangent space spanned by
the corresponding CLVs, which are geometrically very degenerate. Such CLVs
correspond to two different classes of ocean/atmosphere coupled modes. The
tangent space spanned by the CLVs corresponding to the positive and negative
LEs has, instead, a non-pathological behaviour, and one can construct robust
large deviations laws for the finite time LEs, thus providing a universal model
for assessing predictability on long to ultra-long scales along such
directions. It is somewhat surprising to find that the tangent space of the
unstable manifold has strong projection on both atmospheric and oceanic
components. Our results underline the difficulties in using hyperbolicity as a
conceptual framework for multiscale chaotic dynamical systems, whereas the
framework of partial hyperbolicity seems better suited, possibly indicating an
alternative definition for the chaotic hypothesis. Our results suggest the need
for accurate analysis of error dynamics on different time scales and domains
and for a careful set-up of assimilation schemes when looking at coupled
atmosphere-ocean models.

We compare the first local Lyapunov vector (LLV) and the leading optimal vectors in a T10/18 level truncated version of the National Centers for Environmental Prediction global spectral model. The leading LLV is a vector toward which all other perturbations turn and hence it is characterized by the fastest possible growth over infinitely long time periods, while the optimal vectors are perturbations that maximize growth for a finite time period, with respect to a chosen norm. Linear tangent model breeding experiments without convection at T10 resolution show that arbitrary random perturbations converge within a transition period of 3 to 4 days to a single LLV. We computed optimal vectors with the Lanczos algorithm, using the total energy norm. For optimization periods shorter than the transition period (about 3 days), the horizontal structure of the leading initial optimal vectors differs substantially from that of the leading LLV, which provides maximum sustainable growth. There are also profound differences between the two types of vectors in their vertical structure. While the 24- h optimal vectors rapidly become similar to the LLV in their vertical structure, changes in their horizontal structure are very slow. As a consequence, their amplification factor drops and stays well below that of the LLV for an extended period after the optimization period ends. This may have an adverse effect when optimal vectors with short optimization periods are used as initial perturbations for medium-range ensemble forecasts. The optimal vectors computed for 3 days or longer are different. In these vectors, the fastest growing initial perturbation has a horizontal structure similar to that of the leading LLV, and its major difference from the LLV, in the vertical structure, tends to disappear by the end of the optimization period. Initially, the optimal vectors are highly unbalanced and the rapid changes in their vertical structure are associated with geostrophic adjustment. The kinetic energy of the initial optimal vectors peaks in the lower troposphere, whereas in the LLV the maximum is around the jet level. During the integration the phase of the streamfunction field of the optimal vectors, with respect to their corresponding temperature field, is rapidly shifted 180°. And, due to drastic changes that also take place in the vertical temperature distribution, the maximum baroclinic shear shifts from the lower troposphere to just below the jet level. Just after initial time, when the geostrophic adjustment dominates, the leading optimal vectors exhibit a growth rate significantly higher than that of the LLV. By the end of the period of optimization, however, the growth rate associated with the leading optimal vectors drops to or below the level of the Lyapunov exponent. The transient super-Lyapunov growth associated with the leading optimal vectors is due to a one-time-only rapid rotation of the optimal vectors toward the leading LLVs. The nature of this rapid rotation depends on the length of the optimization period and the norm chosen. We speculate that the initial optimal vectors computed with commonly used norms may not be realizable DOI: 10.1034/j.1600-0870.1997.00004.x

The deterministic equations describing the dynamics of the atmosphere (and of the climate system) are known to display the property of sensitivity to initial conditions. In the ergodic theory of chaos this property is usually quantified by computing the Lyapunov exponents. In this review, these quantifiers computed in a hierarchy of atmospheric models (coupled or not to an ocean) are analyzed, together with their local counterparts known as the local or finite-time Lyapunov exponents. It is shown in particular that the variability of the local Lyapunov exponents (corresponding to the dominant Lyapunov exponent) decreases when the model resolution increases. The dynamics of (finite-amplitude) initial condition errors in these models is also reviewed, and in general found to display a complicated growth far from the asymptotic estimates provided by the Lyapunov exponents. The implications of these results for operational (high resolution) atmospheric and climate modelling are also discussed.

The principal properties of initial condition and of model errors along with their repercussions on atmospheric predictability are reviewed. A general nonlinear dynamics-inspired approach is developed, from which generic trends are derived. The main ideas are illustrated on selected low-order models capturing the principal qualitative aspects of the phenomena of interest.

: The predictability of the quasi-geostrophic ocean model is considered in this paper. This is a simple dynamical model which assumes the ocean depth to be divided into n layers of different water density. This model is driven by the wind stress on the surface, it includes effects of lateral and bottom friction, of the Earth rotation and the nonlinear interaction between adjacent layers. Internal instability of the system leads to the rapid divergence of its trajectories limiting time of deterministic prediction of the system. To estimate the divergence rate of trajectories we compute Lyapunov exponents of this system. These exponents show us the average growth rate of a small possible perturbation of initial data on infinite time scale. Along with Lyapunov exponents, their generalization, local Lyapunov exponents, are computed as a measure of divergence rate on finite time scales. These exponent provide us with the information about principal mechanisms of local instability. They can ...

The divergence of initially close trajectories sets the limit of dynamical predictability for infinitesimally small errors; its global average measure is given by the first Liapunov exponent. It is shown, within the framework of low-order dynamical systems, that global average error evolution is subject to transient growth. Random errors and analogs are studied and both are found to exhibit transient behavior. The definition of average error that gives the correct asymptotic exponential growth rate is shown to be the one introduced by Lorenz. Transient superexponential growth reduces the predictability time when errors have a finite initial size and explains the apparent dependence of average error growth on the initial error size. The consequences upon short-range forecasting are discussed.

In this paper we suggest that the longevity of the enhanced
predictability periods often observed in the weather and general
circulation models can he quantified by a study of the statistical
moments of error growth rates as has been demonstrated for dynamical
systems. As an illustration, it is shown how this approach can he
pursued in simple cases. For the Lorenz model, the probability density
distribution of error growth is close to log-normal and the average
growth rate is two times shorter than the most probable. In general, we
argue that the ratio of the average growth rate to the most probable is
a measure of enhanced predictability.

In meteorological models, the logistic growth law has been used traditionally to describe the error growth due to sensitivity to the initial conditions. A detailed analysis obtained from long range forecasting experiments using a GCM model, as well as from simulations based on a simple 3-variable model, has revealed significant deviations from the logistic law. A natural generalization is proposed, giving a law that has been used previously for the description of biological growth. A new characteristic parameter, which can be interpreted as a saturation rate for error growth, is identified. Further studies, based on a simple 3-variable model for different magnitudes of the initial error, reveal a more complex behaviour having a transient initial regime that is independent of the error magnitude, a regime of exponential growth, and a “deceleration regime”. The deceleration regime as defined here includes both the phases of linear and saturated error growth in time. For the case of large initial errors, the vanishing of the exponential regime, as a result of the coalescence of the initial and deceleration regimes, gives a continuous decrease in error growth rate with time, which can be well represented by the Gompertz growth law.

An extended formulation of sensitivity to initial conditions applicable to (small) finite errors and finite times is developed. It is shown that the first stages of error growth are neither exponential nor driven by the Lyapunov exponent.

Based on the Lyapunov characteristic exponents, the ergodic property of
dissipative dynamical systems with a few degrees of freedom is studied
numerically by employing, as an example, the Lorenz system. The Lorenz
system shows the spectra of (+,0,-) type concerning the 1-dimensional
Lyapunov exponents, and the exponents take the same values for orbits
starting from almost of all initial points on the attractor.
This result suggests that the ergodic property for general dynamical
systems not necessarily belonging to the category of the axiom-A may
also be characterized in the framework of the spectra of the Lyapunov
characteristic exponents.

The dynamics of error growth in a two-layer nonlinear quasigeostrophic model was studied to gain an understanding of the mathematical theory of atmospheric predictability. The growth of random errors of varying initial magnitudes was studied and the relation between this classical approach and the concepts of the nonlinear dynamical systems theory was explored. The local and global growths of random errors were expressed in terms of the properties of an error ellipsoid and the Lyapunov exponents determined by linear error dynamics. The local growth of small errors was initially governed by several modes of the evolving error ellipsoid but becomes dominated by the longest axis. The average global growth of small errors was exponential with a growth rate consistent with the largest Lyapunov exponent. The duration of the exponential growth phase depended on the initial magnitude of the errors. The subsequent large errors underwent a nonlinear growth with a steadily decreasing growth rate and attained saturation that defined the limit of predictability. The degree of chaos and the largest Lyapunov exponent showed variation with change in the forcing, which implied that the time variation in the external forcing could introduce variable character to the predictability. For sufficiently large initial errors, the exponential growth phase was shown to be absent, indicating that the growth was governed completely by nonlinear error dynamics. During a part of the initial growth phase, the growth rate was higher than the largest Lyapunov exponent. It was shown that the estimations of the growth rates of small errors, obtained from the well-known empirical formula using the error data in their nonlinear growth phase, are inaccurate. The use of Lyapunov exponents to estimate growth rate and predictability was valid only for initially small errors. 16 refs., 18 figs., 2 tabs.

A probabilistic approach accounting for the variability of error growth in the atmosphere is developed and applied to an idealized low-order atmospheric circulation model. Error growth in a single-realization, ensemble-averaged error versus time and the time-dependent as well as the asymptotic probability distribution of errors are determined. A wide dispersion around the mean is found, showing clearly the inadequacy of a description that is limited to averaged properties only. A self-consistent one-variable, two-parameter model of error dynamics is constructed in the form of a stochastically driven logistic equation, and the results are shown to be in good quantitative agreement with the simulation of the full equations. A generalization to more realistic atmospheric models is also outlined.

The finite-time instability and associated predictability of atmospheric of atmospheric circulations are defined in terms of the largest singular values, and associated singular vectors, of the linear evolution operator determined form given equations of motion. These quantities are calculated in both a barotropic and a three-level quasi-geostrophic model, using as basic states realistic large-scale northern wintertime flows that represent the climatological state, regime composites, and specific realizations of these regimes. for time-invariant basic states, the singular vectors are compared with the corresponding normal-mode solutions; it is shown that the perturbations defined (at the initial time) by the singular vectors have much larger growth rates than the normal modes, and possess a more localized spatial structure.
The regimes studied have opposite values of the Pacific/North American (PNA) index, and growth rates for the barotropic basis states appear to confirm earlier studies that the barotropic instability of the negative PNA states may be larger than the corresponding positive PNA states. The evolution of the singular-vector perturbations, with emphasis on the vertical structure, is compared for time-evolving and time-invariant baroclinic basis states; the effects of nonlinearity are also discusses. It is shown that, in the baroclinic model, interactions between synoptic-scale eddies in the time-evolving basic state and in the perturbation field are fundamental for studying the predictability of transitions in the large-scale circulation. Consequently results obtained from linear calculations using very smooth basic states cannot properly account for such predictability.
These results form the basis of a technique used to initialize ensembles of forecasts made with a primitive-equation model, and are described in the companion paper (Mureau et al. 1993).

We introduce the idea of local Lyapunov exponents which govern the way small perturbations to the orbit of a dynamical system grow or contract after afinite number of steps,L, along the orbit. The distributions of these exponents over the attractor is an invariant of the dynamical system; namely, they are independent of the orbit or initial conditions. They tell us the variation of predictability over the attractor. They allow the estimation of extreme excursions of perturbations to an orbit once we know the mean and moments about the mean of these distributions. We show that the variations about the mean of the Lyapunov exponents approach zero asL and argue from our numerical work on several chaotic systems that this approach is asL
–v. In our examplesv 0.5–1.0. The exponents themselves approach the familiar Lyapunov spectrum in this same fashion.

We consider the application of braid and knot theory to single-degree-of-freedom driven oscillators, giving emphasis to the braids of periodic orbits contained in horseshoes. Using such concepts as braid type, relative rotations, Nielsen equivalence, knot polynomials, the reduced Burau representation and positive, regular and ambient isotopy, we illustrate how these can be put together to gain some understanding of bifurcation structure.

The third edition of Van Kampen's standard work has been revised and updated. The main difference with the second edition is that the contrived application of the quantum master equation in section 6 of chapter XVII has been replaced with a satisfactory treatment of quantum fluctuations. Apart from that throughout the text corrections have been made and a number of references to later developments have been included. From the recent textbooks the following are the most relevant. C.W.Gardiner, Quantum Optics (Springer, Berlin 1991) D.T. Gillespie, Markov Processes (Academic Press, San Diego 1992) W.T. Coffey, Yu.P.Kalmykov, and J.T.Waldron, The Langevin Equation (2nd edition, World Scientific, 2004) * Comprehensive coverage of fluctuations and stochastic methods for describing them * A must for students and researchers in applied mathematics, physics and physical chemistry.

Average predictability and error growth in a simple realistic two-level general circulation model (GCM) were investigated using a series of Monte Carlo experiments for fixed external forcing (perpetual winter in the Northern Hemisphere). It was found that, for realistic initial errors, the dependence of the limit of dynamic predictability on total wavenumber was similar to that found for the ECMWF model for the 1980/1981 winter conditions, with the lowest wavenumbers showing significant skill for forecast ranges of more than 1 month. On the other hand, for very small amplitude errors distributed according to the climate spectrum, the total error growth was superexponential, reaching a maximum growth rate (2-day doubling time) in about 1 week. A simple empirical model of error variance, which involved two broad wavenumber bands and incorporating a 3/2 power saturation term, was found to provide an excellent fit to the GCM error growth behavior.

The Lorenz equations are cast in the form of a single stochastic differential equation in which a 'deterministic' part representing a bistable dynamical system is forced by a 'noise' process. The properties of this effective noise are analyzed numerically. An analytically derived fluctuation-dissipation-like relationship linking the variance of the noise to the system's parameters provides a satisfactory fitting of the numerical results. The connection between the onset of chaotic dynamics and the breakdown of the separation between the characteristic time scales of the variables of the original system is discussed.

Analytic expressions of the mean error and of the error probability versus time are derived for chaotic attractors. The expressions are studied for the logistic and Bernoulli maps and for the Roessler flow. In all cases mean error growth follows a logisticlike curve, the characteristics of which are related to the intrinsic properties of the attractor.

Predictability and finite-time instability of the northern winter circulation Probabilistic aspects of error growth in atmospheric dynamics dynamical system

- Tellus
- a

Tellus, 36A, 98-110 Predictability and finite-time instability of the northern winter circulation. Q. J. R. Meteorol. Soc., 119, 169-298 Probabilistic aspects of error growth in atmospheric dynamics. Q. J. R. Meteorol. Soc., 118, 553-568 Nonlinear Sci., 1, 175-199 3595-3598 dynamical system. J. Atmos. Sci., 50, 2215-2229 model. Tellus, 40A, 81-95

A fundamental property of the atmosphere

- Irregularity

Irregularity. A fundamental property of the atmosphere.

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Nicolis, C.
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A predictability study of Lorenz's 28-variable model as a Short-range evolution of small perturbations in a barotropic Deterministic nonperiodic flow

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Nicolis, C.
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