APRIL 2017 A T E N C I A A N D Z A W A D Z K I 1381 Analogs on the Lorenz Attractor and Ensemble Spread AITOR ATENCIA AND ISZTAR ZAWADZKI Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada (Manuscript received 31 March 2016, in final form 3 October 2016) ABSTRACT Intrinsic predictability is defined as the uncertainty in a forecast due to small errors in the initial con- ditions. 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. 1. Introduction an open question and different methods are still used and continuously developed (Pazó et al. 2013). How- From the seminal paper of Lorenz (1963) about non- ever, all these methodologies are focused on the fastest linear chaotic systems, predictability theory has evolved growth direction while compromising other properties from the quantification of the quality of a forecast to a of the initial perturbations. Nowadays, the perturba- novel perspective where sources of uncertainty and its tion methods applied in many forecasting centers are consequences for the limit of predictability are studied. based on ensemble data assimilation introducing other Among the main sources of uncertainty, infinitesimal properties, such as the background covariance matrix errors in the initial conditions and their growth due to the (Buizza et al. 2005). nonlinearities of the system equations are defined as the To study more complex behaviors from the initial intrinsic predictability of the system or first-kind pre- uncertainties in a chaotic system, low-order nonlinear dictability (Chu 1999). systems were studied. Even though these models lack Two main perturbation methodologies are vastly realism in comparison with the atmosphere, they used to perturb the initial conditions: singular vectors compensate by providing a better framework to obtain (Lorenz 1965; Palmer 1993) and bred vectors (Toth statistically significant results and having an easier in- and Kalnay 1993; Kalnay et al. 2002). Magnusson et al. terpretation (Farrell 1990). (2008) compared both perturbation methodologies The growth of initial errors, or the forecast spread (and a new bred vector methodology by applying among an ensemble, depends on several factors: lo- principal component analysis) to determine the dif- cation in the phase space (Nese 1989), direction of ferent fastest perturbation growths. The optimal meth- the initial perturbation (Lorenz 1965), and the fore- odology for perturbing the initial conditions remains cast lead time (Trevisan 1995). Despite the high var- iance of these errors, Lacarra and Talagrand (1988) Corresponding author e-mail: Aitor Atencia, aitor. found a transient growth that affects the short-range [email protected] average of errors and Trevisan and Legnani (1995) DOI: 10.1175/MWR-D-16-0123.1 Ó 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 09/25/21 09:27 AM UTC 1382 MONTHLY WEATHER REVIEW VOLUME 145 related this behavior to the amplitude of the initial variables that tended to zero, obtaining a system of three error. However, Nicolis (1992) showed that the first coupled differential equations: two moments of the distribution (mean and variance) are not enough to characterize the predictability dX 5 s(Y 2 X) (1) due to a bimodal distribution of errors in the Lorenz dt 63 system. dY 52XZ 1 rX 2 Y (2) To entirely understand the difficulty of the problem, dt the structure of the errors in the attractor of the cha- dZ 5 XY 2 bZ, (3) otic system has to be studied (Judd et al. 2008). A way dt to obtain this structure is through the use of analogs in theattractor.Theideaofanalogswasintroducedby where the three parameters s, r, and b are positive and Lorenz (1969) to compensate for the unknown real are the Prandtl number, the Rayleigh number, and a atmosphere system of equations. Analogs were used in physical proportion, respectively. theLorenz63system(Trevisan 1993)tocomparetheir In the early 1960s, Lorenz discovered the chaotic be- properties to those of the random perturbations. havior of his simplified three-dimensional system by set- However, as stated in Trevisan (1993, p. 1017): ‘‘The ting the parameters to s 5 10, r 5 28, and b 5 8/3. This limited total time of the model integration, dictated by chaotic behavior, which appears for a wide range of pa- computational costs, did not allow, even in such a rameter values, is often adopted as a low-order test bed for simple model, one to find analogs sufficiently close to atmospheric predictability studies (Palmer 1993; Evans one another so that the initial error could be consid- et al. 2004; Magnusson et al. 2008; among others). A time ered small.’’ Nowadays, the computer power has in- series of the variables in the Lorenz system can be obtained creased and small enough error analogs can be by using numerical methods. In this work, the fourth-order obtained. To ensure the validity of the analogs used in Runge–Kutta forward method has been used with a time this study, the multiplicative ergodic theory (Oseledets step of Dt 5 0.01. Any random point of the three- 1968) is verified within our states. The main assump- dimensional phase space composed of the variables X, Y, tion of this theory is the equivalence between temporal and Z evolves into the system attractor. The Lorenz system averages of a given observable and the average of attractor has a dimension of around 2.07, which according identical processes at a given time. This ensures the to Ruelle and Takens (1971) is called strange attractor analogs have the desired properties and do not require because its fractal structure has a noninteger dimension. the use of shadowing filtering (Judd and Stemler 2010) The attractor A and the realm of attraction r(A) are to create perturbations compatible with the attractor two subsets in the phase space of variables M. The of the chaotic system. mathematical definition of the attractor can be found in The main goal of the present study is to compare the Milnor (2004), and it is divided into two conditions for a results obtained with previous perturbation tech- closed subset A M: niques with analogs obtained from a long enough d The realm of attraction r(A), consisting of all points dataset that guarantees the ergodicity assumption to x 2 M for which w(x)1 A, must have a strictly be true. The model used in the paper is the Lorenz 63 positive measure. model, and it is introduced in section 2 together with 0 0 d There is no strictly smaller closed set A A,sor(A ) the constructed dataset of observations. The pertur- coincides with r(A) up to a set of measure zero. bation methodologies and the details of the definition of analogs are described in section 3. An in-depth The rigorous demonstration of the existence of an comparison of several properties of the different per- attractor, such as the one in the Lorenz system, requires turbations is presented in section 4 followed by a dis- solving the system of equations using topological ordinary cussion. Finally, the conclusions of the paper are differential equation (ODE) properties as shown in Tucker summarized. (2002). From the point of view of practical applications, the use of density in the phase space is a less rigorous but practically affordable way to determine the realm 2. The model of attraction and attractor. Figure 1 shows three cross The model used in this article to obtain the temporal data used as observations is the well-known chaotic system first introduced by Lorenz (1963). Lorenz trun- 1 The omega limit set [w(x)] is the collection of all accumulation cated the solution of the finite-amplitude convection points for the sequence x;G(x);G2(x);... of successive images of x system derived by Saltzman (1962) and removed the being G a nonlinear mapping. Unauthenticated | Downloaded 09/25/21 09:27 AM UTC APRIL 2017 A T E N C I A A N D Z A W A D Z K I 1383 FIG. 1. Density of states for the Lorenz system in different cross sections. (a) The surface for (left to right) X is equal to 0, Y is equal to 0, and Z is equal to 40 on the plane Y–Z, X–Z, and X–Y, respectively. (b) An initial point is chosen from inside the attractor obtained in the study shown in (a).