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Fluctuations of Finite-Time Lyapunov Exponents in an Intermediate-Complexity Atmospheric Model

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Fluctuations of Finite-Time Lyapunov Exponents in an Intermediate-Complexity Atmospheric Model Nonlin. Processes Geophys., 26, 195–209, 2019 https://doi.org/10.5194/npg-26-195-2019 © Author(s) 2019. This work is distributed under the Creative Commons Attribution 4.0 License. Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective Frank Kwasniok Department of Mathematics, University of Exeter, Exeter, UK Correspondence: Frank Kwasniok ([email protected]) Received: 23 April 2018 – Discussion started: 2 May 2018 Revised: 7 July 2019 – Accepted: 8 July 2019 – Published: 31 July 2019 Abstract. The stability properties as characterized by 1 Introduction the fluctuations of finite-time Lyapunov exponents around their mean values are investigated in a three-level quasi- The atmosphere is a high-dimensional non-linear chaotic dy- geostrophic atmospheric model with realistic mean state and namical system; its time evolution is characterized by sensi- variability. Firstly, the covariance structure of the fluctuation tivity to initial conditions (Lorenz, 1963; Kalnay, 2003). As a field is examined. In order to identify dominant patterns of consequence predictability is limited; small errors in the ini- collective excitation, an empirical orthogonal function (EOF) tial states progressively grow under the time evolution until analysis of the fluctuation field of all of the finite-time Lya- the forecast eventually becomes useless, that is, it is indistin- punov exponents is performed. The three leading modes are guishable from the invariant measure or climatology of the patterns where the most unstable Lyapunov exponents fluc- system. Understanding the structure of this inherent instabil- tuate in phase. These modes are virtually independent of the ity is key to improve forecasts at all timescales. integration time of the finite-time Lyapunov exponents. Sec- Sensitivity to initial conditions and perturbation growth in ondly, large-deviation rate functions are estimated from time non-linear dynamical systems are often quantified using Lya- series of finite-time Lyapunov exponents based on the prob- punov exponents (LEs; e.g. Eckmann and Ruelle, 1985; Ott, ability density functions and using the Legendre transform 2002; Pikovsky and Politi, 2016). They describe the asymp- method. Serial correlation in the time series is properly ac- totic growth or decay of infinitesimal perturbations. A system counted for. A large-deviation principle can be established is chaotic if it has at least one positive Lyapunov exponent. for all of the Lyapunov exponents. Convergence is rather However, the predictability properties may vary substantially slow for the most unstable exponent, becomes faster when across state space. Finite-time (or local) Lyapunov exponents going further down in the Lyapunov spectrum, is very fast for (FTLEs) allow a characterization of the stability of a partic- the near-neutral and weakly dissipative modes, and becomes ular initial state with respect to a predefined prediction hori- slow again for the strongly dissipative modes at the end of the zon. Lyapunov spectrum. The curvature of the rate functions at the LEs have been calculated for various geophysical fluid minimum is linked to the corresponding elements of the dif- systems, ranging from highly truncated atmospheric mod- fusion matrix. Also, the joint large-deviation rate function for els (Legras and Ghil, 1985), to intermediate-complexity at- the first and the second Lyapunov exponent is estimated. mospheric models (Vannitsem and Nicolis, 1997; Schubert and Lucarini, 2015) and coupled atmosphere–ocean models (Vannitsem and Lucarini, 2016). A review has been pub- lished recently by Vannitsem (2017). Models tuned to real- istic conditions were found to possess quite a large number of positive LEs corresponding to a high-dimensional chaotic attractor. Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union. 196 F. Kwasniok: Fluctuations of finite-time Lyapunov exponents The present paper investigates the fluctuations of FTLEs where H is a scale height set to 8 km, and f0 is the Corio- in an intermediate-complexity atmospheric model with real- lis parameter at an average geographic latitude taken to be istic mean state and variability. It focuses on two aspects that 45◦ N. have found little attention in the context of geophysical fluid The dissipative terms are given as follows: systems thus far. Firstly, the covariance structure of the fluc- D D τ −1R−2.9 − 9 / − k r8qO (5) tuation field of the FTLEs is studied by means of a princi- 1 N 1;2 1 2 H 1 D − −1 −2 − C −1 −2 − pal component (PC) or empirical orthogonal function (EOF) D2 τN R1;2.91 92/ τN R2;3.92 93/ analysis (Kuptsov and Politi, 2011). Secondly, we look at the − r8 O large-deviation behaviour of the FTLEs (Kuptsov and Politi, kH q2 (6) D − −1 −2 − − −1r2 − r8 O 2011; Laffargue et al., 2013; Johnson and Meneveau, 2015). D3 τN R2;3.92 93/ τE 93 kH q3: (7) A large-deviation principle links the FTLEs at long integra- They are Newtonian temperature relaxation with a radiative tion times to the global LEs by providing a universal law for timescale of τ D 25 d, Ekman damping on the lowest level the probability density of fluctuations of the FTLEs around N with a spin-down timescale of τ D 1:5 d, and a strongly the mean value. It can be expected to hold for Axiom A dy- E scale-selective horizontal diffusion of vorticity and temper- namical systems and, invoking the chaotic hypothesis, also ature. The qO is the time-dependent part of the potential vor- for certain types of non-Axiom A systems. In particular, a i ticity at level i, that is to say qO D q −f −δ f h. The coeffi- large-deviation law allows one to determine the probability i i i3 0 cient of horizontal diffusion k D τ −1Tn .n C1/U−4 is such of very stable or very unstable atmospheric states. H H m m that harmonics of total wave-number n D 21 are damped at The paper is organized as follows: in Sect. 2 the atmo- m a timescale of τ D 1:5 d. The terms S D S (λ;µ) are dia- spheric model is described; the methodology, which consists H i i batic sources of potential vorticity which are independent of of calculating LEs, the multivariate fluctuation analysis and time but spatially varying. the large-deviation theory, is outlined in Sects. 3, 4 and 5; The model is considered on the Northern Hemisphere. The the results are presented and discussed in Sect. 6; and some boundary condition of no meridional flow, v (λ;0/ D 0, that conclusions are drawn in Sect. 7. i is to say vanishing stream function, 9i(λ;0/ D 0, is applied at the Equator on all three model levels. The horizontal dis- 2 The atmospheric model cretization is spectral, triangularly truncated at total wave- number nm D 21. The number of degrees of freedom is 231 A quasi-geostrophic (QG) three-level model on the sphere, for each level and N D 693 in total. The model is integrated formulated in pressure coordinates, is used here as dynami- in time using the third-order Adams–Bashforth scheme with cal framework. The model is identical to that introduced by a constant step size of 1 h. Kwasniok (2007) except for the horizontal resolution and the The variables of the QG model are listed in Table 1; the coefficient of hyperviscosity. A very similar model was intro- model parameters are listed in Table 2 with their dimensional duced by Marshall and Molteni (1993). The dynamical equa- and non-dimensional values. tions are as follows: In order to get a model behaviour close to that of the @q real atmosphere, the forcing terms Si are determined from i C J .9 ;q / D D C S ; i D 1;2;3; (1) @t i i i i the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis data by requiring that when comput- where 9i and qi are the stream function and the potential vorticity at level i, respectively, and J denotes the Jacobian ing potential vorticity tendencies for a large number of ob- operator on the sphere. All variables are non-dimensional us- served atmospheric fields, the average of these tendencies ing the radius of the Earth as the unit of length and the inverse must be zero (Roads, 1987); this is done in order for the en- of the angular velocity of the Earth as the unit of time. The semble of reanalysis data states to be representative of a sta- three pressure levels are located at 250, 500 and 750 hPa. Po- tistically stable long-term behaviour of the QG model. The tential vorticity and the stream function are related by timescale of horizontal diffusion τH is determined such that the slope of the kinetic energy spectrum at the truncation D r2 − −2 − C q1 91 R1;2.91 92/ f (2) level in the model matches that in the reanalysis data. See − − Kwasniok (2007) for details on the parameter tuning proce- q D r29 C R 2.9 − 9 / − R 2.9 − 9 / C f (3) 2 2 1;2 1 2 2;3 2 3 dure. The QG model exhibits a remarkably realistic mean D r2 C −2 − C C q3 93 R2;3.92 93/ f f0h; (4) state and variability pattern of stream function and potential vorticity in a long-term integration (see Table 3). where r is the horizontal gradient operator, and f is the Coriolis parameter. The Rossby deformation radii R1;2 and R2;3 have dimensional values of 575 and 375 km, 3 Lyapunov exponents respectively. The function h D h(λ;µ) represents a non- dimensional topography which is related to the actual dimen- We consider a non-linear autonomous dynamical system ∗ ∗ ∗ T sional topography of the Earth h D h (λ;µ) by h D h =H, with state vector x D .x1;:::;xN / governed by the evolu- Nonlin.
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