Predictability of large-scale atmospheric motions: Lyapunov exponents and error dynamics

Vannitsem, Stéphane

doi:10.1063/1.4979042

Cited by 57 publications

(101 citation statements)

References 190 publications

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“…The difference spectrum at low k for an EDQNM approximation was found to be k 4 [18], whilst in a single run of DNS it was k 2 with large error [26]. Similar difference spectra as ours at all scales have been seen in atmospheric models [38].…”

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confidence: 85%

Chaotic Properties of a Turbulent Isotropic Fluid

Berera

2018

Phys. Rev. Lett.

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By tracking the divergence of two initially close trajectories in phase space in an Eulerian approach to forced turbulence, the relation between the maximal Lyapunov exponent λ, and the Reynolds number Re is measured using direct numerical simulations, performed on up to 2048 3 collocation points. The Lyapunov exponent is found to solely depend on the Reynolds number with λ ∝ Re 0.53 and that after a transient period the divergence of trajectories grows at the same rate at all scales. Finally a linear divergence is seen that is dependent on the energy forcing rate. Links are made with other chaotic systems.In Press Physical Review Letters 2018 Using the Eulerian approach, we track the divergence of fluid field trajectories, which initially differ by a small perturbation. We do a model independent analysis, evolving the Navier-Stokes equations for three dimensional homogeneous isotropic turbulence (HIT) using direct numerical simulation (DNS). The Eulerian approach to the study of the chaotic properties of turbulence has received only limited numerical tests prior to this Letter. Amongst approximate models, there have been EDQNM closure approximations [18] and shell model studies [19][20][21]. Amongst exact DNS studies, there have been some in two dimensions [22][23][24] and single runs in three dimensions at comparatively small box sizes [25,26], all more than a decade and a half ago. This Letter tests the theory of Ruelle [27] relating the maximal Lyapunov exponent λ and Re in DNS of HIT in a Eulerian sense. The paper also examines the time history of the divergence and finds a uniform exponential growth rate across all scales at an intermediate time and to show a linear growth for late time in three dimensional HIT. The simulations are also the largest yet for measuring the Eulerian aspects of chaos in HIT for DNS, performed on up to 2048 3 collocation points and reach an integral scale Reynolds number of 6200. This allows a more accurate measurement of the Re dependence of λ.For a chaotic system, an initially small perturbation |δu 0 | should grow according to |δu(t)| ≃ |δu 0 |e λt where t is time. It is theoretically predicted that the Lyapunov exponent should depend on the Reynolds number according to the rule [27,28] The Holder exponent, h, is given by |u(x + r) − u(x)| ∼ V l h , where V is the rms velocity, l the size of the eddy, Re = V L/ν the integral scale Reynolds number, L = (3π/4E) (E(k)/k)dk the integral length scale, E the energy, ν the viscosity, T 0 = L/V the large eddy turnover time, τ = (ν/ǫ) 1/2 the Kolmogorov time scale, and ǫ the dissipation rate. In the Kolmogorov theory, h is predicted to be 1/3 and so α is predicted to be 1/2 [27][28][29].Some of the new results found in this Letter from the Eulerian approach are inaccessible to the Lagrangian approach, such as the linear growth rate of the divergence at late times which has no direct Lagrangian counterpart. The paper also highlights different results from the two approaches. For instance, within the Lagrangian approach, the relation...

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confidence: 85%

Chaotic Properties of a Turbulent Isotropic Fluid

Berera

2018

Phys. Rev. Lett.

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“…The decrease in λ 1 also appears to suggest an enhanced predictability for models which have a larger ocean-atmosphere coupling parameter d, but this feature is not so clear for higher resolution versions. Vannitsem (2017) studied the dependence of the predictability on this coupling parameter in the low-order 36-variable model that lies at the basis of MAOOAM. Two distinct mechanisms were identified to drive the increase in predictability with increasing d. To a first approximation, the mechanical coupling of the fast atmosphere to the slow ocean corresponds to an effective friction term which reduces error growth in the atmosphere.…”

Section: Maooammentioning

confidence: 99%

“…This sensitivity property affects not only the dynamics of errors in the initial conditions but also the errors that are present either in the model parametrizations or in the boundary conditions (Nicolis, 2007;Nicolis et al, 2009). Clarifying the nature of this sensitivity is therefore crucial in the perspective of improving forecasts at short, medium and long term (Vannitsem, 2017).…”

Section: Introductionmentioning

confidence: 99%

Exploring the Lyapunov instability properties of high-dimensional atmospheric and climate models

Cruz¹,

Schubert²,

Demaeyer³

et al. 2018

Preprint

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Abstract. The stability properties of intermediate-order climate models are investigated by computing their Lyapunov exponents (LEs). The two models considered are PUMA (Portable University Model of the Atmosphere), a primitive-equation simple general circulation model, and MAOOAM (Modular Arbitrary-Order Ocean-Atmosphere Model), a quasi-geostrophic coupled ocean-atmosphere model on a β-plane. We wish to investigate the effect of the different levels of filtering on the instabilities and dynamics of the atmospheric flows. Moreover, we assess the impact of the oceanic coupling, the dissipation 5 scheme and the resolution on the spectra of LEs.The PUMA Lyapunov spectrum is computed for two different values of the meridional temperature gradient defining the Newtonian forcing to the temperature field. The increase of the gradient gives rise to a higher baroclinicity and stronger instabilities, corresponding to a larger dimension of the unstable manifold and a larger first LE. The Kaplan-Yorke dimension of the attractor increases as well. The convergence rate of the rate functional for the large deviation law of the finite-time 10Lyapunov exponents (FTLEs) is fast for all exponents, which can be interpreted as resulting from the absence of a clear-cut atmospheric time-scale separation in such a model.The MAOOAM spectra show that the dominant atmospheric instability is correctly represented even at low resolutions. However, the dynamics of the central manifold, which is mostly associated to the ocean dynamics, is not fully resolved because of its associated long time scales, even at intermediate orders. As expected, increasing the mechanical atmosphere-ocean coupling 15 coefficient or introducing a turbulent diffusion parametrization reduces the Kaplan-Yorke dimension and Kolmogorov-Sinai entropy. In all considered configurations, it is possible to robustly define large deviations laws describing the statistics of the FTLEs corresponding to the strongly damped modes, while the opposite holds for near-zero LEs and, somewhat unexpectedly, also for the positive LEs. This paper highlights the need to investigate the natural variability of the atmosphere-ocean coupled dynamics by associating 20 rate of growth and decay of perturbations to the physical modes described using the formalism of the covariant Lyapunov vectors and to consider long integrations in order to disentangle the dynamical processes occurring at all time scales.

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“…The coupling between the two components includes momentum transfer (wind forcing), as well as radiative and heat exchanges. Examinations of the Lyapunov spectrum of the coupled configuration of the MAOOAM were conducted by Vannitsem et al (2015), Vannitsem and Lucarini (2016), Vannitsem (2017), and De Cruz et al (2018). Vannitsem and Lucarini (2016) also examined MAOOAM's covariant Lyapunov vectors.…”

Section: Introductionmentioning

confidence: 99%

Strongly Coupled Data Assimilation in Multiscale Media: Experiments Using a Quasi‐Geostrophic Coupled Model

Penny

Bach

Bhargava

et al. 2019

J Adv Model Earth Syst

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Strongly coupled data assimilation (SCDA) views the Earth as one unified system. This allows observations to have an instantaneous impact across boundaries such as the air‐sea interface when estimating the state of each individual component. Operational prediction centers are moving toward Earth system modeling for all forecast timescales, ranging from days to months. However, there have been few studies that examine fundamental aspects of SCDA and the transition from traditional approaches that apply data assimilation only to a single component, whether forecasts were derived from a coupled model or an uncoupled forced model. The SCDA approach is examined here in detail using numerical experiments with a simple coupled atmosphere‐ocean quasi‐geostrophic model. The impact of coupling is explored with respect to its impact on the Lyapunov spectrum and on data assimilation system stability. Different data assimilation methods are compared within the context of SCDA, including the 3‐D and 4‐D Variational methods, the ensemble Kalman filter, and the hybrid gain method. The impact of observing system coverage is also investigated. We find that SCDA is generally superior to weakly coupled or uncoupled approaches. Dynamically defined background error covariance estimates are essential for SCDA to achieve an accurate coupled state estimate as the observing system becomes sparser. As a clarification of seemingly contradictory findings from previous studies, it is shown that ocean observations can adequately constrain atmospheric state estimates provided that the analysis‐observing frequency is sufficiently high and the ensemble size determining the background error covariance is sufficiently large.

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Predictability of large-scale atmospheric motions: Lyapunov exponents and error dynamics

Cited by 57 publications

References 190 publications

Chaotic Properties of a Turbulent Isotropic Fluid

Chaotic Properties of a Turbulent Isotropic Fluid

Exploring the Lyapunov instability properties of high-dimensional atmospheric and climate models

Strongly Coupled Data Assimilation in Multiscale Media: Experiments Using a Quasi‐Geostrophic Coupled Model

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