Frank Kwasniok

The reduction of complex dynamical systems using principal interaction patterns

Abstract A method of constructing low-dimensional nonlinear models capturing the main features of complex dynamical systems with many degrees of freedom is described. The system is projected onto a linear subspace spanned by only a few characteristic spatial structures called Principal Interaction Patterns (PIPs). The expansion coefficients are assumed to be governed by a nonlinear dynamical system. The optimal low-dimensional model is determined by identifying spatial modes and interaction coefficients describing their time evolution simultaneously according to a nonlinear variational principle. The algorithm is applied to a two-dimensional geophysical fluid system on the sphere. The models based on Principal Interaction Patterns are compared to models using Empirical Orthogonal Functions (EOFs) as basis functions. A PIP-model using 12 patterns is capable of capturing the long-term behaviour of the complete system monitored by second-order statistics, while in the case of EOFs 17 modes are necessary.

Empirical low-order models of barotropic flow

Empirical low-order models are constructed in the framework of a quasigeostrophic barotropic spectral model, truncated at T21. The spectral model is projected onto a linear subspace spanned by only a limited number of spatial structures called principal interaction patterns. The expansion coefficients of these modes are assumed to be governed by a forced, dissipative dynamical system with a quadratic nonlinearity that conserves turbulent kinetic energy and turbulent enstrophy. A simple empirical scheme for modeling the energy and enstrophy cascade assuming localization of the nonlinear interactions with respect to spatial scale is applied. The optimal low-order model, that is, the optimal basis functions and the optimal interaction coefficients, is determined by minimizing the mean-squared error between trajectories of the reduced model and trajectories of the T21 model. A model with 40 degrees of freedom succeeds in well capturing, in a long-term integration, the behavior of the T21 model monitored by the mean state, the variance pattern, autocorrelation functions, and probability distributions of the streamfunction. The model based on principal interaction patterns offers a substantial improvement on a model based on empirical orthogonal functions with the same number of degrees of freedom in capturing the autocorrelation function, the probability distribution, and the response of the system to a change in the forcing.

Reduced atmospheric models using dynamically motivated basis functions

Abstract Nonlinear deterministic reduced models of large-scale atmospheric dynamics are constructed. The dynamical framework is a quasigeostrophic three-level spectral model with realistic mean state and variability as well as Pacific–North America (PNA) and North Atlantic Oscillation (NAO) patterns. The study addresses the problem of finding appropriate basis functions for efficiently capturing the dynamics and a comparison between different choices of basis functions; it focuses on highly truncated models, keeping only 10–15 modes. The reduced model is obtained by a projection of the equations of motion onto a truncated basis spanned by empirically determined modes. The total energy metric is used in the projection; the nonlinear terms of the low-order model then conserve total energy. Apart from retuning the coefficient of horizontal diffusion, no empirical terms are fitted in the dynamical equations of the low-order model in order to properly preserve the physics of the system. Using the methodology o...

Analysis and modelling of glacial climate transitions using simple dynamical systems

Glacial climate variability is studied integrating simple nonlinear stochastic dynamical systems with palaeoclimatic records. Different models representing different dynamical mechanisms and modelling approaches are contrasted; model comparison and selection is based on a likelihood function, an information criterion as well as various long-term summary statistics. A two-dimensional stochastic relaxation oscillator model with proxy temperature as the fast variable is formulated and the system parameters and noise levels estimated from Greenland ice-core data. The deterministic part of the model is found to be close to the Hopf bifurcation, where the fixed point becomes unstable and a limit cycle appears. The system is excitable; under stochastic forcing, it exhibits noisy large-amplitude oscillations capturing the basic statistical characteristics of the transitions between the cold and the warm state. No external forcing is needed in the model. The relaxation oscillator is much better supported by the data than noise-driven motion in a one-dimensional bistable potential. Two variants of a mixture of local linear stochastic models, each associated with an unobservable dynamical regime or cluster in state space, are also considered. Three regimes are identified, corresponding to the different phases of the relaxation oscillator: (i) lingering around the cold state, (ii) rapid shift towards the warm state, (iii) slow relaxation out of the warm state back to the cold state. The mixture models have a high likelihood and are able to capture the pronounced time-reversal asymmetry in the ice-core data as well as the distribution of waiting times between onsets of Dansgaard–Oeschger events.

Low-dimensional models of the Ginzburg-Landau equation

Low-dimensional numerical approximations for two boundary value problems of the complex Ginzburg--Landau equation in a chaotic regime are constructed. Spatial structures called principal interaction patterns are extracted from the system according to a nonlinear variational principle and used as basis functions in a Galerkin approximation. The dynamical description in terms of principal interaction patterns requires fewer modes than more conventional approaches using Fourier modes or Karhunen--Loeve modes as basis functions.

Forecasting critical transitions using data-driven nonstationary dynamical modeling.

An approach to predicting critical transitions from time series is introduced. A nonstationary low-order stochastic dynamical model of appropriate complexity to capture the transition mechanism under consideration is estimated from data. In the simplest case, the model is a one-dimensional effective Langevin equation, but also higher-dimensional dynamical reconstructions based on time-delay embedding and local modeling are considered. Integrations with the nonstationary models are performed beyond the learning data window to predict the nature and timing of critical transitions. The technique is generic, not requiring detailed a priori knowledge about the underlying dynamics of the system. The method is demonstrated to successfully predict a fold and a Hopf bifurcation well beyond the learning data window.

Detecting, anticipating, and predicting critical transitions in spatially extended systems

A data-driven linear framework for detecting, anticipating, and predicting incipient bifurcations in spatially extended systems based on principal oscillation pattern (POP) analysis is discussed. The dynamics are assumed to be governed by a system of linear stochastic differential equations which is estimated from the data. The principal modes of the system together with corresponding decay or growth rates and oscillation frequencies are extracted as the eigenvectors and eigenvalues of the system matrix. The method can be applied to stationary datasets to identify the least stable modes and assess the proximity to instability; it can also be applied to nonstationary datasets using a sliding window approach to track the changing eigenvalues and eigenvectors of the system. As a further step, a genuinely nonstationary POP analysis is introduced. Here, the system matrix of the linear stochastic model is time-dependent, allowing for extrapolation and prediction of instabilities beyond the learning data window. The methods are demonstrated and explored using the one-dimensional Swift-Hohenberg equation as an example, focusing on the dynamics of stochastic fluctuations around the homogeneous stable state prior to the first bifurcation. The POP-based techniques are able to extract and track the least stable eigenvalues and eigenvectors of the system; the nonstationary POP analysis successfully predicts the timing of the first instability and the unstable mode well beyond the learning data window.

Surface dynamics of hexagonal close-packed metals

Theoretical investigations on the dynamics at surfaces of hexagonal close-packed metals are reported. Lattice-dynamical as well as continuum theoretical calculations have been performed using a Greens function formalism. The lattice-dynamical studies are based on force constant models which are modified in the first layers close to the surface. The calculations presented in this paper refer to the (0001) surface of beryllium, magnesium and zinc. In all cases the results of the lattice-dynamical calculations in the limit of long wavelength are in qualitative agreement with those obtained from elastic theory. For beryllium and magnesium the (0001) surface is found to be very stable against variations of the surface parameters. One should not expect any surface reconstruction. For zinc the ideal (0001) surface is unstable, but there are particular slight changes in the surface parameters which make it stable. These changes correspond with experimental data.

Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective

The stability properties as characterized by the fluctuations of finite-time Lyapunov exponents around their mean values are investigated in a three-level quasigeostrophic atmospheric model with realistic mean state and variability. Firstly, the covariance structure of the fluctuation field is examined. In order to identify dominant patterns of collective excitation, an empirical orthogonal function (EOF) analysis of the fluctuation field of all of the finite-time Lyapunov exponents is performed. The three leading modes are patterns where the most unstable Lyapunov exponents fluctuate in phase. These modes are virtually independent of the integration time of the finite-time Lyapunov exponents. Secondly, large-deviation rate functions are estimated from time series of finite-time Lyapunov exponents based on the probability density functions and using the Legendre transform method. Serial correlation in the time series is properly accounted for. A large-deviation principle can be established for all of the Lyapunov exponents. Convergence is rather slow for the most unstable exponent, becomes faster when going further down in the Lyapunov spectrum, is very fast for the near-neutral and weakly dissipative modes, and becomes slow again for the strongly dissipative modes at the end of the Lyapunov spectrum. The curvature of the rate functions at the minimum is linked to the corresponding elements of the diffusion matrix. Also, the joint large-deviation rate function for the first and the second Lyapunov exponent is estimated.

Bifurcation analysis of geophysical time series

We propose a general synthetic framework, combining analytical and experimental techniques, for studying climatic bifurcations and transitions by means of the time series analysis. The method employs three major techniques: (i) derivation of potential from time series using unscented Kalman Filter (UKF); (ii) studying possible bifurcations and transitions of the obtained potential; (iii) projection of the time series according to the estimated perturbation. The method is tested on artificial data and then applied to observed records, in particular, a Greenland temperature proxy.