Rafail V. Abramov

Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems

In a recent paper the authors developed and tested two novel computational algorithms for predicting the mean linear response of a chaotic dynamical system to small changes in external forcing via the fluctuation-dissipation theorem (FDT): the short-time FDT (ST-FDT), and the hybrid Axiom A FDT (hA-FDT). Unlike the earlier work in developing fluctuation-dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, these two new methods are based on the theory of Sinai–Ruelle–Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. These two algorithms take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian (qG-FDT) approximation of the fluctuation-dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. It has been discovered that the ST-FDT algorithm is an extremely precise linear response approximation for short response times, but numerically unstable for longer response times. On the other hand, the hA-FDT method is numerically stable for all times, but is less accurate for short times. Here we develop blended linear response algorithms, by combining accurate prediction of the ST-FDT method at short response times with numerical stability of qG-FDT and hA-FDT methods at longer response times. The new blended linear response algorithms are tested on the nonlinear Lorenz 96 model with 40 degrees of freedom, chaotic behaviour, forcing, dissipation, and mimicking large-scale features of real-world geophysical models in a wide range of dynamical regimes varying from weakly to strongly chaotic, and to fully turbulent. The results below for the blended response algorithms have a high level of accuracy for the linear response of both mean state and variance throughout all the different chaotic regimes of the 40-mode model. These results point the way towards the potential use of the blended response algorithms in operational long-term climate change projection.

Information Theory and Predictability for Low-Frequency Variability

Abstract A predictability framework, based on relative entropy, is applied here to low-frequency variability in a standard T21 barotropic model on the sphere with realistic orography. Two types of realistic climatology, corresponding to different heights in the troposphere, are used. The two dynamical regimes with different mixing properties, induced by the two types of climate, allow the testing of the predictability framework in a wide range of situations. The leading patterns of empirical orthogonal functions, projected onto physical space, mimic the large-scale teleconnections of observed flow, in particular the Arctic Oscillation, Pacific–North American pattern, and North Atlantic Oscillation. In the ensemble forecast experiments, relative entropy is utilized to measure the lack of information in three different situations: the lack of information in the climate relative to the forecast ensemble, the lack of information by using only the mean state and variance of the forecast ensemble, and informati...

A New Algorithm for Low-Frequency Climate Response

Abstract The low-frequency response to changes in external forcing for the climate system is a fundamental issue. In two recent papers the authors developed a new blended response algorithm for predicting the response of a nonlinear chaotic forced-dissipative system to small changes in external forcing. This new algorithm is based on the fluctuation–dissipation theorem and combines the geometrically exact general response formula using integration of a linear tangent model at short response times and the classical quasi-Gaussian response algorithm at longer response times. This algorithm overcomes the inherent numerical instability in the geometric formula arising because of positive Lyapunov exponents at longer times while removing potentially large systematic errors from the classical quasi-Gaussian approximation at moderate times. Here the new blended method is tested on the low-frequency response for a T21 barotropic truncation on the sphere with realistic topography in two dynamical regimes correspon...

An improved algorithm for the multidimensional moment-constrained maximum entropy problem

The maximum entropy principle is a versatile tool for evaluating smooth approximations of probability density functions with the least bias beyond specified constraints. In the recent paper we introduced new computational framework for the moment-constrained maximum entropy problem in a multidimensional domain, and developed a simple numerical algorithm capable of computing maximum entropy problem in a two-dimensional domain with moment constraints of order up to 4. Here we design an improved numerical algorithm for computing the maximum entropy problem in a two- and higher-dimensional domain with higher order moment constraints. The algorithm features multidimensional orthogonal polynomial basis in the dual space of Lagrange multipliers to achieve numerical stability and rapid convergence of Newton iterations. The new algorithm is found to be capable of solving the maximum entropy problem in the two-dimensional domain with moment constraints of order up to 8, in the three-dimensional domain with moment constraints of order up to 6, and in the four-dimensional domain with moment constraints of order up to 4, corresponding to the total number of moment constraints of 44, 83 and 69, respectively. The two- and higher-dimensional maximum entropy test problems in the current work are based upon long-term statistics of numerical simulation of the real-world geophysical model for wind stress driven oceanic currents such as the Gulf Stream and the Kuroshio.

The multidimensional moment-constrained maximum entropy problem: A BFGS algorithm with constraint scaling

In a recent paper we developed a new algorithm for the moment-constrained maximum entropy problem in a multidimensional setting, using a multidimensional orthogonal polynomial basis in the dual space of Lagrange multipliers to achieve numerical stability and rapid convergence of the Newton iterations. Here we introduce two new improvements for the existing algorithm, adding significant computational speedup in situations with many moment constraints, where the original algorithm is known to converge slowly. The first improvement is the use of the BFGS iterations to progress between successive polynomial reorthogonalizations rather than single Newton steps, typically reducing the total number of computationally expensive polynomial reorthogonalizations for the same maximum entropy problem. The second improvement is a constraint rescaling, aimed to reduce relative difference in the order of magnitude between different moment constraints, improving numerical stability of iterations due to reduced sensitivity of different constraints to changes in Lagrange multipliers. We observe that these two improvements can yield an average wall clock time speedup of 5-6 times compared to the original algorithm.

Statistically relevant conserved quantities for truncated quasigeostrophic flow

Systematic applications of ideas from equilibrium statistical mechanics lead to promising strategies for assessing the unresolved scales of motion in many problems in science and engineering. A scientific debate over more than the last 25 years involves which conserved quantities among the formally infinite list are statistically relevant for the large-scale equilibrium statistical behavior. Here this important issue is addressed by using suitable discrete numerical approximations for geophysical flows with many conserved quantities as a numerical laboratory. The results of numerical experiments are presented here for these truncated geophysical flows with topography in a suitable regime. These experiments establish that the integrated third power of potential vorticity besides the familiar constraints of energy, circulation, and enstrophy (the integrated second power) is statistically relevant in this regime for the coarse-grained equilibrium statistical behavior at large scales. Furthermore, the integrated higher powers of potential vorticity larger than three are statistically irrelevant for the large-scale equilibrium statistical behavior in the examples studied here.

High skill in low-frequency climate response through fluctuation dissipation theorems despite structural instability

Climate change science focuses on predicting the coarse-grained, planetary-scale, longtime changes in the climate system due to either changes in external forcing or internal variability, such as the impact of increased carbon dioxide. The predictions of climate change science are carried out through comprehensive, computational atmospheric, and oceanic simulation models, which necessarily parameterize physical features such as clouds, sea ice cover, etc. Recently, it has been suggested that there is irreducible imprecision in such climate models that manifests itself as structural instability in climate statistics and which can significantly hamper the skill of computer models for climate change. A systematic approach to deal with this irreducible imprecision is advocated through algorithms based on the Fluctuation Dissipation Theorem (FDT). There are important practical and computational advantages for climate change science when a skillful FDT algorithm is established. The FDT response operator can be utilized directly for multiple climate change scenarios, multiple changes in forcing, and other parameters, such as damping and inverse modelling directly without the need of running the complex climate model in each individual case. The high skill of FDT in predicting climate change, despite structural instability, is developed in an unambiguous fashion using mathematical theory as guidelines in three different test models: a generic class of analytical models mimicking the dynamical core of the computer climate models, reduced stochastic models for low-frequency variability, and models with a significant new type of irreducible imprecision involving many fast, unstable modes.

Low-Frequency Climate Response of Quasigeostrophic Wind-Driven Ocean Circulation

AbstractLinear response to external perturbation through the fluctuation–dissipation theorem has recently become a popular topic in the climate research community. It relates an external perturbation of climate dynamics to climate change in a simple linear fashion, which provides key insight into physics of the climate change phenomenon. Recently, the authors developed a suite of linear response algorithms for low-frequency response of large-scale climate dynamics to external perturbation, including the novel blended response algorithm, which combines the geometrically exact general response formula using integration of a linear tangent model at short response times and the classical quasi-Gaussian response algorithm at longer response times, overcoming numerical instability of the tangent linear model for longer times due to positive Lyapunov exponents. Here, the authors apply the linear response framework to several leading empirical orthogonal functions (EOFs) of a quasigeostrophic model of wind-driven...

Suppression of chaos at slow variables by rapidly mixing fast dynamics through linear energy-preserving coupling

Chaotic multiscale dynamical systems are common in many areas of science, one of the examples being the interaction of the low-frequency dynamics in the atmosphere with the fast turbulent weather dynamics. One of the key questions about chaotic multiscale systems is how the fast dynamics affects chaos at the slow variables, and, therefore, impacts uncertainty and predictability of the slow dynamics. Here we demonstrate that the linear slow-fast coupling with the total energy conservation property promotes the suppression of chaos at the slow variables through the rapid mixing at the fast variables, both theoretically and through numerical simulations. A suitable mathematical framework is developed, connecting the slow dynamics on the tangent subspaces to the infinite-time linear response of the mean state to a constant external forcing at the fast variables. Additionally, it is shown that the uncoupled dynamics for the slow variables may remain chaotic while the complete multiscale system loses chaos and becomes completely predictable at the slow variables through increasing chaos and turbulence at the fast variables. This result contradicts the common sense intuition, where, naturally, one would think that coupling a slow weakly chaotic system with another much faster and much stronger mixing system would result in general increase of chaos at the slow variables.

A Simple Stochastic Parameterization for Reduced Models of Multiscale Dynamics

Multiscale dynamics are frequently present in real-world processes, such as the atmosphere-ocean and climate science. Because of time scale separation between a small set of slowly evolving variables and much larger set of rapidly changing variables, direct numerical simulations of such systems are difficult to carry out due to many dynamical variables and the need for an extremely small time discretization step to resolve fast dynamics. One of the common remedies for that is to approximate a multiscale dynamical systems by a closed approximate model for slow variables alone, which reduces the total effective dimension of the phase space of dynamics, as well as allows for a longer time discretization step. Recently, we developed a new method for constructing a deterministic reduced model of multiscale dynamics where coupling terms were parameterized via the Fluctuation-Dissipation theorem. In this work we further improve this previously developed method for deterministic reduced models of multiscale dynamics by introducing a new method for parameterizing slow-fast interactions through additive stochastic noise in a systematic fashion. For the two-scale Lorenz 96 system with linear coupling, we demonstrate that the new method is able to recover additional features of multiscale dynamics in a stochastically forced reduced model, which the previously developed deterministic method could not reproduce.