Cyclic Markov chains with an application to an intermediate ENSO model

Pasmanter, R.A.; Timmermann, Axel

doi:10.5194/npg-10-197-2003

Cited by 19 publications

(21 citation statements)

References 44 publications

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“…In Pasmanter and Timmermann (2003), Markov chains are used for studying ENSO predictability. The influence of the seasonal cycle is accounted for by employing so-called cyclic Markov chains: twelve different stochastic matrices m i (i = 1, ..., 12) are constructed (or in fact estimated), each specifying the transition probabilities of the various states from month i to month i + 1.…”

Section: Beyond Diffusion Processesmentioning

confidence: 99%

Stochastic Climate Theory

Gottwald¹,

Crommelin²,

Franzke³

2016

Nonlinear and Stochastic Climate Dynamics

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In this chapter we review stochastic modelling methods in climate science. First we provide a conceptual framework for stochastic modelling of deterministic dynamical systems based on the Mori-Zwanzig formalism. The Mori-Zwanzig equations contain a Markov term, a memory term and a term suggestive of stochastic noise. Within this framework we express standard model reduction methods such as averaging and homogenization which eliminate the memory term. We further discuss ways to deal with the memory term and how the type of noise depends on the underlying deterministic chaotic system. Secondly, we review current approaches in stochastic data-driven models. We discuss how the drift and diffusion coefficients of models in the form of stochastic differential equations can be estimated from observational data. We pay attention to situations where the data stems from multi scale systems, a relevant topic in the context of data from the climate system. Furthermore, we discuss the use of discrete stochastic processes (Markov chains) for e.g. stochastic subgrid-scale modeling and other topics in climate science.

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Section: Beyond Diffusion Processesmentioning

confidence: 99%

Stochastic Climate Theory

Gottwald¹,

Crommelin²,

Franzke³

2016

Nonlinear and Stochastic Climate Dynamics

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“…One approach might be to attempt to work in a state space of drastically reduced dimension (e.g. Egger 2001;Pasmanter and Timmermann 2003), making it feasible to tabulate g. The reduced state space might be based, for example, on a set of leading Empirical Orthogonal Functions (EOFs). However, either ∂M/∂p or ∇g could project strongly onto directions orthogonal to the leading EOFs, and the dynamics can be sensitive to apparently minor EOFs (e.g.…”

Section: Discussionmentioning

confidence: 99%

Climate sensitivities via a Fokker–Planck adjoint approach

Thuburn

2005

Q. J. R. Meteorol. Soc.

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SUMMARYThe traditional method for computing the sensitivity of a particular climate diagnostic to some input parameter, using a climate model, is to carry out integrations of the climate model for control and perturbed values of the parameter. However, when the sensitivity of the diagnostic to many parameters is required, this method becomes very expensive. Motivated by the need to calculate such sensitivities more cheaply, a new approach is developed here. Theoretically, it is based on considering the climate to be described by the probability density function that satisfies the relevant steady Fokker-Planck equation. The climate sensitivity (i.e. the derivative of the diagnostic with respect to the input parameters) can then be expressed in terms of the solution of the adjoint of the steady Fokker-Planck equation. This adjoint solution could be computed in practice via an ensemble of climate model integrations. Currently the theory is valid only for climate models that include stochastic forcing. A practical demonstration is given of two methods based on the new theory, applied to a stochastic version of the famous Lorenz equations. Some further refinements are needed, however, to achieve a method that would be computationally practicable for a full climate model.

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“…For example, singular vector decomposition of a linearized version of the ZC model (Thompson and Battisti, 2000), as well as the ZC forward tangent model along a trajectory in reduced EOF space (Xue et al, 1997), result in singular values that have a strong seasonal dependence, with growth of the singular vectors peaking in boreal winter. Similary, cyclic Markov models derived from the ZC model (Pasmanter and Timmermann, 2003), an anomaly coupled GCM (Kallummal and Kirtman, 2008), and observations (Johnson et al, 2000), reveal a stong seasonality in the internal dynamics of the equatorial Pacific coupled ocean-atmosphere system. It has been suggested that this internal seasonality is su cient to produce the observed the ENSO seasonal variance, without the need for nonlinear dynamics or seasonality in the noise forcing (Thompson and Battisti, 2000;Kallummal and Kirtman, 2008;Stein et al, 2010).…”

Section: Introductionmentioning

confidence: 99%

“…It has been suggested that this internal seasonality is su cient to produce the observed the ENSO seasonal variance, without the need for nonlinear dynamics or seasonality in the noise forcing (Thompson and Battisti, 2000;Kallummal and Kirtman, 2008;Stein et al, 2010). Moreoever, the Markov models can be explicitly related to Floquet analysis (Pasmanter and Timmermann, 2003), which can be used to show that the dynamics of most unstable mode of the ZC model with a seasonally varying background are the same as in the annual average case (Jin et al, 1996;Thompson and Battisti, 2000), which forms the basis of our dynamical understanding of ENSO (Philander et al, 1984;Hirst, 1986;Neelin and Jin, 1993a,b).…”

Section: Introductionmentioning

confidence: 99%

ENSO Seasonal Synchronization Theory

Stein

2014

Journal of Climate

102

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One of the key characteristics of El Niño–Southern Oscillation (ENSO) is its synchronization to the annual cycle, which manifests in the tendency of ENSO events to peak during boreal winter. Current theory offers two possible mechanisms to account the for ENSO synchronization: frequency locking of ENSO to periodic forcing by the annual cycle, or the effect of the seasonally varying background state of the equatorial Pacific on ENSO’s coupled stability. Using a parametric recharge oscillator (PRO) model of ENSO, the authors test which of these scenarios provides a better explanation of the observed ENSO synchronization. Analytical solutions of the PRO model show that the annual modulation of the growth rate parameter results directly in ENSO’s seasonal variance, amplitude modulation, and 2:1 phase synchronization to the annual cycle. The solutions are shown to be applicable to the long-term behavior of the damped model excited by stochastic noise, which produces synchronization characteristics that agree with the observations and can account for the variety of ENSO synchronization behavior in state-of-the-art coupled general circulation models. The model also predicts spectral peaks at “combination tones” between ENSO and the annual cycle that exist in the observations and many coupled models. In contrast, the nonlinear frequency entrainment scenario predicts the existence of a spectral peak at the biennial frequency corresponding to the observed 2:1 phase synchronization. Such a peak does not exist in the observed ENSO spectrum. Hence, it can be concluded that the seasonal modulation of the coupled stability is responsible for the synchronization of ENSO events to the annual cycle.

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Cyclic Markov chains with an application to an intermediate ENSO model

Cited by 19 publications

References 44 publications

Stochastic Climate Theory

Stochastic Climate Theory

Climate sensitivities via a Fokker–Planck adjoint approach

ENSO Seasonal Synchronization Theory

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