A delay differential model of ENSO variability, Part 2: Phase locking, multiple solutions, and dynamics of extrema

We consider a highly idealized model for El Nino/Southern Oscillation (ENSO) variability, as introduced in an earlier paper. The model is governed by a delay differential equation for sea surface temperature in the Tropical Pacific, and it combines two key mechanisms that participate in ENSO dynamics: delayed negative feedback and seasonal forcing. We perform a theoretical and numerical study of the model in the three-dimensional space of its physically relevant parameters: propagation period of oceanic waves across the Tropical Pacific, atmosphere-ocean coupling, and strength of seasonal forcing. Phase locking of model solutions to the periodic forcing is prevalent: the local maxima and minima of the solutions tend to occur at the same position within the seasonal cycle. Such phase locking is a key feature of the observed El Nino (warm) and La Nina (cold) events. The phasing of the extrema within the seasonal cycle depends sensitively on model parameters when forcing is weak. We also study co-existence of multiple solutions for fixed model parameters and describe the basins of attraction of the stable solutions in a one-dimensional space of constant initial model histories.Comment: Nonlin. Proc. Geophys., 2010, accepte

Endogenous Business Cycles and the Economic Response to Exogenous Shocks

In this paper, we investigate the macroeconomic response to exogenous shocks, namely natural disasters and stochastic productivity shocks. To do so, we make use of an endogenous business cycle model in which cyclical behavior arises from the investment–profit instability; the amplitude of this instability is constrained by the increase in labor costs and the inertia of production capacity and thus results in a finite-amplitude business cycle. This model is found to exhibit a larger response to natural disasters during expansions than during recessions, because the exogenous shock amplifies pre-existing disequilibria when occurring during expansions, while the existence of unused resources during recessions allows for damping the shock. Our model also shows a higher output variability in response to stochastic productivity shocks during expansions than during recessions. This finding is at odds with the classical real-cycle theory, but it is supported by the analysis of quarterly U.S. Gross Domestic Product series; the latter series exhibits, on average, a variability that is 2.6 times larger during expansions than during recessions.Business cycles, Natural disasters, Productivity shocks, Output variability

Cluster analysis of multiple planetary flow regimes

A modified cluster analysis method was developed to identify spatial patterns of planetary flow regimes, and to study transitions between them. This method was applied first to a simple deterministic model and second to Northern Hemisphere (NH) 500 mb data. The dynamical model is governed by the fully-nonlinear, equivalent-barotropic vorticity equation on the sphere. Clusters of point in the model's phase space are associated with either a few persistent or with many transient events. Two stationary clusters have patterns similar to unstable stationary model solutions, zonal, or blocked. Transient clusters of wave trains serve as way stations between the stationary ones. For the NH data, cluster analysis was performed in the subspace of the first seven empirical orthogonal functions (EOFs). Stationary clusters are found in the low-frequency band of more than 10 days, and transient clusters in the bandpass frequency window between 2.5 and 6 days. In the low-frequency band three pairs of clusters determine, respectively, EOFs 1, 2, and 3. They exhibit well-known regional features, such as blocking, the Pacific/North American (PNA) pattern and wave trains. Both model and low-pass data show strong bimodality. Clusters in the bandpass window show wave-train patterns in the two jet exit regions. They are related, as in the model, to transitions between stationary clusters

A delay differential model of ENSO variability: Parametric instability and the distribution of extremes

We consider a delay differential equation (DDE) model for El-Nino Southern Oscillation (ENSO) variability. The model combines two key mechanisms that participate in ENSO dynamics: delayed negative feedback and seasonal forcing. We perform stability analyses of the model in the three-dimensional space of its physically relevant parameters. Our results illustrate the role of these three parameters: strength of seasonal forcing $b$, atmosphere-ocean coupling $\kappa$, and propagation period $\tau$ of oceanic waves across the Tropical Pacific. Two regimes of variability, stable and unstable, are separated by a sharp neutral curve in the $(b,\tau)$ plane at constant $\kappa$. The detailed structure of the neutral curve becomes very irregular and possibly fractal, while individual trajectories within the unstable region become highly complex and possibly chaotic, as the atmosphere-ocean coupling $\kappa$ increases. In the unstable regime, spontaneous transitions occur in the mean ``temperature'' ({\it i.e.}, thermocline depth), period, and extreme annual values, for purely periodic, seasonal forcing. The model reproduces the Devil's bleachers characterizing other ENSO models, such as nonlinear, coupled systems of partial differential equations; some of the features of this behavior have been documented in general circulation models, as well as in observations. We expect, therefore, similar behavior in much more detailed and realistic models, where it is harder to describe its causes as completely.Comment: 22 pages, 9 figure

Transport on river networks: A dynamical approach

This study is motivated by problems related to environmental transport on river networks. We establish statistical properties of a flow along a directed branching network and suggest its compact parameterization. The downstream network transport is treated as a particular case of nearest-neighbor hierarchical aggregation with respect to the metric induced by the branching structure of the river network. We describe the static geometric structure of a drainage network by a tree, referred to as the static tree, and introduce an associated dynamic tree that describes the transport along the static tree. It is well known that the static branching structure of river networks can be described by self-similar trees (SSTs); we demonstrate that the corresponding dynamic trees are also self-similar. We report an unexpected phase transition in the dynamics of three river networks, one from California and two from Italy, demonstrate the universal features of this transition, and seek to interpret it in hydrological terms.Comment: 38 pages, 15 figure

Low-frequency variability and heat transport in a low-order nonlinear coupled ocean-atmosphere model

We formulate and study a low-order nonlinear coupled ocean-atmosphere model with an emphasis on the impact of radiative and heat fluxes and of the frictional coupling between the two components. This model version extends a previous 24-variable version by adding a dynamical equation for the passive advection of temperature in the ocean, together with an energy balance model. The bifurcation analysis and the numerical integration of the model reveal the presence of low-frequency variability (LFV) concentrated on and near a long-periodic, attracting orbit. This orbit combines atmospheric and oceanic modes, and it arises for large values of the meridional gradient of radiative input and of frictional coupling. Chaotic behavior develops around this orbit as it loses its stability; this behavior is still dominated by the LFV on decadal and multi-decadal time scales that is typical of oceanic processes. Atmospheric diagnostics also reveals the presence of predominant low- and high-pressure zones, as well as of a subtropical jet; these features recall realistic climatological properties of the oceanic atmosphere. Finally, a predictability analysis is performed. Once the decadal-scale periodic orbits develop, the coupled system's short-term instabilities --- as measured by its Lyapunov exponents --- are drastically reduced, indicating the ocean's stabilizing role on the atmospheric dynamics. On decadal time scales, the recurrence of the solution in a certain region of the invariant subspace associated with slow modes displays some extended predictability, as reflected by the oscillatory behavior of the error for the atmospheric variables at long lead times.Comment: v1: 41 pages, 17 figures; v2-: 42 pages, 15 figure

Endogenous Business Cycles and the Economic Response to Exogenous Shocks

In this paper, we investigate the macroeconomic response to exogenous shocks, namely natural disasters and stochastic productivity shocks. To do so, we make use of an endogenous business cycle model in which cyclical behavior arises from the investmentprofit instability; the amplitude of this instability is constrained by the increase in labor costs and the inertia of production capacity and thus results in a finite-amplitude business cycle. This model is found to exhibit a larger response to natural disasters during expansions than during recessions, because the exogenous shock amplifies pre-existing disequilibria when occurring during expansions, while the existence of unused resources during recessions allows for damping the shock. Our model also shows a higher output variability in response to stochastic productivity shocks during expansions than during recessions. This finding is at odds with the classical real-cycle theory, but it is supported by the analysis of quarterly U.S. Gross Domestic Product series; the latter series exhibits, on average, a variability that is 2.6 times larger during expansions than during recessions

Intrinsic and climatic factors in North-American animal population dynamics

BACKGROUND: Extensive work has been done to identify and explain multi-year cycles in animal populations. Several attempts have been made to relate these to climatic cycles. We use advanced time series analysis methods to attribute cyclicities in several North-American mammal species to abiotic vs. biotic factors. RESULTS: We study eleven century-long time series of fur-counts and three climatic records – the North Atlantic Oscillation (NAO), the El-Niño-Southern Oscillation (ENSO), and Northern Hemisphere (NH) temperatures – that extend over the same time interval. Several complementary methods of spectral analysis are applied to these 14 times series, singly or jointly. These spectral analyses were applied to the leading principal components (PCs) of the data sets. The use of both PC analysis and spectral analysis helps distinguish external from intrinsic factors that influence the dynamics of the mammal populations. CONCLUSIONS: Our results show that all three climatic indices influence the animal-population dynamics: they explain a substantial part of the variance in the fur-counts and share characteristic periods with the fur-count data set. In addition to the climate-related periods, the fur-count time series also contain a significant 3-year period that is, in all likelihood, caused by biological interactions

A multi-model ensemble Kalman filter for data assimilation and forecasting

Data assimilation (DA) aims to optimally combine model forecasts and observations that are both partial and noisy. Multi-model DA generalizes the variational or Bayesian formulation of the Kalman filter, and we prove that it is also the minimum variance linear unbiased estimator. Here, we formulate and implement a multi-model ensemble Kalman filter (MM-EnKF) based on this framework. The MM-EnKF can combine multiple model ensembles for both DA and forecasting in a flow-dependent manner; it uses adaptive model error estimation to provide matrix-valued weights for the separate models and the observations. We apply this methodology to various situations using the Lorenz96 model for illustration purposes. Our numerical experiments include multiple models with parametric error, different resolved scales, and different fidelities. The MM-EnKF results in significant error reductions compared to the best model, as well as to an unweighted multi-model ensemble, with respect to both probabilistic and deterministic error metrics.Comment: 23 pages, 10 figure