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To characterize local finite-time properties associated with transient chaos in open dynamicalsystems, we introduce an escape rate and fractal dimensions suitable for this purpose in acoarse-grained description. We numerically illustrate that these quantifiers have a considerablespread across the domain of the dynamics, but their spatial variation, especially on long butnon-asymptotic integration times, is approximately consistent with the relationship that wasrecognized by Kantz and Grassberger for temporally asymptotic quantifiers. In particular,deviations from this relationship are smaller than differences between various locations, whichconfirms the existence of such a dynamical law and the suitability of our quantifiers to representunderlying dynamical properties in the non-asymptotic regime.

We study the forced response of the teleconnection between the El Niño–Southern Oscillation (ENSO) and the Indian summer monsoon (IM) in the Max Planck Institute Grand Ensemble, a set of Earth system ensemble simulations under historical and RCP forcing. The forced response of the teleconnection, or a characteristic of it, is defined as the time dependence of a correlation coefficient evaluated over the ensemble. We consider the temporal variability of spatial averages and that with respect to dominant spatial modes in the sense of Maximal Covariance Analysis, Canonical Correlation Analysis and Empirical Orthogonal Function analysis across the ensemble. A further representation of the teleconnection that we define here takes the point of view of the predictability of the complete spatiotemporal variability of the Indian summer monsoon. We find that the strengthening of the ENSO-IM teleconnection is robustly or consistently featured in view of various teleconnection representations, whether sea surface temperature (SST) or sea level pressure (SLP) is used to characterise ENSO, and both in the historical period and under the RCP8.5 forcing scenario. It is found to be associated dominantly with the principal mode of ENSO variability. Concerning representations that involve an autonomous characterisation of the Pacific, in terms of a linear regression model, the main contributor to the strengthening} is the regression coefficient, which can outcompete even a declining ENSO variability when it is represented by SLP. We also find that the forced change of the teleconnection is typically nonlinear by (1) formally rejecting the hypothesis that ergodicity holds, i.e., that expected values of temporal correlation coefficients with respect to the ensemble equal the ensemble-wise correlation coefficient itself, and also showing that (2) the trivial contributions of the forced changes in means and standard deviations are insignificant here. We also provide, in terms of the test

We consider transport in a fluid flow of arbitrary complexity but with a dominant flow direction. One of the situations in which this occurs is when describing by an effective flow the dynamics of sufficiently small particles immersed in a turbulent fluid and vertically sinking because of their weight. We develop a formalism characterizing the dynamics of particles released from one layer of fluid and arriving in a second one after traveling along the dominant direction. The main ingredient in our study is the definition of a two-layer map that describes the Lagrangian transport between both layers. We combine geometric approaches and probabilistic network descriptions to analyze the two-layer map. From the geometric point of view, we express the properties of lines, surfaces, and densities transported by the flow in terms of singular values related to Lyapunov exponents, and define a specific quantifier, the finite depth Lyapunov exponent. Within the network approach, degrees and an entropy are considered to characterize transport. We also provide relationships between both methodologies. The formalism is illustrated with numerical results for a modification of the ABC flow, a model commonly studied to characterize three-dimensional chaotic advection.

We study the vertical dispersion and distribution of negatively buoyant rigid microplastics within a realistic circulation model of the Mediterranean sea. We first propose an equation describing their idealized dynamics. In that framework, we evaluate the importance of some relevant physical effects (inertia, Coriolis force, small-scale turbulence and variable seawater density), and we bound the relative error of simplifying the dynamics to a constant sinking velocity added to a large-scale velocity field. We then calculate the amount and vertical distribution of microplastic particles on the water column of the open ocean if their release from the sea surface is continuous at rates compatible with observations in the Mediterranean. The vertical distribution is found to be almost uniform with depth for the majority of our parameter range. Transient distributions from flash releases reveal a non-Gaussian character of the dispersion and various diffusion laws, both normal and anomalous. The origin of these behaviors is explored in terms of horizontal and vertical flow organization.

We study the forced response of the teleconnection between the El Niño–Southern Oscillation (ENSO) and the Indian summer monsoon (IM) in the Max Planck Institute Grand Ensemble, a set of Earth system ensemble simulations under historical and Representative Concentration Pathway (RCP) forcing. The forced response of the teleconnection, or a characteristic of it, is defined as the time dependence of a correlation coefficient evaluated over the ensemble. We consider the temporal variability of spatial averages and that with respect to dominant spatial modes in the sense of Maximal Covariance Analysis, Canonical Correlation Analysis and Empirical Orthogonal Function analysis across the ensemble. A further representation of the teleconnection that we define here takes the point of view of the predictability of the spatiotemporal variability of the Indian summer monsoon. We find that the strengthening of the ENSO-IM teleconnection is robustly or consistently featured in view of various teleconnection representations, whether sea surface temperature (SST) or sea level pressure (SLP) is used to characterize ENSO, and both in the historical period and under the RCP8.5 forcing scenario. It is found to be associated dominantly with the principal mode of ENSO variability. Concerning representations that involve an autonomous characterisation of the Pacific, in terms of a linear regression model, the main contributor to the strengthening is the regression coefficient, which can outcompete even a declining ENSO variability when it is represented by SLP. We also find that the forced change of the teleconnection is typically nonlinear by 1) formally rejecting the hypothesis that ergodicity holds, i.e., that expected values of temporal correlation coefficients with respect to the ensemble equal the ensemble-wise correlation coefficient itself, and also showing that 2) the trivial contributions of the forced changes in means and standard deviations are insignificant here. We also pro

We study the forced response of the teleconnection between the El Nino-Southern Oscillation (ENSO) and global precipitation in general and the Indian summer monsoon (IM) in particular in the Max Planck Institute Grand Ensemble. The forced response of the teleconnection is defined as the time-dependence of a correlation coefficient evaluated over the ensemble. The ensemble-wise variability is taken either wrt. spatial averages or dominant spatial modes in the sense of Maximal Covariance Analysis or Canonical Correlation Analysis or EOF analysis. We find that the strengthening of the ENSO-IM teleconnection is robustly or consistently featured in view of all four teleconnection representations, whether sea surface temperature (SST) or sea level pressure (SLP) is used to characterise ENSO, and both in the historical period and under the RCP8.5 forcing scenario. The main contributor to this strengthening in terms of a linear regression model is the regression coefficient, which can outcompete even a declining ENSO variability in view of using the SLP. We also find that the forced change of the teleconnection is typically nonlinear by (1) formally rejecting the hypothesis that ergodicity holds, i.e., that expected values of temporal correlation coefficients with respect to the ensemble equal the ensemble-wise correlation coefficient itself, and also showing that (2) the trivial contributions of the forced changes of e.g. the mean SST and/or precipitation to temporal correlations are insignificant here. We also provide, in terms of the test statistics, global maps of the degree of nonlinearity/nonergodicity of the forced change of the teleconnection between local precipitation and ENSO.

To characterize local finite-time properties associated with transient chaos in open dynamical systems, we introduce an escape rate and fractal dimensions suitable for this purpose in a coarse-grained description. We numerically illustrate that these quantifiers have a considerable spread across the domain of the dynamics, but their spatial variation, especially on long but non-asymptotic integration times, is approximately consistent with the relationship that was recognized by Kantz and Grassberger for temporally asymptotic quantifiers. In particular, deviations from this relationship are smaller than differences between various locations, which confirms the existence of such a dynamical law and the suitability of our quantifiers to represent underlying dynamical properties in the non-asymptotic regime.

The availability of a large ensemble enables one to evaluate empirical orthogonal functions (EOFs)with respect to the ensemble without relying on temporal variability at all. Variability across theensemble at any given time is supposed to represent the most relevant probability distribution forclimate-related studies, and this distribution is presumably subject to temporal changes in thepresence of time-dependent forcing. Such changes may be observable in spatial patterns ofensemble-based EOFs and associated eigenvalues. Unfortunately, estimates of these changescome with a considerable error due to the finite size of the ensemble, so that associating asignificance level with the presence of a change (with respect to a null hypothesis about theabsence of any change) should be the first step of analyzing the time evolution.It turns out, however, that the conditions for the applicability of usual hypothesis tests aboutstationarity are not satisfied for the above-mentioned quantities. What proves to be feasible is toestimate an upper bound on the significance level for nonstationarity. This means that the truesignificance level would ideally be lower or equal to what is estimated, which would preventunjustified confidence in the detection of nonstationarity (i.e., falsely rejecting the null hypothesiscould not become more probable than claimed). Most importantly, one would avoid seriouslyoverconfident conclusions about the sign of the change in this way. Notwithstanding, the estimatefor the upper bound on the significance level is also affected by the finite number of the ensemblemembers. It nevertheless becomes more and more precise for increasing ensemble size and mayserve as a first guidance for currently available ensemble sizes.The details of the estimation are presented in the example of the EOF-based analysis of the ElNiño–Southern Oscillation (ENSO) as it appears in the historical and RCP8.5 simulations of the MaxPlanck Institute Grand Ensemble. A comparison bet

We study the teleconnection between El Niño–Southern Oscillation (ENSO) and the Indian summer monsoon (IM) in large ensemble simulations, the Max Planck Institute Earth System Model (MPI-ESM), and the Community Earth System Model (CESM1). We characterize ENSO by the June–August Niño-3 box-average SST and the IM by the June–September average precipitation over India, and define their teleconnection in a changing climate as an ensemble-wise correlation. To test robustness, we also consider somewhat different variables that can characterize ENSO and the IM. We utilize ensembles converged to the system’s snapshot attractor for analyzing possible changes in the teleconnection. Our main finding is that the teleconnection strength is typically increasing on the long term in view of appropriately revised ensemble-wise indices. Indices involving a more western part of the Pacific reveal, furthermore, a short-term but rather strong increase in strength followed by some decrease at the turn of the century. Using the station-based Southern Oscillation index (SOI) as opposed to area-based indices leads to the identification of somewhat more erratic trends, but the turn-of-the-century “bump” is well detectable with it. All this is in contrast, if not in contradiction, to the discussion in the literature of a weakening teleconnection in the late twentieth century. We show here that this discrepancy can be due to any of three reasons: 1) ensemble-wise and temporal correlation coefficients used in the literature are different quantities; 2) the temporal moving correlation has a high statistical variability but possibly also persistence; or 3) MPI-ESM does not represent the Earth system faithfully.

In radiative-convective equilibrium (RCE), radiative cooling of the troposphere is roughly balanced by the vaporization enthalpy set free by precipitating moist convection. Many earlier studies restricted the investigation of RCE to the dynamics of the atmosphere with constant boundary conditions including prescribed surface temperature. We investigate a GCM setup where a slab ocean is coupled to the atmosphere, and we explore a wide range of CO2 concentrations. We obtain reliable statistical quantities from thousand-year-long simulations. For moderate CO2 concentrations, we find unskewed temporal variations of 1–2 K in global mean surface temperature, with an almost constant climate sensitivity of 2 K. At CO2 concentrations beyond four times the preindustrial value, the climate sensitivity decreases to nearly zero as a result of episodic global cooling events as large as 10 K. The dynamics of these cooling events are investigated in detail and shown to be associated with an increase in large-scale low-level stratiform cloudiness in the subsiding region, which is a result of penetrative shallow convection being capped by an inversion and thus not ventilating the lower troposphere. These dynamics depend on the CO2 concentration: both through the effect of temperature on stratification and through the changing spatial scale of organization of the flow, which determines the spatial scale and temporal coherence of the stratiform cloud sheets.

We study the effect of turbulence on a sedimenting layer of particles by means of direct numerical simulations. A Lagrangian model in which particles are considered as tracers with an additional downward settling velocity is integrated together with an isotropic homogeneous turbulent flow. We study the spatial distribution of particles when they are collected on a plane at non-asymptotic times. We relate the resulting coarse-grained particle density to the history of the stretching rate along the particle trajectory and the projection of the density onto the accumulation plane and analyze the deviation from homogeneity in terms of the Reynolds number and the settling velocity. We identify two regimes that arise during the early and well-mixed stages of advection. In the former regime, more inhomogeneity in the particle distribution is introduced for decreasing settling velocity or increasing Reynolds number, while the tendencies are opposite in the latter regime. A resonant-like crossover is found between these two regimes where inhomogeneity is maximal.

Sedimentation of particles in the ocean leads to inhomogeneous horizontal distributions at depth, even if the release process is homogeneous. We study this phenomenon considering a horizontal sheet of sinking particles immersed in an oceanic flow and determine how the particles are distributed when they sediment on the seabed (or are collected at a given depth). The study is performed from a Lagrangian viewpoint attending to the properties of the oceanic flow and the physical characteristics (size and density) of typical biogenic sinking particles. Two main processes determine the distribution, the stretching of the sheet caused by the flow, and its projection on the surface where particles accumulate. These mechanisms are checked, besides an analysis of their relative importance to produce inhomogeneities, with numerical experiments in the Benguela region. Faster (heavier or larger) sinking particles distribute more homogeneously than slower ones.

In an incompressible flow, fluid density remains invariant along fluid element trajectories. This implies that the spatial distribution of non-interacting noninertial particles in such flows cannot develop density inhomogeneities beyond those that are already introduced in the initial condition. However, in certain practical situations, density is measured or accumulated on (hyper-) surfaces of dimensionality lower than the full dimensionality of the flow in which the particles move. An example is the observation of particle distributions sedimented on the floor of the ocean. In such cases, even if the initial distribution of noninertial particles is uniform but its support is finite, advection in an incompressible flow will give rise to inhomogeneities in the observed density. In this paper, we analytically derive, in the framework of an initially homogeneous particle sheet sedimenting toward a bottom surface, the relationship between the geometry of the flow and the emerging distribution. From a physical point of view, we identify the two processes that generate inhomogeneities to be the stretching within the sheet and the projection of the deformed sheet onto the target surface. We point out that an extreme form of inhomogeneity, caustics, can develop for sheets. We exemplify our geometrical results with simulations of particle advection in a simple kinematic flow, study the dependence on various parameters involved, and illustrate that the basic mechanisms work similarly if the initial (homogeneous) distribution occupies a more general region of finite extension rather than a sheet. Sedimentation of small particles in complex flows is an outstanding problem in science and technology. In particular, the sinking of biogenic particles from the marine surface to the bottom is a fundamental process of the biological carbon pump, playing a key role in the global carbon cycle. A complete understanding of this problem is still lacking. It has been recently shown that de

Sedimentation of particles in the ocean leads to inhomogeneous horizontal distributions at depth. The present dataset contains density and trajectory data for particle sedimentation computed from the velocity field given by a ROMS simulation at 1/12 x 1/12 degrees of horizontal resolution in the region of Benguela (south east Atlantic). Direct particle sampling as well as the result of a geometrical approach are included.

Ensemble approaches are becoming widely used in climate research. In contrast to weather forecast, however, in the climatic context one is interested in long-time properties, those arising on the scale of several decades. The well-known strong internal variability of the climate system implies the existence of a related dynamical attractor with chaotic properties. Under the condition of climate change this should be a snapshot attractor, naturally arising in an ensemble-based framework. Although ensemble averages can be evaluated at any instant of time, results obtained during the process of convergence of the ensemble towards the attractor are not relevant from the point of view of climate. In simulations, therefore, attention should be paid to whether the convergence to the attractor has taken place. We point out that this convergence is of exponential character, therefore, in a finite amount of time after initialization relevant results can be obtained. The role of the time scale separation due to the presence of the deep ocean is discussed from the point of view of ensemble simulations.