Sequential Data Assimilation Research Articles

Uncertainty assessment of hydrologic model states and parameters: Sequential data assimilation using the particle filter

Two elementary issues in contemporary Earth system science and engineering are (1) the specification of model parameter values which characterize a system and (2) the estimation of state variables which express the system dynamic. This paper explores a novel sequential hydrologic data assimilation approach for estimating model parameters and state variables using particle filters (PFs). PFs have their origin in Bayesian estimation. Methods for batch calibration, despite major recent advances, appear to lack the flexibility required to treat uncertainties in the current system as new information is received. Methods based on sequential Bayesian estimation seem better able to take advantage of the temporal organization and structure of information, so that better compliance of the model output with observations can be achieved. Such methods provide platforms for improved uncertainty assessment and estimation of hydrologic model components, by providing more complete and accurate representations of the forecast and analysis probability distributions. This paper introduces particle filtering as a sequential Bayesian filtering having features that represent the full probability distribution of predictive uncertainties. Particle filters have, so far, generally been used to recursively estimate the posterior distribution of the model state; this paper investigates their applicability to the approximation of the posterior distribution of parameters. The capability and usefulness of particle filters for adaptive inference of the joint posterior distribution of the parameters and state variables are illustrated via two case studies using a parsimonious conceptual hydrologic model.

On the eddy‐Kuroshio interaction: Meander formation process

The Kuroshio flows along the southern coast of Japan during its non‐large‐meandering state, then separates from the coast near the Kii Peninsula and attaches at the north of the Izu Ridge. The amplitude of the offshore displacement of the Kuroshio changes as a result of mesoscale perturbations. Satellite SSH and SST observations (TOPEX/Poseidon: T/P and NOAA AVHRR) have suggested that the short‐term Kuroshio meander formation is triggered by anticyclonic eddies originating in the Kuroshio Extension. To simulate such an event, a numerical experiment using a high‐resolution regional GCM was conducted by initializing the eddy with the observed T/P sea level anomaly. An anticyclonic eddy was injected to the south of the Kuroshio (140°E, 30°N) by means of sequential data assimilation of the T/P data of October 1992 for 30 days. The volume transport of the model Kuroshio was kept constant at 25 Sv, a condition which would not cause large meander of the Kuroshio. The simulation successfully reproduced the four phases of the interaction: (1) westward propagation of the eddy; (2) advection of the eddy by the Kuroshio; (3) meander formation; and (4) detachment of the eddy from the Kuroshio and their repetition. The analyses revealed that the inshore high potential vorticity (PV) water is generated at the sharp coastal topography (the Kii peninsula). The cyclonic eddies shed will eventually coalesce with the existing inshore cyclonic circulation and the meander grows. During the growth of the meander, the necessary barotropic kinetic energy is produced through the shallowing of the thermocline of the anticyclonic eddy as it elongates and splits. The growth of the meander ceases when the split anticyclonic eddies merge, the thermocline deepens, and the eddy detaches itself from the Kuroshio as a result of its own westward thrust. Simultaneously, the accumulated high PV inshore is released to the Kuroshio Extension region as cyclonic eddies. This study provides evidence for the active role of the anticyclonic eddy in causing the variability of the Kuroshio path and suggests a mechanism of rapid discharge/recharge of the available potential energy of the eddy and production/release of high PV inshore Kuroshio to cause the short‐term Kuroshio meander.

A demonstration of ensemble-based assimilation methods with a layered OGCM from the perspective of operational ocean forecasting systems

A demonstration study of three advanced, sequential data assimilation methods, applied with the nonlinear Miami Isopycnic Coordinate Ocean Model (MICOM), has been performed within the European Commission-funded DIADEM project. The data assimilation techniques considered are the Ensemble Kalman Filter (EnKF), the Ensemble Kalman Smoother (EnKS) and the Singular Evolutive Extended Kalman (SEEK) Filter, which all in different ways resemble the original Kalman Filter. In the EnKF and EnKS an ensemble of model states is integrated forward in time according to the model dynamics, and statistical moments needed at analysis time are calculated from the ensemble of model states. The EnKS, as opposed to the EnKF, update the analysis also backward in time whenever new observations are available, thereby improving the estimated states at the previous analysis times. The SEEK filter reduces the computational burden of the error propagation by representing the errors in a subspace which is initially calculated from a truncated EOF analysis. A hindcast experiment, where sea-level anomaly and sea-surface temperature data are assimilated, has been conducted in the North Atlantic for the time period July until September 1996. In this paper, we describe the implementation of ensemble-based assimilation methods with a common theoretical framework, we present results from hindcast experiments achieved with the EnKF, EnKS and SEEK filter, and we discuss the relative merits of these methods from the perspective of operational marine monitoring and forecasting systems. We found that the three systems have similar performances, and they can be considered feasible technologically for building preoperational prototypes.

Assimilation of ocean colour data into a biochemical model of the North Atlantic: Part 1. Data assimilation experiments

An advanced multivariate sequential data assimilation method, the ensemble Kalman filter (EnKF), has been investigated with a three-dimensional biochemical model of the North Atlantic, utilizing real chlorophyll data from the from the Sea-viewing Wide Field-of-view Sensor (SeaWiFS). The approach chosen here differs significantly from conventional parameter estimation techniques. We keep the parameters fixed, and instead update the actual model state, allowing for unknown errors in the dynamical formulation. In the ensemble Kalman filter, estimates of the true dynamical error covariances are provided from an ensemble of model states. The physical ocean is described through the Miami Isopycnic Coordinate Ocean Model (MICOM). Its output, e.g. fields of temperature, velocities and layer thicknesses, is used to force the ecosystem model, which contains 11 biochemical components. The system is driven by realistic ECMWF atmospheric forcing fields. A simple demonstration experiment was performed for April and May 1998, that is, the early part of the North Atlantic spring bloom is included. It is shown that the ensemble Kalman filter analysis estimate is consistent with the choice of prior error variances. Furthermore, it is illustrated that the multivariate analysis scheme also affects the unobserved model compartments, and that the variance fields for all the variables decrease during the assimilation. A discussion of the number of ensemble members needed to ensure proper representations of the true error covariances is also given. In a companion paper [J. Mar. Syst. 40/41 (2003)], hereafter referred to as the Part 2 paper, we illustrate some useful approaches to monitor the evolution (i.e., time series) of the ensemble of model states.

An Ensemble Kalman Filter with a complex marine ecosystem model: hindcasting phytoplankton in the Cretan Sea

Abstract. The purpose of this paper is to examine the use of a complex ecosystem model along with near real-time in situ data and a sequential data assimilation method for state estimation. The ecosystem model used is the European Regional Seas Ecosystem Model (ERSEM; Baretta et al., 1995) and the assimilation method chosen is the Ensemble Kalman Filer (EnKF). Previously, it has been shown that this method captures the nonlinear error evolution in time and is capable of both tracking the observations and providing realistic error estimates for the estimated state. This system has been used to assimilate long time series of in situ chlorophyll taken from a data buoy in the Cretan Sea. The assimilation of this data using the EnKF method results in a marked improvement in the ability of ERSEM to hindcast chlorophyll. The sensitivity of this system to the type of data used for assimilation, the frequency of assimilation, ensemble size and model errors is discussed. The predictability window of the EnKF appears to be at least 2 days. This is an indication that the methodology might be suitable for future operational data assimilation systems using more complex three-dimensional models. Key words. Oceanography: general (numerical modelling; ocean prediction) – Oceanography: biological and chemical (plankton)

Sequential Data Assimilation Techniques in Oceanography

SummaryWe review recent developments of sequential data assimilation techniques used in oceanography to integrate spatio‐temporal observations into numerical models describing physical and ecological dynamics. Theoretical aspects from the simple case of linear dynamics to the general case of nonlinear dynamics are described from a geostatistical point‐of‐view. Current methods derived from the Kalman filter are presented from the least complex to the most general and perspectives for nonlinear estimation by sequential importance resampling filters are discussed. Furthermore an extension of the ensemble Kalman filter to transformed Gaussian variables is presented and illustrated using a simplified ecological model. The described methods are designed for predicting over geographical regions using a high spatial resolution under the practical constraint of keeping computing time sufficiently low to obtain the prediction before the fact. Therefore the paper focuses on widely used and computationally efficient methods.

Improving global chemical simulations of the upper troposphere—lower stratosphere with sequential assimilation of MOZAIC data

The quantity and variety of atmospheric measurements of chemical constituents are strongly increasing, both for in situ and remote-sensing observing systems. This is an incentive to develop data assimilationschemes for atmospheric chemistry models, in the perspective of building realistic three-dimensionaltime-dependent distributions of observed and chemically related non-observed compounds. Sequentialdata assimilation experiments within the global Chemistry and Transport Model MOCAGE have beenconducted, in the perspective of using the MOZAIC database and meteorological analyses to drivea global simulation of ozone and related tracers, focussing specially on the upper troposphere—lowerstratosphere; the chemistry and dynamics in this region of the atmosphere are of great environmentalinterest, yet they are currently very difficult to model. Assimilation of subsonic flight-level observationsof ozone is anticipated to provide a significant contribution to improve numerical simulations. Resultsobtained with two different sequential data assimilation schemes are presented. The experiments havebeen set up on an 8-d period in February 1997. The “non-local” data assimilation technique appearspreferable to the “local” technique tested, since the benefits of assimilation for the former appear toremain for longer, possibly up to 3–4 d in the model integration after the data assimilation phase.

Mantle-circulation models with sequential data assimilation: inferring present-day mantle structure from plate-motion histories.

Data assimilation is an approach to studying geodynamic models consistent simultaneously with observables and the governing equations of mantle flow. Such an approach is essential in mantle circulation models, where we seek to constrain an unknown initial condition some time in the past, and thus cannot hope to use first-principles convection calculations to infer the flow history of the mantle. One of the most important observables for mantle-flow history comes from models of Mesozoic and Cenozoic plate motion that provide constraints not only on the surface velocity of the mantle but also on the evolution of internal mantle-buoyancy forces due to subducted oceanic slabs. Here we present five mantle circulation models with an assimilated plate-motion history spanning the past 120 Myr, a time period for which reliable plate-motion reconstructions are available. All models agree well with upper- and mid-mantle heterogeneity imaged by seismic tomography. A simple standard model of whole-mantle convection, including a factor 40 viscosity increase from the upper to the lower mantle and predominantly internal heat generation, reveals downwellings related to Farallon and Tethys subduction. Adding 35% bottom heating from the core has the predictable effect of producing prominent high-temperature anomalies and a strong thermal boundary layer at the base of the mantle. Significantly delaying mantle flow through the transition zone either by modelling the dynamic effects of an endothermic phase reaction or by including a steep, factor 100, viscosity rise from the upper to the lower mantle results in substantial transition-zone heterogeneity, enhanced by the effects of trench migration implicit in the assimilated plate-motion history. An expected result is the failure to account for heterogeneity structure in the deepest mantle below 1500 km, which is influenced by Jurassic plate motions and thus cannot be modelled from sequential assimilation of plate motion histories limited in age to the Cretaceous. This result implies that sequential assimilation of past plate-motion models is ineffective in studying the temporal evolution of core-mantle-boundary heterogeneity, and that a method for extrapolating present-day information backwards in time is required. For short time periods (of the order of perhaps a few tens of Myr) such a method exists in the form of crude 'backward' convection calculations. For longer time periods (of the order of a mantle overturn), a rigorous approach to extrapolating information back in time exists in the form of iterative nonlinear optimization methods that carry assimilated information into the past through the use of an adjoint mantle convection model.

Stratospheric transport by planetary wave mixing as observed during CRISTA‐2

Planetary waves drive the mean meridional circulation of the stratosphere and at the same time facilitate quasi‐horizontal mixing of trace gases. This paper presents significant day‐to‐day variability of stratospheric trace gas fields associated with large planetary wave activity observed during the second mission of the Cryogenic Infrared Spectrometers and Telescopes for the Atmosphere (CRISTA) experiment. Geopotential height data of the UK Met Office show that the CRISTA‐2 observations in the Southern Hemisphere winter were made during a period of extremely large amplitudes of both wave‐1 and wave‐2. The planetary wave‐1, usually a quasi‐stationary feature, moved eastward with the traveling planetary wave‐2. The large amplitudes of both wave‐1 and wave‐2 led to a significant displacement of the edge of the polar vortex toward the tropics (down to 30°S). As a result of the large wave amplitudes and favorable phase alignment, the anticyclone drawing up tropical air was unusually strong, and thus considerable wave‐induced trace gas flux from the tropics toward midlatitudes was observed, mainly in the form of a pronounced planetary‐scale tongue advected out of the tropics around the vortex and into the anticyclone. Quantitative transport calculations based on a sequential data assimilation system highlight the importance of such transport events for trace gas eddy‐flux in the Southern Hemisphere winter stratosphere.

On the eddy‐Kuroshio interaction: Evolution of the mesoscale eddy

Presented are results from a numerical study of the interaction of a mesoscale eddy and the Kuroshio using a high‐resolution regional general circulation model (GCM). The distinct evolution of the mesoscale eddies is not new in the literature, but most of the previous studies were conducted with simpler model configurations. To our knowledge the present study is the first to use a high‐resolution GCM, proven to replicate a realistic ocean circulation. An anticyclonic eddy was injected to the south of Kuroshio (140°E, 30°N) by means of sequential data assimilation of TOPEX/Poseidon altimeter data of October 1992 for 30 days. The strength of the assimilation was varied to produce five eddies (Rossby number ϵ ∼ 0.012–0.014) comparable in scale to those observed in this region. The two results (ϵ = 0.0131, 0.0141) resemble the observed onset of short‐term meandering events in 1993 and 1998. The westward propagating anticyclonic eddy collides with the Kuroshio southeast of Kyushu, propagates downstream, and triggers the short‐term Kuroshio meander (occurring between the Kii Peninsula and Izu ridge with duration of half a year). The sequence resembles the scenario hypothesized on the basis of altimeter observations and other in situ measurements during Tokyo Ogasawara Line Experiment. The description of these cases is important, but we focus here on the differentiation between the strong eddy cases (ϵ = 0.0131, 0.0141) and the weak eddy cases (ϵ = 0.0118, 0.0124, 0.0125) classified on the basis of their subsequent evolutions: the strong geddy (meandering case) propagates west as it elongates zonally and barotropic tripolar vortices form, and the weak eddy (nonmeandering case) makes an abrupt southward migration while the eddy core splits as a result of advection by lower layer geostrophic motions. Such distinct evolutions are conjectured a result of a competition of the effects of nonlinearity, dispersion, and barotropicity of which their relative importance varied among the simulated eddies. Because the scale variation of the injected eddies are natural, we expect that even the weak‐eddy case exists in nature; in fact, such a case was observed in 1994.

An Ensemble Kalman filter with a 1-D marine ecosystem model

The Ensemble Kalman Filter (EnKF) has been examined in a data assimilation experiment with a one-dimensional three-component ecosystem model. The model is an extension of the zero-dimensional model developed by Evans and Parslow [Biol. Oceanogr. 3 (1985) 327.]. The purpose of this paper is to examine the possibilities of using a sequential data assimilation method for state estimation in a biological model, an approach which differs from the more traditional parameter estimation studies. The method chosen is the Ensemble Kalman Filer (EnKF), and it has been shown that this method captures the nonlinear error evolution in time and is capable of both tracking the reference solution and to provide realistic error estimates for the estimated state. This is an indication that the methodology might be suitable for future operational data assimilation systems using more complex three-dimensional models.

Developments in Operational Shelf Sea Modelling in Danish Waters

An initial implementation of an operational system for the Danish waters has been carried out. The system consists of an observational network of water level stations, a two-dimensional model that computes simultaneously areas with different grid resolution, and a sequential data assimilation method. For assimilation of water level measurements an approximate Kalman filter algorithm, the ensemble Kalman filter, has been implemented. The ensemble Kalman filter has a computational cost much lower than the cost associated with a full Kalman filter application, and therefore it is suitable to be applied in an operational system. The error covariance matrix in this specific case tends to a quasi-steady state after a few days of assimilation. Thus, a Kalman filter with a constant weighting matrix has been applied during a 13-day test period to assimilate observed water levels in a model covering the entire North Sea and Baltic Sea area. The corrections achieved by the assimilation procedure are significant in most of the validation stations. Moreover, the error estimation provided by the filter is very accurate, especially in the inner Danish waters.

Stochastic Methods for Sequential Data Assimilation in Strongly Nonlinear Systems

This paper considers several filtering methods of stochastic nature, based on Monte Carlo drawing, for the sequential data assimilation in nonlinear models. They include some known methods such as the particle filter and the ensemble Kalman filter and some others introduced by the author: the second-order ensemble Kalman filter and the singular extended interpolated filter. The aim is to study their behavior in the simple nonlinear chaotic Lorenz system, in the hope of getting some insight into more complex models. It is seen that these filters perform satisfactory, but the new filters introduced have the advantage of being less costly. This is achieved through the concept of second-order-exact drawing and the selective error correction, parallel to the tangent space of the attractor of the system (which is of low dimension). Also introduced is the use of a forgetting factor, which could enhance significantly the filter stability in this nonlinear context.

Error covariance modeling in sequential data assimilation

The efficiency of a sequential data assimilation scheme relies on the capability to describe the error covariance. This aspect is all the more relevant if one needs accurate statistics on the estimation error. Frequently an ad hoc function depending on a few parameters is proposed, and these parameters are tuned, estimated or updated. This usually requires that the covariance is second-order stationary (i.e. depends only on the distance between two points). In this paper, we discuss this feature and show that even in simple applications (such as one-dimensional hydrodynamics), this assumption does not hold and may lead to poorly described estimation errors. We propose a method relying on the analysis of the error term and the use of the hydrodynamical model to generate one part of the covariance function, the other part being modeled using a second-order stationary approach. This method is discussed using a twin experiment in the case where a physical parameter is erroneous, and improves significantly the results: the model bias is strongly reduced and the estimation error is well described. Moreover, it enables a better adaptation of the Kalman gain to the actual estimation error.

A Data Assimilation Method Used with an Ocean Circulation Model and Its Application to the Tropical Atlantic Data Analysis

During the past several years, there has been an increase in the development of data assimilation methods and their application. The first data assimilation applications were in meteorology, and now the data assimilation technique is an important component of numerical weather forecasting. The data revolution in oceanography is bringing the daily practice of physical oceanography closer to that of dynamic meteorology. There are now large observational data sets of temperature and salinity in the Atlantic from global projects such as WOCE, SECTIONS, TOGA, COARE, PIRATA, and others. Improved assimilation techniques are needed in order to have ocean models fully exploit the new observational facilities. Data assimilation improves the estimate of the ocean and atmospheric physical state by extracting as much information as possible from both the measurements and the dynamic model, combining them in an optimal way. Assimilation may be used to improve initial and/or boundary conditions and to evaluate poorly known model parameters. Data assimilation in ocean models has been ongoing in the scientific literature for approximately 30 years (for a review, see [8]). It is necessary to understand which processes can be realistically described by the ocean models and which is the role of measurements. Two extreme opinions exist. One considers that the ocean is like a quasi-stationary medium and that it is sufficient to make only a few measurements in order to have a good description of the ocean state. From this point of view, the numerical models are useless (or simply wrong) and data assimilation techniques are restricted to a statistical correction process. The second opinion is that the ocean is a highly turbulent fluid that does not have a memory about its previous states. In this case, information generated by measurements becomes meaningless in a short time interval, and it is necessary to continuously monitor the ocean. Thus, only well-designed resolution models would be able to describe real processes in the ocean and data assimilation would be useful only to obtain correct initial (and possible boundary) conditions. Reality, however, lies somewhere in between these two extreme points of view. Thus, the scientific question is how to represent this situation numerically. The answer depends on both the model and the data assimilation techniques as well as on the local dynamic. There are two general concepts which have been discussed for data assimilation. The first is the “variational/adjoint” method, which has been the most popular scheme (see, e.g., [22, 24, 28]). An example of this technique is to assume that the model initial (and/or boundary) conditions are unknown parameters and that there is observation data distributed over some time interval. The technique seeks the optimal initial conditions by comparing the model trajectory with the measurements with respect to some criteria. This can be formulated as a constrained minimization problem. The present paper does not deal with this class of methods so there is no need to go into the details of this problem. The other class of methods is the “sequential data assimilation.” Starting from some initial condition the model solution is sequentially updated whenever measurements are available. The model solution will approach the observed state under certain conditions. This group of methods requires an updating scheme, which combines the model solution and the measurements to find the “best” state estimate. The Kalman-filter method belongs to this class of data assimilation technique. The Kalman filter is derived in a number of books on control theory (e.g., [1, 11]). In oceanography, the Kalmanfilter has been used in [7, 16]. The main idea of this method is to solve dynamic equations using the error covariance matrix equation, where an error is the difference between the model and observation values. These applications have been presented in [9, 16] and are commonly used. In this paper, another method for the definition of the error covariance matrix is presented. It continues the studies published in our paper by K. Belyaev, S. Meyers, and J. J. O’Brien.

Horizontal and Vertical Structure of the Representer Functions for Sea Surface Measurements in a Coastal Circulation Model

Abstract The representer functions used to propagate the information from observations onto model variables in a sequential data assimilation scheme are calculated with a statistical method. The Princeton Ocean Model is used in a simple configuration simulating the North Current flowing along the French coasts in the northwestern Mediterranean Sea. The influence of sea level measurements on the models’ three-dimensional temperature and velocity fields is investigated. Inhomogeneities and directions of anisotropy are evidenced. A comparison with simplified reduced order assimilation methods suggests that such schemes will not allow for consistent assimilation of nearshore altimetric data.

Tests of sequential data assimilation for retrieving profile soil moisture and temperature from observed L-band radiobrightness

Sequential data assimilation (Kalman filter optimal estimation) techniques are applied to the problem of retrieving near-surface soil moisture and temperature state from periodic terrestrial radiobrightness observations that update soil heat and moisture diffusion models. The retrieval procedure uses a time-explicit numerical model to continuously propagate the soil state profile, its error of estimation, and its interdepth covariances through time. The model's coupled soil moisture and heat fluxes are constrained by micrometeorology boundary conditions drawn from observations or atmospheric modeling. When radiometer data are available, the Kalman filter updates the state profile estimate by weighing the propagated state, error, and covariance estimates against an a priori estimate of radiometric measurement error. The Kalman filter compares predicted and observed radiobrightnesses directly, so no inverse algorithm relating brightness to physical parameters is required. The authors demonstrate Kalman filter model effectiveness using field observations and a simulation study. An observed 1 m soil state profile is recovered over an eight-day period from daily L-band observations following an intentionally poor initial state estimate. In a four-month simulation study, they gauge the longer term behavior of the soil state retrieval and Kalman gain through multiple rain events, soil dry-downs, and updates from radiobrightnesses.

Spatial regression and multiscale approximations for sequential data assimilation in ocean models

Effects of spatial regularity and locality assumptions in the extended Kalman filter are examined for oceanic data assimilation problems. Biorthogonal wavelet bases are used to implement spatial regularity through multiscale approximations, while a Markov random field (MRF) is used to impose locality through spatial regression. Both methods are shown to approximate the optimal Kalman filter estimates closely, although the stability of the estimates can be dependent on the choice of basis functions in the wavelet case. The observed filter performance is nearly constant over a wide range of values for the scalar weights (uncertainty variances) given to the model and data examined here. The MRF‐based method, with its inhomogeneous and anisotropic covariance parameterization, has been shown to be particularly effective and stable in assimilation of simulated TOPEX/POSEIDON altimetry data into a reduced‐gravity, shallow‐water equation model.

Application of advanced data assimilation methods for the initialisation of storm surge models

A sequential data assimilation technique (utilising a reduced-rank square-root filter) has been incorporated into the complete non-linear equations of a hydrodynamical module of a standard two dimensional hydrodynamic modelling tool. The deterministic model output is corrected using the available data and the best model initial conditions are generated. The deterministic model is then used to generate the water level forecast. A description of the general technique that is applied and its adaptation to this particular problem are provided. The application to a real storm that occurred during February 1993 in the North sea is presented.

Convergence of data assimilation by periodic updating in simple Hamiltonian and dissipative systems

In this paper, we study the influence of the interval between data insertion events on the convergence of sequential data assimilation problems. An example of a conservative Hamiltonian system is presented (that of Hénon and Heiles (1964)) where sequential assimilation with periodic data insertion every Δt achieves a more rapid convergence if data is not inserted at the smallest possible update interval, Δt. It is shown analytically that this is true for all Hamiltonian systems when the updated variables produce convergence of the assimilation, because the resolvent matrix then varies as O(Δt2) to highest order. The theory successfully predicts the turnover point for the He´non and Heiles system when a larger Δt leads to slower convergence and also the assimilation interval at which convergence may cease altogether. The application to a simplified low order shallow water model describing coupled Rossby and gravity waves and with a forced-dissipative perturbation extends the previous result to systems which are a more realistic model for the atmosphere and the ocean. Formally, the same behaviour still holds when a realistic dissipation scheme is applied with increasing amplitudes or when strongly dissipative systems, which are not forced-dissipative perturbations of Hamiltonians, are used.