Time-dependent Forcing Research Articles

Predictions of critical transitions with non-stationary reduced order models

Here we demonstrate the ability of stochastic reduced order models to predict the statistics of non-stationary systems undergoing critical transitions. First, we show that the reduced order models are able to accurately predict the autocorrelation function and probability density functions (PDF) of higher dimensional systems with time-dependent slow forcing of either the resolved or unresolved modes. Second, we demonstrate that whether the system tips early or repeatedly jumps between the two equilibrium points (flickering) depends on the strength of the coupling between the resolved and unresolved modes and the time scale separation. Both kinds of behaviour have been found to precede critical transitions in earlier studies. Furthermore, we demonstrate that the reduced order models are also able to predict the timing of critical transitions. The skill of various proposed tipping indicators are discussed.

Analysis of time discretization methods for Stokes equations with a nonsmooth forcing term

We consider a time dependent Stokes problem that is motivated by two-phase incompressible flow problems with surface tension. The surface tension force results in a right-hand side functional in the momentum equation with poor regularity properties. As a strongly simplified model problem we treat a Stokes problem with a similar time dependent nonsmooth forcing term. We consider the implicit Euler and Crank-Nicolson methods for time discretization. The regularity properties of the data are such that for the Crank-Nicolson method one can not apply error analyses known in the literature. We present a convergence analysis leading to a second order error bound in a suitable negative norm that is weaker that the $$L^2$$ L 2 -norm. Results of numerical experiments are shown that confirm the analysis.

Study of forced turbulence and its modulation by finite-size solid particles using the lattice Boltzmann approach

The paper describes application of the mesoscopic lattice Boltzmann (LB) method to the simulations of both single-phase turbulence and particle-laden turbulence which are maintained by a large-scale forcing. The disturbance flows around finite-size solid particles are resolved, providing the opportunity to study the detailed interactions between fluid turbulence and solid particles at the particle–fluid interfaces. Specifically, a nonuniform time-dependent stochastic forcing scheme is implemented within the mesoscopic multiple-relaxation-time LB approach. The statistics of single-phase forced turbulence obtained from the LB approach are found to be in excellent agreement with those from the pseudo-spectral simulations, provided that the grid resolution in the LB simulation is doubled. It is shown that the flow statistics is not sensitive to the velocity scale used for the LB simulation. Preliminary results on forced turbulence laden with non-sedimenting solid particles at a particle-to-fluid density ratio of 5, solid volume fraction of 0.102, and particle diameter to Kolmogorov length ratio of 8.05 are interpreted, using a systematic analysis conducted at three levels: whole-field, phase-partitioned, and profiles as a function of distance from the surface of solid particles. It is found that the particle-laden turbulence is much more dissipative in terms of the non-dimensional dissipation rate, due to both reduction of the effective flow Reynolds number and the viscous boundary layer on the surfaces of solid particles. The thickness of the boundary layer is found to be about 0.4rp. While this boundary layer region accounts for 19.5% of the space within the fluid, it contributes to 57.5% of total viscous dissipation. The vorticity magnitude exhibits a maximum inside the boundary layer and a minimum outside the boundary layer, showing detachment of the vorticity structure from the solid surface. The sharp gradients near the particle surface contribute dominantly to the value of velocity derivative flatness, making the flatness in particle-laden flow much larger than that of single-phase turbulence. In the spectral space, presence of solid particles attenuates energy at large scales including the forcing shells and augments energy at the small scales. The pivot wavenumber is found to be very similar to the value previously found in decaying particle-laden turbulence under the similar parameter setting.

Intermittency in relation with 1/f noise and stochastic differential equations

One of the models of intermittency is on-off intermittency, arising due to time-dependent forcing of a bifurcation parameter through a bifurcation point. For on-off intermittency, the power spectral density (PSD) of the time-dependent deviation from the invariant subspace in a low frequency region exhibits 1/f power-law noise. Here, we investigate a mechanism of intermittency, similar to the on-off intermittency, occurring in nonlinear dynamical systems with invariant subspace. In contrast to the on-off intermittency, we consider the case where the transverse Lyapunov exponent is zero. We show that for such nonlinear dynamical systems, the power spectral density of the deviation from the invariant subspace can have 1/fβ form in a wide range of frequencies. That is, such nonlinear systems exhibit 1/f noise. The connection with the stochastic differential equations generating 1/fβ noise is established and analyzed, as well.

Driving a conceptual model climate by different processes: Snapshot attractors and extreme events

In a low-order chaotic model of global atmospheric circulation the effects of driving, i.e., time-dependent (periodic, chaotic, and noisy) forcing, are investigated, with particular interest in extremal behavior. An approach based on snapshot attractors formed by a trajectory ensemble is applied to represent the time-dependent likelihood of extreme events in terms of a physical observable. A single trajectory-based framework, on the other hand, is used to determine the maximal value and the kurtosis of the distribution of the same observable. We find the most significant effect of the driving on the magnitude, relative frequency, and variability of extreme events when its characteristic time scale becomes comparable to that of the model climate. Extreme value statistics is pursued by the method of block maxima, and found to follow Weibull distributions. Deterministic drivings result in shape parameters larger in modulus than stochastic drivings, but otherwise strongly dependent on the particular type of driving. The maximal effects of deterministic drivings are found to be more pronounced, both in magnitude and variability of the extremes, than white noise, and the latter has a stronger effect than red noise.

Lie Group Analysis of a Forced KdV Equation

The Korteweg-de Vries (KdV) equation considered in this work contains a forcing term and is referred to as forced KdV equation in the sequel. This equation has been investigated recently as a mathematical model for waves on shallow water surfaces under the influence of external forcing. We employ the Lie group analysis approach to specify the time-dependent forcing term.

Optimal spatiotemporal reduced order modeling, Part II: application to a nonlinear beam

A geometrically nonlinear, simply supported beam under the influence of time-dependent external forcing serves as a testbed to demonstrate application of the optimal spatiotemporal reduced order modeling (OPSTROM) framework proposed in Part I of this work. Fully resolved simulations, which are relatively expensive to perform, are used to accurately predict the beam response for a few forcing parameters. More affordable simulations are achieved with a conventional finite-difference scheme by coarsening the computational grid in space and time. Discretization errors are reduced with OPSTROM as subgrid-scale models are designed to account for the underlying space-time statistical structure using principles of mean-square error minimization, conditional expectations and stochastic estimation. When included in the under-resolved simulations, these optimal subgrid-scale models are shown to significantly improve the accuracy of predictions for both periodic and chaotic response types. This improved accuracy is further demonstrated through a set of numerical experiments designed to capture the complex bifurcation behavior of the beam response.

Sediment mixtures, coastal bedforms and grain sorting phenomena: An overview of the theoretical analyses

Recent studies, which investigate the processes leading to the formation of rhythmic morphological features in the coastal region, are reviewed. Emphasis is given to morphodynamic stability analyses which consider the presence of a heterogeneous sediment. The selective transport of the different fractions of a sediment mixture leads to grain sorting phenomena, which significantly affect the dynamics of the bottom forms. Small scale bedforms (ripples), intermediate bedforms (sorted bedforms and tidal dunes) and large scale bedforms (shoreface-connected ridges and sand banks) are considered. References are given to papers describing the new aspects of the stability analyses arising from the simultaneous presence of a time-dependent forcing flow and a sediment mixture.

Arrays of vortices in shallow fluids: Three-dimensional structure and dispersion

The role of three-dimensional (3D) flow structures in an electromagnetically generated array of vortices in a single shallow (electrolytic) fluid layer with and without a vertical no-slip wall has been investigated by stereoscopic particle image velocimetry. This array of vortices is a result of a continuous, however, time-dependent periodic forcing. Additionally, the dispersion of passive tracers by this linear array of vortices is explored with 3D numerical simulations. As the parameter regime is quite extensive (fluid layer depth, current density, forcing frequency, etc.), we have restricted this study to two specific cases with a fixed fluid-layer depth: the quasi-two-dimensional (quasi-2D) laminar regime (low current density, or weak forcing) and a regime with substantial 3D secondary flows (high current density, or strong forcing). In all cases, the forcing frequency is taken similar to the typical eddy turnover time of the quasi-2D vortices. The low-forcing regime typically results in quasi-2D laminar flows which can hardly be considered turbulent. However, the compressible horizontal free-surface flow strongly affects the spatial distribution of passive tracers. The high-forcing regime gives rise to a strong inertia-dominated flow, including locally strong 3D secondary circulations and rapid vertical mixing of passive tracers, despite the shallowness of the fluid layer. Finally, the present investigation suggests that forcing too close to the vertical (no-slip) walls results in a substantial reduction of the horizontal integral length scale and enhanced production of small-scale 3D flows.

Modeling seasonal and diurnal pCO 2 variations in the northern South China Sea

This paper describes the simulated temporal variation of surface seawater CO 2 partial pressure ( pCO 2) in the northern South China Sea. We produced the simulations with a one-dimensional (1-D) coupled physical–biogeochemical model that had high-frequency, time-dependent atmospheric forcing and that were validated with field observations. We also examined the associated processes that modulate seawater pCO 2 at different time scales, from diurnal to seasonal, using a series of process-oriented experiments. At seasonal time scales, we revealed that the sea–air CO 2 exchange was a primary process that modulated surface pCO 2 and exceeded the role of sea surface temperature (SST) even though the phase of the pCO 2 variation generally followed the strong seasonal cycle of SST. This was because sea–air CO 2 exchange is a slow process and has an accumulative effect on surface water pCO 2 due to the buffering effect of the carbonate system once CO 2 has dissolved in the seawater, which leads to a long equilibration time of CO 2 between the atmosphere and seawater. The mixing effect on pCO 2 induced by total alkalinity and dissolved inorganic carbon variations was, generally, positively correlated with the seasonal evolution of wind speed. Biological processes were the smallest contributors to pCO 2 variations at the seasonal scale because of the oligotrophic characteristic of the region. At diurnal time scales, the dominant pCO 2 controlling factor was mainly associated with the local physical and biological conditions. Temperature and wind-induced vertical mixing played major roles in pCO 2 when the winter heat flux and upward transport of low temperature and high pCO 2 in deep water were intensified. Phytoplankton blooms generally occur after a period of strong wind, as a result, biological metabolism becomes the most important pCO 2 regulator when the surface chlorophyll-a reached its highest level. Unlike that in the seasonal scale, the effect of sea–air CO 2 exchange was minor at diurnal time scales due to the long equilibration time of CO 2 between the atmosphere and seawater. We also found that the frequency of the model driving force was important in reproducing the sea surface pCO 2. The high frequency forcing was important in controlling the pCO 2 variation through the feedback effect to the corresponding physical and biogeochemical responses.

State space decomposition for non-autonomous dynamical systems

The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

Simulating idealized Dansgaard-Oeschger events and their potential impacts on the global methane cycle

Ice-core records indicate that during the last glacial period, atmospheric CH 4 and Greenland temperature rose abruptly in a series of Dansgaard-Oeschger (D-O) events. These increases attained up to two-thirds of the glacial-interglacial amplitude for CH 4. We use these major changes as possible constraints on the mechanisms of D-O variability. A series of simulations are performed with a coupled atmosphere-ocean general circulation model with a time-dependent freshwater forcing which induces rapid variations in the Atlantic meridional overturning circulation (AMOC). The transient climate output is then used to drive a dynamic vegetation model which simulates wetland CH 4 emissions. During freshwater input, emissions are reduced and, in sensitivity simulations, demonstrate strong dependence on the background climate (orbital insolation or ice sheet extent). In the reverse situation, the strengthening of the AMOC leads to warming over Northern Eurasia, but only minor change to the hydrological cycle in the tropics where most CH 4 emissions occur. This is a robust result from the climate models examined and is of central importance to our results, which suggest that AMOC driven CH 4 variations are considerably smaller than observed D-O events. Modelled emissions of volatile organic compounds (VOC) lead to a significant effect on the CH 4 lifetime during these events, but both the VOC emissions model and VOC effects on CH 4 lifetime are subject to uncertainty. The model results can only be reconciled with some of the larger changes in the ice-core record by including the effect of VOCs and by taking the total amplitude of AMOC perturbation from weak to very strong. Our results suggest that either the model is too insensitive to change, or that further mechanisms may be important for the large CH 4 changes during D-O events. In particular, a strengthening of the AMOC alone cannot reproduce the observed abrupt CH 4 increases. Comparisons with independent models would help to identify possible avenues for further work.

Accurate low dimensional models for deterministic fluid systems driven by uncertain forcing

This paper addresses the model reduction of high-order linear systems within the framework of the incompressible Navier-Stokes equations. We look for reduced-order models that capture the response of some specific sensor whatever the initial flow condition and in the presence of any time-dependent external forcing. Namely, this work deals with the accurate modeling of the input-output dynamics of a fluid system when considering each degree of freedom of the system as an input, and the given measurement as the output. In the case of complex or realistic flows, the number of inputs is too large to apply the standard balanced truncation procedure. To alleviate this problem, we introduce a method called input projection. Input projection is shown to be analogous to the output projection procedure introduced by Rowley, Int. J. Bifurcation. Chaos Appl. Sci. Eng. 15, 997 (2005). To illustrate the model reduction, we consider the dynamics of a globally stable flow over a rounded backward-facing step. Reduced-order models are obtained by projecting the full original system onto: (i) the basis of the leading balanced modes computed from the input-projected systems and (ii) the most observable modes. The balanced models are observed to accurately capture the transient growths along the separated flow whatever the input while outperforming the models based on the most observable modes.

Effects of time-dependent large-scale forcing, solar zenith angle, and sea surface temperature on time-mean rainfall: a partitioning analysis based on surface rainfall budget

Effects of time-dependent large-scale forcing, solar zenith angle, and sea surface temperature on time-mean rainfall during the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) are examined through a partitioning analysis of a series of sensitivity cloud-resolving model experiment data based on surface rainfall budget. The model is forced by time-dependent large-scale forcing (LSF), solar zenith angle (SZA), and sea surface temperature (SST) in the control experiment and is forced only by either time-dependent LSF, SZA, or SST while others are replaced with their time averages in the sensitivity experiments. The rainfall associated with water vapor divergence and local atmospheric drying and hydrometeor loss/convergence has the largest contribution to total rainfall among eight rainfall types. The largest rainfall contribution is increased in the simulations where either time-dependent LSF, SZA, or SST is replaced with its average, whereas it is decreased in the simulation where COARE-derived large-scale vertical velocity is replaced with zero vertical velocity. The contribution of the rainfall associated with water vapor convergence to total rainfall is decreased in the simulations with time-mean LSF, SZA, and SST, whereas it is increased in the simulation without large-scale vertical velocity.

Simulating the biogeochemical effects of volcanic CO2 degassing on the oxygen-state of the deep ocean during the Cenomanian/Turonian Anoxic Event (OAE2)

Cretaceous anoxic events may have been triggered by massive volcanic CO2 degassing as large igneous provinces (LIPs) were emplaced on the seafloor. Here, we present a comprehensive modeling study to decipher the marine biogeochemical consequences of enhanced volcanic CO2 emissions. A biogeochemical box model has been developed for transient model runs with time-dependent volcanic CO2 forcing. The box model considers continental weathering processes, marine export production, degradation processes in the water column, the rain of particles to the seafloor, benthic fluxes of dissolved species across the seabed, and burial of particulates in marine sediments. The ocean is represented by twenty-seven boxes. To estimate horizontal and vertical fluxes between boxes, a coupled ocean–atmosphere general circulation model (AOGCM) is run to derive the circulation patterns of the global ocean under Late Cretaceous boundary conditions. The AOGCM modeling predicts a strong thermohaline circulation and intense ventilation in the Late Cretaceous oceans under high pCO2 values. With an appropriate choice of parameter values such as the continental input of phosphorus, the model produces ocean anoxia at low to mid latitudes and changes in marine δ13C that are consistent with geological data such as the well established δ13C curve. The spread of anoxia is supported by an increase in riverine phosphorus fluxes under high pCO2 and a decrease in phosphorus burial efficiency in marine sediments under low oxygen conditions in ambient bottom waters. Here, we suggest that an additional mechanism might contribute to anoxia, an increase in the C:P ratio of marine plankton which is induced by high pCO2 values. According to our AOGCM model results, an intensively ventilated Cretaceous ocean turns anoxic only if the C:P ratio of marine organic particles exported into the deep ocean is allowed to increase under high pCO2 conditions. Being aware of the uncertainties such as diagenesis, this modeling study implies that potential changes in Redfield ratios might be a strong feedback mechanism to attain ocean anoxia via enhanced CO2 emissions. The formation of C-enriched marine organic matter may also explain the frequent occurrence of global anoxia during other geological periods characterized by high pCO2 values.

Two Limits of Initial-Value Decadal Predictability in a CGCM

Abstract When the climate system experiences time-dependent external forcing (e.g., from increases in greenhouse gas and aerosol concentrations), there are two inherent limits on the gain in skill of decadal climate predictions that can be attained from initializing with the observed ocean state. One is the classical initial-value predictability limit that is a consequence of the system being chaotic, and the other corresponds to the forecast range at which information from the initial conditions is overcome by the forced response. These limits are not caused by model errors; they correspond to limits on the range of useful forecasts that would exist even if nature behaved exactly as the model behaves. In this paper these two limits are quantified for the Community Climate System Model, version 3 (CCSM3), with several 40-member climate change scenario experiments. Predictability of the upper-300-m ocean temperature, on basin and global scales, is estimated by relative entropy from information theory. Despite some regional variations, overall, information from the ocean initial conditions exceeds that from the forced response for about 7 yr. After about a decade the classical initial-value predictability limit is reached, at which point the initial conditions have no remaining impact. Initial-value predictability receives a larger contribution from ensemble mean signals than from the distribution about the mean. Based on the two quantified limits, the conclusion is drawn that, to the extent that predictive skill relies solely on upper-ocean heat content, in CCSM3 decadal prediction beyond a range of about 10 yr is a boundary condition problem rather than an initial-value problem. Factors that the results of this study are sensitive and insensitive to are also discussed.

Stochastic amplification in an epidemic model with seasonal forcing

We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle is annual, and after a period doubling when it is biennial. The comprehensive nature of our approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in whooping cough and measles epidemics.

Phase description of nonlinear dissipative waves under space–time-dependent external forcing

Based on the model system undergoing phase separation and chemical reactions, we investigate the dynamics of propagating dissipative waves under external forcing which is periodic both in space and time. A phase diagram for the entrained and non-entrained states under the external forcing is obtained numerically. Theoretical analysis in terms of phase description of the traveling waves is carried out to show that the transition between the entrained and the non-entrained states by changing the external frequency occurs either through a saddle–node bifurcation or through a Hopf bifurcation and that these two bifurcation lines are connected at a Bogdanov–Takens bifurcation point.

Chaotic mixing in a Joule-heated glass melt

A numerical study of two-dimensional thermal convection of a highly viscous fluid driven by volumetric heating has been carried out to investigate the effect of time-dependent thermal forcing on the degree of mixing. The problem is relevant to electrically heated glass melting furnaces which are traditionally operated with time-independent heating. The numerical computations carried out are for two model fluids representing semitransparent and opaque melts, respectively. Lagrangian motion of passive tracers is numerically simulated and degree of mixing in the glass melt is quantified in terms of mixing entropy. The computed flow patterns indicate that time-dependent streamlines intersect transversely and, thereby, result in well-mixed regions. The stirring mechanism involves the rotational stretching and folding produced by two oscillating vortices with varying size and circulation. The model predictions indicate that the chaotic mixing is strongly dependent on the period of electrode firing cycle. The present investigation demonstrates that time-dependent thermal forcing improves the mixing characteristics of both semitransparent and opaque glass melts significantly.

Stability of the Slow Manifold in the Primitive Equations

We show that, with some regularity hypotheses, the solution of the forced-dissipative rotating primitive equations of the ocean loses most of its fast, inertia-gravity, component in the small Rossby number limit as $t\to\infty$. At leading order in the Rossby number, the solution approaches what is known as “geostrophic balance” even under ageostrophic, slowly time-dependent forcing. Results of higher orders in the Rossby number can be obtained if one further assumes that the forcing is time-independent and sufficiently smooth. If the forcing lies in some Gevrey space, the solution will be exponentially close to a finite-dimensional “slow manifold” after some time.