Climate change is a complex, multidisciplinary problem which relates our physical understanding of the consequences of greenhouse gas emissions with economic and socio-political actions to mitigate and adapt to those consequences. An important role that the mathematics of climate change can play involves utilising and developing understanding of nonlinear systems in such a way as to guide the design of ensembles of Global Climate and Earth System Models (ESMs), as well as integrated assessment and economic models. To this end it is informative to view these computer models as high-dimensional nonlinear systems and ask what we can learn about ensemble design from somewhat related, low-dimensional nonlinear systems.

This talk will discuss what it means to make a prediction of climate change within a computer model as well as how we can design ensembles to reflect our uncertainty in the real-world, physical climate system. The Lorenz ’84/Stommel ’61 (L84-S61) system will be introduced as a valuable tool for studying issues of ensemble design and will be used to illustrate key sources of uncertainty and sensitivity.

First amongst these senstitivities is initial value sensitivity of the sort explored in a variety of single model large ensembles (see session CL4.10/NH11/OS4) - these are known as micro-initial-condition ensembles. However, the results of such ensembles can themselves be dependent on large scale features of the starting conditions - so-called macro-initial-condition uncertainty. Lastly, the sensitivity of ensemble results to model structure and parameter value selection is crucial. How can we identify how close to the target system a model has to be to make useful probabilistic forecasts at different lead times? This question raises the prospect that climate predictions could be vulnerable to the “hawkmoth effect” - the potential for probabilistic forecasts based on initial condition ensembles to be highly sensitive to the finest details of model formulation.

Here the different types of initial value and model parameter sensitivities will be illustrated with the L84-S61 system. Based on these, a series of design questions will be raised - questions which suitably-designed ensembles of low-dimensional systems could help us understand and answer, and which could be extremely valuable in improving the design of ensembles of GCMs and ESMs.

How to cite: Stainforth, D. A.: Ensemble Design: Sensitivity Beyond Initial Values, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-14148, https://doi.org/10.5194/egusphere-egu23-14148, 2023.

Large and/or long-lived convective clusters are associated with extreme weather, drive the global circulation by forcing atmospheric waves, and affect the energy budget of the atmosphere by modulating outgoing longwave radiation in their vicinity. The majority of tropical clusters follow scale-free occurrence frequency distributions for cluster sizes and the rainfall integrated over a cluster (intensity). The relationships between intensity and area, and circumference and area also follow scaling laws. The exponents of all of these four scaling laws follow when we assume that precipitation clusters inherit their properties from the geometry of the integrated column water vapor field. Specifically, the column water vapor field would have to be a self-affine surface with a roughness exponent H=0.4. Coincidentally, H=0.4 is the prediction of the Kardar-Parisi-Zhang universality class in two dimensions.

I analyze the statistics of precipitation clusters and the column water vapor field in observations (using data from CMORPH and ERA5) and thirteen one-year global simulations performed with the ICON model at a horizontal resolution of 10 km. The simulations differ for example in their forcing (RCE or realistic forcing), in their rotation (no rotation, real rotation, constant Coriolis parameter), in their sea surface temperatures (SSTs; realistic and with land, zonal mean with land, constant without land, latitudinal gradient without land) etc. They are designed to test how robust the scaling laws of precipitation and column water vapor are.

What changes drastically between the simulations is the probability density distribution of points in the phase space of column water vapor and tropospheric bulk temperature. This distribution occupies a very narrow space in the RCE simulations, but a very wide space in the realistic simulation with land. The critical column water vapor, where precipitation starts to occur, is approximately a linear function of temperature. It turns out that the column water vapor axes and the temperatures axes can be rescaled so that the onset curves of all simulations collapse onto one line (approximately). The results show that there is a good match with the observed scaling in most simulations, with the control simulation (realistic SSTs and land) showing the closest match. I speculate what the results may imply for interpreting observed scalings based on the Kardar-Parisi-Zhang equation.

How to cite: Stephan, C.: Testing the robustness of precipitation cluster scalings with an ensemble of aquaplanet simulations, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-3901, https://doi.org/10.5194/egusphere-egu23-3901, 2023.

The state-of-the-art General Circulation Models or Earth System Models are based on conservation equations like the conservation of mass, momentum (Navier-Stokes), energy, and water... These equations are written in the form of partial derivative equations and are resolved on a grid whose spatial increment is a few tens or hundreds of kilometers and whose time increment is a few minutes. This means that phenomena acting below the numerical resolution are not computed. But because of the nonlinearity of equations, large scales are not independent of small scales, therefore the cutoff in resolution induces errors. For example, the linear relation between energy fluxes and temperature gradients (Fourier law) is not true for a grid of this size. To overcome this issue, it is usual to add new equations in order to close the conservation equations. In these new equations, new parameters are added and are generally tuned to fit observations. Though they are all based on the same physics, every climate model has a different set of "closure equations" and tuned parameters, leading to different results. For instance, while model comparisons are satisfying when looking at temperatures, results may differ significantly between two models when looking at precipitations.

Now, I am going to present an alternative way of resolving the climate system using zero tunable parameters. To achieve this, a paradigm change is needed. Partial derivative equations are no longer used, and variables are resolved with an optimization problem: maximizing a function under constraints (of conservations). The maximized function is the entropy production due to energy transfers and depends on temperatures only. Because solving the optimization problem isn't straightforward, the climate system is for now reduced to a vertical atmosphere, with only vertical energy fluxes. Such kind of model is sometimes called "radiative-convective" model and can be compared to tropical atmospheric observations because horizontal fluxes are less important there. The constraints imposed are the conservation of energy, the conservation of mass, and the conservation of water. Surprisingly, adding this last constraint to the model enables us to predict precipitations of about 1.2 m/year, in the good order of magnitude of average tropical precipitations. Theoretically, this means that precipitations depend mostly on the radiative transfer in the atmosphere.

The maximization of entropy production is probably not a generic "law of Nature" and might not apply to any out-of-equilibrium system. Here, we choose not to enter the debate whether it should be true for the climate or not, but only to show that this procedure can be a useful and efficient tool to close equations without introducing any tunable parameters, even when applied to precipitations. Though the optimization problem may rapidly become intractable, we can still envision building a more complete model of the atmospheric water cycle on these premises.

How to cite: Pikeroen, Q., Paillard, D., Dubrulle, B., and Watrin, K.: Computing precipitations with a vertical radiative-convective model with no adjustable parameters, using the maximum entropy production hypothesis., EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-15021, https://doi.org/10.5194/egusphere-egu23-15021, 2023.

In the analysis and interpretation of climate data, both from model simulations and observations, it is often of interest to establish relations between the responses of different observables to a global forcing. This problem in its generality is relevant in the context of the identification of emergent constraints for the climate system, detection and attribution studies, and the analysis of proxy data. Recently it has been discussed how in linear response theory it is possible to build proxy response operators, that allow to use the response of one observable to a forcing to predict the response of another observable. The spectral properties of the proxy response functions determine then the properties of statistical predictability at different time scales for the pair of observables. The skill and feasibility of this approach for complex climate data has however not been fully tested yet. In this work we analyse the properties of proxy response in experiments with the coupled general circulation model MPI-ESM v.1.2. We consider ensemble simulations of abrupt CO₂ doubling and 1% per year CO₂ increase scenarios. We study the response of different atmospheric and oceanic variables, and we compute proxy response functions for different pairs of observables. We analyse the predictive power for the different cases, and interpret differences in skills in terms of causal relations among observables. We also study the relation between statistical variability and long term sensitivity, and we discuss differences between ensemble and internal variability in unforced and forced states. We then link our results to the discussion on the interpretation of emergent constraints in climate change simulations.

How to cite: Ragone, F., Bastiaansen, R., Lembo, V., and Lucarini, V.: Analysis of proxy response and sensitivities in a coupled general circulation model, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-16258, https://doi.org/10.5194/egusphere-egu23-16258, 2023.

Various studies have identified possible drivers of extreme Arctic sea ice reduction, as observed in the summers of 2007 and 2012, including preconditioning, the oceanic heat transport and the synoptic-scale to large-scale atmospheric circulation. However, a quantitative statistical assessment of these drivers and a better understanding of the seasonal predictability of these events are hindered by the poor statistics of extremes in observations and in numerical simulations with computationally expensive climate models. Recent studies have addressed the problem of sampling extreme events in climate models by using rare event algorithms, computational techniques developed in statistical physics to increase the sampling efficiency of rare events in numerical models. In this work, we study the statistics of summer seasons with extremely low pan-Arctic sea ice area under pre-industrial greenhouse gas conditions, applying a rare event algorithm to the intermediate complexity coupled climate model PlaSim. Using the rare event algorithm, we oversample dynamical trajectories leading to events with extremely low summer and September mean pan-Arctic sea ice area. Compared to standard simulations of the same computational cost, we increase the sample size of the extremes by several orders of magnitude, which allows to perform statistically robust composite analyses of dynamical quantities conditional on these events. In addition, we have access to ultra-rare events with return times of up to 10⁵ years. We exploit the improved statistics of summers with extremely low pan-Arctic sea ice area to study precursors of these events, including a surface energy budget analysis to disentangle the oceanic and atmospheric forcing on the sea ice. Particularly, we investigate the linkage between the extremes in summer Arctic sea ice area and the preceding states of the Arctic Oscillation and of the Arctic Dipole Anomaly pattern, as well as between the extremes and the preconditioning in the sea ice-ocean system during the onset of the melt season.

How to cite: Sauer, J., Ragone, F., Massonnet, F., Demaeyer, J., and Zappa, G.: Drivers and predictability of extreme summer Arctic sea ice reduction with rare event simulation methods, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-7813, https://doi.org/10.5194/egusphere-egu23-7813, 2023.

Rare and extreme events are notoriously hard to handle in any complex stochastic system: They are simultaneously too rare to be reliably observable in numerics or experiment, but at the same time too important to be ignored if they have a large impact. This is a particular complication in climate science, atmosphere and ocean dynamics that deals with a large number of strongly coupled degrees of freedom. Often these rare events come in the form of a stochastically induced transition between different viable macrostates. Examples include atmospheric jets, oceanic currents, etc, that correspond to large coherent structures which are long live-lived, but might ultimately disappear. In this talk, I discuss rare events algorithms based on instanton calculus and large deviation theory that are capable of computing probabilities of such transitions happening, as well as their most likely pathway of occurrence.

How to cite: Grafke, T.: Sample Path Large Deviations for Climate, Ocean, and Atmosphere, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-9799, https://doi.org/10.5194/egusphere-egu23-9799, 2023.

Following Hasselmann’s ansatz, the climate system may be viewed as a multistable dynamical system internally driven by noise. Its long-term evolution will then feature noise-induced critical transitions between the competing attracting states. In the weak-noise limit, large deviation theory allows predicting the transition rate and most probable transition path of these tipping events. However, the limit of zero noise is never obtained in reality. In this work we show that, even for weak finite noise, sample transition paths may disagree with the large deviation prediction – the minimum action path, or instanton – if multiple timescales are at play. We illustrate this behavior in selected box models of the bistable Atlantic Meridional Overturning Circulation (AMOC), where different restoring times of temperature and salinity induce a fast-slow characteristic. While the minimum action path generally crosses the basin boundary at a saddle point, we demonstrate cases in which ensembles of sample transition paths cross far away from the saddle. We discuss the conditions for saddle avoidance and relate this to the flatness of the quasipotential, a central object of large deviation theory. We further probe the vicinity of the weak-noise limit by applying a pathspace method that generates transition samples for arbitrarily weak noise. Our results highlight that predictions by large deviation theory must be treated cautiously in multiscale dynamical systems.

How to cite: Börner, R., Deeley, R., Nesbitt, C., Römer, R., Grafke, T., Feudel, U., and Lucarini, V.: Limits of large deviation theory in predicting transition paths of climate tipping events, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-16458, https://doi.org/10.5194/egusphere-egu23-16458, 2023.

There have been five major mass extinction events and a number of smaller extinction events throughout geological time. Each of these events characterises a widespread decrease in species diversity. The largest of these was the End-Permian extinction which saw about 90% of species go extinct. Extinction may be caused by a variety of factors such as asteroid impacts, CO₂ driven ocean acidification, large igneous provinces, global warming/cooling, and oceanic anoxic events. All of these factors cause stress on the environment.

The ability of a species to avoid extinction is dependent on its environmental tolerances, i.e., the ability of a species to tolerate, or survive, changes in environmental conditions.

In population biology one way in which species may become extinct is through competition. The classical theory of competitive exclusion does not consider the type of interaction between species. We create a new mathematical model of competition between species in which the maximum population of a species is dependent on the availability of resources (or food supply) and competition is in the form of competition for these resources. We find this model always leads to stable coexistence. Another way in which populations can go extinct is through extreme oscillations in predator-prey systems; we explain how this can occur and illustrate this with a specific realistic predator-prey model that we then couple to our competition model.

How to cite: Garvey, R. and Fowler, A.: A Resource Dependent Competition Model, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-9628, https://doi.org/10.5194/egusphere-egu23-9628, 2023.

In recent years, the applications of stochastic partial differential equations to geophysical fluid dynamics has increased massively, as there are several complex dynamic models which can be represented using systems of SPDEs. An important problem to be adressed in this context is the correct noise calibration such that the resulting stochastic model efficiently incorporates the a priori unrepresented sub-scale geophysical processes. In this talk I will present a new method of stochastic calibration which can be applied to a class of stochastic fluid dynamics models. I will focus on an application specifically tailored for the stochastic rotating shallow water model. 

How to cite: Lang, O., Crisan, D., and Lobbe, A.: A new calibration method for the stochastic rotating shallow water model, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-8616, https://doi.org/10.5194/egusphere-egu23-8616, 2023.

An important question of climate science is the effect of a changing climate on the long term statistical properties of the atmosphere and ocean dynamics. Mathematically speaking, the question is whether and how statistical quantities of the dynamics (e.g. correlations, averages, variabilities etc) react to changes in the external forcing of the system.

A (stochastic or deterministic) dynamical system is said to exhibit linear response if the statistical quantities describing the long term behaviour of the system depend differentiably on the relevant parameter (i.e. the forcing), and therefore a small change in the forcing will result in a small and proportional change of the statistical quantity. A methodology to establish response theory for a class of nonlinear stochastic partial differential equations has recently been provided in [1]. This contribution will discuss the ``ingredients'' necessary for this methodology on an intuitive level. In particular, the required mathematical properties of the system are related to their physical counterparts. The results are applied to stochastic single-layer and two-layer quasi-geostrophic models which are popular in the geosciences to study atmosphere and ocean dynamics.

[1] G. Carigi, T. Kuna and J. Bröcker, Linear and fractional response for nonlinear dissipative SPDEs, arXiv, doi = 10.48550/ARXIV.2210.12129, 2022.

How to cite: Broecker, J., Carigi, G., and Kuna, T.: Linear response for stochastic models of geophysical fluid dynamics with medium complexity, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-9798, https://doi.org/10.5194/egusphere-egu23-9798, 2023.

The mathematical stochastic energy balance models (SEBMs) pioneered by Hasselmann and Mitchell  have long been known to climate scientists to be important aids to gaining both qualitative insight and quantitative information about global mean temperatures. SEBMs are now much more widely visible, after the award of the 2021 Physics Nobel Prize to Hasselmann,  Manabe and Parisi. The earliest univariate SEBMs were, however, built around the simplest linear and Markovian stochastic process, enabling Hasselmann and his successors to exploit their equivalence to the Langevin equation of 1908. Multivariate SEBMs have now been extensively studied  but this presentation focuses on the continuing value of univariate SEBMs, especially when coupled to economic models, or when used to study longer-ranged memory than the exponential type seen in Hasselmann's Markovian case.

I will highlight how we and others are now going beyond the first SEBMs to incorporate more general temporal dependence, motivated by increasing evidence of non-Markovian, and in particular long-ranged, memory in the climate system. This effort has brought new and interesting challenges, both in mathematical methods and physical interpretation. I will highlight our recent paper [Calel et al, Nature Communications, 2021] on using a Markovian Hasselmann-type EBM to study the economic impacts of climate change and variability and our other ongoing work on generalisations (in particular fractional ones) of Hasselmann SEBMs.

This presentation updates our preprints [Watkins et al, arXiv; Watkins et al, in preparation for submission to Chaos] to show how the overdamped generalised Langevin equation can be mapped onto an SEBM that generalises Lovejoy et al's FEBE and I will give a progress report on this work. I will also briefly discuss  the relation of such non-Markovian SEBMs to fluctuation-dissipation relations.

How to cite: Watkins, N. W., Calel, R., Chapman, S., Chechkin, A., Ford, I., Klages, R., and Stainforth, D.: Progess in non-Markovian (and Fractional) StochasticClimate Modelling: A GLE-based perspective, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-9433, https://doi.org/10.5194/egusphere-egu23-9433, 2023.

Many systems in nature are characterized by the coexistence of different stable states for a given set of environmental parameters and external forcing. Examples for such behavior can be found in different fields of Earth system, e.g. ecosystems and climate dynamics. As a consequence of the coexistence of a multitude of stable states, the final state of the system depends strongly on the initial condition.  The set of initial conditions which all converge to the same stable state is called the basin of attraction. In addition, the dynamics of complex systems is often characterized by the different time scales on which certain processes act. We show that the interplay of these different time scales is important particularly for the case of rate-induced tipping. This tipping phenomenon occurs when the rate of change of an internal parameter or an external forcing is varying on a different timescale as the intrinsic timescale of the system.  The system can track its original stable state under such time-dependent forcing as long as the rate of environmental change is slow. If this rate is larger than a critical rate the system will tip and obey a rather different dynamical behavior, either by approaching a different stable state or by visiting temporarily different parts of the state space.  We study the role of the relative size of the basins of attraction and the location of their boundaries in rate-induced tipping and demonstrate that the decision whether a trajectory tips or tracks the original stable state depends crucially on the changes in the basins of attraction, in particular their size and, more importantly on their boundaries, that also “move” in state space under a time-dependent variation of intrinsic parameters/external forcing.  This dependence is discussed for the two cases of smooth basin boundaries made up by the stable manifolds of saddle points and fractal basin boundaries where chaotic saddles embedded in the boundary influence the tipping of trajectories. 

How to cite: Feudel, U.: The role of multiple time scales for rate-induced tipping phenomena, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-15095, https://doi.org/10.5194/egusphere-egu23-15095, 2023.

Many components of the Earth system are thought to be prone to dangerous transitions, presenting a big challenge for human societies. Known as tipping elements, those form an intricate network of interacting subsystems, creating the possibility of cascading critical transitions. The presence of those interacting tipping events makes it hard to predict the outcome of climate change. In this research, we investigate those phenomena above the usual approach of linearly interacting bistable components.

We propose to study generic nonlinear systems under generic nonlinear interaction. As a first step, we focus on unilaterally coupled components, where a leading and a following subsystem are naturally identified. Using singular perturbation methods, we show how the stability landscape can be approached semi-analytically when considering the weak and strong coupling limit. With only limited knowledge about the system's structure, this method applies to a wide class of interacting systems and allows for approaching steady states with a controlled error. This provides information on important structural features of the bifurcation diagram such as the presence of steady branches, their stability, and bifurcations. Finally, we illustrate our results using climate relevant conceptual models.

How to cite: Sinet, S., Kuehn, C., Bastiaansen, R., von der Heydt, A. S., and Dijkstra, H. A.: Cascading Transitions in the Weak and Strong Coupling Limit, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-3509, https://doi.org/10.5194/egusphere-egu23-3509, 2023.

The climate system can be regarded as a non-equilibrium dynamical system that relaxes toward a steady state under the continuous input of solar radiation and dissipative mechanisms over a multitude of temporal and spatial scales. The steady state is not necessarily unique. A useful tool to describe the possible steady states of the climate system is the bifurcation diagram, where the long-term behaviour of a state variable (like surface air temperature) is plotted as a function of force intensity. This diagram reveals the regions of multi-stability, the position of B-tipping (bifurcation points at critical forcing values giving rise to an abrupt and irreversible climate change), the range of stability of each attractor and the intensity of climate variability needed to induce transitions between states (N-tipping).

The construction of the bifurcation diagram requires to run long simulations from a huge ensemble of initial conditions until convergence to a steady state is attained (standard method). This procedure has prohibitive computational costs in general circulation models of the climate that include deep ocean dynamics relaxing on timescales of the order of thousand years, or other feedback mechanisms with even longer time scales, like continental ice or carbon cycle.

Using a coupled setup of the MIT general circulation model, we propose two techniques that require lower computational costs and show complementary advantages. We test them in a numerical setup that includes deep ocean dynamics and we compare the resulting bifurcation diagram with the one obtained with the standard method. The first technique is based on the introduction of random fluctuations in the forcing and allows one to explore a large part of the phase space. The second, based on the estimate of internal variability and relaxation time, is more precise in finding B-tipping.

How to cite: Brunetti, M. and Ragon, C.: Steady states in complex climate models and different methods for the construction of the bifurcation diagram, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-6502, https://doi.org/10.5194/egusphere-egu23-6502, 2023.

Oceanic flow comprises of a fast and a slow evolving component. Decomposing the flow field into these components is necessary to understand processes like mesoscale eddy dissipation and spontaneous wave emission. These processes are potentially important wave sources and lead to an energy transfer between the slow and the fast component. The first order approach is to decompose in geostrophic and non-geostrophic components. Since a part of the non-geostrophic component evolves slowly due to nonlinear interactions between both component, this approach is not precise enough to quantify energy transfers. To obtain higher accuracy in decomposing the flow field, more precise methods are required, such as optimal balance or nonlinear normal mode decomposition. However, their application is limited to idealized model settings that neither include topography nor a varying Coriolis parameter. Here, we modified the optimal balance method with a time averaging procedure, such that it is applicable in more realistic ocean models. We compared the new modified method with existing methods in a shallow water model and in a non-hydrostatic model. For longer time averaging periods, the modified optimal balance method converges against the original method.

How to cite: Rosenau, S., Chouksey, M., and Eden, C.: On flow decomposition in realistic ocean models, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-8133, https://doi.org/10.5194/egusphere-egu23-8133, 2023.

Parametrizing physical processes in the ocean is a universal approach to overcome resolution limitations across different scales. Parametrizations represent the mean effect of processes occurring on the scales less than the grid scale (i.e. on the subgrid) on the resolved mean flow through parametric equations. In this work, a viscous momentum closure, equipped with a backscatter operator that returns excessively dissipated energy back to the system, is used to parametrize mesoscale range processes on eddy-permitting mesh resolutions.

The part of the variability that is not represented by the deterministic backscatter operator is modelled stochastically. We propose a stochastic field component, based on the patterns of variability extracted from the output of model simulations with different grid resolutions.

As a continuation of this work, we propose an interaction of the backscatter parametrization with the Gent-McWilliams parametrization which is generally applied for coarser grids corresponding to non-eddy resolving resolutions. This connection is relevant to link kinetic and potential energy backscatter.

The implementations are tested on two intermediate complexity setups of the global ocean model FESOM2: a doubly-periodic channel and a double-gyre box model. In this contribution, we present an increase of eddy activity and show that a greater complexity of setup enhances response to the implementation.  

Keywords: mesoscale eddies, parametrization, backscatter, stochastic parametrization, GM parametrization.

How to cite: Bagaeva, E., Juricke, S., Danilov, S., and Oliver, M.: Towards a subgrid momentum closure via stochastic backscatter and its linkages with the Gent-McWilliams parametrization, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-15627, https://doi.org/10.5194/egusphere-egu23-15627, 2023.