Taking inspiration from the Mathematics of Planet Earth 2013 initiative, this session aims at bringing together contributions from the growing interface between the geophysical, the mathematical, and the theoretical physical communities. Specific topics include: PDEs, numerical methods, extreme events, statistical mechanics, pattern formation and emergence, (random and non-autonomous ) dynamical systems, large deviation theory, response theory, tipping points, model reduction techniques, coarse graining, stochastic processes, parametrizations, data assimilation, and thermodynamics. We invite talks and poster both related to specific applications as well as more speculative and theoretical investigations. We particularly encourage early career researchers to present their interdisciplinary work in this session.

Co-organized by AS4/CL4/NH1
Convener: Valerio Lucarini | Co-conveners: Peter Ashwin, Niklas Boers, Vera Melinda Galfi, Michel Crucifix, Hansjoerg Seybold, Piotr Szymczak
| Attendance Mon, 04 May, 14:00–18:00 (CEST)

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Download all presentations (183MB)

Chat time: Monday, 4 May 2020, 14:00–15:45

D2921 |
| solicited
Georg Gottwald and Caroline Wormell

The long-term average response of observables of chaotic systems to dynamical perturbations can often be predicted using linear response theory, but not all chaotic systems possess a linear response. Macroscopic observables of complex dissipative chaotic systems, however, are widely assumed to have a linear response even if the microscopic variables do not, but the mechanism for this is not well-understood.

We present a comprehensive picture for the linear response of macroscopic observables in high-dimensional coupled deterministic dynamical systems, where the coupling is via a mean field and the microscopic subsystems may or may not obey linear response theory. We derive stochastic reductions of the dynamics of these observables from statistics of the microscopic system, and provide conditions for linear response theory to hold in finite dimensional systems and in the thermodynamic limit. In particular, we show that for large systems of finite size, linear response is induced via self-generated noise.

We present examples in the thermodynamic limit where the macroscopic observable satisfies LRT, although the microscopic subsystems individually violate LRT, as well a converse example where the macroscopic observable does not satisfy LRT despite all microscopic subsystems satisfying LRT when uncoupled. This latter, maybe surprising, example is associated with emergent non-trivial dynamics of the macroscopic observable. We provide numerical evidence for our results on linear response as well as some analytical intuition.

How to cite: Gottwald, G. and Wormell, C.: Linear response theory for macroscopic observables in high-dimensional systems: when is it valid and when not?, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-1670, https://doi.org/10.5194/egusphere-egu2020-1670, 2020.

D2922 |
Francesco Ragone and Freddy Bouchet

Extreme events are a major topic of interest in climate science. Studying rare extreme events with complex numerical climate models is computationally challenging, since in principle very long simulations are needed to sample a sufficient number of events to provide a reliable statistics. This problem can be tackled using rare event algorithms, numerical tools developed in the past decades in mathematics and statistical physics, dedicated to the reduction of the computational effort required to sample rare events in dynamical systems. Typically they are designed as genetic algorithms, in which a set of suppression and cloning rules are applied to an ensemble simulation in order to focus the computational effort on the trajectories leading to the events of interest. Recently we showed the great potential of rare event algorithms for climate modelling, applying a rare event algorithm to study extremes of European surface temperature in Plasim, an intermediate complexity model, in absence of external time dependent forcings (no seasonal and daily cycles). Here we go beyond these limitations, studying extreme heat waves and warm summers in the Northern extratropics in fully realistic conditions including daily and seasonal cycles, both in Plasim and in the state of the art Earth system model CESM. We show how the algorithm allows to sample extreme events characterised by persistency on different time scales, discussing links with large deviation theory. We show how one can characterise the statistics of heat waves and warm summers with extremely large return times, with computational costs orders of magnitude smaller than with direct sampling, and reach ultra rare events that would have been impossible to observe otherwise. We analyse the emergence of teleconnection patterns during the extreme events and their relation to the dynamics of planetary waves. Finally we discuss how these results open the way to the systematic application of these techniques to a vast range of applicative and theoretical studies.

How to cite: Ragone, F. and Bouchet, F.: Studying heat waves and warm summers in numerical climate models with a rare event algorithm, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-19177, https://doi.org/10.5194/egusphere-egu2020-19177, 2020.

D2923 |
Jakob Runge

Detecting causal relationships from observational time series datasets is a key problem in better understanding the complex dynamical system Earth. Recent methodological advances have addressed major challenges such as high-dimensionality and nonlinearity, e.g., PCMCI (Runge et al. Sci. Adv. 2019), but many more remain. In this talk I will give an overview of challenges and methods and present a novel algorithm to identify causal directions among contemporaneous (or instantaneous) relationships. Such contemporaneous relations frequently appear when time series are aggregated (e.g., at a monthly resolution). Then approaches such as Granger Causality and PCMCI fail because they currently only address time-lagged causal relations.
We present extensive numerical examples and results on the causal relations among major climate modes of variability. The work overcomes a major drawback of current causal discovery methods and opens up entirely new possibilities to discover causal relations from time series in climate research and other fields in geosciences.

Runge et al., Detecting and quantifying causal associations in large nonlinear time series datasets, Science Advances eeaau4996 (2019).

How to cite: Runge, J.: Recent progress and new methods for detecting causal relations in large nonlinear time series datasets, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-9554, https://doi.org/10.5194/egusphere-egu2020-9554, 2020.

D2924 |
Lea Oljaca, Jochen Broecker, and Tobias Kuna

Data assimilation is a term used to describe efforts to improve our knowledge
of a system by combining incomplete observations with imperfect models.
This is more generally known as filtering, which is ’optimal’ estimation of
the state of a system as it evolves over time, in the mean square sense. In
a Bayesian framework, the optimal filter is therefore naturally a sequence of
conditional probabilities of a signal given the observations and can be up-
dated recursively with new observations with Bayes’ formula. When, the
dynamics and observations errors are linear, this is equivalent to the Kalman
filter. In the nonlinear case, deriving an explicit form for the posterior dis-
tribution is in general not possible.
One of the important difficulties with applying the nonlinear filter in practice
is that the initial condition, the prior, is required to initialise the filtering.
However we are unlikely to know the correct initial distribution accurately
or at all. A filter is called stable if it is insensitive with respect to the
prior, that is, it converges to the same distribution, regardless of the initial
A body of work exists showing stability of the filter which rely on the stochas-
ticity of the underlying dynamics. In contrast, we show stability of the op-
timal filter for a class of nonlinear and deterministic dynamical systems and
our result relies on the intrinsic chaotic properties of the dynamics. We build
on the considerable knowledge that exists on the existence of SRB measures
in uniformly hyperbolic dynamical systems and we view the conditional prob-
abilities as SRB measures ‘conditional on the observation’ which are shown
to be absolutely continuous along the unstable manifold. This is in line with
the result of Bouquet, Carrassi et al [1] regarding data assimilation in the
“unstable subspace”, where they show stability of the filter if the unstable
and neutral subspaces are uniformly observed.

[1] M. Bocquet et al. “Degenerate Kalman Filter Error Covariances and
Their Convergence onto the Unstable Subspace”. In: SIAM/ASA Jour-
nal on Uncertainty Quantification 5.1 (2017), pp. 304–333. url: https:

How to cite: Oljaca, L., Broecker, J., and Kuna, T.: Insensitivety to initial condition/prior in data assimilation for the case of the optimal filter and deterministic model, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-19640, https://doi.org/10.5194/egusphere-egu2020-19640, 2020.

D2925 |
| Highlight
Julien Brajard, Alberto Carrassi, Marc Bocquet, and Laurent Bertino

Can we build a machine learning parametrization in a numerical model using sparse and noisy observations?

In recent years, machine learning (ML) has been proposed to devise data-driven parametrizations of unresolved processes in dynamical numerical models. In most of the cases, ML is trained by coarse-graining high-resolution simulations to provide a dense, unnoisy target state (or even the tendency of the model).

Our goal is to go beyond the use of high-resolution simulations and train ML-based parametrization using direct data. Furthermore, we intentionally place ourselves in the realistic scenario of noisy and sparse observations.

The algorithm proposed in this work derives from the algorithm presented by the same authors in https://arxiv.org/abs/2001.01520.The principle is to first apply data assimilation (DA) techniques to estimate the full state of the system from a non-parametrized model, referred hereafter as the physical model. The parametrization term to be estimated is viewed as a model error in the DA system. In a second step, ML is used to define the parametrization, e.g., a predictor of the model error given the state of the system. Finally, the ML system is incorporated within the physical model to produce a hybrid model, combining a physical core with a ML-based parametrization.

The approach is applied to dynamical systems from low to intermediate complexity. The DA component of the proposed approach relies on an ensemble Kalman filter/smoother while the parametrization is represented by a convolutional neural network.  

We show that the hybrid model yields better performance than the physical model in terms of both short-term (forecast skill) and long-term (power spectrum, Lyapunov exponents) properties. Sensitivity to the noise and density of observation is also assessed.

How to cite: Brajard, J., Carrassi, A., Bocquet, M., and Bertino, L.: Data-driven parametrizations in numerical models using data assimilation and machine learning., EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-13794, https://doi.org/10.5194/egusphere-egu2020-13794, 2020.

D2926 |
Giulia Carigi, Jochen Bröcker, and Tobias Kuna

In the Climate Sciences, there is great interest in understanding the long term average behaviour of the climate system. In the context of climate models, this behaviour can be expressed intrinsically by concepts from the theory of dynamical systems such as attractors and invariant measures. In particular to ensure long term statistics of the model are well defined from a mathematical perspective, the model needs to admit a unique ergodic invariant probability measure.

In this work we consider a classic two layer quasi geostrophic model, with the upper layer perturbed by additive noise, white in time and coloured in space, in order to account for random forcing, for instance through wind shear. Existence and uniqueness of an ergodic invariant measure is established using a technique from stochastic analysis called asymptotic coupling as described in [1]. The main difficulty in the proof is to show that two copies of the system that are driven by the same noise realisation can be synchronised through a coupling. This coupling has to be finite dimensional and act only on the upper layer. 

Our results will be a key step to allow rigorous investigation of the response theory for the system under concern.


[1] Glatt-Holtz, N., Mattingly, J.C. & Richards, G. J Stat Phys (2017) 166: 618.  

How to cite: Carigi, G., Bröcker, J., and Kuna, T.: Ergodicity of a stochastic Two Layer Quasi Geostrophic Model, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-17313, https://doi.org/10.5194/egusphere-egu2020-17313, 2020.

D2927 |
| solicited
| Highlight
Michael Ghil, Gisela D. Charó, Denisse Sciamarella, and Mickael D. Chekroun

Chekroun et al. (Physica D, 240, 2011) studied the global random attractor associated with the Lorenz (1963) model driven by multiplicative noise; they dubbed this time-evolving attractor LORA for short. The present talk examines the topological structure of the snapshots that approximate LORA’s evolution in time. 

Sciamarella & Mindlin (Phys. Rev. Lett., 82, 1999; Phys. Rev. E, 64, 2001) introduced the methodology of Branched Manifold Analysis through Homologies (BraMAH) to the study of chaotic flows. Here, the BraMAH methodology is extended for the first time, to the best of our knowledge, from deterministically chaotic flows to nonlinear noise-driven systems. 

The BraMAH algorithm starts from a cloud of points given by a large number of orbits and it builds a rough skeleton of the underlying branched manifold on which the points lie. This construction is achieved by local approximations of the manifold that use Euclidean closed sets; the nature of these sets depends on their topological dimension, e.g., segments or disks.  The skeleton is mathematically expressed as a complex of cells, whose algebraic topology is analyzed by computing its homology groups. 

The analysis is performed for a fixed realization of the driving noise at different time instants. We show that the topology of LORA changes in time and that it is quite distinct from the time-independent one of the classical Lorenz (1963) convection model, for the same values of the parameters. Topological tipping points are also studied by varying the parameter values from the classical ones.

How to cite: Ghil, M., Charó, G. D., Sciamarella, D., and Chekroun, M. D.: The Lorenz convection model's random attractor (LORA) and its robust topology, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-10973, https://doi.org/10.5194/egusphere-egu2020-10973, 2020.

D2928 |
| solicited
Manita Chouksey

Geophysical flows such as the atmosphere and the ocean are characterized by rotation and stratification, which together give rise to two dominant motions: the slow balanced and the fast unbalanced motions. The interaction between the balanced and unbalanced motions and the energy transfers between them impact the energy and momentum cycle of the flow, and is therefore crucial to understand the underlying energetics of the atmosphere and the ocean. Balanced motions, for instance mesoscale eddies, can transfer their energy to unbalanced motions, such as internal gravity waves, by spontaneous loss of balance amongst other processes. The exact mechanism of wave generation, however, remain less understood and is hindered to an extent by the challenge of separating the flow field into balanced and unbalanced motions.

This separation is achieved using two different balancing procedures in an identical model setup and assess the differences in the obtained balanced state and the resultant energy transfer to unbalanced motions. The first procedure we implement is a non-linear initialisation procedure based on Machenhauer (1977) but extended to higher orders in Rossby number. The second procedure implemented is the optimal potential vorticity balance to achieve the balanced state. The results show that the numerics of the model affect the obtained balanced state from the two procedures, and thus the residual signal which we interpret as the unbalanced motions, i.e. internal gravity waves.  A further complication is the presence of slaved modes, which appear along the unbalanced motions but are tied to the balanced motions, for which we need to extend the separation to higher orders in Rossby number. Further, we assess the energy transfers between balanced and unbalanced motions in experiments with different Rossby numbers and for different orders in Rossby number. We find that it is crucial to consider the effect of the numerics in models and make a suitable choice of the balancing procedure, as well as diagnose the unbalanced motions at higher orders to precisely detect the unbalanced wave signal.

How to cite: Chouksey, M.: Energy Transfers Between Balanced and Unbalanced Motions in Geophysical Flows, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-9994, https://doi.org/10.5194/egusphere-egu2020-9994, 2020.

D2929 |
| Highlight
Maximilian Gelbrecht, Jürgen Kurths, and Frank Hellmann

Many high-dimensional complex systems such as climate models exhibit an enormously complex landscape of possible asymptotic state. On most occasions these are challenging to analyse with traditional bifurcation analysis methods. Often, one is also more broadly interested in classes of asymptotic states. Here, we present a novel numerical approach prepared for analysing such high-dimensional multistable complex systems: Monte Carlo Basin Bifurcation Analysis (MCBB).  Based on random sampling and clustering methods, we identify the type of dynamic regimes with the largest basins of attraction and track how the volume of these basins change with the system parameters. In order to due this suitable, easy to compute, statistics of trajectories with randomly generated initial conditions and parameters are clustered by an algorithm such as DBSCAN. Due to the modular and flexible nature of the method, it has a wide range of possible applications. While initially oscillator networks were one of the main applications of this methods, here we present an analysis of a simple conceptual climate model setup up by coupling an energy balance model to the Lorenz96 system. The method is available to use as a package for the Julia language. 

How to cite: Gelbrecht, M., Kurths, J., and Hellmann, F.: Analysing Conceptual Climate Models with Monte Carlo Basin Bifurcation Analysis, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-18301, https://doi.org/10.5194/egusphere-egu2020-18301, 2020.

D2930 |
Shaun Lovejoy, Lenin Del Rio Amador, and Roman Procyk

The conventional 1-D energy balance equation (EBE) has no vertical coordinate so that radiative imbalances between the earth and outer space are redirected in the horizontal in an ad hoc manner.  We retain the basic EBE but add a vertical coordinate so that the imbalances drive the system by imposing heat fluxes through the surface.   While this is theoretically correct, it leads to (apparently) difficult mixed boundary conditions.  However, using Babenko’s method, we directly obtain simple analytic equations for (2D) surface temperature anomalies Ts(x,t): the Half-order Energy Balance Equation (HEBE) and the Generalized HEBE, (GHEBE) [Lovejoy, 2019a].  The HEBE anomaly equation only depends on the local climate sensitivities and relaxation times.  We analytically solve the HEBE and GHEBE for Ts(x,t) and provide evidence that the HEBE applies at scales >≈10km.  We also calculate very long time diffusive transport dominated climate states as well as space-time statistics including the cross-correlation matrix needed for empirical orthogonal functions.

The HEBE is the special H = 1/2 case of the Fractional EBE (FEBE) [Lovejoy, 2019b], [Lovejoy, 2019c] and has a long (power law) memory up to its relaxation time t.  Several papers have empirically estimated H ≈ 0.5, as well as t ≈ 4 years, whereas the classical zero-dimensional EBE has H = 1 and t ≈ 4 years.   The former values permit accurate macroweather forecasts and low uncertainty climate projections; this suggests that the HEBE could apply to time scales as short as a month.  Future generalizations include albedo-temperature feedbacks and more realistic treatments of past and future climate states.



Lovejoy, S., The half-order energy balance equation, J. Geophys. Res. (Atmos.), (submitted, Nov. 2019), 2019a.

Lovejoy, S., Weather, Macroweather and Climate: our random yet predictable atmosphere, 334 pp., Oxford U. Press, 2019b.

Lovejoy, S., Fractional Relaxation noises, motions and the stochastic fractional relxation equation Nonlinear Proc. in Geophys. Disc., https://doi.org/10.5194/npg-2019-39, 2019c.

How to cite: Lovejoy, S., Del Rio Amador, L., and Procyk, R.: Correcting Budyko-Sellers boundary conditions: The Half-order Energy Balance Equation (HEBE), EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-11557, https://doi.org/10.5194/egusphere-egu2020-11557, 2020.

D2931 |
Michael Coughlan, Ian Hewitt, Sam Howison, and Andrew Wells

Arctic sea ice forms a thin but significant layer at the ocean surface, mediating key climate feedbacks. During summer, surface melting produces considerable volumes of water, which collect on the ice surface in ponds. These ponds have long been suggested as a contributing factor to the discrepancy between observed and predicted sea ice extent. When viewed at large scales ponds have a complicated, approximately fractal geometry and vary in area from tens to thousands of square meters. Increases in pond depth and area lead to further increases in heat absorption and overall melting, contributing to the ice-albedo feedback.

Previous modelling work has focussed either on the physics of individual ponds or on the statistical behaviour of systems of ponds. We present a physically-based network model for systems of ponds which accounts for both the individual and collective behaviour of ponds. Each pond initially occupies a distinct catchment basin and evolves according to a mass-conserving differential equation representing the melting dynamics for bare and water-covered ice. Ponds can later connect together to form a network with fluxes of water between catchment areas, constrained by the ice topography and pond water levels.

We use the model to explore how the evolution of pond area and hence melting depends on the governing parameters, and to explore how the connections between ponds develop over the melt season. Comparisons with observations are made to demonstrate the ways in which the model qualitatively replicates properties of pond systems, including fractal dimension of pond areas and two distinct regimes of pond complexity that are observed during their development cycle. Different perimeter-area relationships exist for ponds in the two regimes. The model replicates these relationships and exhibits a percolation transition around the transition between these regimes, a facet of pond behaviour suggested by previous studies. Our results reinforce the findings of these studies on percolation thresholds in pond systems and further allow us to constrain pond coverage at this threshold - an important quantity in measuring the scale and effects of the ice-albedo feedback.

How to cite: Coughlan, M., Hewitt, I., Howison, S., and Wells, A.: Network models for ponding on sea ice, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-10021, https://doi.org/10.5194/egusphere-egu2020-10021, 2020.

D2932 |
Michael Berhanu, Raphael Dubourg, Arthur Walbecq, Cyril Ozouf, Adrien Guerin, Julien Derr, and Sylvain Courrech du Pont

Erosion by dissolution is a decisive process shaping small-scale landscape morphology [1]. For fast dissolving minerals, the erosion rate is controlled by the solute transport [2] and characteristic erosion patterns can appear due to hydrodynamics mechanisms. Among the diversity of the dissolution patterns, the scallops are small depressions in a dissolving wall, appearing as cups with sharp edges. Their size varies from few millimeters to around ten centimeters. The scallops occur typically as the final steady form of ripple patterns created by the action of a turbulent flow on a dissolving surface [3,4]. Moreover, very similar shapes are also met, without imposed external flow, when the fluid motion results from the solutal convection induced by the dissolution [2,5,6]. Finally, scallop patterns resulting from similar mechanisms appear also on ice surfaces by melting in presence of a turbulent flow [7] or a convection flow [6].
Using three-dimensional surface reconstruction, we characterize quantitatively the scallop patterns mainly for experimental samples patterned by solutal convection. The temporal evolution of the scallop shape, of their spatial distribution and of the induced roughness are specifically investigated, in order to determine mechanisms explaining the generic aspects of dissolution patterns.

[1] P. Meakin and B. Jamtveit, Geological pattern formation by growth and dissolution in aqueous systems, Proc. R. Soc. A 466 659-694 (2010)

[2] J. Philippi, M. Berhanu, J. Derr and S. Courrech du Pont, Solutal convection induced by dissolution, Phys. Rev. Fluids, 4, 103801 (2019)

[3] P.N. Blumberg and R.L. Curl, Experimental and theoretical studies of dissolution roughness,  J. Fluid Mech. 65, 735 (1974)

[4] P. Claudin, O. Durán and B. Andreotti, Dissolution instability and roughening transition,  J. Fluid Mech. 832, R2  (1974)

[5] T.S. Sullivan, Y. Liu and R. E. Ecke, Turbulent solutal convection and surface patterning in solid dissolution, Phys. Rev. E 54, (1) 486, (1996)

[6] C. Cohen, M. Berhanu, J. Derr and S. Courrech du Pont, Erosion patterns on dissolving and melting bodies (2015 Gallery of Fluid motion), Phys. Rev. Fluids, 1, 050508 (2016)

[7] M. Bushuk, D. M. Holland, T. P. Stanton, A. Stern and C. Gray. Ice scallops: a laboratory investigation of the Ice-water interface, J. Fluid Mech. 873, 942 (2019)

How to cite: Berhanu, M., Dubourg, R., Walbecq, A., Ozouf, C., Guerin, A., Derr, J., and Courrech du Pont, S.: Morphology of scallop patterns in erosion by dissolution, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-8689, https://doi.org/10.5194/egusphere-egu2020-8689, 2020.

D2933 |
Pauline Delorme, Giles Wiggs, Matthew Baddock, Joanna Nield, James Best, Kenneth Christensen, Nathaniel Bristow, Andrew Valdez, and Philippe Claudin

Early-stage aeolian bedforms develop into sand dunes through complex interactions between flow, sediment transport and surface topography. Depending on the specific environmental and wind conditions the mechanisms of dune formation, and ultimately the shape of the nascent dunes, may differ. Here, we investigate the formation of a proto-dune-field, located in the Great Sand Dunes National Park ( Colorado, USA), using a three dimensional linear stability analysis.

We use in-situ measurements of wind and sediment transport, collected during a one-month field campaign, as part of a linear stability analysis to predict the orientation and wavelength of the proto-dunes.

We find that the output of the linear stability analysis compares well to high-resolution Digital Elevation Models measured using terrestrial laser scanning. Our findings suggest that the bed instability mechanism is a good predictor of proto-dune development on sandy surfaces with a bimodal wind regime.

How to cite: Delorme, P., Wiggs, G., Baddock, M., Nield, J., Best, J., Christensen, K., Bristow, N., Valdez, A., and Claudin, P.: Proto-dune formation under a bimodal wind regime, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-182, https://doi.org/10.5194/egusphere-egu2020-182, 2020.

D2934 |
Milad Hooshyar, Sara Bonetti, Arvind Singh, Efi Foufoula-Georgiou, and Amilcare Porporato

The channelization cascade observed in terrestrial landscapes describes the progressive formation of large channels from smaller ones starting from diffusion-dominated hillslopes. This behavior is reminiscent of other non-equilibrium complex systems, particularly fluids turbulence, where larger vortices break down into smaller ones until viscous dissipation dominates. Based on this analogy, we show that topographic surfaces emerging between parallel zero-elevation boundaries present a logarithmic scaling in the mean-elevation profile, which resembles the well-known logarithmic velocity profile in wall-bounded turbulence. Within this region of elevation fluctuation, the power spectrum exhibits a power-law decay resembling the Kolmogorov -5/3 scaling of turbulence. We also demonstrate that similar scaling behaviors emerge in surfaces from a laboratory experiment, natural basins, and constructed following optimality principles. In general, we show that the steady-state solutions of the governing equations of landscape evolution are the stationary surfaces of a functional defined as the average domain elevation. Depending on the exponent of the specific drainage area in the erosion term (m), the steady-state surfaces are local minimum (m<1) or maximum (m>1) of the average domain elevation.

How to cite: Hooshyar, M., Bonetti, S., Singh, A., Foufoula-Georgiou, E., and Porporato, A.: Optimality in landscape channelization and analogy with turbulence, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-11309, https://doi.org/10.5194/egusphere-egu2020-11309, 2020.

D2935 |
Manuel Santos Gutiérrez and Valerio Lucarini

Dynamical systems are often subject to forcing or changes in their governing parameters and it is of interest to study

how this affects their statistical properties. A prominent real-life example of this class of problems is the investigation

of climate response to perturbations. In this respect, it is crucial to determine what the linear response of a system is

as a quantification of sensitivity. Alongside previous work, here we use the transfer operator formalism to study the

response and sensitivity of a dynamical system undergoing perturbations. By projecting the transfer operator onto a

suitable finite dimensional vector space, one is able to obtain matrix representations which determine finite Markov

processes. Further, using perturbation theory for Markov matrices, it is possible to determine the linear and nonlinear

response of the system given a prescribed forcing. Here, we suggest a methodology which puts the scope on the

evolution law of densities (the Liouville/Fokker-Planck equation), allowing to effectively calculate the sensitivity and

response of two representative dynamical systems.

How to cite: Santos Gutiérrez, M. and Lucarini, V.: Response and Sensitivity Using Markov Chains, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-4854, https://doi.org/10.5194/egusphere-egu2020-4854, 2020.

D2936 |
Valerio Lembo, Valerio Lucarini, and Francesco Ragone

Global Climate Models are key tools for predicting the future response of the climate system to a variety of natural and anthropogenic forcings. Typically, an ensemble of simulations is performed considering a scenario of forcing, in order to analyse the response of the climate system to the specific forcing signal. Given that the the climate response spans a very large range of timescales, such a strategy often requires a dramatic amount of computational resources. In this paper we show how to use statistical mechanics to construct operators able to flexibly predict climate change for a variety of climatic variables of interest, going beyond the limitation of having to consider specific time patterns of forcing. We perform our study on a fully coupled GCM - MPI-ESM v.1.2 - and for the first time we prove the effectiveness of response theory in predicting future climate response to CO2 increase on a vast range of temporal scales. We specifically treat atmospheric  (surface temperature) and oceanic variables (strength of the Atlantic Meridional Overturning Circulation and of the Antarctic Circumpolar Current), as well as the global ocean heat uptake.

How to cite: Lembo, V., Lucarini, V., and Ragone, F.: Predicting Climate Change through Response Operators in a Coupled General Circulation Model, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-9174, https://doi.org/10.5194/egusphere-egu2020-9174, 2020.

D2937 |
Hege-Beate Fredriksen

We investigate a new algorithm for estimating time-evolving global forcing in climate models. The method is an extension of a previous method by Forster et al. (2013), but here we also allow for a globally nonlinear feedback. We assume the nonlinearity of this global feedback can be explained as a time-scale dependence, associated with linear temperature responses to the forcing on different time scales, as in Proistosescu and Huybers (2017). With this method we obtain stronger forcing estimates than previously believed for the representative concentration pathway experiments in CMIP5 models. The reason for the higher future forcing is that the global feedback has a higher magnitude at the smaller time scales than at the longer time scales – this is closely related to provided arguments for the equilibrium climate sensitivity showing changes with time.

We find also that the linear temperature response to our new forcing predicts the modelled response quite well, although the response is a little overestimated for some CMIP5 models. Finally, we discuss the assumptions made in our study, and consistency of our assumptions with other leading hypotheses for why the global feedback is nonlinear.



Forster, P. M., T. Andrews, P. Good, J. M. Gregory, L. S. Jackson, and M. Zelinka (2013), Evaluating adjusted forcing and model spread for historical and future scenarios in the cmip5 generation of climate models, Journal of Geophysical Research, 118, 1139–1150, doi:10.1002/jgrd.50174.

Proistosescu, C., and P. J. Huybers (2017), Slow climate mode reconciles historical and model-based estimates of climate sensitivity, Sci. Adv., 3, e1602, 821, doi:10.1126/sciadv.1602821

How to cite: Fredriksen, H.-B.: Effective forcing in CMIP5 assuming nonconstant feedback parameter and linear response, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-17967, https://doi.org/10.5194/egusphere-egu2020-17967, 2020.

D2938 |
Chiara Cecilia Maiocchi, Valerio Lucarini, Andrey Gritsun, and Grigorios Pavliotis

Unstable periodic orbits (UPOs) have been proved to be a relevant mathematical tool in the study of Climate Science. In a recent paper Lucarini and Gritsun [1] provided an alternative approach for understanding the properties of the atmosphere. Climate can be interpreted as a non-equilibrium steady state system and, as such, statistical mechanics can provide us with tools for its study.

UPOs decomposition plays a relevant role in the study of chaotic dynamical systems. In fact, UPOs densely populate the attractor of a chaotic system, and can therefore be thought as building blocks to construct the dynamic of the system itself. Since they are dense in the attractor, it is always possible to find a UPO arbitrarily near to a chaotic trajectory: the trajectory will remain close to the UPO, but it will never follow it indefinitely, because of its instability. Loosely speaking, a chaotic trajectory is repelled between neighbourhoods of different UPOs and can thus be approximated in terms of these periodic orbits. The characteristics of the system can then be reconstructed from the full set of periodic orbits in this fashion.

The sampling of UPOs is therefore a relevant problem for describing chaotic dynamical systems and can represent an interesting topic for the study of Climate Science. In this work we address this problem and present an algorithm to numerically extract UPOs from the attractor of a simple Climate Model such as Lorenz-63.

[1] V. Lucarini and A. Gritsun, “A new mathematical framework for atmospheric blocking events,” Climate Dynamics, vol. 54, pp. 575–598, Jan 2020.

How to cite: Maiocchi, C. C., Lucarini, V., Gritsun, A., and Pavliotis, G.: Unstable Periodic Orbits Sampling in Climate Models , EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-18823, https://doi.org/10.5194/egusphere-egu2020-18823, 2020.

D2939 |
Michel Crucifix, Dmitri Alexandrov, irina Bashkirtseva, and Lev Ryashko

Glacial-interglacial cycles are global climatic changes which have characterised the last 3 million years. The eight latest
glacial-interglacial cycles represent changes in sea level over 100 m, and their average duration was around 100 000 years. There is a
long tradition of modelling glacial-interglacial cycles with low-order dynamical systems. In one view, the cyclic phenomenon is caused by
non-linear interactions between components of the climate system: The dynamical system model which represents Earth dynamics has a limit cycle. In an another view, the variations in ice volume and ice sheet extent are caused by changes in Earth's orbit, possibly amplified by feedbacks.
This response and internal feedbacks need to be non-linear to explain the asymmetric character of glacial-interglacial cycles and their duration. A third view sees glacial-interglacial cycles as a limit cycle synchronised on the orbital forcing.

The purpose of the present contribution is to pay specific attention to the effects of stochastic forcing. Indeed, the trajectories
obtained in presence of noise are not necessarily noised-up versions of the deterministic trajectories. They may follow pathways which
have no analogue in the deterministic version of the model.  Our purpose is to
demonstrate the mechanisms by which stochastic excitation may generate such large-scale oscillations and induce intermittency. To this end, we
consider a series of models previously introduced in the literature, starting by autonomous models with two variables, and then three
variables. The properties of stochastic trajectories are understood by reference to the bifurcation diagram, the vector field, and a
method called stochastic sensitivity analysis.  We then introduce models accounting for the orbital forcing, and distinguish forced and
synchronised ice-age scenarios, and show again how noise may generate trajectories which have no immediate analogue in the determinstic model. 

How to cite: Crucifix, M., Alexandrov, D., Bashkirtseva, I., and Ryashko, L.: Nonlinear Climate Dynamics: from Deterministic Behavior to Stochastic Excitability and Chaos, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-11345, https://doi.org/10.5194/egusphere-egu2020-11345, 2020.

D2940 |
Nikolas Porz

The representation of cloud processes in weather and climate models is crucial for their feedback on atmospheric flows. Since there is no general macroscopic theory of clouds, the parameterization of clouds in corresponding simulation software depends fundamentally on the underlying modeling assumptions. We present a new model of intermediate complexity (a one-and-a-half moment scheme) for warm clouds, which is derived from physical principles. Our model consists of a system of differential-algebraic equations which allows for supersaturation and thus avoids the commonly used but somewhat outdated concept of so called 'saturation adjustment'. This is made possible by a non-Lipschitz right-hand side, which allows for nontrivial solutions. In a recent effort we have proved under mild assumptions on the external forcing that this system of equations has a unique physically consistent solution, i.e., a solution with a nonzero droplet population in the supersaturated regime. For the numerical solution of this system we have developed a semi-implicit integration scheme, with efficient solvers for the implicit parts. The model conserves air and water (if one accounts for the precipitation), and it comes with eight parameters that cannot be measured since they describe simplified processes, so they need to be fitted to the data. For further studies we implemented our cloud micro physics model into ICON, the weather forecast model operated by the German forecast center DWD.

How to cite: Porz, N.: Unique solvability of a system of ordinary differential equations modeling a warm cloud parcel and avoiding saturation adjustment, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-20030, https://doi.org/10.5194/egusphere-egu2020-20030, 2020.

D2941 |
Robert Malte Polzin, Annette Müller, Peter Nevir, Henning Rust, and Peter Koltai

The presented work contains an investigation of the stochastic aggregation of convective structures on different scales in the atmosphere. A
computational framework is applied that provides highly scalable identification of reduced Bayesian models. The deterministic large scale
flow variables are reduced into latent states, whereas the stochastic small scale convective structures are affiliated to these. The analysis of
the latent states in number and maximization reduction improves the understanding for the large scale forcing of convective processes. The
convective structures are determined by vertical velocities. Different variables of the large-scale flow, such as the convective available
potential energy, available moisture, vertical windshear and the Dynamic State Index (DSI), a diabaticity indicator, are investigated. Our approach
does not require a distributional assumption but works instead with a discretised and categorised state vector.

How to cite: Polzin, R. M., Müller, A., Nevir, P., Rust, H., and Koltai, P.: Reduced stochastic aggregation of convection conditioned by large scale dynamics in the atmosphere, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-22524, https://doi.org/10.5194/egusphere-egu2020-22524, 2020.

Chat time: Monday, 4 May 2020, 16:15–18:00

D2942 |
| Highlight
Jeroen Wouters, Vera Melinda Galfi, and Valerio Lucarini

We use large deviation theory to study persistent extreme events of temperature, like heat waves or cold spells. We consider the mid-latitudes of a simplified yet Earth-like general circulation model of the atmosphere and numerically estimate large deviation rate functions of near-surface temperature averages over different spatial scales. We find that, in order to represent persistent extreme events based on large deviation theory, one has to look at temporal averages of spatially averaged observables. The spatial averaging scale is crucial, and has to correspond with the scale of the event of interest. Accordingly, the computed rate functions indicate substantially different statistical properties of temperature averages over intermediate spatial scales (larger, but still of the order of the typical scale), as compared to the ones related to any other scale. Thus, heat waves (or cold spells) can be interpreted as large deviations of temperature averaged over intermediate spatial scales. Furthermore, we find universal characteristics of rate functions, based on the equivalence of temporal, spatial, and spatio-temporal rate functions if we perform a re-normalisation by the integrated auto-correlation.

How to cite: Wouters, J., Galfi, V. M., and Lucarini, V.: On the connection between heat waves and large deviations of temperature, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-7336, https://doi.org/10.5194/egusphere-egu2020-7336, 2020.

D2943 |
Tobias Kuna, Valerio Lucarini, Davide Faranda, Jerouen Wouters, and Viviane Baladi

Extremes are related to high impact and serious hazard events and hence their study and prediction have been and continue to be highly relevant for all kind of applications in geoscience and beyond. Extreme value theory is promising to be able to predict them reliably and robustly. In the last fifteen years the classical extreme value theory for stochastic processes has been extended to dynamical systems and has been related to properties of physical measure (statistical properties of the system), return and hitting times. We will review what one can say for highly dimensional perfectly chaotic systems.  We will concentrate on relations between the index of the extreme distribution and invariants of the underlying dynamical system which are stable, in the sense that they will continuously depend on changing parameters in the dynamics.  Furthermore, we explore whether there exists a response theory for extremes, that is, whether the change of extremes can be quantitatilvely expressed  in terms of changing parameters. 


How to cite: Kuna, T., Lucarini, V., Faranda, D., Wouters, J., and Baladi, V.: Extremes for high dimensional chaotic systems, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-16406, https://doi.org/10.5194/egusphere-egu2020-16406, 2020.

D2944 |
Paul Bowen

Approximations in the moist thermodynamics of atmospheric/weather models are often inconsistent. Different parts of numerical models may handle the thermodynamics in different ways, or the approximations may disagree with the laws of thermodynamics. In order to address these problems, we may derive all relevant thermodynamic quantities from a defined thermodynamic potential; approximations are then instead made to the potential itself — this guarantees self-consistency. This concept is viable for vapor and liquid water mixtures in a moist atmospheric system using the Gibbs function but on extension to include the ice phase an ambiguity presents itself at the triple-point. In order to resolve this the energy function must be utilised instead; constrained maximisation methods can then be used on the entropy in order to solve the system equilibrium state. Once this is done however, a further extension is necessary for atmospheric systems. In the Earth’s atmosphere many important non-equilibrium processes take place; for example, freezing of super-cooled water, evaporation, and precipitation. To fully capture these processes the equilibrium method must be reformulated to involve finite rates of approach towards equilibrium. This may be done using various principles of non-equilibrium thermodynamics, principally Onsager reciprocal relations. A numerical scheme may then be implemented which models competing finite processes in a moist thermodynamic system.

How to cite: Bowen, P.: Consistent Modelling of Non-Equilibrium Thermodynamic Processes in the Atmosphere, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-20739, https://doi.org/10.5194/egusphere-egu2020-20739, 2020.

D2945 |
Charline Ragon, Valerio Lembo, Valerio Lucarini, Jérôme Kasparian, and Maura Brunetti

The climate system can be seen as a thermal engine that generates entropy by irreversible processes and achieves a steady state by redistributing the input solar energy among its different components (ocean, atmosphere, etc) and by balancing the energy, water mass and entropy budgets over all the spatial scales. Biases in modern climate models are generally related to the fact that their statistical properties are not well represented, giving rise to imperfect closures of the energy cycle. Thus, a proper measurement of the efficiency of the thermal engine in each climate model is needed. Moreover, possible steady states (attractors) that can be approached at climate tipping-points are characterised by different feedbacks becoming dominant in the thermal engine.

We apply the Thermodynamic Diagnostic Tool (TheDiaTo) [1] to the attractors recently obtained using the MIT general circulation model (MITgcm) in a coupled aquaplanet [2], a planet where the ocean covers the entire globe. Such coupled aquaplanets, where nonlinear interactions between atmosphere, ocean and sea ice are fully taken into account, provide a relevant framework to understand the role of the main feedbacks at play in the climate system. Five attractors have been found, ranging from snowball (where ice covers the entire planet) to hot state conditions (where ice completely disappears) [2].

Using TheDiaTo, we analyse the five climate attractors by estimating: a) the energy budgets and meridional energy transports; b) the water mass and latent energy budgets and respective meridional transports; c) the Lorenz Energy Cycle; d) the material entropy production. We consider different coupled atmosphere-ocean-sea ice configurations and cloud parameterizations of MITgcm where the energy balance at the top of the atmosphere is progressively better closed in order to understand the occurrence of possible biases in the statistical properties of each attractor.

Our contribution will help clarify the thermodynamic differences in climate attractors and their stability to external perturbations that could shift the climate from a steady state to the other.

[1] Lembo V., Lunkeit F., Lucarini V., TheDiaTo (v1.0) – a new diagnostic tool for water, energy amd entropy budgets in climate models, Geosci. Model Dev. 12, 3805-3834 (2019)

[2] Brunetti M., Kasparian J., Vérard C., Co-existing climate attractors in a coupled aquaplanet, Climate Dynamics 53, 6293-6308 (2019)

How to cite: Ragon, C., Lembo, V., Lucarini, V., Kasparian, J., and Brunetti, M.: Comparing water, energy and entropy budgets of aquaplanet climate attractors, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-9572, https://doi.org/10.5194/egusphere-egu2020-9572, 2020.

D2946 |
Stan Schymanski and Martijn Westhoff

Thermodynamic optimality principles, such as maximum entropy production or maximum power extraction, hold a great promise to help explain self-organisation of various compartments of planet Earth, including the climate system, catchments and ecosystems. There is a growing number of examples for more or less successful use of these principles in earth system science, but a common systematic approach to the formulation of the relevant system boundaries, state variables and exchange fluxes has not yet emerged. Here we present a blueprint for the thermodynamically consistent formulation of box models and rigorous testing of optimality principles, in particular the maximum entropy production (MEP) and the maximum power (MP) principle. We investigate under what conditions these principles can be used to predict energy transfer coefficients across internal system boundaries and demonstrate that, contrary to common perception, these principles do not lead to similar predictions if energy and entropy balances are explicitly considered for the whole system and the defined sub-systems. We further highlight various pitfalls that may result in thermodynamically inconsistent models and potentially wrong conclusions about the implications of thermodynamic optimality principles. 
The analysis is performed in an open source mathematical framework, using the notebook interface Jupyter, the programming language Python, Sympy and a newly developed package for Python, "Environmental Science using Symbolic Math" (ESSM, https://github.com/environmentalscience/essm). This ensures easy verifiability of the results and enables users to re-use and modify variable definitions, equations and mathematical solutions to suit their own thermodynamic problems. 

How to cite: Schymanski, S. and Westhoff, M.: A blueprint for thermodynamically consistent box models and a test bed for thermodynamic optimality principles, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-8084, https://doi.org/10.5194/egusphere-egu2020-8084, 2020.

D2947 |
Bianca Kretz, Willi Freeden, and Volker Michel

The aspect of poroelasticity is anywhere interesting where a solid material and a fluid come into play and have an effect on each other. This is the case in many applications and we want to focus on geothermics. It is useful to consider this aspect since the replacement of the water in the reservoir below the Earth's surface has an effect on the sorrounding material and vice versa. The underlying physical processes can be described by partial differential equations, called the quasistatic equations of poroelasticity (QEP). From a mathematical point of view, we have a set of three (for two space and one time dimension) partial differential equations with the unknowns u (displacement) and p (pore pressure) depending on the space and the time.

Our aim is to do a decomposition of the data given for u and p in order that we can see underlying structures in the different decomposition scales that cannot be seen in the whole data.
For this process, we need the fundamental solution tensor of the QEP (cf. [1],[5]).
That means we assume that we have given data for u and p (they can be obtained for example by a method of fundamental solutions, cf. [1]) and want to investigate a post-processing method to these data. Here we follow the basic approaches for the Laplace-, Helmholtz- and d'Alembert equation (cf. [2],[4],[6]) and the  Cauchy-Navier equation as a tensor-valued ansatz (cf. [3]). That means we want to modify our elements of the fundamental solution tensor in such a way that we smooth the singularity concerning a parameter set τ=(τxt). 
With the help of these modified functions, we construct scaling functions which have to fulfil the properties of an approximate identity.
They are convolved with the given data to extract more details of u and p.


[1] M. Augustin: A method of fundamental solutions in poroelasticity to model the stress field in geothermal reservoirs, PhD Thesis, University of Kaiserslautern, 2015, Birkhäuser, New York, 2015.
[2] C. Blick, Multiscale potential methods in geothermal research: decorrelation reflected post-processing and locally based inversion, PhD Thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern, 2015.
[3] C. Blick, S. Eberle, Multiscale density decorrelation by Cauchy-Navier wavelets, Int. J. Geomath. 10, 2019, article 24.
[4] C. Blick, W. Freeden, H. Nutz: Feature extraction of geological signatures by multiscale gravimetry. Int. J. Geomath. 8: 57-83, 2017.
[5] A.H.D. Cheng and E. Detournay: On singular integral equations and fundamental solutions of poroelasticity. Int. J. Solid. Struct. 35, 4521-4555, 1998.
[6] W. Freeden, C. Blick: Signal decorrelation by means of multiscale methods, World of Mining, 65(5):304-317, 2013.

How to cite: Kretz, B., Freeden, W., and Michel, V.: Poroelastic aspects in geothermics, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-7100, https://doi.org/10.5194/egusphere-egu2020-7100, 2020.

D2948 |
Matteo Bernard Bertagni and Carlo Camporeale

The interactions between water and rocks create an extensive variety of marvelous patterns, which span on several classes of time and space scales. In this work, we provide a mathematical model for the formation of longitudinal erosive patterns commonly found in karst and alpine environments. The model couples the hydrodynamics of a laminar flow of water (Orr-Somerfield equation) to the concentration field of the eroded-rock chemistry. Results show that an instability of the plane rock wetted by the water film leads to a longitudinal channelization responsible for the pattern formation. The spatial scales predicted by the model span over different orders of magnitude depending on the flow intensity and this may explain why similar patterns of different sizes are observed in nature (millimetric microrills, centimetric rillenkarren, decametric solution runnels).

How to cite: Bertagni, M. B. and Camporeale, C.: A model for the longitudinal patterns shaped by water on erodible rocks, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-7031, https://doi.org/10.5194/egusphere-egu2020-7031, 2020.

D2949 |
Imogen Dell

There exists a coupling mechanism between the troposphere and the stratosphere, which plays a fundamental role in weather and climate. The coupling is highly complex and rests upon radiative and chemical feedbacks, as well as dynamical coupling by Rossby waves. The troposphere acts as a source of Rossby waves which propagate upwards in to the stratosphere, affecting the zonal mean flow. Rossby waves are also likely to play a significant role in downward communication of information via reflection from the stratosphere in to the troposphere. A mechanism for this reflection could be from a so-called critical layer. A shear flow exhibits a critical layer where the phase speed equals the flow velocity, where viscous and nonlinear effects become important. A wave incident upon a critical layer may be absorbed, reflected or overreflected, whereby the amplitude of the reflected wave is larger than that of the incident wave. In the case of troposphere-stratosphere coupling, the concept of critical layer overreflection is key to understanding atmospheric instability.

Motivated by this, a mathematical framework for understanding the coupling will be presented together with an investigation in to the role of nonlinearity versus viscosity inside the critical layer.

How to cite: Dell, I.: Troposphere-Stratosphere Coupling and the Role of Critical Layer Nonlinearity, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-11841, https://doi.org/10.5194/egusphere-egu2020-11841, 2020.

D2950 |
Gözde Özden and Marcel Oliver

Consider the motion of a rotating fluid governed by the Boussinesq equations with full Coriolis parameter. This is contrary to the so-called ''traditional approximation'' in which the horizontal part of the Coriolis parameter is zero. The model is obtained using variational principle which depends on Lagrangian dynamics. The full Coriolis force is used since the horizontal component of the angular velocity has a crucial role in that it introduces a dependence on the direction of the geostrophic flow in the horizontal geostrophical plane. We aim that singularity near the equatorial region can be solved with this assumption. This gives a consistent balance relation for any latitude on the Earth. We follow the similar strategy to that Oliver and Vasylkevych (2016) for the system to derive the Euler-Poincaré equations. Firstly, the system is transformed into desired scale giving the differences with the other scales. We derive the balance model Lagrangian as called L1 model, R. Salmon, using Hamiltonian principles. Near identity transformation is applied to simplify the Hamiltonian. Whole calculations are done considering the smallness assumption of the Rossby number. Long term, we aim that results help to understand the global energy cycle with the goal of validity and improving climate models.

How to cite: Özden, G. and Oliver, M.: Variational Model Reduction for Rotating Geophysical Flows with Full Coriolis Force, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-11603, https://doi.org/10.5194/egusphere-egu2020-11603, 2020.

D2951 |
Stephen Griffiths

The Kelvin wave is perhaps the most important of the equatorially trapped waves in the terrestrial atmosphere and ocean, and plays a role in various phenomena such as tropical convection and El Nino. Theoretically, it can be understood from the linear dynamics of a stratified fluid on an equatorial β-plane, which, with simple assumptions about the disturbance structure, leads to wavelike solutions propagating along the equator, with exponential decay in latitude. However, when the simplest possible background flow is added (with uniform latitudinal shear), the Kelvin wave (but not the other equatorial waves) becomes unstable. This happens in an extremely unusual way: there is instability for arbitrarily small nondimensional shear λ, and the growth rate is proportional to exp(-1/λ^2) as λ → 0. This in contrast to most hydrodynamic instabilities, in which the growth rate typically scales as a positive power of λ-λc as the control parameter λ passes through a critical value λc.

This Kelvin wave instability has been established numerically by Natarov and Boyd, who also speculated as to the underlying mathematical cause by analysing a quantum harmonic oscillator perturbed by a potential with a remote pole. Here we show how the growth rate and full spatial structure of the Kelvin wave instability may be derived using matched asymptotic expansions applied to the (linear) equations of motion. This involves an adventure with confluent hypergeometric functions in the exponentially-decaying tails of the Kelvin waves, and a trick to reveal the exponentially small growth rate from a formulation that only uses regular perturbation expansions. Numerical verification of the analysis is also interesting and challenging, since special high-precision solutions of the governing ordinary differential equations are required even when the nondimensional shear is not that small (circa 0.5). 

How to cite: Griffiths, S.: The strange instability of the equatorial Kelvin wave , EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-9302, https://doi.org/10.5194/egusphere-egu2020-9302, 2020.

D2952 |
Richard Blender and Joscha Fregin

We consider recharge-discharge processes in a forced wave-mean flow interaction model and in a forced Rossby wave triad. Such processes are common in atmospheric dynamics and are typically modelled by nonlinear oscillators, for example for mid-latitude storms by Ambaum and Novak (2013) and for convective cycles by Yano and Plant (2012). A similar behaviour can be seen in the simulation of a forced wave number triad by Lynch (2009). Here we construct noncanonical Hamiltonian and Nambu representations in three-dimensional phase space for available and prescribed conservation laws during the recharge and discharge regimes. Divergence in phase space is modelled by a pre-factor. The approach allows the design of conservative and forced dynamical systems.

How to cite: Blender, R. and Fregin, J.: Wave-mean flow interaction, forced triads, and recharge-discharge Processes as noncanonical Hamiltonian Systems, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-13284, https://doi.org/10.5194/egusphere-egu2020-13284, 2020.

D2953 |
Stuart Grieve, Simon Mudd, Fiona Clubb, Michael Singer, Katerina Michaelides, and Shiuan-An Chen

The topology of fluvial networks has long been studied, with Horton's laws describing relationships between stream order, stream density, and stream length often cited as fundamental governing principles of drainage basin development. Building upon these principles, small scale studies have identified patterns of self-similarity in drainage networks in the continental USA, suggesting that to some extent, river networks self-organise in a scale invariant manner. More stringent measures of self-similarity have also been developed, which quantify the fractal nature of side branching structures in fluvial networks. Using such metrics, studies have identified similarities between leaf vein structures and fluvial networks, and have identified a potential climatic signature in North American river topology.

The appeal of such techniques over traditional methods of channel analysis using topographic data is that in self-similar networks, the precise location of channel heads is unimportant, allowing analysis to be performed at unprecedented scales, and in locations where data quality is limited. Here, we attempt to reconcile these two suites of techniques to understand the potential and limitations of network topology as an indicator of broader landscape dynamics. We achieve this through the analysis of fluvial networks extracted at a global scale from the Shuttle Radar Topography Mission dataset alongside other global earth observation data.

How to cite: Grieve, S., Mudd, S., Clubb, F., Singer, M., Michaelides, K., and Chen, S.-A.: Inverting fluvial network topology to understand landscape dynamics, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-8740, https://doi.org/10.5194/egusphere-egu2020-8740, 2020.

D2954 |
Nathaniel Bristow, James Best, Kenneth Christensen, Matthew Baddock, Giles Wiggs, Pauline Delorme, and Joanna Nield

Understanding the initiation of aeolian dunes poses significant challenges due to the strong couplings between turbulent fluid flow, sediment transport, and bedform morphology. While much is known concerning the dynamics of more mature bedforms, open questions remain as to how protodunes are formed, as well as the mechanisms by which they continue to evolve. The structure of the turbulent flow field drives the mobilization or deposition of sediment, thus controlling the initial formation of sand patches, yet is also strongly influenced itself by local conditions such as surface roughness and moisture. Furthermore, an additional feedback on the flow and transport is exerted by the sand patches themselves once they begin to form.

As protodunes begin to develop from this initial deposition, their morphologies possess unique characteristics involving a reverse asymmetry of the stoss and lee sides, wherein the crest begins upstream, close to the toe, and gradually shifts downstream toward the "regular" asymmetric profile exhibited by more mature dunes. However, these early stages of development also involve very gentle slopes and low profiles which make field measurements of the associated flow particularly challenging.

The current research effort involves a combination of field measurements, documenting the initiation and morphological development of sand patches and protodunes, in concert with detailed measurements of the flow-form interactions in a laboratory flume. The work presented herein focuses primarily on experiments conducted in a unique flow facility wherein high-resolution measurements of the turbulent flow field associated with the early stages of protodune development are obtained utilizing particle-image velocimetry (PIV) in a refractive-index-matched (RIM) environment. The RIM technique facilitates flow measurements extremely close to model surfaces as well as unimpeded optical access which are critical to understanding the flow-form coupling. A series idealized, fixed-bed models are fabricated to mimic the key morphological characteristics of early protodune development observed in the field, and the flow measurements associated with them are analyzed to reveal the mechanisms controlling the bedform dynamics.

How to cite: Bristow, N., Best, J., Christensen, K., Baddock, M., Wiggs, G., Delorme, P., and Nield, J.: The Origin of Aeolian Dunes – PIV measurements of flow structure over early stage protodunes in a refractive-index-matching flume, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-4259, https://doi.org/10.5194/egusphere-egu2020-4259, 2020.

D2955 |
Tjebbe Hepkema, Huib de Swart, and Henk Schuttelaars
Tidal bars are bed forms in tidal channels that have a wave-like structure in both the along-channel and cross-channel direction. They are found in tidal channels all around the globe, for example, in the Western Scheldt in the Netherlands, the Exe Estuary in England, the Ord River Estuary in Australia and the Venice Lagoon in Italy. Typically, tidal bars are several meters high, have wavelengths of 1-15 km and migration speeds of meters per day. Understanding their dynamics is important as they are invaluable for many living organisms (e.g., migrating birds) but they hamper marine traffic.
It has been shown, by means of a linear stability analysis, that these bars emerge due to inherent feedbacks between the tidal currents and the erodible bed. When the bars mature, their dynamics becomes nonlinear. Schramkowski et al. (2004) applied a bifurcation analysis to analyse the bar dynamics, but their method was limited to small bottom friction. Here, we developed a numerical (time integration) model that simulates the nonlinear dynamics and the corresponding (stable) equilibrium patterns for realistic parameter values.
Using the output of the numerical model we derive a reduced order model with a method called SINDy (Brunton et al., 2016). Loiseau and Brunton (2018) showed that from output of complex numerical models simulating fully nonlinear fluid flows, SINDy can identify small systems of equations which govern the complex flows. Here we show that, for parameters regimes where the dynamics is weakly nonlinear, SINDy finds a Landau type equation that reproduces the tidal bar dynamics well. The Landau equation is a nonlinear ordinary differential equation in terms of the Fourier amplitude of the pattern that initially has the largest growth rate. The form of this equation corresponds with the one that is expected from the symmetry of the patterns. Also, the application of SINDy to the fully nonlinear dynamics of tidal bars will be discussed.
Brunton, S.L., Proctor, J.L., and Kutz, J.N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15):3932-3937.
Loiseau, J.-C. and Brunton, S.L. (2018). Constrained sparse Galerkin regression. Journal of Fluid Mechanics, 838:42-67.
Schramkowski, G.P., Schuttelaars, H.M., and de Swart, H.E. (2004). Nonlinear channel-shoal dynamics in long tidal embayments. Ocean Dynamics, 54(3):399-407.

How to cite: Hepkema, T., de Swart, H., and Schuttelaars, H.: Data-driven reduced order modelling of tide-induced sand bars in confined channels, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-10258, https://doi.org/10.5194/egusphere-egu2020-10258, 2020.

D2956 |
Dmitry Dukhovskoy

Increasing Greenland discharge has contributed more than 5000 km3 of surplus fresh water to the Subpolar North Atlantic since the early 1990s. The volume of this freshwater anomaly is projected to cause freshening in the North Atlantic leading to changes in the intensity of deep convection and thermohaline circulation in the subpolar North Atlantic. This is roughly half of the freshwater volume of the Great Salinity Anomaly of the 1970s that caused notable freshening in the Subpolar North Atlantic. In analogy with the Great Salinity Anomaly, it has been proposed that, over the years, this additional Greenland freshwater discharge might have a great impact on convection driving thermohaline circulation in the North Atlantic with consequent impact on climate. Previous numerical studies demonstrate that roughly half of this Greenland freshwater anomaly accumulates in the Subpolar Gyre. However, time scales over which the Greenland freshwater anomaly can accumulate in the subpolar basins is not known. This study estimates the residence time of the Greenland freshwater anomaly in the Subpolar Gyre by approximating the process of the anomaly accumulation in the study domain with a first order autonomous dynamical system forced by the Greenland freshwater anomaly discharge. General solutions are obtained for two types of the forcing function. First, the Greenland freshwater anomaly discharge is a constant function imposed as a step function. Second, the surplus discharge is a linearly increasing function. The solutions are deduced by utilizing results from the numerical experiments that tracked spreading of the Greenland fresh water with a passive tracer. The residence time of the freshwater anomaly is estimated to be about 10–15 years. The main differences in the solutions is that under the linearly increasing discharge rate, the volume of the accumulated Greenland freshwater anomaly in the Subpolar Gyre does not reach a steady state. By contrast, solution for the constant discharge rate reaches a steady state quickly asymptoting the new steady state value for time exceeding the residence time. Estimated residence time is compared with the numerical experiments and observations.

How to cite: Dukhovskoy, D.: Using a first-order autonomous dynamical system to evaluate residence time of the Greenland freshwater anomaly in the Subpolar Gyre, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-1958, https://doi.org/10.5194/egusphere-egu2020-1958, 2020.

D2957 |
Tom Manzocchi, Deirdre Walsh, Carneiro Marcus, Javier López-Cabrera, and Soni Kishan

Irrespective of the specific technique (variogram-based, object-based or training image-based) applied, geostatistical facies models usually use facies proportions as the constraining input parameter to be honoured in the output model. The three-dimensional interconnectivity of the facies bodies in these models increases as the facies proportion increases, and the universal percolation thresholds that define the onset of macroscopic connectivity in idealized statistical physics models define also the connectivity of these facies models. Put simply, the bodies are well connected when the model net:gross ratio exceeds about 30%, and because of the similar behaviour of different geostatistical approaches, some researchers have concluded that the same threshold applies to geological systems.

In this contribution we contend that connectivity in geological systems has more degrees of freedom than it does in conventional geostatistical facies models, and hence that geostatistical facies modelling should be constrained at input by a facies connectivity parameter as well as a facies proportion parameter. We have developed a method that decouples facies proportion from facies connectivity in the modelling process, and which allows systems to be generated in which both are defined independently at input. This so-called compression-based modelling approach applies the universal link between the connectivity and volume fraction in geostatistical modelling to first generate a model with the correct connectivity but incorrect volume fraction using a conventional geostatistical approach, and then applies a geometrical transform which scales the model to the correct facies proportions while maintaining the connectivity of the original model. The method is described and illustrated using examples representative of different geological systems. These include situations in which connectivity is both higher (e.g. fluid-driven injectite or karst networks) and lower (e.g. many depositional systems) than can be achieved in conventional geostatistical facies models.

How to cite: Manzocchi, T., Walsh, D., Marcus, C., López-Cabrera, J., and Kishan, S.: Explicit inclusion of connectivity in geostatistical facies modelling., EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-21710, https://doi.org/10.5194/egusphere-egu2020-21710, 2020.

D2958 |
Peter Bossew

The asymptotic shape of the marginal frequency distribution of geochemical variables has been proposed as indicator of multi-fractality. Transition into a certain statistical regime as inferred from the distribution function may be considered as criterion to delineate geochemical anomalies, including mineral resources or pollutants such as radioactive fallout or geogenic radon.

The argument is that asymptotic linearity in log-log scale, log(F(z)) = a - b log(z) as z→∞, b>0 a constant, indicates multi-fractality.

We discuss this with respect to two issues:

(1) What are the consequences of estimating the slope b for non-ergodic, in particular non-representative and preferential sampling schemes, as often the case in geochemical or pollution surveys?

(2) Frequently in geochemistry, multiplicative cascades are considered valid generators of multifractal fields, corroborated by observed f(α) functions and variograms (Matèrn or power, for low lags). This generator leads to marginally asymptotically (high cascade orders) log-normal distributions, which in log-log scale are asymptotically (high z) parabolic, not linear.

Theoretical aspects are addressed as well as examples given.

How to cite: Bossew, P.: Log-log linearity of the asymptotic distribution - a valid indicator of multi-fractality?, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-6447, https://doi.org/10.5194/egusphere-egu2020-6447, 2020.

D2959 |
Tommaso Alberti, Giuseppe Consolini, Peter D. Ditlevsen, Reik V. Donner, and Virgilio Quattrociocchi

Several attempts have been made in characterizing the multiscale nature of fluctuations from nonlinear and nonstationary time series. Particularly, the study of their fractal structure has made use of different approaches like the structure function analysis, the evaluation of the generalized dimensions, and so on. Here we report on a different approach for characterizing phase-space trajectories by using the empirical modes derived via the Empirical Mode Decomposition (EMD) method. We show how the derived Intrinsic Mode Functions (IMFs) can be used as source of local (in terms of scales) information allowing us in deriving multiscale measures when looking at the behavior of the generalized fractal dimensions at different scales. This formalism is applied to three pedagogical examples like the Lorenz system, the Henon map, and the standard map. We also show that this formalism is readily applicable to characterize both the behavior of the Earth’s climate during the past 5 Ma and the dynamical properties of the near-Earth electromagnetic environment as monitored by the SYM-H index.

How to cite: Alberti, T., Consolini, G., Ditlevsen, P. D., Donner, R. V., and Quattrociocchi, V.: Multiscale measures of phase-space trajectories, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-268, https://doi.org/10.5194/egusphere-egu2020-268, 2020.

D2960 |
Mark McGuinness and Emma Greenbank

A Surtseyan volcanic eruption involves a bulk interaction between water and hot magma, mediated by the return of ejected ash. Surtsey Island, off the coast of Iceland, was born during such an eruption process in the 1940s. Mount Ruapehu in New Zealand also undergoes Surtseyan eruptions, due to its crater lake. 

One feature of such eruptions is ejected lava bombs, trailing steam, with evidence that watery slurry was trapped inside them during the ejection process. Simple calculations indicate that the pressures developed due to boiling inside such a bomb should shatter it. Yet intact bombs are routinely discovered in debris piles. In an attempt to crack this problem, and provide a criterion for fragmentation of Surtseyan bombs, a transient mathematical model of the flashing of water to steam inside one of these hot erupted lava balls is developed, with a particular focus on the maximum pressure attained, and how it depends on magma and fluid properties. Numerical and asymptotic solutions provide some answers for volcanologists.

How to cite: McGuinness, M. and Greenbank, E.: Fragmentation of steaming Surtseyan bombs, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-2299, https://doi.org/10.5194/egusphere-egu2020-2299, 2020.

D2961 |
Michal Kuraz and Petr Mayer

Modeling the kinematic wave equation and sediment transport equation enables a deterministic approach for predicting surface runoff and resulting sediment transport. Both the kinematic wave equation and the sediment transport equation are first order differential equations. Moreover the kinematic wave equation is a quasilinear problem. In many engineering applications this set of equations is solved on one-dimensional representative cross-sections. However, a proper selection of representative cross-section(s) is  cumbersome. On the other hand integrating this set of equations on real catchment topography  yields difficulties for standard variational methods such as continous Galerkin method. These difficulties are two-fold (1) the nonlinearity of the kinematic wave, and (2) the absence of diffusion term, which acts as a stabilization term for convection-diffusion equation. In a theory, the Peclet number of numerical stability reaches infinity. 

In this contribution we will focus on a stable numerical approximation of this convection-only problem using least square method. With this method we are able to reliably solve both the kinematic wave equation and the sediment transport equation on computational  domains representing real catchment topography. Several examples representing real-world scenarios will be given.

How to cite: Kuraz, M. and Mayer, P.: Solving the erosion transport equation on three dimensional catchments, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-20491, https://doi.org/10.5194/egusphere-egu2020-20491, 2020.

D2962 |
Elena Malinovskaya and Otto Chkhetiani

One of the important characteristics of the wind process of dust removal is a critical or threshold wind velocity [1]. Saltating flow grows with increasing of the effective roughness [2] that affecting shear stress and friction velocity [3]. The drag coefficient increases depending on the density of the coating by particles of the surface [4]. The location of particles in the aeolian structure, their size and relative position determine their resistance to wind influence. Aeolian structures change the structure of flows and the balance of mass transfer of particles deposited and rising from the surface [5]. The surface microstructures and ripples significantly affect of sand removal.
The flow of particles with a size of 100 μm on the surface has been considered using the OPEN FOAM with LES model. The calculation area has sizes of 5x5x2 mm. For the velocity at the upper boundary, 2.8 m/s select in accordance with the experimental data [6]. It should be noted that with a relative increase in the distance between pairs of particles and a change in the level of the upper surface, the pressure difference between the base and top of the particle increases by 10-30 percents. Depending on the distance between the particles, the buoyant force acting from the side of the air flow, the critical velocity, and the departure velocity of the particle also change. When the distances between the surfaces of the particles are close to its size, the buoyant force is greater than the adhesion and gravity forces. As a result, areas with different probability for the sand removal by wind, due to which, in particular, the occurrence of aeolian ripples occurs.
The average critical velocity increases when moving up the windward slope of the dune [7, 8]. This phenomenon is possibly associated with the influence of ripples on the air flow. The flow around of the micro-ripples with a height of 0.1-1 mm was considered for air flow velocity of 2-4 m/s at a height of 1-2 cm. The addition of supplementary elements of heterogeneity at the apex near the rough surface of the streamlined aeolian structure leads to a displacement of the separation point of the ascending flows. Also we have a change in the length of the recirculation zone and the time intervals of the strengthening of the wind at the apex, which was observed in [6].
The study was supported by the RFBR project 19-05-50110 and partial support of the program of the Presidium of the Russian Academy of Sciences No. 12.
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2. Martin R.L., Kok J.F. J.Geophys.Res.2018.123(7).1546-1565.
3. Turpin C et al. Earth Surf. Proc. and Land.2010.35(12). 1418-1429.
4. Yang X.I.A. et al. J. Fluid Mech.2019.880. 992-1019.
5. Luna M.C.M.M. et al. Geomorph.2011.129(3-4). 215-224.
6. Semenov O.E. Introduction to experimental meteorology and climatology of the sand storms. Almaty. 2011. p.580 (in Russian).
7. Neuman C.M.K et al. Sediment. 2000. 47(1). 211-226.
8. Malinovskaya E.A. Izv. Atmos. Oceanic Phys. 2019. 55(2). 86-92.

How to cite: Malinovskaya, E. and Chkhetiani, O.: Conditions for the emergence and growth of aeolian sand structures , EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-2040, https://doi.org/10.5194/egusphere-egu2020-2040, 2020.