Climate Fluctuations and Non-equilibrium Statistical Mechanics: An Interdisciplinary Dialogue

We study here the tangent space dynamics of the 2-level Lorenz 96 model under two different set of boundary conditions, with and without fast forcing. Our aim is to find whether one can find any signature of the non-trivial interaction between scales in the geometric properties of the so-called Covariant Lyapunov Vectors (CLVs). This would allow an educated parameterization of the fast scales through the identification of the relevant physical modes. We first display results relative to the localization properties of the vectors, their relative angles and projection on the slow variables. We show that only the vectors associated with near 0 Lyapunov exponents have a significant projection on the slow variables, a possible signature of the existence of Hydrodynamics Lyapunov Modes, and investigate how the coupling between scales affects this property.

(Authors: Adrian Odenweller, Reik V. Donner) Over the last decade, complex network methods have been frequently used for characterizing spatio-temporal patterns of climate variability from a complex systems perspective, yielding new insights into time-dependent teleconnectivity patterns and couplings between different components of the Earth climate. Among the foremost results reported, network analyses of the synchronicity of extreme events as captured by the so-called event synchronization have been proposed to be powerful tools for disentangling the spatio-temporal organization of particularly extreme rainfall events and anticipating the timing of monsoon onsets or extreme floodings. Rooted in the analysis of spike train synchrony analysis in the neurosciences, event synchronization has the great advantage of automatically classifying pairs of events arising at two distinct spatial locations as temporally close (and, thus, possibly statistically -- or even dynamically -- interrelated) or not without the necessity of selecting an additional parameter in terms of a maximally tolerable delay between these events. This consideration is conceptually justified in case of the original application to spike trains in electroencephalogram (EEG) recordings, where the inter-spike intervals show relatively narrow distributions at high temporal sampling rates. However, in case of climate studies, precipitation extremes defined by daily precipitation sums exceeding a certain empirical percentile of their local distribution exhibit a distinctively different type of distribution of waiting times between subsequent events. This raises conceptual concerns if event synchronization is still appropriate for detecting interlinkages between spatially distributed precipitation extremes. In order to study this problem in more detail, we employ event synchronization together with an alternative similarity measure for event sequences, event coincidence rates, which requires a manual setting of the tolerable maximum delay between two events to be considered potentially related. Both measures are then used to generate climate networks from parts of the satellite-based TRMM precipitation data set at daily resolution covering the Indian and East Asian monsoon domains, respectively, thereby re-analysing previously published results. The obtained spatial patterns of degree densities and local clustering coefficients exhibit marked differences between both similarity measures. Specifically, we demonstrate that there exists a strong relationship between the fraction of extremes occurring at subsequent days and the degree density in the event synchronization based networks, suggesting that the spatial patterns obtained using this approach are strongly affected by the presence of serial dependencies between events. Given that a manual selection of the maximally tolerable delay between two events can be guided by a priori climatological knowledge and even used for systematic testing of different hypotheses on climatic processes underlying the emergence of spatio-temporal patterns of extreme precipitation, our results provide evidence that event coincidence rates are a more appropriate statistical characteristic for similarity assessment and network construction for climate extremes, while results based on event synchronization need to be interpreted with great caution.

The talk introduces the active wind as the deviation from a general local wind balance, the inactive wind. The inactive wind is directed along intersection lines of Bernoulli function and potential temperature surfaces. In climatological steady state, the inactive mass flux cannot participate in net-mass fluxes, because the mean position of the mentioned intersection lines does not change. A conceptual proximity of the zonal-mean active wind to the residual wind as occurring in the transformed Eulerian mean equations suggests itself. The zonal- and time-mean active wind is compared to the residual wind for the Held-Suarez test case. Similarities occur for the meridional components in the zone of Rossby wave breaking in the upper troposphere equatorward of the jet. The vertical components are similar, too. However, the vertical active wind is much stronger in the baroclinic zone. This is due to the missing vertical eddy flux of Ertel's potential vorticity (EPV) in the TEM equations. The largest differences are to be found in the boundary layer, where the active wind exhibits typical pattern of Ekman dynamics. Instantaneous active wind vectors demonstrate mass-inflow for lows and mass-outflow for highs in the boundary layer. An active meridional wind is associated with a filamentation of EPV in the zone of Rossby wave breaking in about 300 hPa. Strong gradients of EPV act as a transport barrier.

Glacial climate variability is studied integrating simple nonlinear stochastic dynamical systems with palaeoclimatic records. Different models representing different dynamical mechanisms and modelling approaches are contrasted; model comparison and selection is based on a likelihood function, an information criterion as well as various long-term summary statistics. A two-dimensional stochastic relaxation oscillator model with proxy temperature as the fast variable is formulated and the system parameters and noise levels estimated from Greenland ice-core data. The deterministic part of the model is found to be close to the Hopf bifurcation, where the fixed point becomes unstable and a limit cycle appears. The system is excitable; under stochastic forcing, it exhibits noisy large-amplitude oscillations capturing the basic statistical characteristics of the transitions between the cold and the warm state. No external forcing is needed in the model. The relaxation oscillator is much better supported by the data than noise-driven motion in a one-dimensional bistable potential. Two variants of a mixture of local linear stochastic models, each associated with an unobservable dynamical regime or cluster in state space, are also considered. Three regimes are identified, corresponding to the different phases of the relaxation oscillator: (i) lingering around the cold state, (ii) rapid shift towards the warm state, (iii) slow relaxation out of the warm state back to the cold state. The mixture models have a high likelihood and are able to capture the pronounced time-reversal asymmetry in the ice-core data as well as the distribution of waiting times between onsets of Dansgaard–Oeschger events.

Finite-time Lyapunov exponents of generic chaotic dynamical systems fluctuate in time. These fluctuations are due to the different degree of stability across the accessible phase-space. An earlier numerical study[1] revealed that the diffusion coefficient $D$ of the Lyapunov exponents (LEs) exhibits a non-trivial scaling behaviour, $D(L) \sim L^{-\gamma}$, with the system size $L$. For chaotic dissipative systems, we show that the wandering exponent $\gamma$ can be expressed in terms of the kinetic roughening exponents associated with the corresponding (logarithm of the) Covariant Lyapunov Vector (CLV) via the universal scaling relation. Our theoretical predictions are supported by the numerical analysis of several spatially-extended systems[2]. In particular, we find that the wandering exponent of the first LE is universal and connected to the Kardar-Parisi-Zhang (KPZ) equation. Furthermore, our simulations reveal that the bulk of the spectrum belongs to a different, but possibly unique, universality class. In all cases, the fluctuation exponent $\gamma_n$ of the $n$-th LE is connected to the $n$-th CLV surface via $\gamma_n = z_n-2\alpha_n$, in agreement with our theory. In contrast, in generic Hamiltonian lattices the diffusion coefficient of the maximum Lyapunov exponent diverges in the thermodynamic limit[3]. We trace the divergence back to the long-range correlations associated with the evolution of the hydrodynamic modes. In the case of normal heat transport, the divergence is even stronger. A similar scenario is expected to arise in the evolution of rough interfaces in the presence of a suitably correlated background noise. [1] P.~V. Kuptsov and A. Politi, Phys. Rev. Lett. 107, 114101 (2011). [2] D. Paz\'o, J.M. L\'opez, and A. Politi, Phys. Rev. E 87, 062909 (2013). [3]D. Paz\'o, J.M. L\'opez, and A. Politi, Phys. Rev. Lett. 117, 034101 (2016).

Denny Gohlke & Richard Blender (Meteorological Institute, University of Hamburg), Bastian Sommerfeld & Almut Gassmann (Leibniz Institute of Atmospheric Physics, Kühlungsborn) The physically consistent representation of turbulence subgrid-scale processes in forced dissipative systems like atmosphere and ocean requires the handling of statistical nonequilibrium fluctuations. The statistics of these fluctuations — as a fingerprint of the chaotic dynamics — provide useful insights into the dynamical response behaviour of a system (transport coefficients) and can be restricted by fluctuation relations. The idea of this project is the analysis and the incorporation of suchlike constraints on fluctuations to modify existing parameterisation schemes, focusing on a stochastic and counter-gradient parameterisation of momentum and heat fluxes which are related to energy dissipation and backscatter.