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Unit# Emerging complexity in physical systems

Laboratory

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The accurate investigation of many geophysical phenomena via direct numerical simulations is computationally not possible nowadays due to the huge range of spatial and temporal scales to be resolved. Therefore advances in this field rely on the development of new theoretical tools and numerical algorithms. In this work we investigate a new mathematical formalism that exploits the property of quasi-linear systems to self-tune towards marginally stable states. The inspiration for this study comes from the asymptotic analysis of strongly stratified flows, performed by Chini et al.. The application of multi-scale analysis to this problem, justified by the presence of scale separation, yields to a simplified quasi-linear model. In this reduced description small-scale instabilities evolve linearly about a large-scale hydrostatic field (whose evolution is fully non-linear) and modify it via a feedback term. From the only assumption of scale separation, two extremely interesting features of this model arise. First the presence of the coupling term between the two dynamics and second the quasi-linearity of the dynamics. The first aspect, generally not present in the hydrostatic approximation, can capture the non-local energy transfer between the small and the large scales, which is key for the quantification of the mixing efficiency in the deep ocean. The second aspect, namely the quasi-linearity, is suggestive of the self-organisation of the dynamics about marginally-stable states. This results in a coupled evolution where the fast dynamics adapts (is slaved) to the mean field, maintaining the marginal stability of the latter. The low-dimensional evolution that arises, enables the integration of the reduced system on temporal scales comparable to the characteristic time scale of the slow dynamics, making this novel approach highly suited to the investigation of the stratified flow problem.Building upon the results obtained by Chini et al. in the present work we extend this methodology addressing three different aspects of the reduced model.As a first case we investigate the twofold nature of the fluctuation feedback, which is not sign-definite and might lead to intense bursting events where the fluctuations exhibit positive growth rates on a fast time scale. In this scenario the scale separation is temporarily lost and the two dynamics have to be co-evolved until a new marginally stable manifold can be approached. Here we propose three different co-evolution techniques and test their efficacy on a one-dimensional model problem.The second aspect we address is the presence of a finite scale separation between the two dynamics. We develop an algorithm that carefully identifies the validity regions of the quasi-linear reduction and determines the relevance of the fluctuation feedback w.r.t. the characteristic time scale of the slow dynamics and the growth rate of the fluctuations.As a third case we derive an efficient extension of the original methodology to two-dimensional model problems, deriving an evolution equation for the wavenumber of the fastest growing mode, which then get slaved to the mean dynamics.Eventually the methodologies derived in the context of the two model problems are applied and discussed for the strongly stratified flow problem.

Omid Ashtari, Sajjad Azimi, Tobias Schneider

Chaotic dynamics in systems ranging from low-dimensional nonlinear differential equations to high-dimensional spatiotemporal systems including fluid turbulence is supported by nonchaotic, exactly recurring time-periodic solutions of the governing equations. These unstable periodic orbits capture key features of the turbulent dynamics and sufficiently large sets of orbits promise a framework to predict the statistics of the chaotic flow. Computing periodic orbits for high-dimensional spatiotemporally chaotic systems remains challenging as known methods either show poor convergence properties because they are based on time-marching of a chaotic system causing exponential error amplification, or they require constructing Jacobian matrices which is prohibitively expensive. We propose a new matrix-free method that is unaffected by exponential error amplification, is globally convergent, and can be applied to high-dimensional systems. The adjoint-based variational method constructs an initial value problem in the space of closed loops such that periodic orbits are attracting fixed points for the loop dynamics. We introduce the method for general autonomous systems. An implementation for the one-dimensional Kuramoto-Sivashinsky equation demonstrates the robust convergence of periodic orbits underlying spatiotemporal chaos. Convergence does not require accurate initial guesses and is independent of the period of the respective orbit.

2022The current research focuses on the prediction of the maximum axial compression load a cylindrical shell is able to bear. This maximum axial compression load is the value at which a cylindrical shell loses stability and abruptly buckles. After the buckling event the shells are permanently damaged and become unfit for any structural function. Hence, buckling is a critical not admissible failure mode in the design of shell structures. Standard methodologies available in the literature such as linear stability analysis fail to deliver an accurate prediction and overestimate the value of the load at which collapse occurs. The reason for this deviation between real shells and theory is the presence of geometric imperfections that are unique for each shell. In fact, nominally identical shells exhibit large variations in their load bearing capability. This variability means that the only reliable approach able to obtain the real loading bearing capability previous to this research is a destructive compression test. In this thesis, a new conceptual approach to describe the behavior of cylindrical shells is introduced. The new description is based on the dynamical systems approach used to study turbulence in the field of fluid dynamics. Using the dynamical systems approach applied to a non-linear formulation of the shell equations, fix points of the dynamical system are calculated and their stability under finite amplitude perturbations characterised. The boundary delimiting the transitions to a buckled state or returning to the unbuckled one is also characterized. The basin enclosed by this boundary, the basin of attraction, becomes smaller as the axial load is increased, vanishing at the compression load at which a cylindrical shell buckles. The shrinking of the basin of attraction can be characterized by evaluating its extension at different axial loads. This can be done by probing cylindrical shells. This probing at different axial compression loads defines a landscape that can be used to extrapolate the load at which the landscape, together with the basin of attractions vanishes. This axial compression load is the buckling load of the shell.The framework described theoretically is implemented in a series of test campaigns to show the predictive capability of stability landscapes in real shells. Stability landscapes are the experimental technique employed to explore and characterize the basin of attraction associated with each fix point. The exploration of the basin of attraction is performed using a single poker that represents only one of the directions to perturb a fix point of the dynamical system towards the boundary of the basin of attraction. The success of extrapolating the buckling load with stability landscapes constructed with a single poker at a single location does not provide a perfect predictive capability in real shells. The reason for this is the complex interaction between the imperfections present in real shells. The complex interaction between imperfections was also studied during this research, revealing that the buckling load of cylindrical shells is a function of all the imperfections present in the shell. The buckling load of a cylindrical shell is not dictated by the strongest imperfection alone, but by the combination of all defects.The key output of the current research is a non-destructive test procedure based on constructing stability landscapes at different locations of a cylindrical shell