When dielectric materials are brought into contact and then separated, they develop static electricity. For centuries, it has been assumed that such contact charging derives from the spatially homogeneous material properties (along the material's surface) and that within a given pair of materials, one charges uniformly positively and the other negatively. We demonstrate that this picture of contact charging is incorrect. Whereas each contact-electrified piece develops a net charge of either positive or negative polarity, each surface supports a random "mosaic" of oppositely charged regions of nanoscopic dimensions. These mosaics of surface charge have the same topological characteristics for different types of electrified dielectrics and accommodate significantly more charge per unit area than previously thought.

Abstract.We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterising transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate in a concrete manner the issues and phenomena that arise in the setting of finitetime dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing and important area of dynamical systems theory.Correspondence to: M. Branicki

The modus operandi of modern applied mathematics in developing very recent mathematical strategies for uncertainty quantification in partially observed high-dimensional turbulent dynamical systems is emphasized here. The approach involves the synergy of rigorous mathematical guidelines with a suite of physically relevant and progressively more complex test models which are mathematically tractable while possessing such important features as the two-way coupling between the resolved dynamics and the turbulent fluxes, intermittency and positive Lyapunov exponents, eddy diffusivity parameterization and turbulent spectra. A large number of new theoretical and computational phenomena which arise in the emerging statistical-stochastic framework for quantifying and mitigating model error in imperfect predictions, such as the existence of information barriers to model improvement, are developed and reviewed here with the intention to introduce mathematicians, applied mathematicians, and scientists to these remarkable emerging topics with increasing practical importance.

This article discusses a range of important mathematical issues arising in applications of a newly emerging stochastic-statistical framework for quantifying and mitigating uncertainties associated with prediction of partially observed and imperfectly modelled complex turbulent dynamical systems. The need for such a framework is particularly severe in climate science where the true climate system is vastly more complicated than any conceivable model; however, applications in other areas, such as neural networks and materials science, are just as important. The mathematical tools employed here rely on empirical information theory and fluctuation-dissipation theorems and it is shown that they seamlessly combine into a concise systematic framework for measuring and optimizing consistency and sensitivity of imperfect models. Here, we utilize a simple statistically exactly solvable 'perfect' system with intermittent hidden instabilities and with time-periodic features to address a number of important issues encountered in prediction of much more complex dynamical systems. These problems include the role and mitigation of model error due to coarse-graining, moment closure approximations, and the memory of initial conditions in producing short, medium and long range predictions. Importantly, based on a suite of increasingly complex imperfect models of the perfect test system, we show that the predictive skill of the imperfect models and their sensitivity to external perturbations is improved by ensuring their consistency on the statistical attractor (i.e., the climate) with the perfect system. Furthermore, the discussed link between climate fidelity and sensitivity via the fluctuation-dissipation theorem opens up an enticing prospect of developing techniques for improving imperfect model sensitivity based on specific tests carried out in the training phase of the unperturbed statistical equilibrium/climate.

The filtering skill for turbulent signals from nature is often limited by model errors created by utilizing an imperfect model for filtering. Updating the parameters associated with unresolved or unknown processes in the imperfect model "on the fly" through stochastic parameter estimation is an efficient way to increase filtering skill and model performance. Here, a suite of filters implementing stochastic parameter estimation is examined on a nonlinear, exactly solvable, stochastic test model mimicking turbulent signals in regimes ranging from configurations with strongly intermittent, transient instabilities to laminar behavior. Stochastic Parameterization Extended Kalman Filter (SPEKF) systematically corrects both multiplicative and additive biases in the observed dynamics and it involves exact formulas for propagating the mean and covariance including the unresolved parameters in the test model. The remaining filters use the same nonlinear test model but they introduce additional model error through different moment closure approximations and/or linear tangent approximation used for computing the second-order statistics in the stochastic forecast model. A comprehensive study of filter performance is carried out in the presence of various sources of model error as the observation time and observation noise levels are varied. In particular, regimes of filter divergence for the linear tangent filter are identified. The estimation skill of the unresolved stochastic parameters by various filters is also discussed and it is shown that the linear tangent filter, despite its popularity, is completely unreliable in many dynamical regimes. The results presented here provide useful guidelines for filtering turbulent, high-dimensional, spatially extended systems with significant model errors. They also provide unambiguous benchmarks for the capabilities of linear and nonlinear extended Kalman filters on a stringent, exactly solvable test bed.

Swarming is a unique manifestation of dynamic self-assembly (DySA) 1-3 in which self-propelling objects not only organize into dissipative structures but also perform collective motions. While swarms are ubiquitous in biological systems (bacteria 4,5 , fish 6 , ants 7,8 , etc.), the examples of artificial collective movers are largely limited to complex robotics systems 9,10. The chief difficulty in making simple components swarm is to engineer interactions that would propel these components while maintaining them at a distance from and in proper orientation with respect to one another. Here, we describe a hydrodynamic system in which swarming is mediated by asymmetric convection "rolls" around small, millimeter-sized gel particles floating at a water/air interface and emitting surface active chemicals. Remarkably, for thin water layers, these convective flows give rise to interparticle attractions that bring the particles close to-but not into contact-with one another. In collections of identical particles, this previously undescribed hydrodynamic interaction leads to the formation of high-symmetry, open-lattice, stationary structures. In contrast, particles of different shapes assemble into lower-symmetry, dynamic formations

Real-time capture of the relevant features of the unresolved turbulent dynamics of complex natural systems from sparse noisy observations and imperfect models is a notoriously difficult problem. The resulting lack of observational resolution and statistical accuracy in estimating the important turbulent processes, which intermittently send significant energy to the large-scale fluctuations, hinders efficient parameterization and real-time prediction using discretized PDE models. This issue is particularly subtle and important when dealing with turbulent geophysical systems with an vast range of interacting spatio-temporal scales and rough energy spectra near the mesh scale of numerical models. Here, we introduce and study a suite of general Dynamic Stochastic Superresolution (DSS) algorithms and show that, by appropriately filtering sparse regular observations with the help of cheap stochastic exactly solvable models, one can derive stochastically 'superresolved' velocity fields and gain insight into the important characteristics of the unresolved dynamics, including the detection of the so-called black swans. The DSS algorithms operate in Fourier domain and exploit the fact that the coarse observation network aliases high-wavenumber information into the resolved waveband. It is shown that these cheap algorithms are robust and have significant skill on a test bed of turbulent solutions from realistic nonlinear turbulent spatially extended systems in the presence of a significant model error. In particular, the DSS algorithms are capable of successfully capturing time-localized extreme events in the unresolved modes, and they provide good and robust skill for recovery of the unresolved processes in terms of pattern correlation. Moreover, we show that DSS improves the skill for recovering the primary modes associated with the sparse observation mesh which is equally important in applications. The skill of the various DSS algorithms depends on the energy spectrum of the turbulent signal and the observation time relative to the decorrelation time of the turbulence at a given spatial scale in a fashion elucidated here. The DSS technique exploiting a simple Gaussian closure of the nonlinear stochastic forecast model emerges as the most suitable trade-off between the superresolution skill and computational complexity associated with estimating the cross-correlations between the aliasing modes of the sparsely observed turbulent signal. Such techniques offer a promising and efficient approach to constraining unresolved turbulent fluxes through stochastic superparameterization and a subsequent improvement in coarse-grained filtering and prediction of the next generation Atmosphere-Ocean System (AOS) models.

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