Evaluating a stochastic parametrization for a fast--slow system using the Wasserstein distance

Constructing accurate, flexible, and efficient parametrizations is one of the great challenges in the numerical modeling of geophysical fluids. We consider here the simple yet paradigmatic case of a Lorenz 84 model forced by a Lorenz 63 model and derive a parametrization using a recently developed statistical mechanical methodology based on the Ruelle response theory. We derive an expression for the deterministic and the stochastic component of the parametrization and we show that the approach allows for dealing seamlessly with the case of the Lorenz 63 being a fast as well as a slow forcing compared to the characteristic timescales of the Lorenz 84 model. We test our results using both standard metrics based on the moments of the variables of interest as well as Wasserstein distance between the projected measure of the original system on the Lorenz 84 model variables and the measure of the parametrized one. By testing our methods on reduced-phase spaces obtained by projection, we find support for the idea that comparisons based on the Wasserstein distance might be of relevance in many applications despite the curse of dimensionality

Stochastic control applied to the ISWEC Wave Energy System

ISWEC (Inertial Sea Wave Energy Converter) is a fl oating marine device able to harvest sea waves energy by the interaction between the pitching motion of a fl oater and a spinning fl ywheel which can drive an electric PTO. In the ISWEC the hull dynamics is governed and controlled by the gyroscopic torque. The optimal control logic results in tuning the fl oater dynamics to the incoming waves in order to maximize the power transfer from the waves to the fl oater. In this paper the control problems of the ISWEC are stated and a control scheme based on the sub-optimal stochastic control logic is presented. The control scheme here presented has been tested using real wave records acquired at the deployment location in Pantelleria Island, which is one of the most energetic sites of the Mediterranean Sea

Evaluating the performance of climate models based on Wasserstein distance

We propose a methodology for intercomparing climate models and evaluating their performance against benchmarks based on the use of the Wasserstein distance (WD). This distance provides a rigorous way to measure quantitatively the difference between two probability distributions. The proposed approach is flexible and can be applied in any number of dimensions; it allows one to rank climate models taking into account all the moments of the distributions. By selecting the combination of climatic variables and the regions of interest, it is possible to highlight specific model deficiencies. The WD enables a comprehensive evaluation of the skill of a climate model. We apply this approach to a selected number of physical fields, ranking the models in terms of their performance in simulating them and pinpointing their weaknesses in the simulation of some of the selected physical fields in specific areas of the Earth

Response formulae for n-point correlations in statistical mechanical systems and application to a problem of coarse graining

Predicting the response of a system to perturbations is a key challenge in mathematical and natural sciences. Under suitable conditions on the nature of the system, of the perturbation, and of the observables of interest, response theories allow to construct operators describing the smooth change of the invariant measure of the system of interest as a function of the small parameter controlling the intensity of the perturbation. In particular, response theories can be developed both for stochastic and chaotic deterministic dynamical systems, where in the latter case stricter conditions imposing some degree of structural stability are required. In this paper we extend previous findings and derive general response formulae describing how n-point correlations are affected by perturbations to the vector flow. We also show how to compute the response of the spectral properties of the system to perturbations. We then apply our results to the seemingly unrelated problem of coarse graining in multiscale systems: we find explicit formulae describing the change in the terms describing parameterisation of the neglected degrees of freedom resulting from applying perturbations to the full system. All the terms envisioned by the Mori-Zwanzig theory - the deterministic, stochastic, and non-Markovian terms - are affected at 1st order in the perturbation. The obtained results provide a more comprehesive understanding of the response of statistical mechanical systems to perturbations and contribute to the goal of constructing accurate and robust parameterisations and are of potential relevance for fields like molecular dynamics, condensed matter, and geophysical fluid dynamics. We envision possible applications of our general results to the study of the response of climate variability to anthropogenic and natural forcing and to the study of the equivalence of thermostatted statistical mechanical systems

Analysis of a bistable climate toy model with physics-based machine learning methods

We propose a comprehensive framework able to address both the predictability of the first and of the second kind for high-dimensional chaotic models. For this purpose, we analyse the properties of a newly introduced multistable climate toy model constructed by coupling the Lorenz ’96 model with a zero-dimensional energy balance model. First, the attractors of the system are identified with Monte Carlo Basin Bifurcation Analysis. Additionally, we are able to detect the Melancholia state separating the two attractors. Then, Neural Ordinary Differential Equations are applied to predict the future state of the system in both of the identified attractors

Mechanics and thermodynamics of a new minimal model of the atmosphere

The understanding of the fundamental properties of the climate system has long benefitted from the use of simple numerical models able to parsimoniously represent the essential ingredients of its processes. Here, we introduce a new model for the atmosphere that is constructed by supplementing the now-classic Lorenz ’96 one-dimensional lattice model with temperature-like variables. The model features an energy cycle that allows for energy to be converted between the kinetic form and the potential form and for introducing a notion of efficiency. The model’s evolution is controlled by two contributions—a quasi-symplectic and a gradient one, which resemble (yet not conforming to) a metriplectic structure. After investigating the linear stability of the symmetric fixed point, we perform a systematic parametric investigation that allows us to define regions in the parameters space where at steady-state stationary, quasi-periodic, and chaotic motions are realised, and study how the terms responsible for defining the energy budget of the system depend on the external forcing injecting energy in the kinetic and in the potential energy reservoirs. Finally, we find preliminary evidence that the model features extensive chaos. We also introduce a more complex version of the model that is able to accommodate for multiscale dynamics and that features an energy cycle that more closely mimics the one of the Earth’s atmosphere