Variational Integrators for Nonvariational Partial Differential Equations

Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods. These methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered problem. Even though for a large class of systems this requirement is fulfilled, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type frequently encountered in fluid dynamics or plasma physics. On the other hand, it is always possible to embed an arbitrary dynamical system into a larger Lagrangian system using the method of formal (or adjoint) Lagrangians. We investigate the application of the variational integrator method to formal Lagrangians, and thereby extend the application domain of variational integrators to include potentially all dynamical systems. The theory is supported by physically relevant examples, such as the advection equation and the vorticity equation, and numerically verified. Remarkably, the integrator for the vorticity equation combines Arakawa's discretisation of the Poisson brackets with a symplectic time stepping scheme in a fully covariant way such that the discrete energy is exactly preserved. In the presentation of the results, we try to make the geometric framework of variational integrators accessible to non specialists.Comment: 49 page

The relationship between the Wigner-Weyl kinetic formalism and the complex geometrical optics method

The relationship between two different asymptotic techniques developed in order to describe the propagation of waves beyond the standard geometrical optics approximation, namely, the Wigner-Weyl kinetic formalism and the complex geometrical optics method, is addressed. More specifically, a solution of the wave kinetic equation, relevant to the Wigner-Weyl formalism, is obtained which yields the same wavefield intensity as the complex geometrical optics method. Such a relationship is also discussed on the basis of the analytical solution of the wave kinetic equation specific to Gaussian beams of electromagnetic waves propagating in a ``lens-like'' medium for which the complex geometrical optics solution is already available.Comment: Extended version comprising two new section

Global transpiration data from sap flow measurements : the SAPFLUXNET database

Plant transpiration links physiological responses of vegetation to water supply and demand with hydrological, energy, and carbon budgets at the land-atmosphere interface. However, despite being the main land evaporative flux at the global scale, transpiration and its response to environmental drivers are currently not well constrained by observations. Here we introduce the first global compilation of whole-plant transpiration data from sap flow measurements (SAPFLUXNET, https://sapfluxnet.creaf.cat/, last access: 8 June 2021). We harmonized and quality-controlled individual datasets supplied by contributors worldwide in a semi-automatic data workflow implemented in the R programming language. Datasets include sub-daily time series of sap flow and hydrometeorological drivers for one or more growing seasons, as well as metadata on the stand characteristics, plant attributes, and technical details of the measurements. SAPFLUXNET contains 202 globally distributed datasets with sap flow time series for 2714 plants, mostly trees, of 174 species. SAPFLUXNET has a broad bioclimatic coverage, with woodland/shrubland and temperate forest biomes especially well represented (80 % of the datasets). The measurements cover a wide variety of stand structural characteristics and plant sizes. The datasets encompass the period between 1995 and 2018, with 50 % of the datasets being at least 3 years long. Accompanying radiation and vapour pressure deficit data are available for most of the datasets, while on-site soil water content is available for 56 % of the datasets. Many datasets contain data for species that make up 90 % or more of the total stand basal area, allowing the estimation of stand transpiration in diverse ecological settings. SAPFLUXNET adds to existing plant trait datasets, ecosystem flux networks, and remote sensing products to help increase our understanding of plant water use, plant responses to drought, and ecohydrological processes. SAPFLUXNET version 0.1.5 is freely available from the Zenodo repository (https://doi.org/10.5281/zenodo.3971689; Poyatos et al., 2020a). The "sapfluxnetr" R package - designed to access, visualize, and process SAPFLUXNET data - is available from CRAN.Peer reviewe

EUROfusion Integrated Modelling (EU-IM) capabilities and selected physics applications

International audienceRecent developments and achievements of the EUROfusion Code Development for Integrated Modelling project (WPCD), which aim is to provide a validated integrated modelling suite for the simulation and prediction of complete plasma discharges in any tokamak, are presented. WPCD develops generic complex integrated simulations, workflows, for physics applications, using the standardized European Integrated Modelling (EU-IM) framework. Selected physics applications of EU-IM workflows are illustrated in this paper

UNIQUENESS OF THE CAUCHY DATUM FOR THE TEMPERED-IN-TIME RESPONSE AND CONDUCTIVITY OPERATOR OF A PLASMA

We study the linear Vlasov equation with a given electric field E ∈ S, where S is the space of Schwartz functions. The associated damped partial differential equation has a unique tempered solution, which fixes the needed Cauchy datum. This tempered solution then converges to the causal solution of the linear Vlasov equation when the damping parameter goes to zero. This result allows us to define the plasma conductivity operator σ, which gives the current density j = σ(E) induced by the electric field E. We prove that σ is continuous from S to its dual S ′. We can treat rigorously the case of uniform non-magnetized non-relativistic plasma (linear Landau damping) and the case of uniform magnetized relativistic plasma (cyclotron damping). In both cases, we demonstrate that the main part of the conductivity operator is a pseudo-differential operator and we give its expression rigorously. This matches the formal results widely used in the theoretical physics community