Verified Decision Procedures for Modal Logics

We describe a formalization of modal tableaux with histories for the modal logics K, KT and S4 in Lean. We describe how we formalized the static and transitional rules, the non-trivial termination and the correctness of loop-checks. The formalized tableaux are essentially executable decision procedures with soundness and completeness proved. Termination is also proved in order to define them as functions in Lean. All of these decision procedures return a concrete Kripke model in cases where the input set of formulas is satisfiable, and a proof constructed via the tableau rules witnessing unsatisfiability otherwise. We also describe an extensible formalization of backjumping and its verified implementation for the modal logic K. As far as we know, these are the first verified decision procedures for these modal logics

Atmospheric jet stream variability reflects vegetation activity in Europe

Jet streams are a key component of the climate system, whose dynamics couple closely to regional climate variability. Yet, the link between jet stream variability and vegetation activity has received little attention. Here, we leverage our understanding of the mid-latitude jet stream dynamics over the Euro-Atlantic sector to probe climate-vegetation interactions across Europe. We link indices related to the meridional location of the jet and the large-scale zonal wind speed with remotely-sensed vegetation greenness anomalies during locally-defined growing seasons. Correlations between greenness anomalies and jet latitude anomalies point to a control of the jet stream's variability on vegetation activity over large parts of Europe. This potential control is mediated by the jet latitude anomalies' correlations with temperature, soil moisture and downward surface solar radiation. The sign and strength of these correlations depend on location and time of the year. Furthermore, jet stream variability modulates conditions at the onset and end of the growing season. The link between jet latitude anomalies and vegetation greenness is not only specific to the climate zone, but also to the landclass and subperiod within the growing season. It is thus important to use a locally-defined growing season for interpreting the atmospheric controls on regional vegetation phenology. Results consistent with the correlation analysis emerge when focussing on local high or low greenness months only or on zonal wind speed anomalies, confirming the relevance of jet variability for vegetation activity

Drought Legacy in Sub-Seasonal Vegetation State and Sensitivity to Climate Over the Northern Hemisphere

Droughts affect ecosystems at multiple time scales, but their sub-seasonal legacy effects on vegetation activity remain unclear. Combining the satellite-based enhanced vegetation index MODIS EVI with a novel location-specific definition of the growing season, we quantify drought impacts on sub-seasonal vegetation activity and the subsequent recovery in the Northern Hemisphere. Drought legacy effects are quantified as changes in post-drought greenness and sensitivity to climate. We find that greenness losses under severe drought are partially compensated by a similar to+5% greening within 2-6 growing-season months following the droughts, both in woody and herbaceous vegetation but at different timings. In addition, post-drought sensitivity of herbaceous vegetation to hydrological conditions increases noticeably at high latitudes compared with the local normal conditions, regardless of the choice of drought time scales. In general, the legacy effects on sensitivity are larger in herbaceous vegetation than in woody vegetation

Early growing season anomalies in vegetation activity determine the large-scale climate-vegetation coupling in Europe

The climate-vegetation coupling exerts a strong control on terrestrial carbon budgets and will affect the future evolution of global climate under continued anthropogenic forcing. Nonetheless, the effects of climatic conditions on such coupling at specific times in the growing season remain poorly understood. We quantify the climate-vegetation coupling in Europe over 1982–2014 at multiple spatial and temporal scales, by decomposing sub-seasonal anomalies of vegetation greenness using a grid-wise definition of the growing season. We base our analysis on long-term vegetation indices (Normalized Difference Vegetation Index and two-band Enhanced Vegetation Index), growing conditions (including 2m temperature, downwards surface solar radiation, and root-zone soil moisture), and multiple teleconnection indices that reflect the large-scale climatic conditions over Europe. We find that the large-scale climate-vegetation coupling during the first two months of the growing season largely determines the full-year coupling. The North Atlantic Oscillation and Scandinavian Pattern phases one-to-two months before the start of the growing season are the dominant and contrasting drivers of the early growing season climate-vegetation coupling over large parts of boreal and temperate Europe. The East Atlantic Pattern several months in advance of the growing season exerts a strong control on the temperate belt and the Mediterranean region. The strong role of early growing season anomalies in vegetative activity within the growing season emphasizes the importance of a grid-wise definition of the growing season when studying the large-scale climate-vegetation coupling in Europe

Representation of precipitation and top-of-atmosphere radiation in a multi-model convection-permitting ensemble for the Lake Victoria Basin (East-Africa)

The CORDEX Flagship Pilot Study ELVIC (climate Extremes in the Lake VICtoria basin) was recently established to investigate how extreme weather events will evolve in this region of the world and to provide improved information for the climate impact community. Here we assess the added value of the convection-permitting scale simulations on the representation of moist convective systems over and around Lake Victoria. With this aim, 10 year present-day model simulations were carried out with five regional climate models at both PARameterized (PAR) scales (12–25 km) and Convection-Permitting (CP) scales (2.5–4.5 km), with COSMO-CLM, RegCM, AROME, WRF and UKMO. Most substantial systematic improvements were found in metrics related to deep convection. For example, the timing of the daily maximum in precipitation is systematically delayed in CP compared to PAR models, thereby improving the agreement with observations. The large overestimation in the total number of rainy events is alleviated in the CP models. Systematic improvements were found in the diurnal cycle in Top-Of-Atmosphere (TOA) radiation and in some metrics for precipitation intensity. No unanimous improvement nor deterioration was found in the representation of the spatial distribution of total rainfall and the seasonal cycle when going to the CP scale. Furthermore, some substantial biases in TOA upward radiative fluxes remain. Generally our analysis indicates that the representation of the convective systems is strongly improved in CP compared to PAR models, giving confidence that the models are valuable tools for studying how extreme precipitation events may evolve in the future in the Lake Victoria basin and its surroundings

Land-atmosphere interactions and regional Earth system dynamics due to natural and anthropogenic vegetation changes

Observation and modelling studies have indicated that the global land surfaces have been undergoing significant changes in the past few decades, driven by both natural and anthropogenic factors, such as changes in ecosystem productivity, fire and land use. Land surface changes can potentially influence local and regional climate through land-atmosphere interactions. Continued greenhouse gas emissions and current socioeconomic trends are expected to drive further land cover changes in the future, thus further understanding of land-atmosphere interactions including different feedback mechanisms is necessary to understand how future climate change will continue unfolding. Land-atmosphere interactions vary under different conditions. The strength of local land-atmosphere interactions depends on the capabilities of different land covers to control surface energy and mass exchanges, including latent and sensible heat, water and carbon. Local feedbacks can also influence regional to global climate, such as circulation changes that affect regional energy and moisture transport, or cloud cover that affects incoming radiation. Regional Earth system models (RESMs) with high resolution dynamical downscaling approaches and incorporating individual-based vegetation dynamics add value to the traditional global climate modelling studies for regions with highly complex topography or/and pronounced seasonal water deficits, potentially allowing for more refined land-atmosphere interactions studies thanks to more realistic vegetation dynamics and biophysical feedbacks, more accurate regional climate dynamics and overall richer spatial detail.In this thesis, I investigated regional land surface changes due to natural and anthropogenic vegetation changes and their impacts on land-atmosphere interactions, by applying a dynamical downscaling approach with RCA-GUESS, a RESM that couples the Rossby Centre regional climate model RCA4 to LPJ-GUESS, an ecosystem model that combines an individual-based representation of vegetation structure and dynamics with process-based physiology and biogeochemistry. Europe, Africa and South America were chosen as research domains. In the land surface study based on LPJ-GUESS simulations, I showed that future changes in the fire regime over Europe, driven by climate and socioeconomic change, were important for projecting future land surface changes. Fire-vegetation interactions and socioeconomic effects emerged as important uncertainties for future burned area. My study on land-atmosphere interactions based on RCA-GUESS simulations indicated that the hydrological cycle in the tropics was sensitive to land cover changes over semi-arid regions in Africa, and that biophysical feedbacks were important through their modulation of regional circulation patterns. A study based on the analysis of empirical datasets and CMIP5 ESMs outputs revealed that simulated climate biases are the main cause of model-data discrepancies. Models and data shared a marked hydrological relationship that suggested that decreased precipitation and land use change constituted the largest threats to the future Amazon forest. A study based on RCA-GUESS simulations with a realistic land use scenario identified both positive and negative impacts of land use on natural ecosystem productivity in the Amazon through its effects on the local and the regional climate

A Bi-Directional Extensible Interface Between Lean and Mathematica

We implement a user-extensible ad hoc connection between the Lean proof assistant and the computer algebra system Mathematica. By reflecting the syntax of each system in the other and providing a flexible interface for extending translation, our connection allows for the exchange of arbitrary information between the two systems. We show how to make use of the Lean metaprogramming framework to verify certain Mathematica computations, so that the rigor of the proof assistant is not compromised. We also use Mathematica as an untrusted oracle to guide proof search in the proof assistant and interact with a Mathematica notebook from within a Lean session. In the other direction, we import and process Lean declarations from within Mathematica. The proof assistant library serves as a database of mathematical knowledge that the CAS can display and explore

A Bi-Directional Extensible Interface Between Lean and Mathematica

We implement a user-extensible ad hoc connection between the Lean proof assistant and the computer algebra system Mathematica. By reflecting the syntax of each system in the other and providing a flexible interface for extending translation, our connection allows for the exchange of arbitrary information between the two systems. We show how to make use of the Lean metaprogramming framework to verify certain Mathematica computations, so that the rigor of the proof assistant is not compromised. We also use Mathematica as an untrusted oracle to guide proof search in the proof assistant and interact with a Mathematica notebook from within a Lean session. In the other direction, we import and process Lean declarations from within Mathematica. The proof assistant library serves as a database of mathematical knowledge that the CAS can display and explore.Comment: arXiv admin note: substantial text overlap with arXiv:1712.0928

Refined Wind Simulation Based on Large Eddy Simulation and Mesoscale Numerical Weather Model

[Introduction] Combining mesoscale numerical model and large eddy simulation (LES) model, numerical sumulation of sub-kilometer-scale project unit placement is carried out, which takes into account atmospheric boundary layer changes. It provides offshore wind turbine projects with high-efficiency power generation placement schemes. [Method] This study converted the mesoscale numerical weather simulation results into boundary conditions for the input of the LES model and introduced model parameters reflecting the operation of an actual wind farm into the LES simulation. The numerical sumulation experiments of the ambient wind field in the wind farm region was carried out under the consideration of the change of the actual atmospheric boundary layer, and the results of the refined simulation scheme of this wind field were evaluated based on the observation data collected from the wind farm. [Result] The simulation results indicate that by converting the results of the mesoscale weather model into the dynamic drive which is read by the LES model and simulating the wind field where the wind farm is located based on the model, the simulation results are able to replicate the changes in the external wind field after passing through the wind farm and the wake generated within the wind turbine fleet, as well as its impact on the internal wind field of the wind farm. The root mean square error of wind speed simulation at the hub of wind turbines is 1.54 m/s. [Conclusion] The refined wind field simulation scheme, which takes into account the variation of mesoscale meteorological elements and the impact of wind farms on the ambient wind field, can provide guidance for the design phase of actual projects