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We present a Parseval tight wavelet frame for the representation and analysis of velocity vector fields of incompressible fluids. Our wavelets have closed form expressions in the frequency and spatial domains, are divergence free in the ideal, analytic sense, have a multi-resolution structure and fast transforms, and an intuitive correspondence to common flow phenomena. Our construction also allows for well defined directional selectivity, e.g. to model the behavior of divergence free vector fields in the vicinity of boundaries or to represent highly directional features like in a von Kármán vortex street. We demonstrate the practicality and efficiency of our construction by analyzing the representation of different divergence free vector fields in our wavelets.

Earth System Models (ESM) are our main tool for projecting the impacts of climate change. However, running these models at sufficient resolution for local-scale risk-assessments is not computationally feasible. Deep learning-based super-resolution models offer a promising solution to downscale ESM outputs to higher resolutions by learning from data. Yet, due to regional variations in climatic processes, these models typically require retraining for each geographical area–demanding high-resolution observational data, which is unevenly available across the globe. This highlights the need to assess how well these models generalize across geographic regions. To address this, we introduce RainShift, a dataset and benchmark for evaluating downscaling under geographic distribution shifts. We evaluate state-of-the-art downscaling approaches including GANs and diffusion models in generalizing across data gaps between the Global North and Global South. Our findings reveal substantial performance drops in out-of-distribution regions, depending on model and geographic area. While expanding the training domain generally improves generalization, it is insufficient to overcome shifts between geographically distinct regions. We show that addressing these shifts through, for example, domain adaptation can improve spatial generalization. Our work advances the global applicability of downscaling methods and represents a step toward reducing inequities in access to high-resolution climate information.

Climate science has been revolutionized by the combined effects of an exponential growth in computing power, which has enabled more sophisticated and higher-resolution simulations to be made of the climate system, and an exponential increase in observations since the first weather satellite was put in orbit. Big data and associated algorithms, coalesced under the field of machine learning (ML), offer the opportunity to study the physics of the climate system in ways, and with an amount of detail, that were previously infeasible. Additionally, ML can ask causal questions to determine whether one or more variables cause or affect one or more outcomes and improve prediction skills beyond classical limits. Furthermore, when paired with modelling experiments or robust research on model parameterizations, ML can accelerate computations, increasing accuracy and generating very large ensembles with a fraction of the computational cost of traditional systems. In this Review, we outline the accomplishments of ML in climate physics. We discuss how ML has been used to tackle long-standing problems in the reconstruction of observational data, representation of sub-grid-scale phenomena and climate (and weather) prediction. Finally, we consider the benefits and major challenges of exploiting ML in studying complex systems.Artificial intelligence techniques, specifically machine learning, are being increasingly applied to climate physics owing to the growing availability of big data and increasing computational power. This Review focuses on key results obtained with machine learning in reconstruction, sub-grid-scale parameterization, and weather or climate prediction.

The cryosphere plays a crucial role in the Earth's climate system, making accurate sea-ice simulation essential for improving climate projections. To achieve higher-resolution simulations, graphics processing units (GPUs) have become increasingly appealing due to their higher floating-point peak performance compared to central processing units (CPUs). However, harnessing the full theoretical performance of GPUs often requires significant effort in redesigning algorithms and careful implementation. Recently, several frameworks have emerged that aim to simplify general-purpose GPU programming. In this study, we evaluate multiple such frameworks, including CUDA, SYCL, Kokkos, and PyTorch, for the parallelization of neXtSIM-DG, a finite-element-based dynamical core for sea ice. Based on our assessment of usability and performance, CUDA demonstrates the best performance while Kokkos is a suitable option for its robust heterogeneous computing capabilities. Our complete implementation of the momentum equation using Kokkos achieves a 6-fold speedup on the GPU compared to our OpenMP-based CPU code, while maintaining competitiveness when run on the CPU. Additionally, we explore the use of lower-precision floating-point types on the GPU, showing that switching to single precision can further accelerate sea-ice codes.

The ability of numerical sea ice models to reproduce localized deformation features associated with fracture processes is key for an accurate representation of the ice dynamics and of dynamically coupled physical processes in the Arctic and Antarctic. Equally key is the capacity of these models to minimize the numerical diffusion stemming from the advection of these features to ensure that the associated strong gradients persist in time, without the need to unphysically re-inject energy for re-localization. To control diffusion and improve the approximation quality, we present a new numerical core for the dynamics of sea ice that is based on higher-order finite-element discretizations for the momentum equation and higher-order discontinuous Galerkin methods for the advection. The mathematical properties of this core are discussed, and a detailed description of an efficient shared-memory parallel implementation is given. In addition, we present different numerical tests and apply the new framework to a benchmark problem to quantify the advantages of the higher-order discretization. These tests are based on Hibler's viscous–plastic sea ice model, but the implementation of the developed framework in the context of other physical models reproducing a strong localization of the deformation is possible.

To manage Earth in the Anthropocene, new tools, new institutions, and new forms of international cooperation will be required. Earth Virtualization Engines is proposed as an international federation of centers of excellence to empower all people to respond to the immense and urgent challenges posed by climate change.

We present a Parseval tight wavelet frame for the representation and analysis of velocity vector fields of incompressible fluids. Our wavelets have closed form expressions in the frequency and spatial domains, are divergence free in the ideal, analytic sense, have a multi-resolution structure and fast transforms, and an intuitive correspondence to common flow phenomena. Our construction also allows for well defined directional selectivity, e.g. to model the behavior of divergence free vector fields in the vicinity of boundaries or to represent highly directional features like in a von Kármán vortex street. We demonstrate the practicality and efficiency of our construction by analyzing the representation of different divergence free vector fields in our wavelets.

We present a local Fourier slice equation that enables local and sparse projection of a signal. Our result exploits that a slice in frequency space is an iso-parameter set in spherical coordinates. Therefore, the projection of suitable wavelets defined separably in these coordinates can be computed analytically, yielding a sequence of wavelets closed under projection. Our local Fourier slice equation then realizes projection as reconstruction with \"sliced\" wavelets with computational costs that scale linearly in the complexity of the projected signal. We numerically evaluate the performance of our local Fourier slice equation for synthetic test data and tomographic reconstruction, demonstrating that locality and sparsity can significantly reduce computation times and memory requirements.

Packed beds are employed in a wide range of industrial processes, making their optimization an important objective. This requires a thorough understanding of the flow characteristics in the interstices of the bed and the influence of parameters such as packing material, fluid type and flow rate. Next to numerical simulations that have been performed, liquid flows in packed beds have been experimentally investigated by Particle Image Velocimetry (PIV), with optical accessibility obtained using refractive index matching and transparent geometries. For gaseous flows through packed beds, where refractive index matching is not available, there is, in contrast, still a lack of experimental work, because the different optical properties of gas and transparent packing geometries generate optical aberrations. The approximation of gaseous flows with results from dynamically similar, liquid flows provide thereby no reliable alternative because they might be inaccurate and lead to incorrect conclusions about the actual gas flow. The high optical aberrations created by transparent packed beds in PIV experiments can be corrected using ray tracing simulations of a geometric-optical model of the viewing media. The present work proposes a novel image correction technique based on this approach that overcomes the considerable optical distortions caused by arrangements of transparent solid spheres. We analyze the influence of spheres along the camera view in three test cases: the first studies optical distortions of target images, the second PIV computations of a known displacement of a printed particle field, and the third PIV measurements of an air-jet flow. The image correction is qualitatively and quantitatively evaluated by computing errors in target images and in velocity fields. We show that the high optical aberrations that are caused by the spheres can mostly be corrected using ray tracing, allowing for accurate optical measurements in complex geometries, such as packed beds. The approach proposed in this work might be extended to even more challenging geometries in order to solve limitations of existing techniques in the literature.