Abstract. The development of highly efficient, robust and scalable numerical algorithms lags behind the rapid increase in massive parallelism of modern hardware. We address this challenge with the accelerated pseudo-transient (PT) iterative method and present a physically motivated derivation. We analytically determine optimal iteration parameters for a variety of basic physical processes and confirm the validity of theoretical predictions with numerical experiments. We provide an efficient numerical implementation of PT solvers on graphical processing units (GPUs) using the Julia language. We achieve a parallel efficiency of more than 96 % on 2197 GPUs in distributed-memory parallelisation weak-scaling benchmarks. The 2197 GPUs allow for unprecedented tera-scale solutions of 3D variable viscosity Stokes flow on 49953 grid cells involving over 1.2 trillion degrees of freedom (DoFs). We verify the robustness of the method by handling contrasts up to 9 orders of magnitude in material parameters such as viscosity and arbitrary distribution of viscous inclusions for different flow configurations. Moreover, we show that this method is well suited to tackle strongly nonlinear problems such as shear-banding in a visco-elasto-plastic medium. A GPU-based implementation can outperform direct-iterative solvers based on central processing units (CPUs) in terms of wall time, even at relatively low spatial resolution. We additionally motivate the accessibility of the method by its conciseness, flexibility, physically motivated derivation and ease of implementation. This solution strategy thus has a great potential for future high-performance computing (HPC) applications, and for paving the road to exascale in the geosciences and beyond.

The transition to a low-carbon economy will increase demand for a wide range of metals, notably copper, which is used extensively in power generation and in electric vehicles. Increased demand will require new, sustainable approaches to copper exploration and extraction. Conventional copper mining entails energy-intensive extraction of relatively low-grade ore from large open pits or underground mines and subsequent ore refining. Most copper derives ultimately from hot, hydrous magmatic fluids. Ore formation involves phase separation of these fluids to form copper-rich hypersaline liquids (or ‘brines') and subsequent precipitation of copper sulfides. Geophysical surveys of many volcanoes reveal electrically conductive bodies at around 2 km depth, consistent with lenses of brine hosted in porous rock. Building upon emerging concepts in crustal magmatism, we explore the potential of sub-volcanic brines as an in situ source of copper and other metals. Using hydrodynamic simulations, we show that 10 000 years of magma degassing can generate a Cu-rich brine lens containing up to 1.4 Mt Cu in a rock volume of a few km 3 at approximately 2 km depth. Direct extraction of metal-rich brines represents a novel development in metal resource extraction that obviates the need for conventional mines, and generates geothermal power as a by-product.

the degree and extent of crustal hydrothermal alteration related to the eruption of large igneous provinces is poorly known and not easily recognizable in the field. We here report a new δ 18 o dataset for dikes and lavas from the Columbia River Basalt Group (16-15 Ma) in the western USA, and document that dikes on average are 1-2‰ more depleted in δ 18 O than basalt flows. We show that this observation is best explained with the involvement of heated meteoric waters during their cooling in the crust. the largest 6-8‰ depletion is found around and inside a 10 m-thick feeder dike that intruded the 125 Ma Wallowa tonalitic batholith. This dike likely operated as a magma conduit for 4-7 years, based on the extent of heating and melting its host rocks. We show that this dike also created a hydrothermal system around its contacts extending up to 100 m into the surrounding bedrock. A model that considers (a) hydrothermal circulation around the dike, (b) magma flow and (c) oxygen isotope exchange rates, suggests that the hydrothermal system operated for ~150 years after the cessation of magma flow. In agreement with a previously published (U-Th)/He thermochronology profile, our model shows that rocks 100 m away from such a dike can be hydrothermally altered. Collectively, our sample set is the first documentation of the widespread hydrothermal alteration of the shallow crust caused by the intrusion of dikes and sills of the columbia River Basalt province. it is estimated that heating and hydrothermal alteration of sediments rich in organic matter and carbonates around the dikes and sills releases 18 Gt of greenhouse gases (cH 4 and co 2). Furthermore, hydrothermal δ 18 o depletion of rocks around dikes covers 500-600 km 3 , which, when scaled to the total CRB province constitutes 31,000 km 3 of low-δ 18 o rocks. these volumes of crust depleted in δ 18 O are sufficient to explain the abundant low-δ 18 o magmas in eastern oregon and western idaho. this work also demonstrates that the width and magnitude of δ 18 O depletion around dikes can identify them as feeders. Given this, we here interpret Paleoproterozoic dikes in Karelia with the world's lowest δ 18 o depletions (−27.8‰) as feeders to the coeval large igneous province aged 2.2-2.4 Ga that operated under the Snowball Earth glaciation conditions.