Meshes approximate the boundaries of a geometry when the boundaries are curved. The accuracy of the mesh then affects the error of computations of initial boundary value problems for partial differential equations, especially when using high order methods. Here, we derive global estimates for the error in solutions of linear hyperbolic systems due to inaccurate boundary geometry. We show that the error is bounded by data and bounded in time when the solutions in the true and approximate domains are bounded. Just evaluating boundary data at the correct location has a secondary effect on the error, whereas the primary errors are from the Jacobian and metric terms. In two space dimensions, specifically, we show that to lowest order the errors are proportional to the errors in the boundary curve locations and their derivatives. Therefore, high order accuracy computations cannot be obtained unless the mesh is also high order. The results illustrate the importance of accurately approximating boundaries and should be helpful guides for high-order mesh generation for advection-dominated problems and the design of optimization algorithms for boundary approximations.

Overset grid methods handle complex geometries by overlapping simpler, geometry-fitted grids to cover the original, more complex domain. However, ensuring their stability—particularly at high orders—remains a theoretical challenge: although overset grid methods perform robustly in extensive practical use, general stability proofs are not available. In this work, we address this gap by developing a discrete counterpart to the recent well-posedness analysis of Kopriva, Gassner, and Nordström for continuous overset domain initial-boundary-value problems. To this end, we introduce the novel concept of sub-cell summation-by-parts (SBP) operators. These discrete derivative operators mimic integration by parts at a sub-cell level. By exploiting this sub-cell SBP property, we develop provably conservative and energy-stable overset grid methods for fixed one-dimensional overset domains that do not change with time or under grid refinement, providing a step toward stability proofs for overset grid methods based on the energy method.

In this work, we develop a new hydrostatic reconstruction procedure to construct well-balanced schemes for one and multilayer shallow water flows, including wetting and drying. Initially, we derive the method for a path-conservative finite volume scheme and combine it with entropy conservative fluxes and suitable numerical dissipation to preserve an entropy inequality in the semi-discrete case. We then combine the novel hydrostatic reconstruction with a collocated nodal split-form discontinuous Galerkin spectral element method, extending the method to high-order and curvilinear meshes. The high-order method incorporates an additional positivity-limiter and is blended with a compatible subcell finite volume method to maintain well-balancedness at wet/dry fronts. We prove entropy stability, well-balancedness, and positivity-preservation for both methods. Numerical results for the high-order method validate the theoretical findings and demonstrate the robustness of the scheme.

Atmospheric rivers (ARs) are long, narrow bands of moisture that propagate poleward from the midlatitudes and occasionally reach the Antarctic Ice Sheet. Despite occurring only ~1% of the time, Antarctic ARs contribute 10% of the annual precipitation and are major drivers for heatwaves, foehn events, and surface melting on ice shelves. While snowfall is currently the dominant impact of ARs over the grounded Antarctic Ice Sheet, the relative contribution of ARs to snowfall, rainfall, and surface melt may change in a warming climate, along with the frequency and intensity of AR events themselves. Here, we use the Community Earth System Model version 2 (CESM2) Large Ensemble to detect ARs during the current period (1980–2014) and future climate (2015–2100) under the SSP370 radiative forcing scenario. We use an AR detection threshold for the current period based on the 98th percentile of the meridional component of integrated vapor transport (vIVT). To account for projected future increases in atmospheric moisture content (Clausius-Clapeyron effect) and its impacts on vIVT, we scale our AR detection threshold for the future period by the relative change in integrated water vapor compared to the present-day climatology. We then describe how the frequency, intensity, and year-to-year variability in Antarctic ARs changes by the end of the 21st century by region, with links to changes in the large-scale atmospheric circulation accompanying ARs. Finally, we quantify AR-attributed precipitation, precipitation variability, and trends in the future climate, ultimately providing an early assessment of future AR-driven changes to Antarctic surface mass balance.

In many regions, extreme temperatures increase faster than background global warming due to complex atmospheric circulation changes. This has been overlooked in Antarctica. Here, we examine extreme temperatures recorded at 16 weather stations since 1950, and identify their changes and drivers. Over the Antarctic Peninsula and the South Pole, 10-year return level temperatures increased by up to 1.19°C per decade since 1980, outpacing mean temperature trends. Most extreme events, including statistically implausible or black swan events, were driven by intense atmospheric rivers linked to Rossby wave trains forced by tropical Pacific convection. In contrast, 10-year return level temperature decreased over most of East Antarctica, despite increasing atmospheric river activity, as the mean cooling muted the impacts of warm intrusion until recently. Our results highlight how the entire continent could become vulnerable in a warmer climate or enhanced tropical convective activity, as illustrated by the March 2022 heatwave.

Antarctic atmospheric rivers (ARs) are a form of extreme weather that transport heat and moisture from the Southern Hemisphere subtropics and/or mid-latitudes to the Antarctic continent. Present-day AR events generally have a positive influence on the Antarctic ice-sheet mass balance by producing heavy snowfall, yet they also cause melt of sea ice and coastal ice sheet areas, as well as ice shelf destabilization. In this Review, we explore the atmospheric dynamics and impacts of Antarctic ARs over their life cycle to better understand their net contributions to ice-sheet mass balance. ARs occur in high-amplitude pressure couplets, and those strong enough to reach the Antarctic are often formed within Rossby waves initiated by tropical convection. Antarctic ARs are rare events (~3 days per year per location) but have been responsible for 50–70% of extreme snowfall events in East Antarctica since the 1980s. However, they can also trigger extensive surface melting events, such as the final ice shelf collapse of Larsen A in 1995 and Larsen B in 2002. Climate change will likely cause stronger ARs as anthropogenic warming increases atmospheric water vapour. Future research must determine how these climate change impacts will alter the relationship among Antarctic ARs, net ice-sheet mass balance and future sea-level rise. Atmospheric rivers provide the majority of water vapour transport to the high latitudes. This Review summarizes Antarctic atmospheric river dynamics and climatology and discusses their impacts on the mass balance of the Antarctic ice sheet.

Recent numerical modeling and theoretical work deduce that potential vorticity (PV) can turn negative in the Northern Hemisphere as a result of localized convective heating embedded in vertical wind shear. It has been further postulated that negative potential vorticity (NPV) may be relevant for the large-scale circulation, as it has been observed to grow in scale into elongated mesoscale bands when in close proximity to the jet stream, accelerating jet stream winds and degrading numerical weather prediction skill. However, these findings are largely confined to case studies. Here, we use a climatological and composite perspective to evaluate the occurrence of elongated bands of NPV over the northwest Atlantic and its implications for jet stream dynamics. This research focuses on synoptic-scale bands (>1650 km) of NPV that are in close proximity (< 100 km) to the jet stream (termed NPV-jet interactions) using ERA5 data from January 2000 to December 2021. Climatological characteristics show that NPV-jet interactions occur most frequently over the coastal western Atlantic during boreal winter along 40 degrees N. This latitude band has also seen an 11 % increase (relative change) in NPV-jet interactions over the 22-year time period. Separating NPV-jet interactions into three distinct large-scale flow patterns using k-means clustering conceptually illustrates the evolution of NPV features from their initial formation along the westward flank of the ridge to the eastern flank of the ridge. The large-scale environment of NPV-jet interactions is characterized by a trough-ridge couplet adjacent to positive integrated vapor transport (IVT) anomalies, conducive to warm conveyor belts and mesoscale convective systems. Even when NPV is positioned in a more adiabatic environment (far away from regions of strong IVT anomalies), robust positive-PV gradient and wind speed anomalies exist along the jet stream. Inspecting three detailed case studies that serve as archetypes of the three clusters, we showed that the presence of NPV near the jet stream adiabatically enhances wave activity flux due to NPV mutually strengthening momentum transport and the ageostrophic flux of the geopotential. The results show that the close proximity of synoptic-scale NPV to the jet stream is conducive to the occurrence of wind speed maxima and could be dynamically relevant in enhancing downstream development despite NPV's theorized origin from submesoscales.

We show that even though the Discontinuous Galerkin Spectral Element Method is stable for hyperbolic boundary-value problems, and the overset domain problem is well-posed in an appropriate norm, the energy of the approximation of the latter is bounded by data only for fixed polynomial order, mesh, and time. In the absence of dissipation, coupling of the overlapping domains is destabilizing by allowing positive eigenvalues in the system to be integrated in time. This coupling can be stabilized in one space dimension by using the upwind numerical flux. To help provide additional dissipation, we introduce a novel penalty method that applies dissipation at arbitrary points within the overlap region and depends only on the difference between the solutions. We present numerical experiments in one space dimension to illustrate the implementation of the well-posed penalty formulation, and show spectral convergence of the approximations when sufficient dissipation is applied.

Although rare, atmospheric rivers (ARs) substantially influence the interannual variability of Antarctic surface mass balance. We identify characteristics unique to AR environments by comparing (1) AR, (2) Analog (environments that feature high-low pressure couplets, similar to AR environments, but no AR), and (3) Top AR (high-precipitation AR timesteps) during 1980–2019 around Antarctica. We find significant differences between AR and Analog environments including more intense and poleward-shifted mid-tropospheric geopotential height couplets as well as larger atmospheric moisture anomalies. We find similar significant enhancement in synoptic-scale dynamic drivers of Top ARs compared to AR environments, but no significant difference in local integrated water vapor anomalies. Instead, our results highlight the importance of large-scale dynamic drivers of Top AR timesteps, including connections between high-precipitation ARs and Rossby waves excited by tropical convection. This deeper understanding of Antarctic AR environments provides context for interpreting future changes to the Antarctic surface mass balance.

We use the framework of upwind summation-by-parts (SBP) operators developed by Mattsson (2017, doi:10.1016/j.jcp.2017.01.042) and study different flux vector splittings in this context. To do so, we introduce discontinuous-Galerkin-like interface terms for multi-block upwind SBP methods applied to nonlinear conservation laws. We investigate the behavior of the upwind SBP methods for flux vector splittings of varying complexity on Cartesian as well as unstructured curvilinear multi-block meshes. Moreover, we analyze the local linear/energy stability of these methods following Gassner, Svärd, and Hindenlang (2022, doi:10.1007/s10915-021-01720-8). Finally, we investigate the robustness of upwind SBP methods for challenging examples of shock-free flows of the compressible Euler equations such as a Kelvin-Helmholtz instability and the inviscid Taylor-Green vortex.

High-order numerical methods for conservation laws are highly sought after due to their potential efficiency. However, it is challenging to ensure their robustness, particularly for under-resolved flows. Baseline high-order methods often incorporate stabilization techniques that must be applied judiciously-sufficient to ensure simulation stability but restrained enough to prevent excessive dissipation and loss of resolution. Recent studies have demonstrated that combining upwind summation-by-parts (USBP) operators with flux vector splitting can increase the robustness of finite difference (FD) schemes without introducing excessive artificial dissipation. This work investigates whether the same approach can be applied to nodal discontinuous Galerkin (DG) methods. To this end, we demonstrate the existence of USBP operators on arbitrary grid points and provide a straightforward procedure for their construction. Our discussion encompasses a broad class of USBP operators, not limited to equidistant grid points, and enables the development of novel USBP operators on Legendre-Gauss-Lobatto (LGL) points that are well-suited for nodal DG methods. We then examine the robustness properties of the resulting DG-USBP methods for challenging examples of the compressible Euler equations, such as the Kelvin-Helmholtz instability. Similar to high-order FD-USBP schemes, we find that combining flux vector splitting techniques with DG-USBP operators does not lead to excessive artificial dissipation. Furthermore, we find that combining lower-order DG-USBP operators on three LGL points with flux vector splitting indeed increases the robustness of nodal DG methods. However, we also observe that higher-order USBP operators offer less improvement in robustness for DG methods compared to FD schemes. We provide evidence that this can be attributed to USBP methods adding dissipation only to unresolved modes, as FD schemes typically have more unresolved modes than nodal DG methods.

Situated in the Upper Midwest, Minnesota's midcontinental location places it in a climate transition zone between eastern U.S. humid conditions and western semiarid conditions as well as between warm, moist air from the Gulf of Mexico to the south and drier, polar air to the north. Potential adverse impacts on ecosystems due to changing climate and precipitation patterns, together with ongoing fl ash fl ooding risks, indicate that heavy rainfall occurrence and distribution are important considerations for Minnesota. This research used ERA5 reanalysis data with 0.258 8 grid spacing during May-September 1959-2021 to investigate the synoptic-scale drivers of Minnesota heavy rainfall. The study utilized a neural network, self-organizing map (SOM) technique to identify sea level pressure patterns and precipitation patterns associated with heavy rainfall and used composite analysis to explore the relationships between synoptic-scale conditions and environmental parameters during heavy rain hours. Six sea level pressure patterns were identified, fi ed, three of which represented advancing surface cyclones and accounted for > 70% of the heavy rain hours. The spatial distribution of heavy rainfall was represented by six precipitation patterns. The greatest frequency of heavy rain hours was associated with the northwest precipitation pattern, followed by the southwest and southeast patterns. Analysis of the frequency of pressure and heavy rain precipitation pattern pairs revealed that the top fi ve most frequent pairs were associated with advancing surface cyclones and > 26% of the total heavy rain hours. Composite analysis of environmental parameters showed that favorable conditions related to moisture and lift were associated with heavy rainfall.

Downslope windstorms are common leeward of the Rocky Mountains across the High Plains of Colorado during October through March and can cause property damage and travel disruptions. This region is renowned for two primary types of strong downslope winds: (1) chinooks that feature warm, dry winds and are often associated with the development of mountain waves, and (2) boras that are colder and often associated with the passage of a mid- and upper-level trough. The National Blend of Models (NBM) is regularly utilized by forecasters at the National Weather Service Forecast Office (WFO) in Boulder and is one tool used by the WFO to forecast downslope windstorms. No study, however, has quantified the performance of the NBM during downslope windstorms along the Colorado Front Range. In this exploratory study, downslope windstorms from October 2020 through May 2021 were identified using ten observation sites along the Colorado Front Range. Windstorms were subsequently analyzed to quantify forecast statistics of wind speeds and gusts during windstorms to provide a better understanding of NBM v4.0 performance. On average, we found that the overall maximum magnitudes of wind speeds and gusts are on average 38% and 35% too low in the NBM output, respectively, when averaged across 24 to 72-h lead times. Weather and Research Forecasting (WRF) model simulations of varying vertical resolution were also completed to determine if WRF, a constituent model of the NBM, could provide a robust representation of select downslope windstorms across the Colorado Front Range.

During atmospheric river (AR) landfalls on the Antarctic ice sheet, the high waviness of the circumpolar polar jet stream allows for sub-tropical air masses to be advected towards the Antarctic coastline. These rare but high-impact AR events are highly consequential for the Antarctic mass balance; yet little is known about the various atmospheric dynamical components determining their life cycle. By using an AR detection algorithm to retrieve AR landfalls at Dumont d’Urville and non-AR analogues based on 700 hPa geopotential height, we examined what makes AR landfalls unique and studied the complete life cycle of ARs to affect Dumont d’Urville. ARs form in the mid-latitudes/sub-tropics in areas of high surface evaporation, likely in response to tropical deep convection anomalies. These convection anomalies likely lead to Rossby wave trains that help amplify the upper-tropospheric flow pattern. As the AR approaches Antarctica, condensation of isentropically lifted moisture causes latent heat release that – in conjunction with poleward warm air advection – induces geopotential height rises and anticyclonic upper-level potential vorticity tendencies downstream. As evidenced by a blocking index, these tendencies lead to enhanced ridging/blocking that persist beyond the AR landfall time, sustaining warm air advection onto the ice sheet. Finally, we demonstrate a connection between tropopause polar vortices and mid-latitude cyclogenesis in an AR case study. Overall, the non-AR analogues reveal that the amplified jet pattern observed during AR landfalls is a result of enhanced poleward moisture transport and associated diabatic heating which is likely impossible to replicate without strong moisture transport.

Antarctic atmospheric rivers (ARs) are rare but high-impact events that have a major influence on the surface mass and energy balance of the ice sheet, by causing strong positive anomalies in temperature and precipitation. Previous studies have shown that the main moisture sources of Antarctic ARs extend from the midlatitudes to the subtropics (equatorward of 30° latitude). Here, we focus on the tropical moisture origins through analyzing tropical cyclones and the Madden-Julian Oscillation, i.e. the most intense tropical convective events. They can act in some cases as precursors of the AR development particularly during the austral summer season, increasing the Antarctic AR probability by a factor of 3–5. Not all intense tropical convective events produce ARs: a precise phase locking between regions of poleward moisture export and extratropical Rossby waves is necessary to channel tropical moisture poleward, and form an AR over the Southern Ocean reaching the Antarctic ice sheet.

Accurate forecasts of weather conditions have the potential to mitigate the social and economic damages they cause. To make informed decisions based on forecasts, it is important to determine the extent to which they could be skillful. This study focuses on subseasonal forecasts out to a lead time of four weeks. We examine the differences between the potential predictability, which is computed under the assumption of a " perfect model, " of integrated vapor transport (IVT) and precipitation under extreme conditions in subseasonal forecasts across the northeast Paci fi c. Our results demonstrate signi fi cant forecast skill of extreme IVT and precipitation events (exceeding the 90th percentile) into week 4 for speci fi c areas, particularly when anomalously wet conditions are observed in the true model state. This forecast skill during weeks 3 and 4 is closely associated with a zonal extension of the North Paci fi c jet. These fi ndings of the source of skillful subseasonal forecasts over the U.S. West Coast could have implications for water management in these regions susceptible to drought and fl ooding extremes. Additionally, they may offer valuable insights for governments and industries on the U.S. West Coast seeking to make informed decisions based on extended weather prediction.

We extend the construction of so-called encapsulated global summation-by-parts operators to the general case of a mesh which is not boundary conforming. Owing to this development, energy stable discretizations of nonlinear and variable coefficient initial boundary value problems can be formulated in simple and straightforward ways using high-order accurate operators of generalized summation-by-parts type. Encapsulated features on a single computational block or element may include polynomial bases, tensor products as well as curvilinear coordinate transformations. Moreover, through the use of inner product preserving interpolation or projection, the global summation-by-parts property is extended to arbitrary multi-block or multi-element meshes with non-conforming nodal interfaces.

We present an entropy stable nodal discontinuous Galerkin spectral element method (DGSEM) for the two-layer shallow water equations on two dimensional curvilinear meshes. We mimic the continuous entropy analysis on the semi-discrete level with the DGSEM constructed on Legendre–Gauss–Lobatto (LGL) nodes. The use of LGL nodes endows the collocated nodal DGSEM with the summation-by-parts property that is key in the discrete analysis. The approximation exploits an equivalent flux differencing formulation for the volume contributions, which generate an entropy conservative split-form of the governing equations. A specific combination of a numerical surface flux and discretization of the nonconservative terms is then applied to obtain a high-order path-conservative scheme that is entropy conservative. Furthermore, we find that this combination yields an analogous discretization for the pressure and nonconservative terms such that the numerical method is well-balanced for discontinuous bathymetry on curvilinear domains. Dissipation is added at the interfaces to create an entropy stable approximation that satisfies the second law of thermodynamics in the discrete case, while maintaining the well-balanced property. We conclude with verification of the theoretical findings through numerical tests and demonstrate results about convergence, entropy stability and well-balancedness of the scheme.