semi-Lagrangian Advection Scheme Research Articles

Improving trajectory calculations by FLEXPART 10.4+ using single-image super-resolution

Abstract. Lagrangian trajectory or particle dispersion models as well as semi-Lagrangian advection schemes require meteorological data such as wind, temperature and geopotential at the exact spatiotemporal locations of the particles that move independently from a regular grid. Traditionally, these high-resolution data have been obtained by interpolating the meteorological parameters from the gridded data of a meteorological model or reanalysis, e.g., using linear interpolation in space and time. However, interpolation is a large source of error for these models. Reducing them requires meteorological input fields with high space and time resolution, which may not always be available and can cause severe data storage and transfer problems. Here, we interpret this problem as a single-image super-resolution task. That is, we interpret meteorological fields available at their native resolution as low-resolution images and train deep neural networks to upscale them to a higher resolution, thereby providing more accurate data for Lagrangian models. We train various versions of the state-of-the-art enhanced deep residual networks for super-resolution (EDSR) on low-resolution ERA5 reanalysis data with the goal to upscale these data to an arbitrary spatial resolution. We show that the resulting upscaled wind fields have root-mean-squared errors half the size of the winds obtained with linear spatial interpolation at acceptable computational inference costs. In a test setup using the Lagrangian particle dispersion model FLEXPART and reduced-resolution wind fields, we find that absolute horizontal transport deviations of calculated trajectories from “true” trajectories calculated with un-degraded 0.5∘ × 0.5∘ winds are reduced by at least 49.5 % (21.8 %) after 48 h relative to trajectories using linear interpolation of the wind data when training on 2∘ × 2∘ to 1∘ × 1∘ (4∘ × 4∘ to 2∘ × 2∘) resolution data.

A Flux-Form Semi-Lagrangian advection scheme for tracer transport on arbitrary meshes

A variant of the flux-form semi-Lagrangian (FFSL) advection scheme is presented suitable for use on arbitrary unstructured meshes. The time-step used with the scheme is not constrained by the Courant number, resulting in model run-times that may be many times faster than Courant-constrained schemes. The scheme also has the property of mass conservation. This makes the scheme a potential candidate for offline simulation of tracers in unstructured sediment transport, biogeochemical or atmospheric chemistry models. We describe the numerics of the scheme and apply it to a number of idealised and real applications to demonstrate its utility. In these test cases we compare results from the offline FFSL scheme with those generated by the original hydrodynamic model which uses high order conservative tracer transport. We assess the FFSL scheme for accuracy, conservation and computational efficiency. Additionally, traditional semi-Lagrangian advection schemes are included in assessments for comparison. Results indicate that the FFSL scheme produces closer agreement with tracer distributions generated by the hydrodynamic model than the semi-Lagrange scheme, and mass conservation was vastly superior. The FFSL scheme was also an order of magnitude faster than the hydrodynamic model in certain cases. The FFSL scheme presented here therefore offers a viable mass-conserving, computationally-efficient option for use in offline transport models on arbitrary unstructured meshes.

A mass-conservative semi-implicit volume of fluid method for the Navier–Stokes equations with high order semi-Lagrangian advection scheme

This paper deals with the development of a semi-implicit numerical method for the Navier–Stokes equations using the non-linear volumes of fluid (VOF) approach and a semi-Lagrangian scheme for the discretization of the advection contribution based on a high order reconstruction of the velocity field. The VOF approach guarantees high flexibility and is able to reproduce several phenomena that appear in real scenarios such as free surface flows, pressurized channels and jets. The discrete velocity field from the momentum conservation law is formally inserted into the discrete continuity equation, hence yielding a mildly non-linear system for the unknown hydraulic head which can be solved through a nested Newton-type algorithm. The computation of the non-linear convective diffusion contribution is then based on a high order reconstruction of the velocity field, which is furthermore constrained to exactly recover the original pointwise values of the numerical solution. As a consequence, the mass conservation is fully preserved while providing information about the main velocity field and its high order moments, later employed in the computation of the Lagrangian trajectories needed for the discretization of the convective and diffusive terms. Furthermore, the bottom friction and the tangential stresses can be directly computed from the high order velocity reconstruction. The method is derived in a general form with the only requirement to be structured in the z− direction, so that it applies to the three- and the two-dimensional cases with unstructured grids in the horizontal space. Convergence studies are carried out to demonstrate the accuracy of the reconstruction operator. Finally, the numerical scheme is validated against several benchmarks that include 2Dxz, 2Dxy and 3D non-hydrostatic flows with complex geometry in order to show the flexibility of the proposed algorithm, including a real-world application.

A New Semi-Lagrangian Finite Volume Advection Scheme Combines the Best of Both Worlds

A New Semi-Lagrangian Finite Volume Advection Scheme Combines the Best of Both Worlds

A Fast Linear Semi-Lagrangian Advection Scheme Coupled with Spectral (bin) Microphysics to Simulate an Idealized Super Cell Storm in WRF

AbstractA new, computationally efficient Semi-Lagrangian advection (SLA) scheme was used to simulate an idealized supercell storm using WRF coupled with Spectral (bin) Microphysics (SBM). SLA was developed to make complicated microphysical schemes more computationally accessible to cloud resolving models. The SLA is a linear combination of Semi-Lagrangian schemes of the first and the second order. It has relatively low numerical diffusion, a high level of mass conservation accuracy, and preserves the sum of multiple advected variables. In addition to idealized tests, comparisons were made with standard WRF higher-order, non-linear advection schemes. Tests of the SLA were performed using different weighting coefficients of γ for the combination of the first and second order components. The results of SLA on grids of 1 km, 500 m, and 250 m agree well with those of the standard WRF advection schemes, with results most similar to simulations with 250 m grid spacing. At the same time, the advection CPU time required by the SLA was 2.2 to 3 times shorter than the WRF advection schemes. The speed-up occurred in part because of the utilization of the same advection matrix for the advection of all hydrometeor mass bins. The findings of this work support the hypothesis that cloud microphysical simulation is more sensitive to the choice of microphysics than to the choice of advection schemes, thereby justifying the use of computationally efficient lower order linear schemes.

Development of a semi-Lagrangian advection scheme for the NEMO ocean model (3.1)

Abstract. As resolutions of ocean circulation models increase, the advective Courant number – the ratio between the distance travelled by a fluid parcel in one time step and the grid size – becomes the most stringent factor limiting model time steps. Some atmospheric models have escaped this limit by using an implicit or semi-implicit semi-Lagrangian formulation of advection, which calculates materially conserved fluid properties along trajectories which follow the fluid motion and end at prescribed grid points. Unfortunately, this formulation is not straightforward in ocean contexts, where the irregular, interior boundaries imposed by the shore and bottom orography are incompatible with traditional trajectory calculations. This work describes the adaptation of the semi-Lagrangian method as an advection module for an operational ocean model. We solve the difficulties of the ocean's internal boundaries by calculating parcel trajectories using a time-exponential formulation, which ensures that all parcel trajectories remain inside the ocean domain despite strong accelerations near the boundary. Additionally, we derive this method in a way that is compatible with the leapfrog time-stepping scheme used in the NEMO-OPA (Nucleus for European Modelling of the Ocean, Océan Parallélisé) ocean model, and we present simulation results for a simplified test case of flow past a model island and for 10-year free runs of the global ocean on the quarter-degree ORCA025 grid.

A space-time semi-Lagrangian advection scheme on staggered Voronoi meshes applied to free surface flows

In this article we present a novel space-time semi-Lagrangian advection scheme for the solution of the nonlinear convective terms in hyperbolic conservation laws. The governing equations are discretized on a three-dimensional mesh, composed of a staggered unstructured Voronoi grid on the horizontal plane which is extruded along the vertical direction with z−layers of non-uniform thickness. A high order space-time reconstruction is carried out for the velocity field, that is used for both tracking backward in time the Lagrangian trajectories of the flow and for the interpolation of the transported quantity at the foot of the characteristics. High order in space is achieved via a constrained least-squares reconstruction technique, whereas the ADER procedure is employed for gaining high order of accuracy in time as well. The high order reconstruction polynomials are expanded onto a set of basis functions that are defined in the physical coordinate system for space and in the reference framework for time, thus improving the computational efficiency of the scheme. The trajectory equation of the flow particles is then solved relying on symplectic-type integrators, which are proven to be structure-preserving ODE solvers, unlike standard explicit Runge–Kutta schemes. Application to hydrostatic free surface flows is proposed, demonstrating accuracy and robustness of the novel numerical method via comparison against analytical solutions.

A Nowcasting Model for Tropical Cyclone Precipitation Regions Based on the TREC Motion Vector Retrieval with a Semi-Lagrangian Scheme for Doppler Weather Radar

Accurate observational data and reliable prediction models are both essential to improve the quality of precipitation forecasts. The spiraling trajectories of air parcels within a tropical cyclone (TC) coupled with the large sizes of these systems brings special challenges in making accurate short-term forecasts, or nowcasts. Doppler weather radars are ideal instruments to observe TCs when they move over land, and traditional nowcasts incorporate radar data. However, data from dozens of radars must be mosaicked together to observe the entire system. Traditional single-radar-based reflectivity tracking methods commonly employed in nowcasting are not suitable for TCs as they are not able to capture the circular motion of these systems. Thus, this paper focuses on improving short-term predictability of TC precipitation with Doppler weather radar observations based on: a multi-scale motion vector retrieval algorithm, an optimization technique and a semi-Lagrangian advection scheme. Motion fields of precipitation regions are obtained by a multi-level motion vector retrieval algorithm, then corrected and smoothed by the optimization technique using mass and smooth constraints. Predicted precipitation regions are then extrapolated using the semi-Lagrangian advection scheme. A case study of Hurricane Isabel (2003) shows that the combination of these methods may increase reliable rainfall prediction to about 5 h as the TC moves over land.

Enviro-HIRLAM online integrated meteorology–chemistry modelling system: strategy, methodology, developments and applications (v7.2)

Abstract. The Environment – High Resolution Limited Area Model (Enviro-HIRLAM) is developed as a fully online integrated numerical weather prediction (NWP) and atmospheric chemical transport (ACT) model for research and forecasting of joint meteorological, chemical and biological weather. The integrated modelling system is developed by the Danish Meteorological Institute (DMI) in collaboration with several European universities. It is the baseline system in the HIRLAM Chemical Branch and used in several countries and different applications. The development was initiated at DMI more than 15 years ago. The model is based on the HIRLAM NWP model with online integrated pollutant transport and dispersion, chemistry, aerosol dynamics, deposition and atmospheric composition feedbacks. To make the model suitable for chemical weather forecasting in urban areas, the meteorological part was improved by implementation of urban parameterisations. The dynamical core was improved by implementing a locally mass-conserving semi-Lagrangian numerical advection scheme, which improves forecast accuracy and model performance. The current version (7.2), in comparison with previous versions, has a more advanced and cost-efficient chemistry, aerosol multi-compound approach, aerosol feedbacks (direct and semi-direct) on radiation and (first and second indirect effects) on cloud microphysics. Since 2004, the Enviro-HIRLAM has been used for different studies, including operational pollen forecasting for Denmark since 2009 and operational forecasting atmospheric composition with downscaling for China since 2017. Following the main research and development strategy, further model developments will be extended towards the new NWP platform – HARMONIE. Different aspects of online coupling methodology, research strategy and possible applications of the modelling system, and fit-for-purpose model configurations for the meteorological and air quality communities are discussed.

Dynamic Flow Model for Real-Time Application in Wind Farm Control

For short-term power predictions and estimations of the available power during curtailment of a wind farm, it is necessary to consider the flow dynamics and aerodynamic interactions of the turbines. In this paper, a control-oriented dynamic two-dimensional wind farm model is introduced that aims to incorporate real-time measurements such as flow velocities at turbine locations to estimate the ambient wind farm flow. The model is intended to derive flow predictions for real-time applications. Since fully resolved computational fluid dynamics are too CPU-intensive for such a task, the dynamic model presented in this paper relies on an approximation of the flow equations in a two-dimensional framework. A semi-Lagrangian advection scheme and a step-wise flow solver together offer fast calculation speed, which scales linearly with the number of grid points. In order to emulate effects of realistic three-dimensional wind farm flow, a relaxation of the two-dimensional continuity equation is presented. Furthermore, with little extra computational expense, additional dynamic state variables for various possible applications can be propagated along the wind flow. For instance, a dynamic confidence parameter can provide estimations of the accuracy of flow predictions, while a turbulence parameter adds the possibility to estimate wake induced loads on downstream turbines. In order to demonstrate the performance and validity of the new model it is compared with other models. At first a two turbine reference case is compared with a steady-state model and secondly with results obtained by the dynamic wind farm flow model WFSim. Finally a small wind farm is simulated in order to show the computational scaling of the model.

Improving the inter-hemispheric gradient of total column atmospheric CO<sub>2</sub> and CH<sub>4</sub> in simulations with the ECMWF semi-Lagrangian atmospheric global model

Abstract. It is a widely established fact that standard semi-Lagrangian advection schemes are highly efficient numerical techniques for simulating the transport of atmospheric tracers. However, as they are not formally mass conserving, it is essential to use some method for restoring mass conservation in long time range forecasts. A common approach is to use global mass fixers. This is the case of the semi-Lagrangian advection scheme in the Integrated Forecasting System (IFS) model used by the Copernicus Atmosphere Monitoring Service (CAMS) at the European Centre for Medium-Range Weather Forecasts (ECMWF).Mass fixers are algorithms with substantial differences in complexity and sophistication but in general of low computational cost. This paper shows the positive impact mass fixers have on the inter-hemispheric gradient of total atmospheric column-averaged CO2 and CH4, a crucial feature of their spatial distribution. Two algorithms are compared: the simple "proportional" and the more complex Bermejo–Conde schemes. The former is widely used by several Earth system climate models as well the CAMS global forecasts and analysis of atmospheric composition, while the latter has been recently implemented in IFS. Comparisons against total column observations demonstrate that the proportional mass fixer is shown to be suitable for the low-resolution simulations, but for the high-resolution simulations the Bermejo–Conde scheme clearly gives better results. These results have potential repercussions for climate Earth system models using proportional mass fixers as their resolution increases. It also emphasises the importance of benchmarking the tracer mass fixers with the inter-hemispheric gradient of long-lived greenhouse gases using observations.

A mass-conserving advection scheme for offline simulation of scalar transport in coastal ocean models

We present a flux-form semi-Lagrangian (FFSL) advection scheme designed for offline scalar transport simulation with coastal ocean models using curvilinear horizontal coordinates. The scheme conserves mass, overcoming problems of mass conservation typically experienced with offline transport models, and permits long time steps (relative to the Courant number) to be used by the offline model. These attributes make the method attractive for offline simulation of tracers in biogeochemical or sediment transport models using archived flow fields from hydrodynamic models. We describe the FFSL scheme, and test it on two idealised domains and one real domain, the Great Barrier Reef in Australia. For comparison, we also include simulations using a traditional semi-Lagrangian advection scheme for the offline simulations. We compare tracer distributions predicted by the offline FFSL transport scheme with those predicted by the original hydrodynamic model, assess the conservation of mass in all cases and contrast the computational efficiency of the schemes. We find that the FFSL scheme produced very good agreement with the distributions of tracer predicted by the hydrodynamic model, and conserved mass with an error of a fraction of one percent. In terms of computational speed, the FFSL scheme was comparable with the semi-Lagrangian method and an order of magnitude faster than the full hydrodynamic model, even when the latter ran in parallel on multiple cores. The FFSL scheme presented here therefore offers a viable mass-conserving and computationally-efficient alternative to traditional semi-Lagrangian schemes for offline scalar transport simulation in coastal models.

Moments Preserving and high-resolution Semi-Lagrangian Advection Scheme

We present a forward semi-Lagrangian numerical method for systems of transport equations able to advect smooth and discontinuous fields with high-order accuracy. The numerical scheme is composed of an integration of the transport equations along the trajectory of material elements in a moving grid and a reconstruction of the fields in a reference regular mesh using a non-linear mapping and adaptive moment-preserving interpolations. The nonlinear mapping allows for the arbitrary deformation of material elements. Additionally, interpolations can represent discontinuous fields using adaptive-order interpolation near jumps detected with a slope-limiter function. Due to the large number of operations during the interpolations, a serial implementation of this scheme is computationally expensive. The scheme has been accelerated in many-core parallel architectures using a thread per grid node and parallel data gathers. We present a series of tests that show the scheme to be an attractive option for simulating adv...

A semi-Lagrangian advection scheme for radioactive tracers in the NCEP Regional Spectral Model (RSM)

Abstract. In this study, the non-iteration dimensional-split semi-Lagrangian (NDSL) advection scheme is applied to the National Centers for Environmental Prediction (NCEP) Regional Spectral Model (RSM) to alleviate the Gibbs phenomenon. The Gibbs phenomenon is a problem wherein negative values of positive-definite quantities (e.g., moisture and tracers) are generated by the spectral space transformation in a spectral model system. To solve this problem, the spectral prognostic specific humidity and radioactive tracer advection scheme is replaced by the NDSL advection scheme, which considers advection of tracers in a grid system without spectral space transformations. A regional version of the NDSL is developed in this study and is applied to the RSM. Idealized experiments show that the regional version of the NDSL is successful. The model runs for an actual case study suggest that the NDSL can successfully advect radioactive tracers (iodine-131 and cesium-137) without noise from the Gibbs phenomenon. The NDSL can also remove negative specific humidity values produced in spectral calculations without losing detailed features.

A Partitioned Global Address Space implementation of the European Centre for Medium Range Weather Forecasts Integrated Forecasting System

Today the European Centre for Medium Range Weather Forecasts (ECMWF) runs a 16 km global T1279 operational weather forecast model using 1536 cores of an IBM Power7. Following the historical evolution in resolution upgrades, the ECMWF could expect to be running a 2.5 km global forecast model by 2030 on an exascale system that should be available and hopefully affordable by then. To achieve this would require the Integrated Forecasting System (IFS) to run efficiently on about 1000 times the number of cores it uses today. In a step towards this goal, the ECMWF have demonstrated the IFS running a 10 km global model efficiently on over 40,000 cores of HECToR a Cray XE6 at the Edinburgh Parallel Computing Centre. However, getting to over a million cores remains a formidable challenge, and many scalability improvements have yet to be implemented. The ECMWF is exploring the use of Fortran2008 coarrays; in particular, it is possibly the first time that coarrays have been used in a world-leading production application within the context of OpenMP parallel regions. The purpose of these optimisations is primarily to allow the overlap of computation and communication, and further, in the semi-Lagrangian advection scheme, to reduce the volume of data communicated. The importance of this research is such that if these and other planned developments are successful, the IFS model may continue to use the spectral transform method to 2030 and beyond on an exascale-sized system. The current status of the coarray scalability developments within the IFS are described together with a brief outline of future developments.

Comparison of spectral characteristics of precipitation from radar estimates and COSMO-model predicted fields

The spectral characteristics of hourly precipitation fields are studied for a number of meteorological events, classified depending on the precipitation process involved, namely stratiform, convective or mixed stratiform–convective. The study focuses on the comparison between spatial spectral characteristics of observed precipitation fields, obtained from hourly radar estimates, and modelled hourly predictions of the same field by the non-hydrostatic model COSMO. The results show that the power spectra of radar hourly precipitation are characterised by invariance within ranges of horizontal spatial scales that are different for the three classes of events. COSMO reproduces some basic characteristics of the radar spectra in all cases. Nevertheless, in stratiform events, COSMO spectra present a well-defined scale break at about 15 km with no counterpart in radar data. It is suggested that this model feature is related to the presence of spurious horizontal smoothing introduced by the semi-lagrangian advection scheme for precipitation. Smoothing affects all scales up to the maximum length scale of precipitation and horizontal advection by the wind. An analysis of wind intensity in the lower troposphere over the region supports this interpretation. Discrepancies in precipitation spectra between radar and COSMO data in convective events are interpreted as a consequence of the inadequacy of the model resolution for a correct representation of convection.

Effective Isentropic Diffusivity of Tropospheric Transport

Abstract Tropospheric transport can be described qualitatively by the slow mean diabatic circulation and rapid isentropic mixing, yet a quantitative understanding of the transport circulation is complicated, as nearly half of the isentropic surfaces in the troposphere frequently intersect the ground. A theoretical framework for the effective isentropic diffusivity of tropospheric transport is presented. Compared with previous isentropic analysis of effective diffusivity, a new diagnostic is introduced to quantify the eddy diffusivity of the near-surface isentropic flow. This diagnostic also links the effective eddy diffusivity directly to a diffusive closure of eddy fluxes through a finite-amplitude wave activity equation. The theory is examined in a dry primitive equation model on the sphere. It is found that the upper troposphere is characterized by a diffusivity minimum at the jet’s center with enhanced mixing at the jet’s flanks and that the lower troposphere is dominated by stronger mixing throughout the baroclinic zone. This structure of isentropic diffusivity is generally consistent with the diffusivity obtained from the geostrophic component of the flow. Furthermore, the isentropic diffusivity agrees broadly with the tracer equivalent length obtained from either a spectral diffusion scheme or a semi-Lagrangian advection scheme, indicating that the effective diffusivity of tropospheric transport is largely dictated by large-scale stirring rather than details of the small-scale diffusion acting on the tracers.

Global mass fixer algorithms for conservative tracer transport in the ECMWF model

Abstract. Various mass fixer algorithms (MFAs) have been implemented in the Integrated Forecasting System (IFS) of the European Centre for Medium-Range Weather Forecasts (ECMWF) to ensure mass conservation of atmospheric tracers within the semi-Lagrangian (SL) advection scheme. Emphasis has been placed in implementing schemes that despite being primarily global in nature adjust the solution mostly in regions where the advected field has large gradients and therefore interpolation (transport) error is assumed larger. The MFAs have been tested in weather forecast, idealised and atmospheric dispersion cases. Applying these fixers to specific humidity and cloud fields did not change the accuracy of 10-day forecasts. In other words, global mass tracer conservation is achieved without deteriorating the solution accuracy. However, for longer forecast timescales or for forecasts in which correlated species are transported, experiments suggest that MFAs may improve IFS forecasts.

Improvement of the semi-Lagrangian advection scheme in the GRAPES model: Theoretical analysis and idealized tests

The Global/Regional Assimilation and PrEdiction System (GRAPES) is the new-generation numerical weather prediction (NWP) system developed by the China Meteorological Administration. It is a fully compressible non-hydrostatical global/regional unified model that uses a traditional semi-Lagrangian advection scheme with cubic Lagrangian interpolation (referred to as the SL_CL scheme). The SL_CL scheme has been used in many operational NWP models, but there are still some deficiencies, such as the damping effects due to the interpolation and the relatively low accuracy. Based on Reich’s semi-Lagrangian advection scheme (referred to as the R2007 scheme), the Re_R2007 scheme that uses the low- and high-order B-spline function for interpolation at the departure point, is developed in this paper. One- and two-dimensional idealized tests in the rectangular coordinate system with uniform grid cells were conducted to compare the Re_R2007 scheme and the SL_CL scheme. The numerical results showed that: (1) the damping effects were remarkably reduced with the Re_R2007 scheme; and (2) the normalized errors of the Re_R2007 scheme were about 7.5 and 3 times smaller than those of the SL_CL scheme in one- and two-dimensional tests, respectively, indicating the higher accuracy of the Re_R2007 scheme. Furthermore, two solid-body rotation tests were conducted in the latitude-longitude spherical coordinate system with nonuniform grid cells, which also verified the Re_R2007 scheme’s advantages. Finally, in comparison with other global advection schemes, the Re_R2007 scheme was competitive in terms of accuracy and flow independence. An encouraging possibility for the application of the Re_R2007 scheme to the GRAPES model is provided.

Should weather and climate prediction models be deterministic or stochastic?

Should weather and climate prediction models be deterministic or stochastic?