This paper details and tests numerical improvements to the ADvanced CIRCulation (ADCIRC) model, a widely used finite-element method shallow-water equation solver, to more accurately and efficiently model global storm tides with seamless local mesh refinement in storm landfall locations. The sensitivity to global unstructured mesh design was investigated using automatically generated triangular meshes with a global minimum element size (MinEle) that ranged from 1.5 to 6 km. We demonstrate that refining resolution based on topographic seabed gradients and employing a MinEle less than 3 km are important for the global accuracy of the simulated astronomical tide.
Our recommended global mesh design (MinEle

Extreme coastal sea levels and flooding driven by storms and tsunamis can be accurately modeled by the shallow-water equations (SWEs). The SWEs are often numerically solved by discretizing the continuous equations using unstructured meshes with either finite-volume methods (FVMs) or finite-element methods (FEMs). These unstructured meshes can efficiently model the large range in length scales associated with physical processes that occur in the deep ocean to the nearshore region

A key practical advantage of ocean models discretized using FEMs compared to FVMs is that they are usually less sensitive to mesh quality (e.g., element skewness). Specifically, ocean models discretized using FVMs often use staggered C-grid arrangements (e.g., Delft-FM) that have strict grid orthogonality requirements for numerical accuracy

In the absence of constraints on orthogonality or element skewness, automatically generating unstructured triangular meshes on the spherical Earth that accurately conform to the coastline and cover a wide range of spatial sales (O(10 m)–O(10 km)) is completely realizable

This study conducts a systematic analysis of unstructured mesh design in order to assess and demonstrate the capabilities of global storm tide modeling using FEMs across multi-resolution scales spanning from the deep ocean to the nearshore coastal ocean environment. One outcome of this study is a recommendation of unstructured triangular-element mesh design of the global ocean that represents the barotropic physics with high fidelity using relatively few elements (Sect.

The ADCIRC storm tide model used in this study is an FEM solver that has been extensively used for detailed hurricane inundation studies at local and regional scales

To properly compute the governing equations on the spherical Earth in the FEM framework used by ADCIRC, we have upgraded the model formulation and code as detailed in Sect. S1 in the Supplement. This involves rotating the Earth so that the pole singularity is removed (Sect. S1.3) before applying a rectilinear mapping projection to transform the governing equations into a Cartesian form with spherically based corrections to the spatial derivatives (Sect. S1.1). Here, the continuity equation is multiplied by a factor dependent on the choice of cylindrical projection used (e.g., Mercator) to produce a conservative form that leads to discrete mass conservation and stability

The global unstructured meshes in this study are generated automatically using scripts with version 3.0.0 of OceanMesh2D

Mesh design is handled through mesh size (resolution distribution) functions that are defined on a regular structured grid, usually that of the topo-bathymetric digital elevation model (DEM). In this study we use functions based on the distance to shoreline, bathymetric depths, and topographic gradients (see Table

The mesh size functions used to spatially distribute element resolution,

Additionally, this study makes use of the OceanMesh2D “plus” function, which seamlessly merges two arbitrary meshes together, keeping the finer resolution in the overlapping region. We use this function to apply local mesh refinement to a global mesh in storm-affected coastal regions to better resolve semi-enclosed bays and lakes, inlets, back bays, channels, and other small-scale shoreline geometries.

The experimental design we pursue is composed of two distinct steps, both with the purpose of maximizing model efficiency while maintaining a threshold of accuracy. First, we begin with a mesh design that we assume is a highly refined discretization of the Earth in a global sense and systematically relax the mesh size parameters to reduce the total number of mesh vertices while trying to minimize any negative impacts on global model accuracy. Second, we take the resulting recommended global mesh design from the previous step and apply local mesh refinement to increase the coastal resolution in the storm landfalling region and potentially improve local model accuracy.

In this step, three mesh size function parameters (MinEle, TLS, and FL) are systematically relaxed to coarsen an initially highly refined discretization of the Earth, termed the reference (Ref) mesh design (see Table

Summary of the global mesh designs. Each row is separate and made up of three mesh designs for each variable mesh size function parameter, in addition to the Ref mesh design, which is the same for each row.

Mesh resolution distribution (defined as the minimum connected element edge length for a mesh vertex) for two global mesh designs:

To assess the effect on the global model accuracy as the mesh designs are coarsened, we compare simulated astronomical tidal solutions to the data-assimilated TPXO9-Atlas. We focus on astronomical tides in this step because they can be reduced to a series of harmonic constituents of well-defined frequencies to make systematic global comparisons

In this step, the recommended global mesh design from Step 1 (Sect.

The first is the western North Atlantic – Hurricane Katrina, 23–31 August 2005. The most severe impact of storm-tide-induced coastal flooding occurred in the Louisiana–Mississippi region of the USA in the northern Gulf of Mexico

The second is the western Pacific – Super Typhoon Haiyan, 3–11 November 2013. The most severe impact of storm-tide-induced coastal flooding occurred in and around Tacloban, Philippines, at the back end of the Leyte Gulf

Comparisons of mesh triangulation and resolution (defined as the element circumradius on a Lambert conformal conic projection) in the Hurricane Katrina landfall region around Louisiana–Mississippi, USA.

Comparisons of mesh triangulation and resolution (defined as the element circumradius on a Lambert conformal conic projection) in the Super Typhoon Haiyan landfall region around Leyte and Samar Island, Philippines.

To assess the accuracy and intercompare mesh designs, we primarily use the output of the maximum simulated storm tide elevation from each event. For validation we compare to surveys of high-water marks (HWMs) in the landfall regions that are located close to the coast (within 1.5 km of the nearest 150 m locally refined mesh vertex) and attributed primarily to surge for both Katrina

The full-resolution Global Self-consistent Hierarchical High-resolution Shorelines (GSHHS) dataset

Maximum 10 m wind velocities and minimum pressure contours over the ocean for Hurricane Katrina (25–31 August 2005) and Super Typhoon Haiyan (4–10 November 2013).

Atmospheric forcings are derived from three different sources in this study (see Sect. S2.5 for details on how to use each source in ADCIRC v55). Hourly global reanalysis datasets, 0.313

When only astronomical tides are simulated, we force the model with only astronomical forcing (

When simulating on the Ref mesh using ADCIRC v55, significant improvements to the prediction of astronomical tidal constituents were measured compared to ADCIRC v54. In particular, M

Global M

Global M

The distribution (ECDF curves) of

Increasing MinEle has a clear but gradual degenerative effect on the solution as it is increased from 1.5 to 3 km. The disparities in the ECDF curves noticeably grow as MinEle is increased to 6 km; the value of

Area-weighted ECDF curves of

The two-sample Kolmogorov–Smirnov test statistic,

In summary, the results demonstrate that decreasing the TLS parameter from 20 to 5 substantially decreases the number of vertices, while it has a relatively small effect on the tidal solution compared to the other experiments.
On the other hand, increasing the FL parameter has a comparatively small impact on vertex count reduction, while increasing MinEle has a relatively large impact on the solution. The final choice of mesh design is dependent on one's tolerance for error, but in general it is preferable to choose a mesh that is plotted close to the bottom left corner of the graph in Fig.

The maximum storm tide elevation due to Hurricane Katrina approaches 8 m in the Hancock and Harrison counties of Mississippi (Fig.

Maximum simulated storm tide elevations on the MinEle

As pointed out by

Comparisons of the observed HWMs to the simulated maximum storm tide height at the nearest vertex on meshes with different local MinEle (1.5 km, 500 m, 150 m).

Time series of Hurricane Katrina at NOAA tide gauges show that the timing of the peak storm tide elevation and the amplitude and phase of the tide signal prior to landfall are well-represented by the model (Fig.

Comparisons of simulated storm tide elevation time series on meshes with different local MinEle (1.5 km, 500 m, 150 m) at the point locations shown in Fig

Our results indicate that there is a tendency for the coarser-resolution meshes to have larger storm tide elevations in the open coastal areas of the landfall region. In the case of Katrina, this is most clearly seen in Lake Borgne where maximum storm tide elevations are at least 0.6 m larger on the MinEle

Mesh-resolution-induced differences in the simulated maximum storm tide elevations due to Hurricane Katrina in the Louisiana–Mississippi landfall region.

In the case of the Haiyan, the predominant maximum storm tide elevation difference (

Mesh-resolution-induced differences in the simulated maximum storm tide elevations due to Super Typhoon Haiyan in the Leyte and Samar Island, Philippines, landfall region.

Lastly, we mention two additional general observations. First, for both storms the far-field effects of local mesh refinement were found to be negligible. Second, storm tide elevation time series show that not only does the peak elevation tend to decrease as mesh refinement is made, but the timing of the peak also tends to occur later (Fig.

For all astronomical tide simulations performed in the global mesh design experiments (Sect.

Computational wall-clock times for the various mesh designs and experiments versus the average number of mesh vertices per computational processor. All simulations were computed on 480 computational processors (i.e., the variation in vertices per processor comes from the variation in the total number of vertices for each mesh design). The computational performance in this study using ADCIRC v55 is contrasted with previous ADCIRC model Katrina storm tide simulations

For the storm tide simulations performed in the local mesh refinement experiments (Sect.

In addition, the computational performance of our Katrina simulations is compared to previous ADCIRC model runs

The new version of ADCIRC (v55) demonstrated improved tidal solutions compared to the previous versions of ADCIRC (denoted as ADCIRC v54). This is because ADCIRC v54 does not solve the correct form of the governing equations in spherical coordinates and is thus technically valid only for sufficiently small regional domains (see Sect. S1.2 for more details on this comparison). For instance, this old form of the governing equations appears to be sufficient for the western North Atlantic Ocean regional domain, which has been thoroughly validated using ADCIRC since

Results from the global mesh design experiments echo those found previously for a regional mesh of the western North Atlantic Ocean

The results also show that the use of the barotropic Rossby-radius-based low-pass filter (FL) in the TLS function is able to reduce mesh vertices without substantially degrading the solution. However, our results suggest that it is more efficient to simply reduce the TLS value to 5 compared to using TLS

The astronomical tide solution differences between global mesh designs were shown to be predominantly in the body of the area-weighted ECDF curves, while the tails were almost identical. This implies that simulated tides over most of the area of the ocean are affected by mesh resolution. However, in regions where the tidal range and error are large, which inevitably occurs on shallow shelves, all of the mesh designs have similarly large errors. This perhaps explains why a

The local mesh refinement technique was demonstrated to be a useful tool to provide high refinement with a trivial addition to the total vertex count. Nevertheless, we found that the numerically stable time step decreases as coastal mesh resolution becomes finer (see Sect. S2.1 for details on setting the time step), which increases computational time. Note that additional tests (not shown) were conducted, and these demonstrated that the computational time step used for the same mesh had a negligible effect on storm tide elevation solutions. However, this may not transfer as well for simulations in which there is significant wetting–drying due to the one element per time step wetting–drying logic used. The impact of mesh refinement clearly tends to decrease open-ocean storm tide elevations in open-ocean areas, and the timing of the peak occurs later. This could be attributed to larger physical approximation errors of the shoreline geometry and bathymetry with mesh coarsening

Last, it is widely recognized that sensitivities to local high-resolution bathymetry datasets, internal tide wave drag, and spatially varying bottom friction and surface ice friction are important

Important upgrades to the FEM SWE solver, ADCIRC, have been presented to improve accuracy and efficiency for global storm tide modeling across multi-resolution unstructured meshes. We systemically tested the new model's (ADCIRC v55) sensitivity to mesh design in the simulation of global astronomical tides and storm tides. These mesh design results are expected to be broadly applicable to other SWE solvers that correctly handle solutions on the sphere.

Based on the results for global mesh design we recommend aiming for a minimum element size less than 3 km and using the TLS function to resolve topographic gradient features with a TLS value of 5–10. Paired with the OceanMesh2D software, the ability to seamlessly apply local refinement allows the user to provide fine coastal resolution in the region of interest (e.g., the storm landfall region) without large increases in the total mesh vertex count (increase of 0.5 %–3 % in this study). We found that in general, peak storm tide elevations along the open coast are decreased (therefore, the coastal flooding potential is decreased) and the timing of the peak occurs later with local coastal mesh refinement. When validated against observed high-water marks measured near the coast, coastal mesh refinement only has a significant positive impact on errors in narrow straits and inlets, as well as in bays and lakes separated from the ocean by these passages.

The new ADCIRC v55 code capable of accurate global storm tide modeling with fine coastal resolution is computationally efficient. For global meshes with nominal minimum resolution as fine as 1.5 km, the computational wall-clock time ranged from 5 to 30 s per simulation day on 480 computational processors for astronomical tide simulations. Some improvements that we made to the numerical stability of the algorithm facilitated the application of relatively large 120 s time steps to achieve this efficiency. However, we found that the locally refined meshes (nominal minimum resolution of 500 and 150 m) often required smaller time steps (25–90 s). Nevertheless, these are still much larger than time steps used in previous studies with older versions of ADCIRC, resulting in computational times 1 to 2 orders of magnitude shorter.

The RMSE

The area-weighted ECDF curves of

To obtain a single number to compare the solutions globally or in certain depths or regions, it is common to use the area-weighted root mean square error of the RMSE

The official release version of ADCIRC is available from the project website at

The current version of OceanMesh2D is available from the project website at

An application of the presented ADCIRC v55 model (on the TLS-B mesh design) providing 5 d forecasts of global storm surge is currently running in real time, and maximum surge elevations are available to view at

The supplement related to this article is available online at:

WJP prepared the paper, designed and implemented the coding upgrades into ADCIRC v55, designed and performed the experiments, and conducted the stability analysis. DW mathematically formulated and initially implemented most of the ADCIRC coding upgrades. KJR and WJP equally contributed to the coding and development of the OceanMesh2D software integral to this study, and KJR provided critical feedback to improve the paper design. JJW provided the research and computing resources as well as the funding necessary to conduct this study.

The authors declare that they have no conflict of interest.

This work was supported by the US Alaska/Integrated Ocean Observing System (AOOS/IOOS) sub-award for US National Oceanic and Atmospheric Administration (NOAA) grant no. NA16NOS0120027, IOOS Ocean Technology Transition Project NOAA grant no. NA18NOS0120164, and NOAA Office of Weather and Air Quality, Joint Technology Transfer Initiative NOAA grant no. NA18OAR4590377.

This research has been supported by the National Oceanic and Atmospheric Administration (grant nos. NA16NOS0120027, NA18NOS0120164, and NA18OAR4590377).

This paper was edited by Claire Levy and reviewed by two anonymous referees.