We have implemented our model on an updated bathymetry of with the additional inclusion of 77 000 digitized sounding points from a set of 34 nautical charts published by the Bangladesh Navy (See Supplement Fig. S1). Our bathymetric dataset is a combination of two digitized sounding datasets in the nearshore zone – one from navigational charts produced by the Bangladesh Navy and another being a bathymetry of the Hooghly estuary provided by the IWAI (Inland Waterways Authority of India). Depending on the sounding points, these observations are 7 to 20 years old. The river bathymetry is composed of a set of cross-sections obtained from the Bangladesh Water Development Board (BWDB), which is further interpolated at about 300 m resolution using a dedicated 1D river modelling tool and GIS techniques. The south-central part of the delta is composed of a high-resolution (50 m) inland topography (see Supplement Fig. S2). We considered the General Bathymetric Chart of the Oceans (GEBCO) 2014 bathymetry to complement in the deeper part of the ocean (https://www.gebco.net/data_and_products/gridded_bathymetry_data/, last access: 20 June 2020) and the Shuttle Radar Topography Mission (SRTM) digital elevation model for the rest of the inland topography as appearing in the GEBCO 2014 dataset (https://www2.jpl.nasa.gov/srtm/cbanddataproducts.html, last access: 20 June 2020). Our unstructured model mesh was developed based on this blended bathymetry, covering the northern Bay of Bengal (11 to 24∘ N) with a variable resolution based on the shallow-water wave propagation and bottom slope criteria. Our model grid consists of about 600 000 nodes and 1 million elements in total. The resulting grid resolution ranges from 250 m in the estuaries and the delta to 15 km in the deeper part of the central Bay of Bengal. The model domain and mesh are identical to those presented in (Fig. ).

To account for the effect of short waves on the mean circulation, the Wind Wave Model III (WWMIII), a third-generation spectral wave model, is coupled at source code level with SCHISM in our modelling framework . In our configuration, the WWMIII solves the wave action equation on the native SCHISM grid using a fully implicit scheme. The source terms in our model comprise the energy input due to wind , the non-linear interaction in deep and shallow water, energy dissipation in deep and shallow water due to white capping and wave breaking, and energy dissipation due to bottom friction. We run the WWMIII with 12 directional and 12 frequency bins. Every 30 min of SCHISM runtime, water level and currents are passed to the WWM for calculating the evolution of the wave fields. Calculated wave radiation stress, total surface stress, and the wave orbital velocity are passed back to SCHISM before computing the next time step. Wave breaking is modelled according to . As the nearshore region of the Bengal delta has a very mild slope, the value for the breaking coefficient α, which controls the rate of dissipation, was reduced from 1 to 0.1 to avoid over-dissipation . The WWMIII was also forced along the model's southern open boundary by time series of directional spectra, computed from a large-scale application of the WaveWatch3 (hereafter WW3) model . The WW3 was implemented at the scale of the whole Indian Ocean with a spatial resolution of 0.5∘ and forced by the final analysis (FNL) reanalysis wind data at a 3 h time step .

Our model is forced over the whole domain by the astronomical tidal potential for the 12 main constituents (2N2, K1, K2, M2, Mu2, N2, Nu2, O1, P1, Q1, S2, T2). At the southern boundary, we have prescribed a boundary tidal water level from 26 harmonic constituents (M2, M3, M4, M6, M8, Mf, Mm, MN4, MS4, Msf, Mu2, N2, Nu2, O1, P1, Q1, R2, S1, S2, S4, Ssa, T2, K2, K1, J1, and 2N2) extracted from the FES2012 global tide model . At open upstream-river boundaries of the Ganges and Brahmaputra, a discharge time series from the BWDB is forced for the benchmark tidal simulation, and a climatologic discharge time series is applied for storm surge simulations during Amphan. A discharge climatology is applied at Hooghly and Karnaphuli . At Meghna and Rupnarayan open river boundaries, we implemented a radiating Flather boundary condition. From a year-long tidal simulation, we found that the updated version of the bathymetry performs 2–4 times better at key locations compared to the global tidal models, which include Finite Element Solution (FES; ), Goddard Ocean Tide (GOT; ), and TOPEX/POSEIDON (TPXO; ) (see Supplement Table S1).

We derived the atmospheric wind and pressure fields for SCHISM from a blending of analytical wind field inferred from a parametric wind model (close to the cyclone track) and background atmospheric field from the Global Forecast System (GFS; https://www.ncdc.noaa.gov/data-access/model-data/model-datasets/global-forcast-system-gfs, last access: 30 May 2020) reanalysis (farther away) following . We used the best-track data of the JTWC provided at a 6 h time step as input for the analytical wind and pressure field. Here we choose the analytical wind profile of and the analytical pressure field profile of . To be consistent with our forecast described in Sect. 5, the background wind field was generated incrementally from an accumulative merging of GFS forecasts for each 6 h forecast window, with a 1 h time step. The analytical and background wind fields were first temporally interpolated every 15 min and overlaid on the background GFS fields using a distance-varying weighting coefficient for the cyclone centre to ensure a smooth transition (Fig. b). We took into account the asymmetry of the cyclonic wind field following . The final resolution after merging the analytical wind field with the interpolated background GFS fields is 0.025∘. For all storm surge simulations, a spinup time of 2 d is considered in this study.

With the above-described model set-up, we ran our model to calculate the water level and sea state evolution during Amphan. The simulation was done a posteriori once the cyclone had passed and with JTWC and GFS hindcasts being already available throughout the cyclone lifetime. Figure  shows comparisons between observed and modelled significant wave height (SWH), peak wave period, and water level. First, we compare the significant wave height derived from the altimetric estimate of Sentinel-3B processed by the Wave Thematic Assembly Center from the Copernicus Marine Environment Monitoring Service (https://scihub.copernicus.eu/, last access: 18 June 2020), which overpassed on 18 May 2020 at 16:03 GMT. For each altimetric data point inside our domain, we interpolated the model output using a nearest-neighbour approach, both spatially and temporally. This comparison shows that SWH is reasonably reproduced, with RMSE typically within 1 m (within 18 % of the mean value). However, we observed an overestimation of SWH of about 2.5 m near the cyclone centre. One of the reasons for such a difference is a limitation of analytical wind field models, as discussed by .

We present an in-situ-observed time series of SWH and mean wave period during the life cycle of Amphan in Fig.  for BD08 (Fig. b–c) and BD11 (Fig. d–e) oceanographic buoys (the dataset is collected from the Indian National Centre for Ocean Information Services (INCOIS) data portal at https://incois.gov.in/portal/datainfo/mb.jsp, last access: 30 May 2020). The two buoys are located on either side of the Amphan track, 250 km to its east and 230 km to its west, respectively. We can see that at both locations, the model reproduces the SWH well. The match is particularly good at BD11, consistently within 1 m of the observed evolution. At BD08, SWH is found to be overestimated by about 2 m at its peak, with a 12 h lead shift in time. Such a difference can come from positional and timing uncertainties in JTWC best-track data, from our assumption of a constant translational velocity in between each 6-hourly JTWC time step, and finally from the intrinsic uncertainty in the analytical wind field itself, as shown in Fig. b. Despite multiple sources of potential errors and uncertainties, our model appears capable of capturing the general features of the sea state, both significant wave height (SWH) and period, relatively well. The overall predictive skills of the model in terms of short waves are only slightly below those from hindcast exercises typically conducted over other ocean basins using similar modelling strategies .

The real-time availability of observed water level time series in this region is relatively scanty. We were able to access only a handful of coastal water level records, most of them located eastward of the landfall location. The right panels of Fig. f–j illustrated the comparison of the recorded water level and modelled storm surge. Among these tide gauges, Chittagong (Fig. i) and Visakhapatnam (Fig. j) time series are retrieved from the UNESCO Intergovernmental Oceanographic Commission (IOC) sea level monitoring service (http://www.ioc-sealevelmonitoring.org/map.php, last access: 18 June 2020), and the rest are provided by the Bangladesh Inland Water Transport Authority (BIWTA; http://www.biwta.gov.bd/, last access: 18 June 2020). The datum for water level time series at Chittagong and Visakhapatnam is changed from in situ datum to MSL by removing long-term mean. The time series provided by the BIWTA are already referenced to MSL and thus kept unchanged. We also plotted the reconstructed tidal water level from the tidal atlas derived from our model. The overall tidal propagation is well simulated in Galachipa and Kuakata, while at Tazumuddin the simulation tidal range is found to be underestimated. The local bathymetric error and friction parameterization might be the source of the discrepancy. Tide gauge at Tazumuddin is situated at the mouth of the Meghna, where bathymetry is rapidly changing . However, our model correctly reproduces the peak of the water level at all locations. The maximum recorded water level at these stations is around 2–2.5 m a.m.s.l., with a storm surge (water level – tide) of the same order of magnitude.

To spatially assess the storm surge generated by Cyclone Amphan, we first take a look into the temporal maximum water level in Fig. a. We can see that the whole coastal region experienced a high water level ranging from 2–5 m for Amphan. To quantify the contribution of the cyclone, we have looked in the surge component, defined as the difference between the total modelled water level and a tide-only simulation. Figure b illustrates the temporal maximum of surge over the delta region. The highest surge level is 5 m around the location of the landfall (21.6∘ N, 88.3∘ E). The maximum non-linear interaction between tide and surge amounts to about 30 cm in the nearshore domain (Fig. d).

To quantify the contribution of the wave set-up, we re-calculated the storm surge without the coupling with the wave model. Figure c shows the maximum of the difference between the two simulated water levels. In general, wave set-up contributes to the Amphan storm surge by about 20 cm all along the Bengal shoreline and locally over 30 cm close to the cyclone landfall. Nevertheless, the spatial resolution at the nearshore region employed in this study (250 m) is still too coarse for capturing the maximum wave set-up that develops along the shoreline, where a resolution of a few tens of metres should be employed . Nonetheless, this comparison shows that wave set-up not only is developed along the shoreline exposed to waves but affects the whole delta up to far upstream, a process already observed over large estuaries .

Our findings reaffirm the necessity of a proper coupling between tide, surge, and wave to forecast the water level evolution. For Amphan, the magnitude of tide–surge interaction amounts to about 10 % of the maximum total water level with spatial variation. Similarly, the magnitude of wave set-up is also spatially varying, with a typical amplitude of 10 %–15 % of the maximum total water level. This non-negligible, non-linear dependency shows the importance of a fully coupled hydrodynamics–wave modelling system for a successful forecast of the water levels in the hydrodynamic setting of the Bengal delta. Our modelling framework showed reasonable accuracy in reproducing observed storm surge water level along the Bengal coastline. The accuracy is in line with the typical level of performance of similar systems applied elsewhere in the world ocean . The hindcast experiment thus shows the viability of our model for a proof of concept of a real-time forecasting exercise.

During Cyclone Amphan, we performed near-real-time storm surge forecasts based on our model. In our forecasts, we relied on the outputs of global models (GFS and HWRF) and advisories (JTWC) for the estimates of the atmospheric forcing. These advisories and global models are updated every 6 h. Similar to the GFS forecast cycle, we updated our forecasts at every 00:00, 06:00, 12:00, and 18:00 UTC.

For each forecast, we derived the atmospheric forcing from a blend of JTWC, HWRF, and GFS data (Table ). The JTWC publishes storm advisories at 03:00, 09:00, 15:00, and 21:00 UTC. The advisory includes an analysis of the storm intensity and position from a satellite fix 3 h prior, followed by a forecast for the following 72 h, with 6 to 12 h time steps. The NOAA published their HWRF model advisory for each GFS forecast cycle at 3 h time steps. The GFS provides the wind and pressure fields in hourly time steps at a 0.25∘ regular spatial grid. The GFS model is initialized 6-hourly at 00:00, 06:00, 12:00, and 18:00 UTC, and the data are available 3 h after the initialized period.

The wind and pressure fields around the centre of the storm are derived analytically from a concatenated JTWC and HWRF. We used GFS data as the background wind and pressure on the outer region of the analytical fields.

We initiated our forecast cycles on 16 March 2020 at 06:00 UTC, utilizing the first advisory from the JTWC published at 03:00 UTC merged with the forecast from the HWRF issued at 18:00 UTC of the preceding day. We took the background wind and pressure field from the GFS forecast published at 00:00 UTC. In the subsequent forecast cycles, we updated the previous track file by first replacing the past time steps with the analysis from the latest JTWC advisory, then appending the forecasts from the HWRF again. For the background wind and pressure fields from the GFS, we retained the first 6 h of forecasts from the previous cycles and updated the remainder of the time series from the latest forecast with new initialization. For each forecast cycle, we kept the starting time of our model the same, on 16 May at 00:00 UTC, and ran it until 1 d after the landfall for a consistent initial condition. This approach is feasible as the storms in the Bay of Bengal typically form, grow, and dissipate in about a week. A graphical representation of our strategy to temporally concatenate JTWC, GFS, and HWRF forecasts over a forecast cycle is shown in Fig. a. The spatial blending of the analytical and GFS wind and pressure field is illustrated in Fig. b.

Since our forecast is contingent on the JTWC advisory of 3 h before, we only had a 3-hour-long window to achieve the pre-processing of the forcing files, to run the model, and to complete the necessary post-processing of the model outputs. The numerical efficiency of our model made it possible to cope with this time constraint of the forecast, using a modest computing resource. The computing environment we used is detailed in Table , which essentially amounts to a single desktop computer fitted with a high-end consumer-grade processor.

Figure  illustrates the evolution of the maximum total water level and surge for the corresponding forecasts issued at T-60 h, T-36 h, and T-12 h, where T is the time of landfall (20 May 2020 at 08:00 to 10:00 UTC). For the forecast issued at T-60 h, the cyclone landfall is located near 90∘ E, associated with the strong surge simulated over the surrounding area. As the atmospheric cyclone forecasts gradually converge towards the actual landfall location more than 100 km westward, so do our storm surge forecasts at T-36 h and T-12 h. The dependency of the timing of the landfall on the forecast range is notable here. At T-36 h, the maximum water level pattern is similar to the hindcast shown in Fig. . By T-12 h, the magnitude of the forecast maximum water level corresponds well with the hindcast estimate.

To substantiate the gradual increase in the quality of the forecasts, we compared the maximum surge level predicted at the various forecast ranges (T-60 h, T-36 h, T-12 h) with the hindcast experiment along a line encompassing the nearshore delta (segment displayed in Fig. ). The results show that, in the T-60 h forecast, the location of maximum surge appears offset eastward by as much as 150 km. The magnitude of the maximum surge is also poorly predicted, with an overestimation of about 3 m. In the T-36 h forecast, the location of the maximum surge appears largely corrected, but the magnitude of the peak remains overestimated by about 3 m. In the T-12 h forecast, both the location and magnitude of the peak surge are in relatively good agreement with the hindcast. Overall, along this landfalling coastal section, the standard error in the maximum surge level amounts to 2.06, 1.73, and 0.66 m for the T-60 h, T-36 h, and T-12 h forecast, respectively.

The water level dynamics over the Bay of Bengal and particularly around the Bengal delta is complex, and the various components of water level have different relative contributions depending on the location. Due to tide–surge interaction, the maximum water level can be lower or higher depending on the tidal phase in a fully non-linear tide–surge coupled simulation compared to its linear counterpart. During Amphan the tide–surge interaction was mostly negative, except within about 10 km of the landfall location. The positive interaction near the landfall location means that the coupled tide–surge estimate was higher than the linearly added tide and surge. In this case, the typical magnitude of tide–surge interaction ranged from -30 to +30 cm. The contribution from the wave-induced set-up along the shoreline increases the maximum water level estimation throughout the coast by about 10 %–15 % during Amphan.

Aside from the necessary inclusion of relevant physical processes – tide, surge, waves, and their non-linear interactions – the performance of storm surge forecasting depends on the wind and pressure forcings. The errors in the wind forcing and in the atmospheric pressure forcing are typically the most important ones among the various sources of error in storm surge modelling . In our forecast environment, the source of this error is twofold – the analytical model used to synthesize the fields from the storm parameters and the numerical weather model used to forecast the storm wind and pressure fields to derive the storm parameters themselves. There have been attempts to find a more accurate formulation from recent satellite scatterometer missions . However, not one formulation works best in all radial distances from the storm centre. Despite the well-known error associated with the parametric wind field, they are widely used in applications due to their computational simplicity and lightweight data requirements . The best way to avoid the error from the analytical wind field might be not using these formulations and relying instead on the full-fledged atmospheric forecasts. However, atmospheric models are costly in terms of computation. It is particularly true for cyclonic storms where high spatial resolution (typically kilometric) is required .

From the validation of water level at a few tide gauge locations for our hindcast simulation, it appears that the modelling framework proposed here could capture the maximum surge level successfully throughout the coast. Our tide gauge sites were limited to the eastern side of the Amphan landfall location and were operational throughout the life cycle of the cyclone. However, it was not possible to confirm the ability of the model to reproduce the water level on the western side of the landfall location due to data unavailability. The proper reproduction of the maximum water level by the model gives strong confidence in the formulation as well as in the coupling strategy of our modelling framework.

One of the potential, but challenging, outcomes of storm surge forecasting is the prediction of inundation. Particularly in the northern Bay of Bengal, the existing modelling studies do not generally consider the inundation process . Some modelling systems take advantage of simplified inundation modelling schemes to tackle this problem . In our modelling framework, the inland inundation is calculated seamlessly by SCHISM, solving the same hydrodynamics over the model domain, thanks to the combination of a cross-scale unstructured grid and an efficient wetting–drying algorithm. While the recent improvements in model bathymetry and the detailed accounting of dikes and coastal defences improved the overall modelling of the inundation, the associated error remains large . As an update to , in our model we have incorporated a novel dike height dataset bearing varying dike crest heights for the numerous polders scattered around the Bengal delta. The assessment of the impact of the updated embankment heights is, however, hard to quantify and validate.

In the absence of any operational network of inundation monitoring, to understand and better characterize the inundation dynamics, we systematically skimmed the Bengal delta local newspapers on the day of Amphan landfall and on the few subsequent days to achieve an as-comprehensive-as-possible mapping of the inundation extent observed in situ. We digitized the reported flooded locations through © Google Earth and categorized the inundation mechanism. We digitized 88 such locations over the Bangladeshi part of the Bengal delta, as shown in Fig.  . Over the Indian side of the delta, the news reporting of inland flooding from dike breaching was not accurate enough for us to be able to geotag the locations. We overlaid the inundated locations over the inundation predicted by the model on a false-colour image derived from the Sentinel-2 satellite using © Google Earth Engine (https://earthengine.google.com, last access: 25 June 2020). Three categories of inundation mechanism are typically observed – inundation by breaching of the dikes (labelled as “embankment breach”), inundation of unprotected low land by increased water level (“unembanked lowland”), and flooding by overflowing of the dikes (“embankment overflow”). From the reported news, it is clear that the major part of the inundation results from the breaching of dikes. We also note two breaching instances in the south-eastern corner of our domain, very far from the cyclone landfall. On the other hand, the flooding from merely increased water level in non-diked areas is mostly concentrated along the Meghna estuary. Along the estuary, designed embankments probably do not exist to protect many populated low-lying char areas (land or island formed though accretion). We could find only one report of the occurrence of overflowing of an embankment, in Maheshwaripur of Koyra Upazila (22.45∘ N, 89.30∘ E). Around 22.4–22.6∘ N, 90.1–90.5∘ E, in Barishal district, our model predicted an extended inundated area, which is known to not be polderized. While we could find some reports of flooding around that region from our newspaper survey, the widespread inundation predicted by our model in this region probably did not occur in reality due to the presence of levees and city protection embankments. Indeed, this kind of small-scale geographic information, to the best of our knowledge, is not systematically available publicly and thus could not be incorporated in our model topography, despite being of utmost importance for localized inundation forecasting. The forecast flooded vs. non-flooded areas show a wealth of spatial scales, demonstrating that our unstructured modelling framework has the intrinsic capability to model the small-scale hydrodynamic gradients. In our modelled forecasts, the contrast between the well-predicted water level temporal evolution and the comparatively limited predictive skills of inundated locations points out the necessity of accurate knowledge and incorporation of reliable topographic and embankment information at the local scale. Our modelling system has the ability and appropriate resolution to ingest such topographic information efficiently due to its unstructured nature.

The coastal polder management has long been a boon and a bane . On one side, the polders have protected the fertile areas from daily tidal flooding of brackish waters, which favoured agricultural production by reducing the soil salinity . On the other hand, these embankments have restricted fresh water and sediment supply, causing significant subsidence inside the embanked areas due to sediment compaction . Over time, the land use inside these polders has also changed from agriculture to aquaculture (namely shrimp and crab farming). Although the embankments were not built to protect from the cyclonic surges, in many instances they functioned as such. The presence of these protective structures also contributed to a false sense of security among the residents living inside . Embankments have become part and parcel of survival in this low-lying region. It has now become essential to monitor the condition and topography of the embankments for efficient management and maintenance. From a forecasting point of view, embankment breaching is extremely hard to model in the state-of-the-art hydrodynamic modelling frameworks. However, consistent periodic monitoring of the embankment conditions can improve the forecast by providing a more objective view of the associated inundation hazard.

Throughout our forecast and hindcast experiments, we have relied on freely available forecast products as the forcing for our storm surge model. These forecasts data are publicly available within 3 h of their initialization time through online portals. The primary input to generate the forcing is the forecast storm track distributed as a text file, which is very lightweight even for a limited internet connection. We have implemented our storm surge model on a freely available open-source framework. The model code is already operationally used in regional forecasting systems and services .

As we have demonstrated, the modelling set-up presented here is tractable in an operational, real-time forecasting scenario. Thanks to advanced and efficient numerics, one can deploy such a system in a consumer-grade computing environment. The computing set-up we have deliberately used (see Table ) essentially amounts to a high-end x86-64 personal desktop computer (e.g. Intel® Core^™ i9-10900 or similar). This computing requirement takes only a portion of the available computing capacity of an institute such as the BMD (15×4 cores at 2.8 GHz) . Furthermore, the model is easily deployable in a cloud computing infrastructure , which could provide additional reliability and cost savings for an event-driven operational forecasting system.