This investigation addresses the tsunami inundation in Lima and Callao caused by the massive 1746 earthquake (

Over the past couple of decades, after the catastrophic 2004

The tuning required to reduce numerical uncertainties beyond those already inherent in natural phenomena can be resolved by comparing the effects of different numerical schemes. According to their purpose, tsunami models can be conditionally divided into two types. The first type includes operational models

Numerical models of the second kind should describe the inundation area with a very high spatial resolution and have reliable numerical wetting/drying schemes, which are associated with relatively high energy consumption in the computational aspect. In addition, often, the time between the occurrence of a tsunami and its approach to the coast is minimal, and then pre-calculated databases of possible scenarios of tsunami sources and numerical modeling

Numerical models for calculating surface gravity waves, including tsunami waves, within the framework of SW theory, depending on various approximations, can be divided into linear, nonlinear, and nonlinear dispersive. In operative models, linear equations are often used to reduce computational costs

The shallow water equations contain three types of nonlinearity: momentum advection, nonlinearity in the continuity equation due to variable water thickness, and nonlinear friction

Several works analyze nonlinearity in tsunami models, mainly focusing on comparing the wave amplitude in linear and nonlinear problems in general. Some works link analytical solutions with practical calculations

The theory of the run-up of long waves onto the shore is of considerable interest. This problem of various long symmetrical or asymmetrical waves with the same steepness of the front and rear slopes of unbroken waves onto a flat slope is quite well developed from a mathematical point of view within the framework of the nonlinear theory of shallow water, allowing for an analytical solution

The nonlinear-dispersion terms qualitatively and quantitatively change the amplitude and shape of the wave as it passes over an underwater obstacle or wave run-up on a vertical barrier

This work has a twofold purpose. The first goal relates to a comparative analysis of two numerical tsunami models, TsunAWI and Tsunami-HySEA, using the example of an inundation assessment caused by a strong destructive earthquake (

This study provides special focus on the analysis of the variability in the solution depending on the sensitivity to the Manning coefficient in the bottom friction term. When solving a system of SW equations, depending on the choice of bottom friction coefficient (Manning parameter), there is significant uncertainty in the solution's response to a given input parameter of the model

The article briefly describes two tsunami models, TsunAWI and Tsunami-HySEA, based on nonlinear SW equations. A distinctive feature of implementing these two models is their spatial discretization. TsunAWI operates on unstructured meshes and solves the equations with the finite element method, while the Tsunami-HySEA uses structured nested meshes and employs the finite volume method. The next section of the work is devoted to setting up the problem for these two models. Section 3 describes the simulation domain, initial conditions, and mesh characteristics. The calculation and comparative analysis of the simulation results of the two models are given in Sect. 4. In Sect. 5, based on the calculations of the TsunAWI model, an analysis of the nonlinearity contained in the equations is performed. Section 6 discusses the results of this work and concludes the study. The Appendix provides extended comparison results of the two models.

The shallow water equations

On the solid part of the boundary,

The problem (Eqs.

The equation for the energy of the external motion mode, whose equations are vertically averaged equations, is obtained by multiplying the first of Eq. (

The TsunAWI model is based on finite element methods. The main reason for choosing such a method is the computational grid, which can be adapted to cover basins with uneven bottom topography and coastline without generating nested meshes. The finite element spatial discretization in the TsunAWI model is based on the approach by

Simulation of tsunami wave propagation benefits from using an explicit time discretization. Numerical accuracy requires relatively small time steps, reducing implicit schemes' main advantage. Furthermore, modeling the inundation processes usually requires very high spatial resolution (up to meters) in coastal regions and, consequently, many nodes, drastically increasing the necessary computational resources in cases of implicit temporal discretization. The leap-frog scheme was chosen as a simple and easy-to-implement method. We rewrite Eqs. (

Advection is one of the essential complexities in calculating the waves of a tsunami in a shallow zone. Advection of momentum in the original formulation by

For modeling wetting and drying, we use a moving boundary technique which utilizes extrapolation through the wet–dry boundary and into the dry region

Because the leap-frog scheme is neutrally stable, it demands horizontal viscosity in places of large gradients. The horizontal viscosity coefficient is determined by a Smagorinsky parameterization

A detailed description of the TsunAWI model and its numerical implementation is given in

Tsunami-HySEA solves the two-dimensional shallow water system using a high-order finite volume method. These methods are mass preserving for arbitrary (nested) bathymetries. Tsunami-HySEA implements several reconstruction operators:

Monotonic Upstream-centered Scheme for Conservation Laws (MUSCL; see

the hyperbolic Marquina's reconstruction (see

the total variation diminishing (TVD) combination of piecewise parabolic and linear 2D reconstructions, which also achieves third order.

The Lima/Callao region is located on the central coast of Peru, characterized by a narrow strip of land between the Pacific Ocean and the Andes. The coastal zone is relatively flat, with sandy beaches and rocky cliffs interspersed along the coastline.

The bathymetry of the area is diverse: shallow water near the coast gradually deepens with distance from the coast. The continental shelf is relatively narrow and the water depth beyond it increases rapidly. The maximum depth in this area is about 2 km, located several hundred kilometers from the coast.

The bottom relief is crucial for numerical simulations of wave propagation and inundation. In the deep ocean, we use the General Bathymetric Chart of the Ocean (GEBCO) dataset at a resolution of up to 15 arcsec. Particularly, the company EOMAP produced high resolution bathymetric and topographic datasets for the nearshore range based on different sources. For the water area down to about 300 m depth, a combination of nautical charts and GEBCO was incorporated. Tandem-X topography data

Bathymetry and topography in a section of the model domain with data processing by EOMAP. Values are given in meters relative to average sea level (m a.s.l.). Basemap © OpenStreetMap contributors 2021. Distributed under the Open Data Commons Open Database License (ODbL) v1.0.

These numerical codes are based on two different meshes. In the case of Tsunami-HySEA, four levels of nested grids are used, and for TsunAWI, triangular meshes are required. To have similar resolutions along all domains between the numerical simulations that are compared, the triangular meshes are created from the nested grids with resolutions from 30 arcsec to about 1 arcsec (level 4). These meshes are shown in Figs.

Topography and bathymetry grids used for Tsunami-HySEA numerical simulations. In

Small section of the triangular mesh used for TsunAWI simulations. The edge length of triangles ranges from 10 km in the deep ocean (beyond the scope of this image) to about 10 m in the highest resolved part. Also included are some of the locations used to analyze the time series. Basemap © OpenStreetMap contributors 2021. Distributed under the Open Data Commons Open Database License (ODbL) v1.0.

The event under consideration is the historic tsunami on 28 October 1746. We use the source parameters proposed by

We compare the outcome of our modeling approach to their results. Our results are roughly similar, although bathymetry, topography data, and mesh resolution are different and complete agreement cannot be expected.

The source is shown in Fig.

Initial conditions for the tsunami numerical simulations used by both models. Parameters modified from

Starting from this initial sea surface elevation and zero velocity, we simulate the tsunami propagation and inundation for 4 h in real time. Besides the resulting inundation in the Lima/Callao region, we investigate and compare tide gauge records in virtual and real offshore positions as well as the temporal evolution in selected land points.

This section compares the simulation results of tsunami wave propagation and run-up modeling for the two numerical models presented above. The spatial and temporal fields show a detailed comparison of the simulation results. Some of the comparison results are given in the Appendix.

The Tsunami-HySEA and TsunAWI models are initialized with the same height fields and integrated over a 4 h time interval. The energy distribution indicated by the maximum wave amplitudes received during this period is shown in Fig.

Maximum wave amplitude (in m a.s.l.) for a simulation of 4 h propagation time in the models Tsunami-HySEA (left panel) and TsunAWI (right panel). Basemap © OpenStreetMap contributors 2021. Distributed under the Open Data Commons Open Database License (ODbL) v1.0.

The historic event of 1746 resulted in considerable inundation in the Lima/Callao region

Maximum flow depth (in meters relative to topography) in the inundated area in Callao obtained by Tsunami-HySEA (left panel) and TsunAWI (right panel) for a Manning value of 0.015 and a simulation time of 4 h. The box indicates the area in which the inundation extent and inundation properties were calculated. Basemap © OpenStreetMap contributors 2021. Distributed under the Open Data Commons Open Database License (ODbL) v1.0.

The inundation extent, as well as the height distribution, depends on the bottom friction parameter in the inundated area. Both models use bottom friction parameterization in Manning form with a constant Manning coefficient throughout the domain. Results for Manning values (

Upper panels: inundation extent for the smallest (

The lower panels of Fig.

We will continue the analysis of the results of the implementations of the two models by comparing the course of the wave height (see Fig.

Wave amplitudes and their spectra for M1–M3 stations in the implementation of two models: Tsunami-HySEA and TsunAWI. SSH: sea surface height.

In comparing the horizontal velocity components shown in Fig.

Comparison of horizontal velocity components at three stations, M1–M3, for two models: Tsunami-HySEA and TsunAWI. The top panel is the

The comparison of the two models presented in this section shows a high degree of agreement between the two solutions, which in turn allows us to choose the solution of TsunAWI for a more detailed analysis of the influence of nonlinearity on the behavior of the tsunami wave in the coastal zone and the inundation area.

The system of Eqs. (

Potential and kinetic energies for four experiments for shelf zone and inundation area. Upper panels: potential energy; bottom panels: kinetic energy; left panels: shelf zone (from 200 m to coast); right panels: inundation area.

We performed four main nonlinear experiments on the propagation of a tsunami wave in the region of interest. The first experiment is the solution to the complete equations with all nonlinearity (full tsunami model (FTM)). The second experiment (without momentum advection (WMA)) is connected with disabling advection in the equations of motion, the second term in Eq. (

In the conclusion of this paper, using various coefficients of bottom friction, an analysis is made of the influence of nonlinear friction on the solution in the inundation zone.

We begin our analysis of the influence of nonlinear terms on tsunami wave propagation by comparing the potential and kinetic energies (Eq.

Figure

Maximum sea surface amplitude in the ocean (in m a.s.l.) and flow depth (in meters relative to topography) for the complete simulation period. Upper left panel: full model (FTM); upper right panel: without momentum advection (WMA); lower left panel: omitting nonlinearity in the continuity equation (WNC); lower right panel: only bottom friction (OBF). Basemap © Google Earth 2018.

The difference in kinetic energy in the coastal zone shows an even greater contrast. The absence of momentum in the equation of motion plays the leading role in kinetic and potential energies. The difference in amplitude compared with that of the basic experiment (FTM) reaches 50 %. At the same time, we note that the calculation with only bottom friction shows an even more significant energy contribution. This behavior is explained by a strong nonlinear character in the velocity field on the background of nonlinear bottom friction. The momentum advection in the equations works in the regions of high nonuniform velocities as a dissipation factor. For this reason, the basic experiment (FTM) has a lower energy contribution.

As shown below, considering nonlinear terms does not increase or decrease the horizontal velocity or elevation in the inundation area but completely changes the dynamic fields' structure.

We continue comparing the nonlinear effect by considering the dynamic characteristics: the maximum wave height and the velocity modulus in the Lima/Callao region. Figure

A remarkable feature is highlighted by analyzing the maximum horizontal velocity modulus in the same region. Figure

Maximum velocity modulus (in m s

Figure

Time series of the sea surface elevation for the six points (M1, M2, M3, I4, S7, and S9) indicated in Fig.

Comparison of the wave heights shows good agreement in the amplitude of the first wave at points M1 and M2 with the amplitude values in

An analysis of the results reveals general patterns in studying nonlinear terms for stations M1–M3 offshore and coastal station I4. So, in the WNC experiments and OBF, the shape of the first incoming wave changes somewhat. In the absence of nonlinearity in the continuity equation, the waveform becomes flatter, which leads to a delay in the maximum of the first wave. At the same time, at stations located at a slightly small distance from the coast, i.e., S7 and S9, the effect of nonlinearity in the continuity equation is almost imperceptible. A similar shift in the maximum was also observed in the potential energy in the inundation area (see Fig.

Finally, we summarize inundation properties for the nonlinearity investigation in Fig.

Inundation area obtained with TsunAWI for the experiments with the deactivation of different nonlinear terms. Basemap: © OpenStreetMap contributors 2021. Distributed under the Open Data Commons Open Database License (ODbL) v1.0.

Properties of the inundation area obtained for different setups with regard to nonlinear terms (refer to Fig.

We examined the tsunami wave variations along the coast of Peru based on the 1746 earthquake scenario using two numerical models: Tsunami-HySEA and TsunAWI. For both models, it was found that there is a slight phase shift in the wave propagation velocity. Such a shift begins to manifest as the distance from the source of the tsunami wave increases. In the Tsunami-HySEA model, the leading edge propagation velocity slightly lags behind the wave velocity in implementing the TsunAWI model. We attribute this to the difference in dispersion errors in models with different spatial implementations. In the frequency spectrum, the wave maxima are redistributed at the main frequency of the tsunami wave for this event (

With an increase in the coefficient of bottom friction, the difference between the solutions of the two models, estimated by the total area and volume of water masses in the flood zone (see Fig.

Accounting for the nonlinear terms of the shallow water equations is numerically complex enough that they are often neglected in models designed to generate warning products remarkably quickly, such as in an early warning system. These terms are relatively insignificant in the deep ocean, and it may become acceptable to neglect them in computations. On the other hand, the contribution of nonlinearity becomes very significant when the tsunami wave reaches the coast and plays a very important role, especially in inundation.

A detailed assessment of the influence of nonlinearity on the solutions' behavior in the coastal and inundation areas has been conducted. The shallow water equations are considered in four formulations: complete equations (FTM); equations without momentum advection in horizontal velocity (WMA); in the absence of nonlinearity in the continuity equation, when velocity divergence is considered without taking into account free surface perturbations (WNC); and in the presence of only nonlinearity in bottom friction terms (OBF).

A preliminary assessment of the bathymetry of the area showed that the sharp bottom slope is located at a considerable distance from the coastline. The shelf zone extends for several kilometers behind it. It could be expected that, in this case, the nonlinearity in the continuity equation would be the maximum difference in the solution. A comparison of the results of wave height measurements at stations along the coast showed that the WNC experiment slightly shifts the beginning of the maximum, making the incoming tsunami wave flatter and, at the same time, practically does not change the wave amplitude. The lack of impulse advection (WMA) introduces the most significant changes in the amplitude of the incoming wave. Apparently, this is due to the orientation of the initial momentum of the free surface perturbation, which initially causes a significant shift in the velocity fields in space with a rather complex configuration of the coastline in the study area. We also attribute a significant increase, almost twofold, in the wave amplitude at coastal stations in the WMA experiment to a decrease in the dissipation of the solution, the role of which is partially played by the momentum advection term.

An analysis of the spatial solution in the flood zone without one or another nonlinearity introduces cardinal differences between the level and velocity fields from the complete problem statement. In the WNC experiment, the maximum wave amplitude on the coast is significantly underestimated, while in the WMA calculation the flood maxima are overestimated.

Overall, we confirm that nonlinearity plays a decisive role in estimating inundated areas, wave heights, current speeds, and the spatial structure of inundation maps. These factors should be considered when conducting numerical simulations of tsunami hazards to ensure that the solution persists in the nearshore and inundated zones.

For the sake of completeness of the analysis, we will summarize some more results regarding the model comparison in this Appendix. The inundation area for a given Manning value is quite similar for the two numerical models. This is consistent with a similar assessment conducted for the Chilean cities Valparaíso, Viña del Mar, and Coquimbo

The inundation properties also depend on the shape of the bottom relief in the offshore domain. In the case of Lima and Callao, the nearshore bathymetry is characterized by a very gentle slope in the bottom relief as shown in Fig.

A collection of all inundation polygons is shown in Fig.

Table

The velocity (modulus) distributions, together with the maximum values in the intersect, are shown in Fig.

Maximum flow depth (in meters relative to topography) in La Punta and the Callao port area for simulations of 4 h propagation in Tsunami-HySEA (left panel) and TsunAWI (right panel). Basemap © OpenStreetMap contributors 2021. Distributed under the Open Data Commons Open Database License (ODbL) v1.0.

Upper panels: maximum wave amplitude in the ocean (in m a.s.l.) and flow depth on land (in meters relative to topography) in the Callao port area. Lower panel: maximum wave amplitude projected to the intersect shown in the upper panels. All results obtained for Manning value of 0.020. Basemap © Google Earth 2018.

Slope in bathymetry and topography in the Callao port area. This gradient is determined from the data shown in Fig.

Inundation area with minimum flow depth of 1 cm for all Manning values obtained with Tsunami-HySEA (left panel) and TsunAWI (right panel). Basemap © OpenStreetMap contributors 2021. Distributed under the Open Data Commons Open Database License (ODbL) v1.0.

Upper panels: maximum absolute velocity (in m s

Properties of the inundation area obtained with both models for the full range of Manning values. The numbers specify the inundation area (with a minimum flow depth of 1 cm) in the bounding box shown in Fig.

Inundation quantities as indicated in Table

The inundation data presented in this study can be accessed from

Conceptualization: AA, SH, and NZ; numerical model development: SH, NZ, AA, and NR; numerical simulations: SH, AA, and NZ; data processing: NZ, SH, and KK; writing the original draft: AA, SH, and NZ; review and editing of the manuscript: AA, SH, NZ, NR, and KK; visualization: SH, NZ, and AA. All authors read and approved the final version of the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This article is part of the special issue “Multi-risk assessment in the Andes region”. It is not associated with a conference.

We are grateful to Jorge Macías and Carlos Sánchez and the Edanya Group at the University of Málaga for sharing the Tsunami-HySEA code. Most figures were generated with QGIS

Part of this research was funded by the German Federal Ministry of Education and Research within the projects RIESGOS and RIESGOS2.0 (grant numbers 03G0876C and 03G0905C). Natalia Zamora was funded by the Marie Skłodowska-Curie grant agreement H2020-MSCA-COFUND-2016-75443 and the ChEESE-2p – Center of Excellence in Solid Earth grant agreement no. 101093038. The article processing charges for this open-access publication were covered by the Alfred-Wegener-Institut Helmholtz-Zentrum für Polar- und Meeresforschung.

This paper was edited by Torsten Riedlinger and reviewed by two anonymous referees.