## Abstract

Numerical simulations of Submarine Mass Failures (SMFs) are performed along the upper US East Coast to assess the effect of slide deformation on predicted tsunami hazard. Tsunami generation is simulated using the three-dimensional non-hydrostatic model NHWAVE. For rigid slumps, the geometry and law of motion are specified as bottom boundary conditions. Deforming slide motion is modeled using a depth-integrated bottom layer of dense Newtonian fluid, fully coupled to the overlying fluid motion. Once the SMFs are no-longer tsunamigenic, tsunami propagation simulations are performed using the Boussinesq wave model FUNWAVE-TVD, using nested grids of increasingly fine resolution towards shore and employing a one-way coupling methodology. Probable maximum tsunamis are simulated for Currituck SMF proxies sited in four areas of the shelf break slope that have enough sediment accumulation to cause large failures. Deforming slides have a slightly larger initial acceleration, but still generate a smaller tsunami than rigid slumps due to their spreading and thinning out during motion, which gradually makes them less tsunamigenic. Comparing the maximum envelope of surface elevations along a 5 m isobath, consistent with earlier work, the bathymetry of the wide shelf is found to strongly control the spatial distribution of tsunami inundation. Overall, tsunamis caused by rigid slumps are worst case scenarios, providing up to 50% more inundation than for deforming slides having a moderate level of viscosity set in the upper range of debris flows. Tsunamis from both types of SMFs are able to cause water withdrawal to the 5 m isobath or deeper. Bottom friction effects are assessed by performing some of the simulations using two different Manning coefficients, one 50% larger than the other. With increased bottom friction, the largest tsunami inundations at the coast are reduced by up to 15%. Selected simulations are rerun by turning off dispersion in the model, which leads to moderate changes in maximum surface elevations nearshore (− 10 to + 5% changes), but to more significant effects in the far field (− 40 to 80% changes). Onshore, dispersion causes the appearance of short period undular bores that eventually break nearshore without significantly affecting inundation at the coast. However, these bores increase wave-induced maximum flow velocity and impulse forces, the latter by up to 40%, which may affect the design of coastal structures.

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## Acknowledgements

This work was supported by the National Tsunami Hazard Mitigation Program (NTHMP), NOAA, through Grants NA-15-NWS4670029 and NA-16-NWS4670034 to the University of Delaware (with subaward to the University of Rhode Island). Additional support at the University of Rhode Island and the University of Delaware came from Grants CMMI-15-35568 and CMMI-15-37568 from the Engineering for Natural Hazards Program, National Science Foundation, respectively. Numerical simulations reported in this work used HPC resources, as part of the Extreme Science and Engineering Discovery Environment (XSEDE) (project BCS-170006), which is supported by the National Science Foundation (NSF) Grant number ACI-1548562. FUNWAVE-TVD is open source software, available at http://github.com/fengyanshi/FUNWAVE-TVD/. NHWAVE is open source software, available at http://github.com/jimkirby/nhwave/. Finally, the authors acknowledge anonymous reviewers for their thorough and constructive reviews of this work.

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## Appendix: SMF Geometry and Slump Law of Motion

### Appendix: SMF Geometry and Slump Law of Motion

For rigid slumps, kinematics is specified based on the analytical laws developed by Grilli and Watts (1999, 2005); Watts et al. (2005). Additionally, as in Enet and Grilli (2007), the SMF geometry is idealized as having a “Quasi-Gaussian” shape (below seafloor) of elevation \(\zeta (x,y)\), whose steepness is controlled by a shape parameter \(\varepsilon\) (here \(\varepsilon = 0.717\)), and elliptical footprint of downslope length *b*, width *w*, and maximum thickness *T* defined as (Fig. 23)

where (\(\xi , \chi\)) are the local downslope and spanwise horizontal coordinates, rotated in the direction of SMF motion \(\theta\), and \(k_{\text {b}}=2C/b\), \(k_{\text {w}}=2C/w\), with \(C = a{\text{cos}} h(1/\varepsilon )\). With this geometry and parameters, the SMF volume is given by

and

[Note that Eqs. 2 and 3 have been corrected and are different from those reported in earlier papers (e.g., Enet and Grilli 2007; Grilli et al. 2015), which resulted from a mistake in the volume calculation.] For the specified \(\varepsilon\), we find, \(C=0.8616\), \(I_1=0.4804\), \(I_2=0.5672\), and \(V_{\text {s}} = 0.3508\, b w T\).

Earlier modeling work (Locat et al. 2009) indicates that, during its tsunamigenic period of motion, the Currituck SMF achieved a relatively small maximum displacement (runout) \(S_\mathbf{f } < b\) in its main direction of motion down the slope, over an unknown time of motion \(t_\mathbf{f }\). The combination of rigid block SMF and small displacement parallel to the slope supports modeling the SMF kinematics as a rigid slump or a deforming slide with moderate deformation achieving a similar runout over the same time. In either cases, one can assume a constant basal friction (i.e., slide-to-substrate friction) and negligible hydrodynamic drag (Grilli and Watts 2005). This type of rigid-body motion kinematics was investigated in earlier work (see above-listed references), leading for the slump to a pendulum-like center of mass motion *S*(*t*) parallel to the local mean slope of angle \(\alpha\). Here, this simple law of motion for rigid slumps is used, which reads

with \(S_0=S_\mathbf{f }/2\) and \(t_0 = t_\mathbf{f }/\pi\), and \(S=S_\mathbf{f }\) for \(t>t_\mathbf{f }\) (assuming SMF triggering occurs at \(t=0\)).

At \(t=0\), the SMF elevation is specified below the current seafloor bathymetry \(h_0(x,y)\). Given the SMF initial center of mass location (\(x_0, y_0\)) in global axes (*x*, *y*) (i.e., coordinates of the center of the elliptical footprint) and azimuth angle of SMF motion \(\theta\), the coordinate transformation to the local SMF slope-parallel coordinate system (\(\xi , \chi\)) (Fig. 23) is defined as

with *S*(*t*) given by Eq. (4).

Then, assuming \(\sin \alpha \simeq 0\) for small slopes, the instantaneous seafloor depth above the SMF is given by

with \(\Delta h = h - h_0\). The seafloor motion described by Eq. (6) is similar to a translation parallel to the average slope of part of the seabed, over the actual bathymetry. The vertical seafloor velocity (used in NHWAVE as a bottom boundary condition) is computed as

which can be easily derived from Eqs. (1) to (6) as

the slump velocity obtained from Eq. (4), with \(U_{\max }=S_0/t_0\) the maximum velocity. Similarly, the slump acceleration is found as

the initial acceleration.

For rigid slumps, hydrodynamic drag can be neglected due to low velocity and small amplitude of motion, and inertia includes both the SMF mass \(M_{\text {s}} = \rho _{\text {s}} V_{\text {s}}\), with \(\rho _{\text {s}}\) denoting the sediment bulk density, and the specific density being defined as \(\gamma = \rho _{\text {s}}/\rho _{\text {w}}\), with \(\rho _{\text {w}}\) the water density, and an added mass \(\Delta M_{\text {s}} = C_{\text {M}} \rho _{\text {w}} V_{\text {s}}\), defined by way of an added mass coefficient \(C_{\text {M}}\). Assuming a constant basal friction, a nearly circular rupture surface of radius *R*, and a small angular displacement \(\Delta \Phi\), Grilli and Watts (2005) derived the characteristic distance and time of motion for rigid slumps as

with *g* denoting the gravitational acceleration. Equation (8), proposed by Watts et al. (2005), is a semi-empirical relationship to estimate the radius of slump motion as a function of slump downslope length and maximum thickness.

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Schambach, L., Grilli, S.T., Kirby, J.T. *et al.* Landslide Tsunami Hazard Along the Upper US East Coast: Effects of Slide Deformation, Bottom Friction, and Frequency Dispersion.
*Pure Appl. Geophys.* **176**, 3059–3098 (2019). https://doi.org/10.1007/s00024-018-1978-7

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DOI: https://doi.org/10.1007/s00024-018-1978-7