From Gerris
Introduction
The simulations below were obtained using the nonlinear shallowwater solver of Gerris. This numerical model provides solutions of the mathematical equations describing the flow of a "single layer" of fluid. It is based on the assumption that the wavelength of the tsunami waves is large compared to the depth of the sea. As can be seen on the accompanying animations, the wavelength of tsunami waves can vary between several tens of kilometres in deep water down to a few tens of metres in shallow water. The assumption that their wavelength is long compared to the depth of the sea is verified in most areas.
Numerical models describe the solution by sampling it at given intervals. The spacing between these samples (the "spatial resolution") controls how accurate the solution is. Short waves (near the coast) need to be described with very short sampling intervals (high resolution) whereas long waves (in deep water) can be described with longer sampling intervals (lower resolution). Gerris automatically adjusts this sampling interval to guarantee accurate solutions everywhere. For the simulations below the sampling interval varies automatically between about one kilometre in deep water down to 60 or 15 metres near the coastline. This allows to capture accurately both the longdistance tsunami wave propagation (for distant earthquakes such as Kuril for example) as well as very detailed local inundation on the coasts of Wallis and Futuna. This "adaptive" numerical method has been validated extensively both using theoretical and experimental test cases [Popinet, 2011] as well as realworld tsunamis [Popinet, 2012].
Aside from the choice of spatial resolution, the accuracy and reliability of the modelled scenarios depend also on the quality of the input data. The bathymetry and topography data are important as they control both the deepwater propagation as well as details of local inundation. For Futuna, we had only limited data for the local topography (contour lines at 0, 2.5, 5, 10, and 20 metres) and highresolution multibeam data but only for depths below 10 metres. We had no data on the fringing reef. The reef is particularly important as the shallow water in this area can break and dissipate tsunami waves before they reach the coastline. To fill this gap, we digitised a single line for the contour of the reef from navigation charts and set its depth to 1 metre everywhere. This is clearly a very crude approach but was the best we could do.
For Wallis, we had acquired highresolution GPS data for most land areas close to the coastline (on the main island only) and also had access to highresolution multibeam data for some small areas in the lagoon, as well as for depths below 10 metres for the outer reef wall. The influence of the reef and lagoon is of course even more important for Wallis than for Futuna. For the outer reef platform, we used the same approach as for Futuna, using digitised reef outlines from navigation charts and setting a constant depth of 0 metres. We also added some contours in the lagoon obtained by analysis of ocean colour on satellite images. While the outer reef platform can be considered adequately represented by this approach, the complex bathymetry of the lagoon is very crudely represented and this needs to be taken into account when interpreting highresolution inundation results. While we had more topographic data for Wallis than for Futuna, we consider the terrain model for Wallis to be significantly less reliable than that of Futuna, because the lagoon bathymetry is much more complex.
The other important input to the model is the initial wave field generated by earthquakes. All the sources below are based on the "Okada elastic deformation model" which assumes simple elastic caracteristics for the deformations of the sea floor due to fault ruptures. The corresponding vertical displacement of the seafloor is assumed to translate directly into a vertical motion of the sea surface. This initial vertical deformation is used as input for the tsunami model. This is a very standard approach which has been shown to be able to reproduce historical tsunamis, with compatible estimates of the intensity of the triggering earthquakes. From a tsunami generation perspective, the most important parameters are the total energy released (i.e. the magnitude), the fault orientation and the "dip angle" of the fault plane. The dip angle is important because it controls the amount of energy released as a vertical displacement of the sea surface. For the scenarios below, these parameters are estimated based on our current knowledge of the particular faults concerned.
References
S. Popinet  Quadtreeadaptive tsunami modelling
 Ocean Dynamics 61(9):12611285, 2011
 Bibtex
S. Popinet  Adaptive modelling of longdistance wave propagation and finescale flooding during the Tohoku tsunami
 Natural Hazards and Earth System Sciences 12(4):12131227, 2012
 Bibtex
Near field
Tonga 2009 (Mw 8.1, 15 m resolution)
Tsunami statistics  Wallis  Futuna

Maximum wave elevation  2.7  8.7

Maximum inundation depth  2.2  8.0

Arrival time  [0h33:2h45]  [1h17:6h47]

 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Vanuatu backarc (Mw 7.96, 15 m resolution)
Fault parameters   Tsunami statistics  Wallis  Futuna

Longitude, Latitude  168.16, 15.3  Maximum wave elevation  0.9  3.3

Length, Width, Depth  200e3, 30e3, 0  Maximum inundation depth  0.4  2.6

Strike, Dip, Rake  170, 40, 90  Arrival time  [2h47:3h13]  [2h26:2h58]

Slip  6

Initial vertical displacement (single Okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Vanuatu north (Mw 8.39)
Fault parameters

Longitude, Latitude  166.25, 12.6

Length, Width, Depth  400e3, 40e3, 0

Strike, Dip, Rake  350, 30, 90

Slip  10

Initial vertical displacement (single Okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

South Vanuatu trench (Mw 8.24)
Fault parameters

Longitude, Latitude  168.7, 19.9

Length, Width, Depth  300e3, 40e3, 0

Strike, Dip, Rake  335, 30, 90

Slip  8

Initial vertical displacement (single Okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Futuna (Mw 6.97, 15 m resolution)
Fault parameters   Tsunami statistics  Wallis  Futuna

Longitude, Latitude  178.13, 14.32  Maximum wave elevation  0.1  4.9

Length, Width, Depth  40e3, 15e3, 0  Arrival time  [0.8:1.1]  [0.0:0.3]

Strike, Dip, Rake  300, 60, 90

Slip  2

Initial vertical displacement (single Okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Tonga trench (Mw 9.06)
Fault parameters

Longitude, Latitude  173.7, 20.5

Length, Width, Depth  1000e3, 80e3, 10e3

Strike, Dip, Rake  195, 30, 90

Slip  20

Initial vertical displacement (single Okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

MSL + 1 metre (15 m resolution)
Fault parameters   Tsunami statistics  Wallis  Futuna

Longitude, Latitude  173.7, 20.5  Maximum wave elevation  6.9  13.4

Length, Width, Depth  1000e3, 80e3, 10e3  Arrival time  [0.4: > 4.1]  [0.8: > 4.1]

Strike, Dip, Rake  195, 30, 90

Slip  20

Initial vertical displacement (single Okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Northern tip Tonga (Mw 8.16)
Fault parameters

Longitude, Latitude  173.75, 14.7

Length, Width, Depth  300e3, 40e3, 2.5e3

Strike, Dip, Rake  100, 60, 90

Slip  6

Initial vertical displacement (single Okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Central Tonga (Mw 8.57)
Fault parameters

Longitude, Latitude  174.5, 21.5

Length, Width, Depth  600e3, 50e3, 2.5e3

Strike, Dip, Rake  200, 30, 90

Slip  10

Initial vertical displacement (single Okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Far field
Tohoku 2011 (Mw 9)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Chile 1960 (Mw 9.29)
Fault parameters

Longitude, Latitude  75.11, 42.17

Length, Width, Depth  920e3, 120e3, 0.

Strike, Dip, Rake  10, 12, 90

Slip  32

Initial vertical displacement (single okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Peru 1868 (Mw 9.13)
Fault parameters

Longitude, Latitude  74.5, 16.5

Length, Width, Depth  900e3, 150e3, 10e3

Strike, Dip, Rake  305, 20, 90

Slip  15

Initial vertical displacement (single okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Cascadia (Mw 9)
Fault parameters

Longitude, Latitude  126.38, 45.47

Length, Width, Depth  1050e3, 70e3, 0

Strike, Dip, Rake  350, 15, 90

Slip  17.5

Initial vertical displacement (single okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Aleutians (Mw 9)
Fault parameters

Longitude, Latitude  175.41, 51.63

Length, Width, Depth  850e3, 150e3, 1e3

Strike, Dip, Rake  70, 15, 90

Slip  10

Initial vertical displacement (single okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Kuril (Mw 9.28, 15m resolution)
Fault parameters   Tsunami statistics  Wallis  Futuna

Longitude, Latitude  153.224, 46.616  Maximum wave elevation  3.9  4.5

Length, Width, Depth  1000e3, 200e3, 1e3  Arrival time  [8.6: > 12.2]  [8.8: > 12.2]

Strike, Dip, Rake  220, 30, 90

Slip  17

Initial vertical displacement (single okada fault)
 Maximum wave elevation. Log colorscale. Contours at 0.1,0.2,0.4,0.8,1.6,3.2... metres.

Futuna: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.
 Wallis: Maximum wave elevation. Linear colorscale. 0.5 metres contour interval.

Summary All Scenarios
Number  Scenario Name  Based on Date  Magnitude (Mw)  Length (m)  Width (m)  Slip (m)  Longitude  Latitude  Max Wave height (m)  Max Runout (m)  First Arrival after EQ  Last Arrival after EQ

1.1  Tonga 2009  29 Sep 2009  8.1  ?  342.5  61 

1.2  Vanuatu North      8.4  400  40  10  166.25  12.6

1.3  Futuna Local  12 Mar 1993  7  40  15  2  178.13  14.32

1.4  Vanuatu BackArc    7.96  200  30  6  168.16  15.3

1.5  South Vanuatu Trench    8.24  300  40  8  168.7  19.9

1.6  Tonga Worst Case    9.06  1000  80  20  173.7  20.5

1.7  Tonga Worst Case +1 m    9.06  1000  80  20  173.7  20.5

1.8  Central Tonga    8.57  600  50  10  174.5  21.5

2.1  Tohuku (Japan)  11 Mar 2011  9  700  0  81  

2.2  Chile  22 May 1960  9.3  920  120  32  75.11  42.17

2.3  Peru  13 Aug 1868  8.3  900  150  15  74.5  16.5

2.4  Aleutian  9 Mar 1957  8.6  850  150  10  175.41  51.63

2.5  Cascadia  worst case  9.1  1050  70  17.5  126.38  45.47

2.6  Kurile  worst case  9.28  1000  200  17  153.22  46.62
