#### 2.1. Model Description

Numerical models for tsunami propagation are a relatively well established methodology which has been widely validated against recorded data from historical events over the world [

26,

27,

28,

29]. For the present study, the selected tsunami propagation model is based on the 2D depth-averaged nonlinear barotropic shallow water equations (see for instance Kowalik and Murty, 1993, [

30]):

where

u and

v are the depth averaged water velocities along the

x and

y axes, h is the depth of water below the mean sea level, ζ is the displacement of the water surface above the mean sea level measured upwards, H = h + ζ is the total water depth, Ω is the Coriolis parameter (Ω = 2

w·sinλ, where

w is the Earth rotational angular velocity and λ is latitude),

g is acceleration due to gravity,

ρ is a mean value of water density and

A is the horizontal eddy viscosity. τ

_{u} and τ

_{v} are friction stresses which have been written in terms of a quadratic law:

where

k is the bed friction coefficient. Essentially, these equations express mass and momentum conservation. They have been written in Cartesian coordinates given the relatively small model domain.

Horizontal viscosity has been set as A = 10 m

^{2}/s and the bed friction coefficient as k = 0.0025. Good model results when estimating wave amplitudes and runups have been obtained with these values. Actually, model results (amplitudes, runups and wave arrival times) have been compared with observations in previous works [

28]. Consequently, these values have been retained in the present application. Moreover, the model sensitivity to the bed friction coefficient has been studied in Periáñez and Abril (2014b) [

31].

Still waters are used as initial conditions. As boundary conditions, water flow towards a dry grid cell is not allowed, and a gravity wave radiation condition is imposed at the open boundaries [

32], which is implemented in an implicit form. Due to the CFL stability condition [

30] time step for model integration was fixed as Δt = 2 s.

A flood/dry algorithm is required since when the tsunami reaches the coast new wet or dry grid cells may be generated due to run-up or rundown. The numerical scheme described in Kampf (2009) [

33] and in Periáñez and Abril (2013) [

28] has been adopted. Wet grid cells are defined as those with a total water depth larger than a threshold value typically set as a few centimeters. Dry cells are defined as cells where depth is smaller than the threshold one. Flooding and drying is implemented in the code via the calculation of the water velocity normal to the interface between wet and dry cells. The calculation is performed when the pressure gradient force is directed towards the dry cell. Otherwise velocity is set to zero at this point. In the case of a non-zero velocity, water level in the dry cell will increase and the cell turns into a wet one once the water depth is larger than the threshold depth, which has been set as 10 cm [

28]. All the equations are solved using explicit finite difference schemes [

30] with second order accuracy. In particular, the MSOU (Monotonic Second Order Upstream) is used for the advective non-linear terms in the momentum equations.

For tsunamis produced by earthquakes in geological faults, the vertical sea-floor deformation is considered as the initial condition for the tsunami calculation, and it is computed using the classical Okada formulae [

34]. Inputs for this equation are fault plane strike, rake, dip, slip, location, length and width, as well as seismic moment and rigidity. Validation and applications of this model can be found in Periáñez and Abril (2013 [

28], 2014a [

25], 2014b [

31]), and Abril

et al. (2013) [

35].

The methodology of Harbitz (1992) [

36] and Cecioni and Bellotti (2010) [

37] has been adopted to simulate the generation of tsunamis by submarine landslides, along with the modification provided by Periáñez and Abril (2014a) [

25] to describe varying slopes and asymmetric velocity profiles. In the first stage, the slide can be described as a solid body whose downslope movement locally modifies the bathymetry, resulting in an almost instantaneous and equal change in the level of the overlapping waters, which propagates as gravity waves. The adopted geometry is a box of length

L, width

B and maximum height Δ

h, and with an exponential smoothing over a distance S in the front and rear and

B/2 on the flanks (see scheme in

Figure 2). The resulting volume is V = 0.90

B·Δ

h(

L + 0.90

S) [

36]. As a practical approach, the former slide at its initial position will be superposed onto the present-day bathymetry of the source area. The greatest part of the energy transfer to the water column takes place during the first stage of the slide displacement (approximately until the time of its maximum velocity) [

38]. Details of the subsequent history (including deformation and breaking up of the slide, and the accurate update of the seafloor bathymetry) will contribute only as second order corrections. Thus, corrections of the present day bathymetry in the depositional areas have been omitted in this modelling approach. Away from the coastal area, changes in bathymetry during the last 4 ka are negligible in this context, and the former paleogeography conditions of the eastern coastline of the Nile Delta have been re-created for this work based upon available geological studies, as explained further below.

**Figure 2.**
Sketch with the geometrical parameters defining the submarine landslide (following the formalism proposed by Harbitz, 1992, [

36]). The adopted geometry is a box of length L, width B and maximum height Δh, and with an exponential smoothing over a distance S in the front and rear, and B/2 on the flanks.

**Figure 2.**
Sketch with the geometrical parameters defining the submarine landslide (following the formalism proposed by Harbitz, 1992, [

36]). The adopted geometry is a box of length L, width B and maximum height Δh, and with an exponential smoothing over a distance S in the front and rear, and B/2 on the flanks.

The motion of the slide can be known after solving the governing dynamic equations [

39,

40], or it can be defined by imposing a prescribed motion based upon a maximum velocity,

U_{max}, and the displacement,

R [

36]. Here we adopt the approach by Harbitz (1992) [

36], in which

U_{max} is estimated as a function of the slope angle, α, the average thickness of the slide,

$\overline{h}$, its density,

$\overline{\text{\rho}}$(~1.7 × 10

^{3} kg·m

^{−3}), the density of turbidity currents,

${\text{\rho}}_{t}$ $\overline{\text{\rho}}$(~1.1 × 10

^{3} kg·m

^{−3}), and the friction (μ) and drag (

${C}_{D}^{u}$) coefficients:

with

${C}_{D}^{u}$, the drag coefficient along the upper surface of the slide, being estimated from the roughness length parameter,

k (in the range of 0.01 to 0.1 m):

The value of

U_{max} strongly depends on the estimation of the Coulomb friction coefficient, μ, within an acceptable range (being its upper limit

${\text{\mu}}_{st}=\mathrm{tan}\text{\alpha}$), and

U_{max} must remain within the range of the reported values in scientific literature [

39,

41,

42,

43].

In many cases a first and large slope angle is involved in the triggering mechanism. After a displacement

R_{1}, the slope angle decreases, but the moving masses still complete a second displacement

R_{2}. For each slope angle the maximum velocity

U_{max,1} and

U_{max,2} are estimated as commented above, and the following function of time is imposed for the slide velocity,

v_{s} (Periáñez and Abril, 2014a, [

25]):

with

and

S(

t) is the instantaneous position of the slide front at time

t.

The “two slope angle” kinematics is a model choice, being more general than the single sinus function used by Harbitz (1992) [

36], but containing it as a particular case, and it allows generating asymmetric velocities profiles (as the ones used by Lastras

et al., 2005, [

39]). Applications of this model can be found in Periáñez and Abril (2014a) [

25] and in Abril and Periáñez (2015) [

38]. Friction stresses over the moving slide are formulated in terms of relative speed, as in Harbitz (1992) [

36], in such a way that a slice moving faster than the water column can transfer energy to it.

The slide model requires four input (but not free) parameters: maximum velocities (governed by μ) and displacements. The candidate source areas will be characterized by depth profiles along their respective transects, what allows the identification of slope angles. The displacements can be partially estimated from the previous profiles, or introduced as plausible values. These displacements have to be understood as effective run-out distances (displacement of the sliding block), over which the transfer of energy to the tsunami takes place. More details will be provided along with the source definitions.

A further validation for the performance of the numerical model with its flood-drying algorithm can be found in Periáñez and Abril (2015) [

44]. The submarine landslide submodel has been tested against independent modelling works and subject to extensive sensitivity tests [

38,

44]. The effects of model resolution and sensitivity to the friction coefficient have been studied in Periáñez and Abril (2014b) [

31].

#### 2.3. Tsunamingenic Sources in the Eastern Nile Delta

Works and reviews on the geodynamics of the eastern Mediterranean can be found, among others, in Barka and Reilinger (1997) [

47], Mascle

et al., (2000) [

48], or Yolsal and Taymaz (2012) [

29]. Its complex tectonic is responsible for intense earthquake and volcanism activity, triggering large tsunamis in the past [

17,

29,

49,

50]. The study by Papadopoulos

et al., (2007) [

51] on tsunami hazards in the Eastern Mediterranean concluded that the mean recurrence of strong tsunamis (most of them were caused by earthquakes) is likely equal to about 142 years. The assessment of the probability for landslide tsunami scenarios is a quite difficult task, due to the insufficient statistics from the recorded events, as discussed in detail by Harbitz

et al., (2014) [

52]. Such probabilities of occurrence must not be confused with the probability of that a given coastal area is affected by a tsunami of certain level (e.g., with a maximum water height).

The first map of the Nile deep-sea fan has not been completed until recent years [

53,

54]. The Messinian salinity crisis led to the deposition of salt and anhydrite throughout the Mediterranean basin, while the proto-Nile river excavated deep canyons and transported offshore large volumes of terrigenous sediments. Thick Plio-Quaternary sediments covered then the ductile evaporitic layers, what triggered some giant gravity-driven salt tectonics [

54,

55,

56,

57,

58]. A series of fault trends have been identified across the stable shelf of the Nile Delta, e.g., Rosetta, Baltim and Temsah fault trends, and the Pelusium Mega-Shear Fault system [

59,

60,

61]. The Nile Delta is a major gas and condensate province with several mud volcanoes and geodynamics associated to fluid seepage [

62,

63].

A general overview on multi-scale slope instabilities along the Nile deep-sea fan can be found in Loncke

et al., (2009) [

57], and in Urgeles and Camerlenghi (2013) [

64]. Garziglia

et al., (2008) [

60] identified and dated seven mass-transport deposits on the Western province, northern to the Rosetta Canyon, with volumes ranging from 3 to 500 km

^{3}, mean thickness from 11 to 77 m and run-out distances from 18 to 150 km, being the youngest deposit older than 8940 ± 30 cal. year BP. Ducassou

et al., (2009), based upon 42 sediment cores collected across the entire Nile deep sea fan, identified several slump deposits and turbidities in the last 2000 ka BP [

55], but none in the Mid and Late Holocene (ca 5 ka to present). Thus, any candidate source area for submarine landslides must be compatible with the “empty spaces” within this cloud of cores. Recently, Ducassou

et al., (2013) [

56] reported four highly mobile debris flows in the Nile deep-sea fan system, with a chronology confined between 5599 and 6915 cal. years BP for the most recent event, in the Rosetta province.

Results from numerical models have proved that the effects of the Minoan Santorini tsunamis (triggered by entry of pyroclastic flows and caldera collapse) were negligible in north-eastern Egypt and Levantine coasts [

17,

25], as already commented. Thus, for any less energetic event in the inner Aegean Sea (as the volcanic eruption in the area of Nisyros and Yale suggested by Sivertsen for the sea parting in the exodus-expulsion) a similar behaviour would be expected. Even if potential tsunami directionality is invoked, the Isle of Rhodes prevents any direct pathway for energy transfer from these sources towards the eastern Nile Delta. Moreover, for the Cyprus AD 1222 earthquake tsunami, with source area in southern Cyprus, model results (Periáñez and Abril, 2014a, [

25]) show that the outer shelf of the Nile Delta acts as a natural barrier and slows waves at these shallow depths (<500 m), thus increasing wave amplitude in this area, but with negligible impacts in the inner shoreline.

Recent mass-wasting events can be recognized in bathymetric maps by the presence of head and footwall scars [

57,

64]. Periáñez and Abril (2014a) [

25] suggested several hypothetical candidate sites in the Nile Delta for submarine landslides of 9–10 km

^{3}, accomplishing for high slopes and being distant enough from the already studied areas, where any mass-wasting deposits in the last 5 ka can be discarded. They showed that tsunamis generated by sources in the western and northern Nile Delta did not significantly affect the coastal zones of Israel and Gaza, and their effects on the eastern area of the delta were equally negligible. Thus, the source area within the Nile Delta able to produce tsunamis potentially linked to the Exodus has to be confined to its eastern zone.

In this work the geometry of the moving boxes and their displacement will be defined with the criteria of being generously large, but compatible with the “empty areas” defined by the studied sediment cores. The value of μ will be subject to sensitivity tests; and 50 m/s has been adopted as the upper limit for

U_{max}, according to Harbitz (1992) [

36]. The tsunamigenic sources selected for this study are summarized in

Table 1 and

Table 2 and briefly discussed further below.

Landslide SL-1 is the one discussed and modelled by Periáñez and Abril (2014a) [

25], but applied here along with the more detailed paleogeography. The slide front is placed at the eastern border of the stable Nile delta, facing a down-slope of 2.8 degrees. The second displacement prevents reaching the sites of cores studied by Ducassou

et al. (2009) [

55], although its numerical value has a minor effect in the tsunamigenic potential of the slide. The slide extends over a large area (B = 80 km; L = 20 km), but with a moderate value for its maximum height (6.0 m), accounting for a total volume of 9.80 km

^{3}. The value of μ has been fixed as 0.1, 0.3, 0.5, 0.8 and 0.9 of its maximum (static, μ

_{st}) limit in model runs R1, R2, R3, R4 and R5, respectively. A second version of this slide uses a larger value for its maximum height (20.0 m), leading to a total volume of 32.7 km

^{3}. The larger height increases the slide speed, which reaches values up to 49.5 m/s for μ = 0.5μ

_{st} (run R6), and of 31.3 m/s for μ = 0.8μ

_{st} (run R7). The displacement of the slide over the second slope, with a smaller angle, has a minor contribution to its tsunamigenic potential; thus, and for the sake of simplicity, a value of μ = 0.75μ

_{st} has been adopted for runs R1 to R7.

**Table 1.**
Source parameters for submarine landslides ^{#}.

**Table 1.**
Source parameters for submarine landslides ^{#}.
Landslide | Run | Geometrical Parameters | Front Position | Direction ^{$} | Kinematics ^{¶} |
---|

L (km) | S (km) | B (km) | h_{m} (m) | V (km^{3}) | λ_{E}° | Φ_{N}° | θ° | R_{1} (km) | α_{1} (°) | µ/µ_{st} | U_{max,1} (m/s) | R_{2} (km) | α_{2} (°) | U_{max,2} (m/s) |
---|

SL-1 | R1 | 20.0 | 3.0 | 80.0 | 6.0 | 9.80 | 32.800 | 31.658 | 45 | 6.26 | 2.8 | 0.1 | 38.6 | 7.73 | 0.16 | 4.6 |

| R2 | | | | | | | | | | | 0.3 | 32.1 | | | |

| R3 | | | | | | | | | | | 0.5 | 27.1 | | | |

| R4 | | | | | | | | | | | 0.8 | 17.1 | | | |

| R5 | | | | | | | | | | | 0.9 | 12.1 | | | |

| R6 | | | | 20.0 | 32.7 | | | | | | 0.5 | 49.5 | | | 8.4 |

| R7 | | | | 20.0 | 32.7 | | | | | | 0.8 | 31.3 | | | 8.4 |

SL-2 | R1 | 8.0 | 3.0 | 40.0 | 26.0 | 10.0 | 32.725 | 31.708 | 45 | 4.36 | 2.8 | 0.6 | 47.7 | 11.73 | 0.6 | 17.5 |

| R2 | | | | | | | | | | | 0.7 | 41.3 | | | |

| R3 | | | | | | | | | | | 0.8 | 33.7 | | | |

| R4 | | | | | | | | | | | 0.9 | 23.9 | | | |

CSL * | R1 | 10.0 | 1.0 | 50.0 | 20.4 | 10.0 | 32.350 | 31.417 | 90 | 3.25 | 3.0 | 0.5 | 48.9 | 3.25 | 3.0 | 48.9 |

| R2 | | | | | | | | | | | 0.7 | 37.9 | | | 37.9 |

| R3 | | | | | | | | | | | 0.9 | 21.9 | | | 21.9 |

**Table 2.**
Fault parameters used in the simulations. Geographical coordinates correspond to the fault center. Rake is 90° in all cases.

**Table 2.**
Fault parameters used in the simulations. Geographical coordinates correspond to the fault center. Rake is 90° in all cases.
Tsunami | λ_{E}° | Φ_{N}° | Length (km) | Width (km) | Slip (m) | Strike (degree) | Dip (degree) | Potential Energy ^{#} (J) |
---|

F-1 | 32.667 | 31.500 | 60.0 | 20.0 | 8.0 | 55.0 | 45.0 | 4.6 × 10^{13} |

F-2 | 32.667 | 31.500 | 80.0 | 24.0 | 12.0 | 65.0 | 35.0 | 1.8 × 10^{14} |

F-2 + SL-2 R2 * | 32.667 | 31.500 | 80.0 | 24.0 | 12.0 | 65.0 | 35.0 | 1.8 × 10^{14} |

Slide SL-2 is also extracted from the present bathymetry. Its front (

Table 1), at the south-eastern border of the stable shelf in the Nile Delta, is initially at 71 m water depth and displaces 4.3 km down a slope with angle 2.8°, and then 11.7 km along a slope of 0.6°. The displacement ends at a plateau. The area covered by this hypothetical mass-wasting does not disagree with the geological surveys above referred. With a maximum height of 26.0 m and a total volume of 10 km

^{3}, the moving slide reaches a maximum speed of 47.7 m/s for μ = 0.6μ

_{st} (SL-2, run R1). Runs R2 to R4 for this slide use increasing values of μ in the first displacement, while the same criteria than for SL-1 has been adopted for the second displacement (

Table 1).

Loncke

et al., (2006) [

54] reported (their

Figure 9) the paths of the Messinian canyons in the Nile Delta. Particularly, at the Damietta branch, a former canyon of some 60 km length and 6 km wide ran eastward. In the GEBCO08 bathymetry, the stable shelf area shows some holes of several hundred meters deep and few km wide (dark spots in

Figure 3), likely remains of the former canyons. Due to its proximity to the former

Shi-Hor Lagoon, and as a potential and extreme case, the source CSL (

Table 1) simulates the effect of a large submarine landslide running into a still partially colmated canyon. The box geometry has been selected as large as possible, and its displacement has to be confined within the width of the former canyon. A maximum height of 20.4 m has been adopted to account for a total volume of 10.0 km

^{3}. For simplicity a uniform slope angle of 3.0 degrees has been selected, with three options for µ values, leading to maximum speeds of 48.9, 37.9 and 21.9 m/s in runs CSL R1, R2 and R3, respectively.

Gamal (2013) [

59] provided a detailed study of the Pelusium Mega-Shear Fault system. The Pelusium line runs north-westwards crossing the south-eastern Nile Delta. Hypothetical earthquakes triggered by this fault with epicentres in this area of the Nile Delta could have been tsunamigenic sources whose potential effects on the nearby former shoreline of the

Shi-Hor Lagoon will be studied here. Thus, sources F-1 and F-2 (

Table 2) are defined with tentative values for length, width and slip, and angular parameters inspired in the studies of Badawy and Abdel Fattath (2001) [

65], and Gamal (2013) [

59]. As the tsunamis triggered by geological faults are less energetic events when compared with the studied submarine landslides, the application of sensitivity tests has been discarded in this case.

Finally, the case of a fault-earthquake that triggers an almost simultaneous submarine landslide is also considered by combining the two most energetic sources, F-2 and SL-2 R2.