The mathematical model implemented in the Landslide-HySEA tsunami code consists of a stratified media of two layers: the first layer is composed of a homogeneous inviscid fluid with constant density r1 (sea water here), and the second layer represents the fluidized granular material with density rs and porosity y0. We assume that the mean density of the fluidized debris is constant and equals rho2 = (1 – y0) rs + y0 r1 and that the two fluids (water and fluidized debris) are immiscible.
The resulting system of equations writes as follows:
(h1)t+(q1,x)x+(q1,y)y = 0
(q1,x)t+(q1,x2/h1 + g h12/2)x+(q1,x q1,y/h1 )y = -gh1(h2)x+ gh1 Hx +Sf1(W)
(q1,y)t+(q1,x q1,y/h1 )x +(q1,y2/h1 + g h12/2 )y = -gh1(h2)y+ gh1 Hy +Sf2(W)
(h2)t+(q2,x)x+(q2,y)y = 0
(q2,x)t+(q2,x2/h2 + g h22/2)x+(q2,x q2,y/h2 )y = -grh2(h1)x+ gh2 Hx +Sf3(W)+tx
(q2,y)t+(q2,x q2,y/h2 )x +(q2,y2/h2 + g h22/2 )y = -grh2(h1)y+ gh2 Hy +Sf4(W)+ty
In these equations, index 1 corresponds to the upper layer and index 2 to the second layer. hi(x,y,t), i=1,2 is the layer thickness at each point (x,y) at time t, therefore h2 stands for the thickness of the slide layer material; H(x,y) is the fixed bathymetry at (x,y) measured from a given reference level, qi(x,y,t), i=1,2 is the discharge and is related to the mean velocity by the equation ui(x,y,t)=qi(x,y,t)/hi(x,y,t), g is the gravitational acceleration and r is the ratio of densities r=r1/r2.
Terms Sfi(W), i = 1, …, 4, model the different effects of the dynamical friction while t = (tx, ty) corresponds to the static Coulomb friction term. The terms Sfi are defined as follows:
Sf1(W) = Scx(W) + Sax(W), Sf2(W) = Scy(W) + Say(W),
Sf3(W) = -r Scx(W) + Sbx(W), Sf4(W) = -r Scy(W) + Sby(W),
where Sc (W) = (Scx(W), Scy(W)) parametrizes the friction between layers and it is defined as follows:
Scx(W) = mf (h1 h2)/(h2 + r h1) (u2,x – u1,x) ||u2 – u1||
Scy(W) = mf (h1 h2)/(h2 + r h1) (u2,y – u1,y) ||u2 – u1||
being mf a positive constant.
Sa (W) = (Sax (W), Say (W)) parametrizes the friction between the water and the fixed bottom topography, if there is no granular material and it is defined by a Manning friction law:
Sax (W) = -(g h1 n12 / h1(4/3)) u1,x ||u1||
Say (W) = -(g h1 n12 / h1(4/3)) u1,y ||u1||
where n1 > 0 is the Manning coefficient, between the water and the fixed bottom topography.
Sb (W) = (Sbx (W), Sby (W)) parametrizes the dynamical friction between the debris layer and the fixed bottom topography and, as in the previous case, it is defined using a Manning law:
Sbx (W) = -(g h2 n22 / h2(4/3)) u2,x ||u2||
Sby (W) = -(g h2 n22 / h2(4/3)) u2,y ||u2||
where n2 > 0 is the corresponding Manning coefficient.
Finally, t = (tx, ty) is the static Coulomb friction term and it is defined by
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if ||t|| >= sc ==> {
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tx = -g (1-r) h2 (q2,x / || q2||) tan (a) |
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ty = -g (1-r) h2 (q2,y / || q2||) tan (a) |
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if ||t|| < sc ==> |
q2,x =0, q2,y =0, |
where sc = g (1-r) h2 tan (a), being a the Coulomb friction angle. The above expression models the fact that a critical slope is needed to trigger the slide movement. It must be taken into account that the effects of hydroplaning may be important for submarine mass failures and tsunami generation highly reducing the expected value for sc.
