Tsunami-HySEA uses the well-known 2D nonlinear one-layer shallow water system in both spherical and Cartesian coordinates. For the sake of brevity and simplicity, only the later system is written:
ht+(qx)x+(qy)y = 0
(qx)t+(qx2/h + g h2/2)x+(qx qy/h )y = ghHx +Sx
(qy)t+(qx qy/h )x +(qy2/h + g h2/2 )y = ghHy +Sy
In the previous set of equations, h(x,t), denotes the thickness of the water layer at point x Î D Ì R2 at time t, being D the horizontal projection of the 3D domain where the tsunami takes place. H(x) is the depth of the bottom at point x measured from a fixed level of reference. Let us also define the function h(x,t)=h(x,t)-H(x) that corresponds to the free surface of the fluid. Let us denote by q(x,t) = (qx(x,t), qy(x,t)) the mass-flow of the water layer at point x at time t. The mass-flow is related to the height-averaged velocity u(x,t) by means of the expression q(x,t) = h(x,t) u(x,t). The notation ( )t, ( )x and ( )y applies for temporal and spatial partial derivatives in the x and y directions, respectively.
The terms Sx and Sy parameterize the friction effects and two different laws are considered:
1. The Manning law:
Sx = -gh Mn2 ux ||u|| /h4/3
Sy = -gh Mn 2 uy ||u|| /h4/3
where Mn >0 is the manning coefficient.
A quadratic law
Sx = -cf ux ||u||, Sy = -cf uy ||u||
where cf >0 is the friction coefficient. In all the numerical tests presented in this report the Manning law is used.
A version of the code including dispersion is also available. The dispersive system implemented can be interpreted as a generalized Yamazaki model (Yamazaki et al., 2009) where the term htw is not neglected in the equation for the vertical velocity. The breaking criteria employed is similar to the criteria presented in Roeber et al. (2010), based on an “eddy viscosity” approach.
