2D modeling of tsunami generation F. Chierici, L. Pignagnoli, D. Embriaco

Sea Water Column – Sediment Layer interaction model: a) 2-D (x-z reference frame)‏ b) Compressible water layer of height h c) Porous sea floor of height h s d) Small Amplitude waves: the wave amplitude x is negligible with respect to the wavelength. x/h << 1. Kinetic Energy << Potential Energy e) No Sea Water Viscosity => (Potential Flow)‏

2-D Vertical X-Z frame of reference

in WATER COLUMN we use Navier-Stokes equations for a compressible fluid into SEDIMENT we use Darcy equations for porous medium

Where: Fluid Velocity Fluid Pressure WATER COLUMN Discharge Velocity Fluid Pressure (into sediment)‏ POROUS MEDIUM μ is the dynamic viscosity, Kp is the intrinsic permeability and n is the volumetric porosity, r is the fluid density and g is the gravity acceleration

Mass Conservation Linearized Bernoulli equation EQUATIONS OF MOTION : Sea Water Column: c is the sound speed

Darcy equation Continuity equation Sediment: EQUATIONS OF MOTION :

Linearised Bernoulli equation Kinematic condition BOUNDARY CONDITIONS: Free Surface:

Stress Continuity Vertical velocity continuity BOUNDARY CONDITIONS: Sea Water – Sediment Bed interface:

Non-permeability condition BOUNDARY CONDITIONS: Sediment Bottom : η is the sea floor displacement

We shall use and combine different kind of sea floor motions to model both permanent displacement and elastic oscillation. Duration, phase, amplitude and different kind of motion are employed together in order to obtain a wide typology of sea floor motion. Sea floor displacement is intended as visco-elastic deformation (assuming volume conservation of the porous layer and non permeability of the bottom of the sediment: i.e. z < -(h+h s ) )

Basic Sea Floor Motions (each motion can be either negative or positive polarized and due to linearity they can be composed with different periods, amplitudes and phases): Time Permanent Displacement Space Positive Elastic Motion (no permanent displacement)‏ Elastic Oscillation (no permanent displacement)‏ + +

We solve by transforming x spatial variable with Fourier and t time variable with Laplace

where with k wave number and ω angular velocity A(ω, k), B(ω, k), C(ω, k) and D(ω, k) are the functions obtained imposing the boundary conditions. For example B is given by the following espression and

Example: Model of Sea Floor Permanent Displacement Fourier (x => K) + Laplace (t => ω)‏

Some comparative examples of incompressible, compressible and compressible +porous simulations

Permanent diplacement (η=3 m )‏

Positive motion with no final displacement

Positive and negative motion with no final displacement

Positive and negative motion with final displacement

Positive and negative motion. The pressure field is computed at 750m depth and 158.5km distance from source

Same pressure field at km distance from source

The acoustic signal is the acoustic signal generated by the sea floor motion (in the compressible water) a Tsunami precursor? Surely it is an indication of strong motion of the sea floor at the source and of high stress into the water column. Its characteristic frequencies give a direct measure of the water depth at the source.

The 2-D model shows relevant amplitude of the acoustic signal at distance. The acoustic signals travel much faster than tsunamis even in deep water. The modulation of the acoustic signal carries information of sea floor motion at source, as we will see.(the next slide shows the comparison of the mean slopes of the modulations produced by source motions that differs only in sea bottom velocity.)‏ The amplitude of the signal normalized for the source distance gives information about power of the injected in the system by the sea floor motion.

The different “envelope slopes” scales like the sea bottom velocities (1m/s, 0.2 m/s, 0.1 m/s)‏

the acoustic signal generated by the 1 m/s motion and its envelope (the 0.2 m/s and 0.1 m/s curves are similar except for the amplitude)‏

Some basic physics The sea floor motion excites the normal modes of vibration of the water layer. Due to the water compressibility these vibrations cause elastic oscillation and acoustic waves. The frequencies of the acoustic waves (caused in the fluid by the sea floor motion are function of the water depth only) are given by: f=c(2k+1)/4H In very rough way we can compute the energy that the sea floor motion puts into the tsunami wave and into the acoustic-elastic oscillation:

The tsunami energy in the approximation of incompressible water and source extension much greater than water depth is given by: The acoustic wave energy, in the approximation of compressible fluid is given by

the range of frequencies of the elastic-acoustic waves spans from about 0.05 hz (computed in the case of 8000 m water layer, rigid and horizontal bottom) to 1 hz (in case of 400 m water layer with a maximum sea floor displacement of 1 m, the limit of validity of the model for 1m sea floor displacement).

Much more work must be done. We should extend the model to 3-D to take into account possible interference effect due to the bathymetry. We must model slope variation of the bottom and its effect over acoustic waves assorbtion. Moreover we must evaluate the environmental noise of the ocean at these frequencies. A acoustic antenna to measure these effect is feasible.

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