Classification and calculation of a Tsunami

Calculation of aTsunami                                                                                            Exit

Classification of Tsunamis

It seems convenient to classify the Tsunamis according to their intensity. Hence, the scientists Imamura (1949) and Iida (1970) defined the Tsunami magnitude (m) by the following formulas: Hm is the maximum height of a Tsunami relating to the coast for an epicentre located between 10 and 300 km further. These magnitudes are defined as follows:

-1: Hm less than 0.5m; minor Tsunami.

0: Hm=1m; no damages.

1: Hm=2m; damages (ships broken and dragged to the coast).

2: Hm=4 à 6m; habitations destroyed; human deaths.

3: Hm=10 to 20m; damages observed in a 400 km radius area.

4: Hm=more than 30m; damages observed in a 500 km radius area.

Iida has also defined an equation linking m wih the earthquake magnitude which triggered the Tsunami: Let’s notice that energy of a Tsunami is about ten percent of the earthquake energy.

Calculation of aTsunami

It depends on several hypothesis:

• the way they are generated (landslide, earthquake, meteorites…)
• importance of either vertical or horizontal movement

The Boussinescq’s theory of long waves makes the hypothesis of an horizontal ground (origin for the z axis) and a bi-dimensional movement in the x and z directions. It is based on the solution of the Laplace’s equation satisfying the potential F (x,z,t) developed through the following series: In which F (x,t) is the potential at the ground sea level (z=0).

The conditions on the free surface at the coordinate h(x,t) above the ground sea level are: Or by traducing the potential F (x,t): which is the cinematic condition.

Regarding the dynamic condition (Bernoulli’s equation): The variable d is the depth at the initial moment (F =0).

Let’s introduce some non-dimensional variables for the rest of the solving: e is the wave amplitude corresponding to the depth d.

L is the wave length.

n(x,t) is the relative movement of the free surface relating to the maximum height

U is the Ursell number very characteristic of the waves propagation.

-U<1, it means the linear theory of long waves is valid.

-U=1, it means that the Boussinescq's equations are valid.

-U>1, it means the theory of finite amplitude long waves must be used.

By introducing the following equation: It implies: Let's now introduce the non-dimensional horizontal velocity u defined as it follows: Thus, the mass equation and the pressure condition lead to the following system to solve: Hence, it also leads to the following first order equation: Finaly, we obtain the Kertweg-de-Vries equation traducing the variation of n according to the variables x and t. 