Energy of a tsunami in the framework of an irreversible deformation of the ocean bottom

The classic approach to tsunami simulation by earthquake sources consists
of computing the vertical static deformation of the ocean bottom due to
the dislocation, using formalisms such as Mansinha and Smylie's [1971] or
Okada's [1985], and of transposing that field directly to the ocean's
surface as the initial condition of the numerical simulation.
We look into the limitations of this approach by developing a very
simple general formula for the energy of a tsunami, expressed as the
work performed against the hydrostatic pressure at the bottom of
the ocean, in excess of the simple increase in potential energy
of the displaced water, due to the irreversibility of the process.
We successfully test our results against the exact analytical solution
obtained by Hammack [1972] for the amplitude of a tsunami generated
by the exponentially-decaying uplift of a circular plug on the ocean
bottom. We define a "tsunami efficiency" by scaling the resulting energy
to its classical value derived, e.g., by Kajiura [1963]. As expected, we
find that sources with shorter rise times are more efficient tsunami
generators; however, an important new result is that the efficiency is
asymptotically limited, for fast sources, to a value depending on the
radius of the source, scaled to the depth of the water column; as this
ratio increases, it becomes more difficult to flush the water out of
the source area during the generation process, resulting in greater
tsunami efficiency. Fortunately, this result should not affect
significantly the generation of tusnamis by mega-earthquakes.