z a Model with a rigid lid b Model with a free surface
Fig. 4.2 Multi-layer models a with a rigid lid or b with a free surface.
surface z = 0, so the original problem of a moving boundary is reduced to one with a fixed boundary. Owing to the existence of a non-zero sea surface level Z = 0 at the flat surface z = 0, the equivalent hydrostatic pressure p = pa is not constant. Using the hydrostatic relation, one can calculate the pressure in the layers beneath. Starting fromp = pa at z = 0 and integrating the hydrostatic relation downward, we obtain the hydrostatic pressure in the upper layer (Fig. 4.2a), pi = Pa - Pigz
Note that pa = const is an unknown pressure at z = 0. In fact, pa is equivalent to pa,0 + PgZ, where pa,0 is the sea-level atmospheric pressure and Z is the unknown free surface elevation. At the base of the upper layer, pressure is p2 = pi = pa + p1gh1. Below the interfaces, pressure in the second and third layers is
Applying the horizontal gradient operator Vh to Eqn. (4.2b) leads to
Vhp3 = Vhpa + PigVhhi + P2gVhh2 - P3Vh(hi + h2) (4.3)
Assuming that the third layer is very thick and motionless; thus Vhp3 = 0, and we obtain z z p
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