184 Kikuro Miyakoda
where, Vb = ! ro (u, v)dz is the barotropic component, and it satisfies
HJ-H
(13)
(14)
Applying ro dz/ H to eq. (7) and (8), the equation for the barotropic component
J-H
becomes,
where p = ps(x, y, t) + PH'
PH = 1pgdZ and Po is a reference value ofdensity, which is a function of depth z. The advection operator ,L, and del-operator, V ,
are represented by
a-l) Streamfunction formulation
The assumption of incompressibility and the rigid lid condition, (14), gives a
stream-function 'l',
HU b = -dql'l',
HVb/m = dA.'I'
(17)
Applying l/mVX to eq. (15), one obtains,
(18)
where
G u = ro L(u)dz - jHV b + gm ro dA. ro pdzdz
J-H
aPoJ-H Jz
(19)
G v = ro L(v)dz-jHUb+~ro dqlropdzdz
J-H
aPoJ-H Jz
(19')
In order to solve eq. (18), boundary conditions on the streamfunction must be
specified. The conditions are that the normal components ofthe barotropic velocities vanish along the coastlines, and, therefore, the streamfunction must be constant
along coastlines. In multiply-connected basins, the streamfuntions have different
values for each island, which change with time. The value of '1' around each island
can be obtained by performing line integrals of eq. (18) around the coasts. This