342
Equatorial Dynamics of the Thermocline: The Equatorial Undercurrent
aBn
(y + Cn)Vn = - ax
where:
( 6.4.1)
(6.4.2a,b)
The relative vorticity, Cm is given entirely in terms of the shear in they direction
of the zonal velocity component. The other contributor, which is the shear of
the meridional velocity in the x direction is smaller by an amount (£/Lt
Similarly, only the kinetic energy associated with the zonal motion enters the
definition of the Bernoulli function Bn. The contribution of the meridional
velocity is negligible by the same ratio, (£/L) 2 •
In the absence of cross-isopycnal motion (6.3.5) shows that the horizontal
mass flux is nondivergent. Thus, expressed in nondimensional units:
hnUn = k X Vl/Jn·
Thus ( 6.4.1) becomes:
aljJn aBn
qn ax = ax
where the potential vorticity, qm is given by:
Y- aunfay
qn =
hn
.
(6.4.3)
( 6.4.4)
( 6.4.5)
The geostrophic relation (6.3.18) can as well be written in terms of Bn and qn as:
aljJn aBn
qn ay = ay
( 6.4.6)
which leads to:
( 6.4.7)
obtained by eliminating Bn by cross-differentiation between (6.4.4) and (6.4.6).
Multiplying (6.4.4) by Un and (6.4.6) by Vn and adding we obtain, with the
aid of (6.4.3):
i1n · VBn = 0
( 6.4.8)
so that Bn, as qn, is conserved along streamlines or:
(6.4.9a,b)
Thus, using (6.4.4) and (6.4.6) it follows that:
dEn
dl/Jn = qn
(6.4.10)
and eliminating l/Jn between (6.4.9a,b) yields: