362
Equatorial Dynamics of the Thermocline: The Equatorial Undercurrent
which the undercurrent is represented by a single layer it is possible only
crudely to model this behavior by choosing a mass flux function M(x) that is
weak in the western part of the basin and grows in strength eastward.
Pedlosky (1988) chose the following distribution for M(x):
{
0,
M(x)= Mo(x-xb)
Xe -Xb
( 6.6.9)
so that the cross-isopycnal flux is absent until the current flows east of the
longitude where x = xb. Mo is chosen so that:
B2
Mo =
0
Y2(Xe- Xb)
(6.6.10)
so that along the equator, from (6.6.7):
B~(x,o+) =B~[1- (~~~J\~(x-xb)l
(6.6.11)
In (6.6.11) 0(x- xb) is the unit Heaviside function which is zero when its
argument, x - xb, is less than zero and equal to unity when x - xb ;::: 0. This
choice forces B2 to vanish on the equator at the eastern boundary. If, as in the
calculations of Pedlosky (1988), the model possesses no shadow zone, the
Bernoulli function just outside the boundary current equals h(xe, 0) given by
the ventilated thermocline solution which equals zero if H 2 = 0. In this case the
transport of the undercurrent is exhausted as the current reaches the eastern
boundary.
Figure 6.6.2 shows the profiles of U2, h, h2 and au2/ ay for the integration of
the system (6.4.26) for the case in which (6.6.11) is the boundary condition at
y = 0 while (6.4.19) has been used to relate h2 to h. The longitude Xb is chosen
as 0.4 while the eastern boundary is at x = 1. Otherwise the parameters are the
same as those used to calculate the current in Fig. 6.4.4. Panel a shows the
solution at x = 0.8 for the case in which there is no cross-isopycnal flux, and
therefore B2 on the equator everywhere equals B0 , which is equal to 1.265.
Note that the maximum velocity, in scaled units, is 1.142. Panel b shows the
solution in which (6.6.11) is used. B2 at the equator has now fallen to 0.943 at
this longitude, and the maximum velocity of the current at y = 0 is similarly
reduced from its adiabatic value to 0.881. Otherwise the structure of the current is similar, as might be expected, since the effect of the cross-isopycnal
velocity is limited to a narrow region around the equator.
Figure 6.6.3a shows the maximum velocity with and without the crossisopycnal flux. In the presence of w. the velocity is reduced significantly. Note,
however, that the current continues to accelerate beyond the longitude where
the cross-isopycnal velocity begins. The entry of fluid into the current from the
sides, tending to accelerate the flow, more than compensates for the deceleration due to w. until well past the longitude of onset of nonadiabatic effects.
Equatorial Dynamics of the Thermocline: The Equatorial Undercurrent
which the undercurrent is represented by a single layer it is possible only
crudely to model this behavior by choosing a mass flux function M(x) that is
weak in the western part of the basin and grows in strength eastward.
Pedlosky (1988) chose the following distribution for M(x):
{
0,
M(x)= Mo(x-xb)
Xe -Xb
( 6.6.9)
so that the cross-isopycnal flux is absent until the current flows east of the
longitude where x = xb. Mo is chosen so that:
B2
Mo =
0
Y2(Xe- Xb)
(6.6.10)
so that along the equator, from (6.6.7):
B~(x,o+) =B~[1- (~~~J\~(x-xb)l
(6.6.11)
In (6.6.11) 0(x- xb) is the unit Heaviside function which is zero when its
argument, x - xb, is less than zero and equal to unity when x - xb ;::: 0. This
choice forces B2 to vanish on the equator at the eastern boundary. If, as in the
calculations of Pedlosky (1988), the model possesses no shadow zone, the
Bernoulli function just outside the boundary current equals h(xe, 0) given by
the ventilated thermocline solution which equals zero if H 2 = 0. In this case the
transport of the undercurrent is exhausted as the current reaches the eastern
boundary.
Figure 6.6.2 shows the profiles of U2, h, h2 and au2/ ay for the integration of
the system (6.4.26) for the case in which (6.6.11) is the boundary condition at
y = 0 while (6.4.19) has been used to relate h2 to h. The longitude Xb is chosen
as 0.4 while the eastern boundary is at x = 1. Otherwise the parameters are the
same as those used to calculate the current in Fig. 6.4.4. Panel a shows the
solution at x = 0.8 for the case in which there is no cross-isopycnal flux, and
therefore B2 on the equator everywhere equals B0 , which is equal to 1.265.
Note that the maximum velocity, in scaled units, is 1.142. Panel b shows the
solution in which (6.6.11) is used. B2 at the equator has now fallen to 0.943 at
this longitude, and the maximum velocity of the current at y = 0 is similarly
reduced from its adiabatic value to 0.881. Otherwise the structure of the current is similar, as might be expected, since the effect of the cross-isopycnal
velocity is limited to a narrow region around the equator.
Figure 6.6.3a shows the maximum velocity with and without the crossisopycnal flux. In the presence of w. the velocity is reduced significantly. Note,
however, that the current continues to accelerate beyond the longitude where
the cross-isopycnal velocity begins. The entry of fluid into the current from the
sides, tending to accelerate the flow, more than compensates for the deceleration due to w. until well past the longitude of onset of nonadiabatic effects.