344
Equatorial Dynamics of the Thermocline: The Equatorial Undercurrent
affected by turbulent mixing. This is consistent with the diagnoses both of
Bryden and Brady (1985) of and Johnson and Luther (1994) which show the
effects of vertical mixing penetrate to at least 50 m in depth. This renders
impossible any simple, invisicid model for the upper layer dynamics.
In an effort to overcome this difficulty while still retaining the essential
simplicity of the inertial model for the undercurrent, Pedlosky (1987) suggested
simply avoiding the issue altogether and as an expedient chose h 1 at each
longitude to be just the value at that longitude of the equatorial limit of the
ventilated thermocline solution for h1• That is, within the equatorial region
Pedlosky took h1 to be a function only of x, and the function was chosen to be
the value of h1 determined by the midlatitude solution at an asymptotic
matching latitude Yn, which is large compared to unity (or I! in dimensional
units). Thus h1 is fixed as
h1 (x, y) = h1 (x, Yn)
(6.4.14)
where h1 (x, Yn) is the limit of the ventilated thermocline solution at y = Yn » 1.
If Y2 is the nondimensional position of the outcrop line this yields:
[ -2(x- Xe)r +Hi] 112
h1 (x, Yn) = [1 - Yn/ Y2] [
] 1/2
1 + r12(1- Yn/Y2) 2
( 6.4.15)
In (6.4.15) the stress is nondimensional and has been scaled by ro. The
assumption has also been made, for simplicity, that the stress is strictly zonal
and independent of x. The depth H2 is the thickness of the second layer at the
eastern boundary scaled by the depth H (6.3.21b) and the nondimensional
distance y 2 = f2/ {JI! where h is the Corio lis parameter at the outcrop line. Thus:
h2(x, y) = h- h1 (x, Yn)
(6.4.16)
in the equatorial zone, and this assumption reduces the system (6.4.13a,b) to
two dependent variables. Note that in this formulation the geostrophic zonal
flow in layer I would be the same as in layer 2 since the absence of a tilt of h1
with latitude would eliminate the zonal thermal wind. This does not mean the
zonal velocity in layer 1 must equal that in layer 2 since a westward wind driven
drift current in layer 1 also exists.
An alternative approach was later suggested by Pedlosky (199lb). The
zonal velocity in the upper layer is observed to be considerably smaller than the
velocity in the undercurrent core. This implies that the Coriolis acceleration in
the upper layer is much less than that of the core. On the other hand, the
scaling for the pressure gradient for the upper layer is the same as for the lower
layer and in the absence of any more refined consideration this would lead to a
pressure gradient in the upper layer much larger than the Coriolis acceleration
there. This implies an unbalanced pressure force at lowest order. Thus for a
balance at lowest order to occur the scaled pressure gradient in the upper layer
satisfies the inequality:
Equatorial Dynamics of the Thermocline: The Equatorial Undercurrent
affected by turbulent mixing. This is consistent with the diagnoses both of
Bryden and Brady (1985) of and Johnson and Luther (1994) which show the
effects of vertical mixing penetrate to at least 50 m in depth. This renders
impossible any simple, invisicid model for the upper layer dynamics.
In an effort to overcome this difficulty while still retaining the essential
simplicity of the inertial model for the undercurrent, Pedlosky (1987) suggested
simply avoiding the issue altogether and as an expedient chose h 1 at each
longitude to be just the value at that longitude of the equatorial limit of the
ventilated thermocline solution for h1• That is, within the equatorial region
Pedlosky took h1 to be a function only of x, and the function was chosen to be
the value of h1 determined by the midlatitude solution at an asymptotic
matching latitude Yn, which is large compared to unity (or I! in dimensional
units). Thus h1 is fixed as
h1 (x, y) = h1 (x, Yn)
(6.4.14)
where h1 (x, Yn) is the limit of the ventilated thermocline solution at y = Yn » 1.
If Y2 is the nondimensional position of the outcrop line this yields:
[ -2(x- Xe)r +Hi] 112
h1 (x, Yn) = [1 - Yn/ Y2] [
] 1/2
1 + r12(1- Yn/Y2) 2
( 6.4.15)
In (6.4.15) the stress is nondimensional and has been scaled by ro. The
assumption has also been made, for simplicity, that the stress is strictly zonal
and independent of x. The depth H2 is the thickness of the second layer at the
eastern boundary scaled by the depth H (6.3.21b) and the nondimensional
distance y 2 = f2/ {JI! where h is the Corio lis parameter at the outcrop line. Thus:
h2(x, y) = h- h1 (x, Yn)
(6.4.16)
in the equatorial zone, and this assumption reduces the system (6.4.13a,b) to
two dependent variables. Note that in this formulation the geostrophic zonal
flow in layer I would be the same as in layer 2 since the absence of a tilt of h1
with latitude would eliminate the zonal thermal wind. This does not mean the
zonal velocity in layer 1 must equal that in layer 2 since a westward wind driven
drift current in layer 1 also exists.
An alternative approach was later suggested by Pedlosky (199lb). The
zonal velocity in the upper layer is observed to be considerably smaller than the
velocity in the undercurrent core. This implies that the Coriolis acceleration in
the upper layer is much less than that of the core. On the other hand, the
scaling for the pressure gradient for the upper layer is the same as for the lower
layer and in the absence of any more refined consideration this would lead to a
pressure gradient in the upper layer much larger than the Coriolis acceleration
there. This implies an unbalanced pressure force at lowest order. Thus for a
balance at lowest order to occur the scaled pressure gradient in the upper layer
satisfies the inequality: