Effect of Entrainment
365
The model dynamics of the upper layer, integrated over its depth consists of a
balance between the Coriolis acceleration and the surface wind stress, the zonal
pressure gradient, and a frictional deceleration proportional to the velocity of
the layer itself. This simple representation of the frictional effect is meant to
model in the easiest possible manner the observed role of lateral friction (see
Johnson and Luther I994) which should depend only on u1. Far from the
equator where this term is dominated by the Coriolis acceleration (6.6.I2)
consists of balance which combines the Ekman layer transport of the mixed
layer and the geostrophic transport of layer I. Near the equator the separation
is no longer valid, and they are lumped together. On the equator the Coriolis
acceleration vanishes, and the upper layer zonal current is driven by the imbalance between the zonal pressure gradient and the wind stress. Because of
( 6.4.I8) this pressure gradient is known from the off-equatorial solution of the
thermocline problem.
Outside the small region of half-width b (in nondimensional units) the
upper layer flow is horizontally nondivergent so that a streamfunction can be
introduced for the transport, i.e.:
(6.6.13a,b)
This allows (6.6.12) to be written as the partial differential equation:
aljf,
aljf,
Y ax +ray= -T+h,P,(x)
(6.6.14)
where P 1 (x) is the known east-west pressure force in the upper layer and is
equal to h + r 12 h1 at the matching latitude outside the equatorial region.
The above equation is linked to the equations in the lower layer through
the boundary condition on l/1 1 at the equator. The flux of mass across the
isopycnals from layer 2 must enter layer 1 in the region of the equator and thus
on y = 0:
I aB~
aljf 1
h2v2 = - - = -h1v1 = - - .
2y2 ax
ax
( 6.6.15)
Thus the layers are linked by the condition at the equator:
B/(x, 0) = -2y2l/1 1 (x, 0)
(6.6.I6)
where it is assumed that both l/1 1 and B2 vanish on the eastern boundary.
The characteristics of (6.6.14) are the parabolas:
y2
y2
2 r+(1-x)= 2 ;
(6.6.17)
each one emanating from a different point, Ye, on the eastern boundary. With
the starting value ljJ 1 = 0, and the values of the layer thicknesses known on the
eastern boundary, (6.6.16) can be integrated along a small step, Llx, westward.
One of these characteristics strikes the equator at:
365
The model dynamics of the upper layer, integrated over its depth consists of a
balance between the Coriolis acceleration and the surface wind stress, the zonal
pressure gradient, and a frictional deceleration proportional to the velocity of
the layer itself. This simple representation of the frictional effect is meant to
model in the easiest possible manner the observed role of lateral friction (see
Johnson and Luther I994) which should depend only on u1. Far from the
equator where this term is dominated by the Coriolis acceleration (6.6.I2)
consists of balance which combines the Ekman layer transport of the mixed
layer and the geostrophic transport of layer I. Near the equator the separation
is no longer valid, and they are lumped together. On the equator the Coriolis
acceleration vanishes, and the upper layer zonal current is driven by the imbalance between the zonal pressure gradient and the wind stress. Because of
( 6.4.I8) this pressure gradient is known from the off-equatorial solution of the
thermocline problem.
Outside the small region of half-width b (in nondimensional units) the
upper layer flow is horizontally nondivergent so that a streamfunction can be
introduced for the transport, i.e.:
(6.6.13a,b)
This allows (6.6.12) to be written as the partial differential equation:
aljf,
aljf,
Y ax +ray= -T+h,P,(x)
(6.6.14)
where P 1 (x) is the known east-west pressure force in the upper layer and is
equal to h + r 12 h1 at the matching latitude outside the equatorial region.
The above equation is linked to the equations in the lower layer through
the boundary condition on l/1 1 at the equator. The flux of mass across the
isopycnals from layer 2 must enter layer 1 in the region of the equator and thus
on y = 0:
I aB~
aljf 1
h2v2 = - - = -h1v1 = - - .
2y2 ax
ax
( 6.6.15)
Thus the layers are linked by the condition at the equator:
B/(x, 0) = -2y2l/1 1 (x, 0)
(6.6.I6)
where it is assumed that both l/1 1 and B2 vanish on the eastern boundary.
The characteristics of (6.6.14) are the parabolas:
y2
y2
2 r+(1-x)= 2 ;
(6.6.17)
each one emanating from a different point, Ye, on the eastern boundary. With
the starting value ljJ 1 = 0, and the values of the layer thicknesses known on the
eastern boundary, (6.6.16) can be integrated along a small step, Llx, westward.
One of these characteristics strikes the equator at: