Effect of Entrainment
361
Thus at the point y = o+:
1 8B 2
- 2
a 2 = (h2v2)y=O+ = -M(x) ·
Y2 X
(6.6.6)
The effect of the concentrated cross-isopycnal flux at the equator is to produce
a reduction of the Bernoulli function along the equator proportional to the
amount of mass transferred between the layers.
Integrating (6.6.6) from the western boundary where the Bernoulli function may be considered set by western boundary current's value of B2:
B~(x, o+) = B~- 2y21x M(x')dx' .
(6.6.7)
The Bernoulli function on the western boundary, at x = 0, is Bo. If there were
no cross-isopycnal flux, this would be the value of the Bernoulli function all
along the equator. The presence of the nonadiabatic mass flux reduces this
value with longitude as a function of the strength of the transfer of mass
between the layers.
The zonal velocity of the undercurrent satisfies (6.4.29b) so that at each
longitude the transport in the undercurrent is equal to:
(u2h2dy=- 2
1 [B~(x,o+)-B~(x,£)].
Jo
Y2
(6.6.8)
The second term in the square bracket on the right side of (6.6.8) is set by the
interior, inviscid thermocline solution. As the first term in the bracket is diminished by the cross-isopycnal flux, both the transport in the undercurrent
and its peak velocity weaken with longitude.
The system (6.4.26a,b,c), which holds everywhere outside of the vanishingly thin region of cross-isopycnal flux, can now be solved again with the new
boundary condition (6.6. 7) applied at y = 0 instead of (6.4.32) to which (6.6. 7)
reduces if M(x) is zero.
In principal, the cross-isopycnal mass flux, which is a measure of the
dissipation in the current, should be related to the properties of the current
itself. This requires a model for the dissipation. Such a model is discussed
below. However, it is useful, as in Chapter 5, to examine the response of the
motion to a given distribution of w* or equivalently to a given longitudinal
distribution of M(x).
When the particular density layer is deep, the cross-isopycnal velocity is
negligible, as described by the analysis of Bryden and Brady (1985). As the
current flows eastward, the uppermost layer is slowly entrained into the mixed
layer, exposing the next lowest isopycnallayer to entrainment until, as the jet
crosses the basin, each of the layers exhausts its flux into the mixed layer. Only
when a particular stratum approaches the upper layer where mixing is important does the cross-isopycnal velocity become significant in that layer. For
layers in the core of the undercurrent the effect of the cross-isopycnal flux does
not occur until a large portion of the basin has been traversed. In a model in
361
Thus at the point y = o+:
1 8B 2
- 2
a 2 = (h2v2)y=O+ = -M(x) ·
Y2 X
(6.6.6)
The effect of the concentrated cross-isopycnal flux at the equator is to produce
a reduction of the Bernoulli function along the equator proportional to the
amount of mass transferred between the layers.
Integrating (6.6.6) from the western boundary where the Bernoulli function may be considered set by western boundary current's value of B2:
B~(x, o+) = B~- 2y21x M(x')dx' .
(6.6.7)
The Bernoulli function on the western boundary, at x = 0, is Bo. If there were
no cross-isopycnal flux, this would be the value of the Bernoulli function all
along the equator. The presence of the nonadiabatic mass flux reduces this
value with longitude as a function of the strength of the transfer of mass
between the layers.
The zonal velocity of the undercurrent satisfies (6.4.29b) so that at each
longitude the transport in the undercurrent is equal to:
(u2h2dy=- 2
1 [B~(x,o+)-B~(x,£)].
Jo
Y2
(6.6.8)
The second term in the square bracket on the right side of (6.6.8) is set by the
interior, inviscid thermocline solution. As the first term in the bracket is diminished by the cross-isopycnal flux, both the transport in the undercurrent
and its peak velocity weaken with longitude.
The system (6.4.26a,b,c), which holds everywhere outside of the vanishingly thin region of cross-isopycnal flux, can now be solved again with the new
boundary condition (6.6. 7) applied at y = 0 instead of (6.4.32) to which (6.6. 7)
reduces if M(x) is zero.
In principal, the cross-isopycnal mass flux, which is a measure of the
dissipation in the current, should be related to the properties of the current
itself. This requires a model for the dissipation. Such a model is discussed
below. However, it is useful, as in Chapter 5, to examine the response of the
motion to a given distribution of w* or equivalently to a given longitudinal
distribution of M(x).
When the particular density layer is deep, the cross-isopycnal velocity is
negligible, as described by the analysis of Bryden and Brady (1985). As the
current flows eastward, the uppermost layer is slowly entrained into the mixed
layer, exposing the next lowest isopycnallayer to entrainment until, as the jet
crosses the basin, each of the layers exhausts its flux into the mixed layer. Only
when a particular stratum approaches the upper layer where mixing is important does the cross-isopycnal velocity become significant in that layer. For
layers in the core of the undercurrent the effect of the cross-isopycnal flux does
not occur until a large portion of the basin has been traversed. In a model in