Nonadiabatic Equations in Characteristic Form
307
cell which is inserted between the ventilated fluid and the eastern wall and is
driven by the cross-isopycnal flux, i.e. by the heating. Careful comparison of
panels b and f shows that the eastern regime is characterized by northward flow
east of the critical characteristic. This northward flow returns as a recirculation
west of the critical characteristic. The recirculation is detached from the flow of
the subducting fluid and spans the shadow zone boundary. That boundary is
marked by kinks in the recirculation streamlines. It is important to re-emphasize the fact that the characteristics are not streamlines, and in particular
the characteristic separating the eastern and western regimes is one along which
his continuously decreasing along the characteristic (5.3.15c). This is consistent
with the northwest flow across the regime boundary observed in panel f. The
streamline representing the eastern boundary of the ventilated fluid lies to the
west of the critical characteristic. The critical characteristic forms the boundary
between the northward and southward branches of the recirculation in the old
shadow zone produced by the cross-isopycnal flux.
Comparing panels e and f we see that the streamlines in this eastern regime
are nearly antiparallel in the two layers. Our preceding discussion of the subpolar gyre and equations (5.3.25), (5.3.26), and (5.3.27) suggests that in this
region the circulation is direct, i.e., that w2 (z2 ) and w. are both positive in this
region. Cold water in layer 2 moves northward as it is heated, rises, and enters
the warm water layer and then flows southward. With w2(z2) positive this
implies that v2 > 0, and this is what is observed on the eastern branch of the
cyclonic gyre in the shadow zone. In Section 5.4 we shall reexamine the motion
in this region analytically to explain the occurrence of southward motion in the
western branch of the cyclonic gyre.
In the western regime where the fluid is ventilated the motion is southward,
and therefore the vertical velocity and the cross-isopycnal velocity have opposite signs. In this case the motion is indirect. Cold water flows southward and
enters the upper layer by laterally crossing the strongly sloping density interface.
Since the specification of the heating function, or cross-isopycnal velocity
w., is made a priori, it is difficult, as has been mentioned, to insist on the
physical validity of the details of the solution structure. The solution, however,
is very suggestive as to how the shadow zone, considered stagnant in the
adiabatic theory, can easily be set into motion by heating and cooling. It also
clearly illustrates the existence of distinct regions of flow where heating or
cooling produces vertical motions that may be counterintuitive, i.e., the regions
of indirect circulation.
The characteristic formalism of Luyten and Stommel is well adapted to
numerical calculation but is limited to the dynamics of the two-layer model. If
more than two layers are present, the characteristics of the complete system
become so complex that no real progress can be made. For this reason an
alternative representation of the effect of heating and cooling is useful, and this
is discussed in the following section.
307
cell which is inserted between the ventilated fluid and the eastern wall and is
driven by the cross-isopycnal flux, i.e. by the heating. Careful comparison of
panels b and f shows that the eastern regime is characterized by northward flow
east of the critical characteristic. This northward flow returns as a recirculation
west of the critical characteristic. The recirculation is detached from the flow of
the subducting fluid and spans the shadow zone boundary. That boundary is
marked by kinks in the recirculation streamlines. It is important to re-emphasize the fact that the characteristics are not streamlines, and in particular
the characteristic separating the eastern and western regimes is one along which
his continuously decreasing along the characteristic (5.3.15c). This is consistent
with the northwest flow across the regime boundary observed in panel f. The
streamline representing the eastern boundary of the ventilated fluid lies to the
west of the critical characteristic. The critical characteristic forms the boundary
between the northward and southward branches of the recirculation in the old
shadow zone produced by the cross-isopycnal flux.
Comparing panels e and f we see that the streamlines in this eastern regime
are nearly antiparallel in the two layers. Our preceding discussion of the subpolar gyre and equations (5.3.25), (5.3.26), and (5.3.27) suggests that in this
region the circulation is direct, i.e., that w2 (z2 ) and w. are both positive in this
region. Cold water in layer 2 moves northward as it is heated, rises, and enters
the warm water layer and then flows southward. With w2(z2) positive this
implies that v2 > 0, and this is what is observed on the eastern branch of the
cyclonic gyre in the shadow zone. In Section 5.4 we shall reexamine the motion
in this region analytically to explain the occurrence of southward motion in the
western branch of the cyclonic gyre.
In the western regime where the fluid is ventilated the motion is southward,
and therefore the vertical velocity and the cross-isopycnal velocity have opposite signs. In this case the motion is indirect. Cold water flows southward and
enters the upper layer by laterally crossing the strongly sloping density interface.
Since the specification of the heating function, or cross-isopycnal velocity
w., is made a priori, it is difficult, as has been mentioned, to insist on the
physical validity of the details of the solution structure. The solution, however,
is very suggestive as to how the shadow zone, considered stagnant in the
adiabatic theory, can easily be set into motion by heating and cooling. It also
clearly illustrates the existence of distinct regions of flow where heating or
cooling produces vertical motions that may be counterintuitive, i.e., the regions
of indirect circulation.
The characteristic formalism of Luyten and Stommel is well adapted to
numerical calculation but is limited to the dynamics of the two-layer model. If
more than two layers are present, the characteristics of the complete system
become so complex that no real progress can be made. For this reason an
alternative representation of the effect of heating and cooling is useful, and this
is discussed in the following section.