Application of the Theory to the Subtropical Gyre
139
(3.9.7)
Figure 3.9.1 shows the geostrophic contours for three values of the
parameter at:= foWo/P 2 L~H which measures the relative contribution to the
potential vorticity of the thickness variation of layer 2 due to the wind forcing
and the planetary vorticity gradient. For small values of this parameter the
isolines depart only slightly from latitude circles, and the whole basin is
covered with geostrophic contours emanating from the eastern boundary.
Panel a shows the case in which at:= 0.1, and the basin is indeed covered with
such blocked contours on which the motion in layer 2 is negligible. In panel b,
in which at: = 0.3, we see the appearance of a region in the northwest corner of
the basin in which the geostrophic contours are so distorted that they curve
around to hit the western boundary twice and are completely disconnected
from the eastern boundary. Information from the eastern boundary cannot
enter this region, which is outlined by the critical CJ2 contour indicated by the
plus signs. This is the outermost q2 contour which avoids the eastern boundary.
Within this region flow can circulate within the interior and through the western
boundary current, and the motion is not turned off by the Rossby wave
emanating from the eastern boundary. The Rossby waves are deflected around
the pool and swept southwestward. As f o Wo / P 2 L~H increases, the extent of the
pool region also increases, as shown in panel c, where at: = 1.
Whether or not the pool region appears depends on the east-west extent of
the basin. For example in panel b, were the basin limited to the region
0 :::; x :::; L, the closed contours would lie west of the western boundary of the
ocean at x = 0 and lie outside the basin.
The critical value of the forcing required to produce a pool region in the
basin can be determined directly from (3.9.5). As y approaches Ye it appears
that x must approach the eastern boundary at x = Xe since the right hand side
of (3.9.5) vanishes at y = Ye· However, consider the expression for Xe- x if Ye
lies asymptotically close to the northern boundary. Ifye is equal to L then the
denominator of the right hand side also vanishes as y approaches Ye since WE
vanishes at y = L. Thus for the isoline emanating from the corner at y = L,
x = Xe, the ratio on the right side of (3.9.5) becomes indeterminate and must be
evaluated by expanding numerator and denominator in a Taylor series around
the point y = Ye = L. Doing so, we obtain for the zonal position, Xn of the
geostrophic contour as it emanates from y = Ye = L:
Xe -xr = foF(&wE/oy).
y=L
(3.9.8)
This yields the intersection of the boundary of the pool region with the
northern latitude of the gyre at y = L, as shown in Fig. 3.9.lb,c. This
outermost contour starts from the critical point (x, y) = (xnL). As long as Xr
139
(3.9.7)
Figure 3.9.1 shows the geostrophic contours for three values of the
parameter at:= foWo/P 2 L~H which measures the relative contribution to the
potential vorticity of the thickness variation of layer 2 due to the wind forcing
and the planetary vorticity gradient. For small values of this parameter the
isolines depart only slightly from latitude circles, and the whole basin is
covered with geostrophic contours emanating from the eastern boundary.
Panel a shows the case in which at:= 0.1, and the basin is indeed covered with
such blocked contours on which the motion in layer 2 is negligible. In panel b,
in which at: = 0.3, we see the appearance of a region in the northwest corner of
the basin in which the geostrophic contours are so distorted that they curve
around to hit the western boundary twice and are completely disconnected
from the eastern boundary. Information from the eastern boundary cannot
enter this region, which is outlined by the critical CJ2 contour indicated by the
plus signs. This is the outermost q2 contour which avoids the eastern boundary.
Within this region flow can circulate within the interior and through the western
boundary current, and the motion is not turned off by the Rossby wave
emanating from the eastern boundary. The Rossby waves are deflected around
the pool and swept southwestward. As f o Wo / P 2 L~H increases, the extent of the
pool region also increases, as shown in panel c, where at: = 1.
Whether or not the pool region appears depends on the east-west extent of
the basin. For example in panel b, were the basin limited to the region
0 :::; x :::; L, the closed contours would lie west of the western boundary of the
ocean at x = 0 and lie outside the basin.
The critical value of the forcing required to produce a pool region in the
basin can be determined directly from (3.9.5). As y approaches Ye it appears
that x must approach the eastern boundary at x = Xe since the right hand side
of (3.9.5) vanishes at y = Ye· However, consider the expression for Xe- x if Ye
lies asymptotically close to the northern boundary. Ifye is equal to L then the
denominator of the right hand side also vanishes as y approaches Ye since WE
vanishes at y = L. Thus for the isoline emanating from the corner at y = L,
x = Xe, the ratio on the right side of (3.9.5) becomes indeterminate and must be
evaluated by expanding numerator and denominator in a Taylor series around
the point y = Ye = L. Doing so, we obtain for the zonal position, Xn of the
geostrophic contour as it emanates from y = Ye = L:
Xe -xr = foF(&wE/oy).
y=L
(3.9.8)
This yields the intersection of the boundary of the pool region with the
northern latitude of the gyre at y = L, as shown in Fig. 3.9.lb,c. This
outermost contour starts from the critical point (x, y) = (xnL). As long as Xr