Determination of the Recirculation
127
d'¥2 [ 1 {A- - } 1 { A - }] d'¥2 1 {A- - }
d?J2 Jc Fun· td£ - Jc fJi · td£ = dq 2 Jc Fun· td£
(3. 7.12)
since the integral around the closed curve C of the dot product of the unit
vector in the x direction, i, with the tangent vector to C is zero.
When (3.7.11) and (3.7.12) are used in (3.7.10) we obtain, as long as the
circulation of the barotropic velocity around C differs from zero (and we note
again that this is a known quantity):
(3.7.13)
A{ fl1 }
F r2 H +A2
It is important to note that in this case the relation between l/1 2 and ?;2 is
linear and depends only on the dissipation parameters. Thus integrating
(3.7.13), we obtain:
,,, _'I' (A ) =
A2
{ q2(x, y) _ ?;20 }
'1'2- 2 q2
H,
F
F
r2 H +A2
(3.7.14)
where q2o is the constant value of ?;2 on the outermost closed geostrophic
contour. On this contour the streamfunction in layer 2 must vanish by
continuity with the resting region outside the pool of closed ?;2 contours. Thus
the lower layer motion is now completely determined. The process has
occurred in two steps. First, the inviscid dynamics shapes the geostrophic
contours which determines the region in which motion in layer 2 can occur.
The paths within that region on which the recirculation can take place are
identified as the isolines of q2, i.e., the geostrophic contours. Then weak
dissipation, acting on the free geostrophic mode, forces it into motion with an
amplitude which is determined by a balance between the weak frictional
coupling and dissipation.
There are two particularly interesting limits of(3.7.14). If bottom friction
is very strong compared to the coupling with layer 1, i.e., if the ratio,
r2Hd A2H, is very large, the streamfunction in the lower layer is expunged by
the bottom friction. We can think of this as a strong coupling with either a
solid boundary at the bottom or as a strong frictional coupling with a very
deep and nearly motionless deeper layer. Then, even though the geostrophic
contours allow motion, it is dissipated by bottom friction faster than it can be
generated by the weak coupling with the directly wind-driven layer above.
On the other hand, if the bottom friction is relatively small so that
r2H!/A2H « I, the amplitude of the streamfunction in layer 2 becomes
independent of the value of the mixing coefficient A2•
In the case of the example in Section 3.6 the outermost closed streamline in
the lower layer just touches the outer circle of the region of forcing at the point
127
d'¥2 [ 1 {A- - } 1 { A - }] d'¥2 1 {A- - }
d?J2 Jc Fun· td£ - Jc fJi · td£ = dq 2 Jc Fun· td£
(3. 7.12)
since the integral around the closed curve C of the dot product of the unit
vector in the x direction, i, with the tangent vector to C is zero.
When (3.7.11) and (3.7.12) are used in (3.7.10) we obtain, as long as the
circulation of the barotropic velocity around C differs from zero (and we note
again that this is a known quantity):
(3.7.13)
A{ fl1 }
F r2 H +A2
It is important to note that in this case the relation between l/1 2 and ?;2 is
linear and depends only on the dissipation parameters. Thus integrating
(3.7.13), we obtain:
,,, _'I' (A ) =
A2
{ q2(x, y) _ ?;20 }
'1'2- 2 q2
H,
F
F
r2 H +A2
(3.7.14)
where q2o is the constant value of ?;2 on the outermost closed geostrophic
contour. On this contour the streamfunction in layer 2 must vanish by
continuity with the resting region outside the pool of closed ?;2 contours. Thus
the lower layer motion is now completely determined. The process has
occurred in two steps. First, the inviscid dynamics shapes the geostrophic
contours which determines the region in which motion in layer 2 can occur.
The paths within that region on which the recirculation can take place are
identified as the isolines of q2, i.e., the geostrophic contours. Then weak
dissipation, acting on the free geostrophic mode, forces it into motion with an
amplitude which is determined by a balance between the weak frictional
coupling and dissipation.
There are two particularly interesting limits of(3.7.14). If bottom friction
is very strong compared to the coupling with layer 1, i.e., if the ratio,
r2Hd A2H, is very large, the streamfunction in the lower layer is expunged by
the bottom friction. We can think of this as a strong coupling with either a
solid boundary at the bottom or as a strong frictional coupling with a very
deep and nearly motionless deeper layer. Then, even though the geostrophic
contours allow motion, it is dissipated by bottom friction faster than it can be
generated by the weak coupling with the directly wind-driven layer above.
On the other hand, if the bottom friction is relatively small so that
r2H!/A2H « I, the amplitude of the streamfunction in layer 2 becomes
independent of the value of the mixing coefficient A2•
In the case of the example in Section 3.6 the outermost closed streamline in
the lower layer just touches the outer circle of the region of forcing at the point