Arctic Ocean Circulation
363
have ψ 2 (σ(s)) = constant. By taking the derivative to s we find
∇ψ 2 · σ
′ =0.
(15.21)
Now because n is orthogonal to the tangent to the curve (σ ′ )a n df o rt h e
geostrophic streamfunction, we have u 2 = e 3 ∧∇ψ 2 , it is found that
ˆ
q 2 u 2 · n ds =ˆ q 2
u 2 · n ds =ˆ q 2
d
ds
ψ 2 (σ(s))ds =0,
(15.22)
the latter step because the contour is closed and ψ 2 is single valued.
Integration of (15.19) therefore gives
A
ǫ 0 ∇
2 ψ 2 + A f ∇∧(u 2 − u 1 )
dxdy =0,
(15.23)
and applying Gauss’ and Stokes’ theorems to the integrals above we obtain
A f
u 1 · ds =(ǫ 0 + A f )
u 2 · ds.
(15.24)
To obtain another relation between u 1 and u 2 we use (15.10) to give
u 1 =
D
H 1
u B −
H 2
H 1
u 2 ,
(15.25)
where u B is the Sverdrup velocity field. Substituting this expression into (15.24)
gives
u 2 · ds =
DA f
H 1 ǫ 0 + DA f
u B · ds.
(15.26)
On the other hand, when friction is negligible we have ψ 2 =Ψ ( ˆ
q 2 ) (because
of conservation of potential vorticity in the lower layer) and hence
u 2 · ds =
e 3 ∧∇ψ 2 · ds =
e 3 ∧
∂Ψ
∂ ˆ
q 2
∇ˆ q 2 · ds.
(15.27)
Again, ∂Ψ/∂ ˆ
q 2 is only a function of ˆ
q 2 and hence constant along C. With ˆ
q 2 =
ˆ
Fψ B + β 0 y, we then find
u 2 · ds =
∂Ψ
∂ ˆ
q 2
e 3 ∧∇( ˆ
Fψ B + β 0 y) · ds =
∂Ψ
∂ ˆ
q 2
ˆ
F u B · ds, (15.28)
since e 3 ∧∇y = −e 1 and its contour integral is zero. Combining (15.26) and
(15.28), one can solve
ψ 2 (x, y)=Ψ(ˆ q 2 )=
DA f
ˆ
F (ǫ 0 H 1 + DA f )
(ˆ q 2 (x, y) − ˆ
q 20 )
(15.29)
where ˆ
q 20 is an integration constant which is determined by the value of ˆ
q 2 on
the outermost closed geostrophic contour where ψ 2 =0 . The conclusion of this
analysis is that weak dissipation (interfacial friction) sets the flow in the lower
layer in the region of closed geostrophic contours.
363
have ψ 2 (σ(s)) = constant. By taking the derivative to s we find
∇ψ 2 · σ
′ =0.
(15.21)
Now because n is orthogonal to the tangent to the curve (σ ′ )a n df o rt h e
geostrophic streamfunction, we have u 2 = e 3 ∧∇ψ 2 , it is found that
ˆ
q 2 u 2 · n ds =ˆ q 2
u 2 · n ds =ˆ q 2
d
ds
ψ 2 (σ(s))ds =0,
(15.22)
the latter step because the contour is closed and ψ 2 is single valued.
Integration of (15.19) therefore gives
A
ǫ 0 ∇
2 ψ 2 + A f ∇∧(u 2 − u 1 )
dxdy =0,
(15.23)
and applying Gauss’ and Stokes’ theorems to the integrals above we obtain
A f
u 1 · ds =(ǫ 0 + A f )
u 2 · ds.
(15.24)
To obtain another relation between u 1 and u 2 we use (15.10) to give
u 1 =
D
H 1
u B −
H 2
H 1
u 2 ,
(15.25)
where u B is the Sverdrup velocity field. Substituting this expression into (15.24)
gives
u 2 · ds =
DA f
H 1 ǫ 0 + DA f
u B · ds.
(15.26)
On the other hand, when friction is negligible we have ψ 2 =Ψ ( ˆ
q 2 ) (because
of conservation of potential vorticity in the lower layer) and hence
u 2 · ds =
e 3 ∧∇ψ 2 · ds =
e 3 ∧
∂Ψ
∂ ˆ
q 2
∇ˆ q 2 · ds.
(15.27)
Again, ∂Ψ/∂ ˆ
q 2 is only a function of ˆ
q 2 and hence constant along C. With ˆ
q 2 =
ˆ
Fψ B + β 0 y, we then find
u 2 · ds =
∂Ψ
∂ ˆ
q 2
e 3 ∧∇( ˆ
Fψ B + β 0 y) · ds =
∂Ψ
∂ ˆ
q 2
ˆ
F u B · ds, (15.28)
since e 3 ∧∇y = −e 1 and its contour integral is zero. Combining (15.26) and
(15.28), one can solve
ψ 2 (x, y)=Ψ(ˆ q 2 )=
DA f
ˆ
F (ǫ 0 H 1 + DA f )
(ˆ q 2 (x, y) − ˆ
q 20 )
(15.29)
where ˆ
q 20 is an integration constant which is determined by the value of ˆ
q 2 on
the outermost closed geostrophic contour where ψ 2 =0 . The conclusion of this
analysis is that weak dissipation (interfacial friction) sets the flow in the lower
layer in the region of closed geostrophic contours.