120
DYNAMICAL OCEANOGRAPHY
Figure 5.13. Annual mean wind-induced Ekman upwelling (in cm/day) over the global ocean
(from http://www.ocgy.ubc.ca/projects/clim.pred/Upwell/index.html).
vortex stretching. The third term is the vorticity production due to lateral friction.
To investigate the physical interpretation of the left hand side of (5.91), we write
it, with v 0 = Dy/dt,as
D
dt
ζ
0 − Fη
0 + η b + βy
=
DΠ
dt
.
(5.92)
If DΠ/dt =0then Π is constant along streamlines.
Now consider the dimensional potential vorticity Π ∗ on the β-plane defined by
Π ∗ =
ζ ∗ + f
H ∗
,
(5.93)
where H ∗ is the total thickness of the water column, f = f 0 + β 0 y ∗ and ζ ∗ is the
vertical component of the relative vorticity. For the case considered above with
small amplitude topography, we have
H ∗ = h ∗ − (−D + h b∗ )=D + h ∗ − h b∗ .
(5.94)
In dimensionless quantities, this becomes
D
f 0
Π ∗ =(1+ǫF η − ǫη b )
−1 (ǫζ +1+βǫy).
(5.95)
Expansion of the denominator in ǫ gives
(1 + ǫ(Fη − η b ))
−1 =1− ǫ(Fη − η b )+O(ǫ
2 ),
(5.96)
DYNAMICAL OCEANOGRAPHY
Figure 5.13. Annual mean wind-induced Ekman upwelling (in cm/day) over the global ocean
(from http://www.ocgy.ubc.ca/projects/clim.pred/Upwell/index.html).
vortex stretching. The third term is the vorticity production due to lateral friction.
To investigate the physical interpretation of the left hand side of (5.91), we write
it, with v 0 = Dy/dt,as
D
dt
ζ
0 − Fη
0 + η b + βy
=
DΠ
dt
.
(5.92)
If DΠ/dt =0then Π is constant along streamlines.
Now consider the dimensional potential vorticity Π ∗ on the β-plane defined by
Π ∗ =
ζ ∗ + f
H ∗
,
(5.93)
where H ∗ is the total thickness of the water column, f = f 0 + β 0 y ∗ and ζ ∗ is the
vertical component of the relative vorticity. For the case considered above with
small amplitude topography, we have
H ∗ = h ∗ − (−D + h b∗ )=D + h ∗ − h b∗ .
(5.94)
In dimensionless quantities, this becomes
D
f 0
Π ∗ =(1+ǫF η − ǫη b )
−1 (ǫζ +1+βǫy).
(5.95)
Expansion of the denominator in ǫ gives
(1 + ǫ(Fη − η b ))
−1 =1− ǫ(Fη − η b )+O(ǫ
2 ),
(5.96)