94
DYNAMICAL OCEANOGRAPHY
The explanation of the surface circulation in the North Atlantic such as in
Fig. 5.1, is one of classical problems in dynamical oceanography and forms the
central problem of this chapter. The question can be formulated as what processes
control the ‘gyres’ and why is there a western amplification of the currents, in
other words, why is there a Gulf Stream?
Additional Material
B: The North Atlantic surface circulation is described in more detail in chapter
14 of Tomczak and Godfrey (1994) and chapter 7 of Knaus (1997). For bedtime reading, The Gulf Stream by Henry Stommel (Stommel, 1977) is recommended.
5.2. The barotropic circulation on the β-plane
The North Atlantic flow as described above is a typical mid-latitude phenomenon. This motivates the consideration of a model set up around a central
latitude θ 0 (case (i) in section 3.3). As a first step, we investigate a situation in
which the ocean water has constant density. In this case, only the pure winddriven flow can be analyzed. Again, dimensional quantities will be indicated by a
∗ subscript.
For a flow with characteristic horizontal and vertical length scale L and D, horizontal and vertical velocity scale U and W = UD/L and a time scale L/U ,the
magnitude of the Coriolis acceleration is U 2Ω sin θ 0 = Uf 0 . From the analysis
in section 3.3, we anticipate that the dominant horizontal momentum balance is
between the Coriolis acceleration and the pressure force for small Rossby number
ǫ = U/(f 0 L). Moreover, we anticipate a hydrostatic vertical momentum balance
as D ≪ L. This motivates us to scale the pressure as
p ∗ = −ρ 0 gDz + ρ 0 Uf 0 Lp.
(5.1)
We are now fully prepared to find proper reductions of the total equations of motion specifically targeted to explain the intensification of western boundary currents.
5.2.1. The β-plane approximation
As a first step in the reduction, local coordinates (x ∗ ,y ∗ ) (Fig. 5.2) are introduced with
x ∗ = xL = φr 0 cos θ 0 ,
(5.2a)
y ∗ = yL =( θ − θ 0 )r 0 ,
(5.2b)
z ∗ = zD = r ∗ − r 0 .
(5.2c)
DYNAMICAL OCEANOGRAPHY
The explanation of the surface circulation in the North Atlantic such as in
Fig. 5.1, is one of classical problems in dynamical oceanography and forms the
central problem of this chapter. The question can be formulated as what processes
control the ‘gyres’ and why is there a western amplification of the currents, in
other words, why is there a Gulf Stream?
Additional Material
B: The North Atlantic surface circulation is described in more detail in chapter
14 of Tomczak and Godfrey (1994) and chapter 7 of Knaus (1997). For bedtime reading, The Gulf Stream by Henry Stommel (Stommel, 1977) is recommended.
5.2. The barotropic circulation on the β-plane
The North Atlantic flow as described above is a typical mid-latitude phenomenon. This motivates the consideration of a model set up around a central
latitude θ 0 (case (i) in section 3.3). As a first step, we investigate a situation in
which the ocean water has constant density. In this case, only the pure winddriven flow can be analyzed. Again, dimensional quantities will be indicated by a
∗ subscript.
For a flow with characteristic horizontal and vertical length scale L and D, horizontal and vertical velocity scale U and W = UD/L and a time scale L/U ,the
magnitude of the Coriolis acceleration is U 2Ω sin θ 0 = Uf 0 . From the analysis
in section 3.3, we anticipate that the dominant horizontal momentum balance is
between the Coriolis acceleration and the pressure force for small Rossby number
ǫ = U/(f 0 L). Moreover, we anticipate a hydrostatic vertical momentum balance
as D ≪ L. This motivates us to scale the pressure as
p ∗ = −ρ 0 gDz + ρ 0 Uf 0 Lp.
(5.1)
We are now fully prepared to find proper reductions of the total equations of motion specifically targeted to explain the intensification of western boundary currents.
5.2.1. The β-plane approximation
As a first step in the reduction, local coordinates (x ∗ ,y ∗ ) (Fig. 5.2) are introduced with
x ∗ = xL = φr 0 cos θ 0 ,
(5.2a)
y ∗ = yL =( θ − θ 0 )r 0 ,
(5.2b)
z ∗ = zD = r ∗ − r 0 .
(5.2c)