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DYNAMICAL OCEANOGRAPHY
Now assume that in a stratified flow, the Coriolis acceleration is the dominant term in the momentum balance with a characteristic geostrophic pressure
ρ 0 Lf 0 U . With the pressure differences P from (8.4) we derive
W/D
U/L
=
U 2
N 2 D 2
f 0 L
U
=
Fr 2
ǫ
.
(8.8)
The parameter Fr 2 /ǫ provides the scale of the vertical motions in a stratified
rotating flow. When Fr increases, the influence of the stratification decreases and
the vertical motions will be mainly determined by the rotation; we approximate
the constant density case (Fig. 8.2). In the regime where Fr 2 /ǫ ≈ ǫ, i.e.
U 2
N 2 D 2 ≈
U 2
f 2
0 L 2 ⇒ L ≈
ND
f 0
= L D ,
(8.9)
the effect of stratification is of the same magnitude as that of rotation. The length
scale L D is the internal Rossby deformation radius (cf. section 3.1).
Figure 8.2. Overview of the different flow regimes in a stratified rotating flow.
Typical values of N 2 in the upper layers of the North Atlantic are N 2 =1 0 −5
s −2 . With D =4× 10 3 mandf 0 =10 −4 s −1 , a typical value of L D is about 100
km (see Table 8.1).