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DYNAMICAL OCEANOGRAPHY
Because the concept of vorticity plays an important role in the interpretation of the results of ocean models, a special section is devoted to the vorticity equation (section 4.1), the different mechanisms of vorticity transport (section 4.2) and the concept of potential vorticity (section 4.3). The
shallow-water equations are presented in section 4.4 to nicely illustrate
vorticity concepts for flows in thin liquid layers. Here we also see a first
example of scaling and the use of dimensionless equations.
4.1. The vorticity equation
As we have seen in chapter 3, the relative vorticity is the local spin of a fluid
parcel and defined mathematically as ω = ∇∧v. In a rotating frame of reference,
it is useful to define the planetary vorticity as 2Ω and the absolute vorticity ω a =
ω +2Ω. Streamlines are the integral curves of the instantaneous velocity field
C
2
ω
C
1
C
2
C
2
a
b
σ
σ '
tangent vector
n
S
2
n
V
S
1
Figure 4.1. A vortex tube consisting of vortex lines through a closed curve C1. The mapping
σ :[a, b] ⊂ R → R
3 is a curve which can parameterize a particular vortex line.
Ex. 4.1
and similarly vortex lines are the integral curves of the instantaneous absolute
vorticity field (or more generally, as integral curves of the vector field ω a /ρ ).
Hence, for fixed t 0 , the range of a curve σ : R → R 3 is a vortex line if
σ
′ (s)=
ω a
ρ
(t 0 , σ(s)).
(4.1)
A vortex tube consists of vortex lines which pass through a closed curve (Fig. 4.1).
Ex. 4.2