192
DYNAMICAL OCEANOGRAPHY
Summary
In a rotating stratified flow, the effect of stratification and rotation on
scale of vertical motions is of similar magnitude when
L ≈ L D =
ND
f
In stratified rotating flows, a useful potential vorticity is
Π s∗ = −
N 2
g
(ζ ∗ + f )
where N is the buoyancy frequency and ζ is the vertical component of
the vorticity vector.
In the quasi-geostrophic approximation, this potential vorticity reduces to
Π s∗ = ∇
2 ψ ∗ +
∂
∂z ∗
(
f 2
0
N 2
∂ψ ∗
∂z ∗
)+β 0 y ∗
where ψ is the geostrophic streamfunction. In the continuously stratified quasi-geostrophic model, this quantity is conserved.
The dispersion relation of Rossby waves in this model is given by
σ = −
β 0 k
n 2 π 2
L 2
D
+ k 2 + l 2
,n =0, 1,...
For L D = 100 km, barotropic (n =0 ) waves cross an Atlantic size
basin (5000 km) in a few days, while for baroclinic waves (n>0)this
takes several years.
Bottom topography slightly modifies the baroclinic (n>0) modes
but strongly modifies the barotropic mode (n =0) to become ‘bottom
trapped’.