Stratification
179
This has the same form as the shallow-water potential vorticity but h now indicates
the thickness of a certain density layer instead of the total layer thickness.
With conservation of Π s we can again predict changes in latitudinal motion
or/and rotation when the stratification is changed. For example, when locally
the stratification is reduced, then N 2 is reduced or equivalently a water mass
of certain density gets a larger thickness with respect to its surroundings. On
the large scale, where f is dominant over the relative vorticity, a decrease in
stratification must lead to poleward motion as f has to increase. On a smaller
scale, where f is nearly constant, a decrease in stratification locally must lead to
an increase in local vorticity and hence in an increased counterclockwise motion.
Additional Material
B: See chapter 9 in Cushman-Roisin (1994) for additional details.
D: A more extensive discussion on the use of the potential vorticity concept can
be found in section 4.7 of Vallis (2006) and the first sections of M ¨
uller (1995).
8.3. The stratified quasi-geostrophic model
From Table 8.1, it appears that on scales of O(100) km, adaptions to the homogeneous quasi-geostrophic theory as presented in chapter 5 are needed: this will
lead to the stratified quasi-geostrophic theory.
We scale the horizontal and vertical velocities with U and W = DU/L,r e -
spectively and for the pressure scale we take
p ∗ =¯ p ∗ +¯ ρ ∗ ULf 0 p,
(8.18)
where ¯
ρ ∗ is the dimensional background density field and ¯
p ∗ the associated hydrostatic pressure field. Because the dynamic density field is also in hydrostatic
equilibrium, it scales as
ρ ∗ =¯ ρ ∗ +
¯
ρ ∗ ULf 0
gD
ρ ⇒ ρ ∗ =¯ ρ ∗ (1 + ǫF ρ).
(8.19)
8.3.1. Model formulation
Use of local coordinates (x ∗ ,y ∗ ,z ∗ ) which are again scaled with L and D,
application of the β-plane approximation and taking the limit δ → 0 and L/r 0 ≪
0 gives (cf. section 5.1)
ǫ
Du
dt
− v(1 + βǫy)+
∂p
∂x
(1 + ǫF ρ)
−1 = E H ∇
2
H u + E V
∂ 2 u
∂z 2 , (8.20a)
179
This has the same form as the shallow-water potential vorticity but h now indicates
the thickness of a certain density layer instead of the total layer thickness.
With conservation of Π s we can again predict changes in latitudinal motion
or/and rotation when the stratification is changed. For example, when locally
the stratification is reduced, then N 2 is reduced or equivalently a water mass
of certain density gets a larger thickness with respect to its surroundings. On
the large scale, where f is dominant over the relative vorticity, a decrease in
stratification must lead to poleward motion as f has to increase. On a smaller
scale, where f is nearly constant, a decrease in stratification locally must lead to
an increase in local vorticity and hence in an increased counterclockwise motion.
Additional Material
B: See chapter 9 in Cushman-Roisin (1994) for additional details.
D: A more extensive discussion on the use of the potential vorticity concept can
be found in section 4.7 of Vallis (2006) and the first sections of M ¨
uller (1995).
8.3. The stratified quasi-geostrophic model
From Table 8.1, it appears that on scales of O(100) km, adaptions to the homogeneous quasi-geostrophic theory as presented in chapter 5 are needed: this will
lead to the stratified quasi-geostrophic theory.
We scale the horizontal and vertical velocities with U and W = DU/L,r e -
spectively and for the pressure scale we take
p ∗ =¯ p ∗ +¯ ρ ∗ ULf 0 p,
(8.18)
where ¯
ρ ∗ is the dimensional background density field and ¯
p ∗ the associated hydrostatic pressure field. Because the dynamic density field is also in hydrostatic
equilibrium, it scales as
ρ ∗ =¯ ρ ∗ +
¯
ρ ∗ ULf 0
gD
ρ ⇒ ρ ∗ =¯ ρ ∗ (1 + ǫF ρ).
(8.19)
8.3.1. Model formulation
Use of local coordinates (x ∗ ,y ∗ ,z ∗ ) which are again scaled with L and D,
application of the β-plane approximation and taking the limit δ → 0 and L/r 0 ≪
0 gives (cf. section 5.1)
ǫ
Du
dt
− v(1 + βǫy)+
∂p
∂x
(1 + ǫF ρ)
−1 = E H ∇
2
H u + E V
∂ 2 u
∂z 2 , (8.20a)