Stratification
183
salt) is very small, we can neglect H ρ and hence the dominant O(ǫ) balance in
(8.20e) becomes
Dρ 0
dt
− w
1 S =0,
(8.28)
where S = FN 2 D/g = N 2 D 2 /(f 2
0 L 2 ) is the Burger number (cf. section 3.1).
With the help of (8.28), we can express w 1 into terms of the pressure p 0 and finally,
from (8.27), we find the quasi-geostrophic stratified potential vorticity equation
(with ψ = p 0 )
(
∂
∂t
+ u
0 ∂
∂x
+ v
0 ∂
∂y
)(∇
2 ψ +
∂
∂z
(
1
S
∂ψ
∂z
)+βy)=0.
(8.29)
The equation (8.29) can be written as DΠ s /dt in the limit of small Rossby number, where Π s is the potential vorticity defined in (8.13).
Ex. 8.3
In the stratified case, the equations for motions on the scale of the Rossby
deformation radius are closed by the density equation while in the constant density
case they can only be closed by the Ekman layers. Note, however, that in the
stratified case we cannot integrate over the layer because the horizontal velocities
are z-dependent and hence we have to consider explicit boundary conditions.
8.3.2. Boundary conditions
How do the Ekman boundary layers change from the constant density formulation in chapter 5 to the stratified case discussed here? To analyze this, we first
consider the flat bottom case at z = −1 as in chapter 5 and introduce a boundary
layer coordinate ξ =( z +1)/E
1/2
V . Using the same expansions as in (5.43), but
now also for ˜
ρ,t h eO(1) system of boundary layer equations is again (5.44a-d),
but (5.44c) has changed to
∂ ˜
p 0
∂ξ
= −E
1/2
V ˜
ρ
0 .
(8.30)
In addition, from (8.28) an estimate for ˜
ρ 0 follows as
˜
ρ
0 = O(
˜
w
ǫ
S)=O(E
1/2
V
S
ǫ
),
(8.31)
such that
∂ ˜
p 0
∂ξ
= O(E V
S
ǫ
).
(8.32)
Hence, if E V ≪ ǫ/S,t h e n˜ ρ 0 is approximately constant over the Ekman layer
and we can use the results from the constant density theory. Because S = O(1)
for motions with a horizontal scale L D , this condition is satisfied when E V ≪ ǫ.
183
salt) is very small, we can neglect H ρ and hence the dominant O(ǫ) balance in
(8.20e) becomes
Dρ 0
dt
− w
1 S =0,
(8.28)
where S = FN 2 D/g = N 2 D 2 /(f 2
0 L 2 ) is the Burger number (cf. section 3.1).
With the help of (8.28), we can express w 1 into terms of the pressure p 0 and finally,
from (8.27), we find the quasi-geostrophic stratified potential vorticity equation
(with ψ = p 0 )
(
∂
∂t
+ u
0 ∂
∂x
+ v
0 ∂
∂y
)(∇
2 ψ +
∂
∂z
(
1
S
∂ψ
∂z
)+βy)=0.
(8.29)
The equation (8.29) can be written as DΠ s /dt in the limit of small Rossby number, where Π s is the potential vorticity defined in (8.13).
Ex. 8.3
In the stratified case, the equations for motions on the scale of the Rossby
deformation radius are closed by the density equation while in the constant density
case they can only be closed by the Ekman layers. Note, however, that in the
stratified case we cannot integrate over the layer because the horizontal velocities
are z-dependent and hence we have to consider explicit boundary conditions.
8.3.2. Boundary conditions
How do the Ekman boundary layers change from the constant density formulation in chapter 5 to the stratified case discussed here? To analyze this, we first
consider the flat bottom case at z = −1 as in chapter 5 and introduce a boundary
layer coordinate ξ =( z +1)/E
1/2
V . Using the same expansions as in (5.43), but
now also for ˜
ρ,t h eO(1) system of boundary layer equations is again (5.44a-d),
but (5.44c) has changed to
∂ ˜
p 0
∂ξ
= −E
1/2
V ˜
ρ
0 .
(8.30)
In addition, from (8.28) an estimate for ˜
ρ 0 follows as
˜
ρ
0 = O(
˜
w
ǫ
S)=O(E
1/2
V
S
ǫ
),
(8.31)
such that
∂ ˜
p 0
∂ξ
= O(E V
S
ǫ
).
(8.32)
Hence, if E V ≪ ǫ/S,t h e n˜ ρ 0 is approximately constant over the Ekman layer
and we can use the results from the constant density theory. Because S = O(1)
for motions with a horizontal scale L D , this condition is satisfied when E V ≪ ǫ.