Stratification
175
vertical motion, there will be vertical displacements with an amplitude Δz with
Δz = Wτ a .
(8.2)
With d¯ ρ/dz < 0, the density difference between a particular fluid element that
has moved upwards with Δz>0 and the background density field (on the same
z−level) is equal to Δρ = −(d¯ ρ/dz)Δz>0.I fΔz<0, then the fluid element
is lighter than its environment and Δρ = −(d¯ ρ/dz)Δz<0. With (8.1-8.2) it
follows that
Δρ = N
2 ρ 0
g
Δz = N
2 ρ 0
g
WL
U
.
(8.3)
Differences in density Δρ cause pressure differences Δp and with the hydrostatic
balance, we have the estimate
Δp ≈−gDΔρ = −N
2 ρ 0 WLD
U
.
(8.4)
Whether these density induced pressure differences (with magnitude P =
N 2 ρ 0 WLD/U) will influence the flow now depends on other factors (such as
the background rotation).
First consider the case in which there is no rotation and where inertia is the
dominant term in the momentum balance, with a characteristic pressure scale
ρ 0 U 2 . With (8.4) it follows that
U
2 =
P
ρ 0
= N
2 WLD
U
⇒
W/D
U/L
=
Δz
D
=
U 2
N 2 D 2 = Fr
2 .
(8.5)
The stratified Froude number Fr is a measure of the relative influence of stratification versus inertia. When Fris small then the stratification is strong and vertical
motions will be very small. For large values of Fr, stratification is unimportant
compared to inertia.
If we go back to chapter 5 and consider the constant density rotating case, then
we know that for the vertical ‘outer’ velocity, we had the expansion
w = w
0 + ǫw
1 + O(ǫ
2 ),
(8.6)
where the first term w 0 =0 . The dimensional vertical velocity is scaled with
DU/L, such that the actual scale of the vertical velocity is equal to ǫDU/L.I t
follows then that
W = ǫDU/L ⇒
W/D
U/L
= ǫ,
(8.7)
and it is exactly the Rossby number that determines the magnitude of the vertical
motion. In a rotating flow with ǫ ≪ 1, vertical motions are very restricted and the
motion is quasi two-dimensional.
175
vertical motion, there will be vertical displacements with an amplitude Δz with
Δz = Wτ a .
(8.2)
With d¯ ρ/dz < 0, the density difference between a particular fluid element that
has moved upwards with Δz>0 and the background density field (on the same
z−level) is equal to Δρ = −(d¯ ρ/dz)Δz>0.I fΔz<0, then the fluid element
is lighter than its environment and Δρ = −(d¯ ρ/dz)Δz<0. With (8.1-8.2) it
follows that
Δρ = N
2 ρ 0
g
Δz = N
2 ρ 0
g
WL
U
.
(8.3)
Differences in density Δρ cause pressure differences Δp and with the hydrostatic
balance, we have the estimate
Δp ≈−gDΔρ = −N
2 ρ 0 WLD
U
.
(8.4)
Whether these density induced pressure differences (with magnitude P =
N 2 ρ 0 WLD/U) will influence the flow now depends on other factors (such as
the background rotation).
First consider the case in which there is no rotation and where inertia is the
dominant term in the momentum balance, with a characteristic pressure scale
ρ 0 U 2 . With (8.4) it follows that
U
2 =
P
ρ 0
= N
2 WLD
U
⇒
W/D
U/L
=
Δz
D
=
U 2
N 2 D 2 = Fr
2 .
(8.5)
The stratified Froude number Fr is a measure of the relative influence of stratification versus inertia. When Fris small then the stratification is strong and vertical
motions will be very small. For large values of Fr, stratification is unimportant
compared to inertia.
If we go back to chapter 5 and consider the constant density rotating case, then
we know that for the vertical ‘outer’ velocity, we had the expansion
w = w
0 + ǫw
1 + O(ǫ
2 ),
(8.6)
where the first term w 0 =0 . The dimensional vertical velocity is scaled with
DU/L, such that the actual scale of the vertical velocity is equal to ǫDU/L.I t
follows then that
W = ǫDU/L ⇒
W/D
U/L
= ǫ,
(8.7)
and it is exactly the Rossby number that determines the magnitude of the vertical
motion. In a rotating flow with ǫ ≪ 1, vertical motions are very restricted and the
motion is quasi two-dimensional.