Afi r s ti m p r e s s i o n
19
and
ρ
W
2 = ρ 1 +
∂ρ
∂z
W
|z=−z 1
Δz + ... = ρ 1 +
−Γ
∂ρ W
∂T
+
∂ρ W
∂p
∂p
∂z
z=−z1
Δz + ....
(1.11)
As the variations in T and S are small, ∂ρ W /∂p ≈ ∂ρ/∂p and we find that as
Δz → 0
a z =
g
ρ 1
∂ρ
∂S
∂S
∂z
+
∂ρ
∂T
(
∂T
∂z
+Γ)
Δz.
(1.12)
The quantity E = −a z /(g Δz) is defined as the static stability of the water
column and with (1.12) it follows (with ρ = ρ 1 )that
E = −
1
ρ
∂ρ
∂S
∂S
∂z
+
∂ρ
∂T
(
∂T
∂z
+Γ)
.
(1.13)
When E>0 (and hence a z < 0), then the water column is statically stable since
the force on the water parcel is in the opposite direction to the initial movement.
The water parcel will return towards its initial position and in fact, oscillatory
Ex. 1.4
motion can result, with the Brunt-V¨ ais¨ al¨ a (or buoyancy) frequency N defined by
N
2 = gE.
(1.14)
Va l u e s o f E in the upper 1000 m of the open ocean are in the range 10 −6 –10 −5
m −1 . Below this depth, values of E can decrease to 10 −7 m −1 in trenches; here
∂S/∂z is small and hence E → 0 implies that ∂T/∂z = −Γ. Thus the in-situ
temperature changes at great depth are mainly due to pressure changes.
The potential temperature is the in-situ temperature corrected for the adiabatic
temperature gradient Γ and by good approximation given by ϑ = T +Γz. Hence,
E = −
1
ρ
∂σ ϑ
∂z
,
(1.15)
where the potential density σ ϑ (kgm −3 )isdefinedas
σ ϑ = ρ(ϑ, S, 0) − 1000.
(1.16)
The concept of potential temperature is important to determine whether the water
column is stably stratified or not. When the relative density σ t as in (1.5) is used,
compressibility effects are completely neglected. The difference between σ t and
σ ϑ is illustrated with the data of the WOCE A16 section in Fig. 1.7d. The profile
of σ t would indicate that the water column is statically unstable at depth, but
σ ϑ correctly indicates that it is indeed statically stable: the in situ temperature
increase at depth is only due compressibility effects.
19
and
ρ
W
2 = ρ 1 +
∂ρ
∂z
W
|z=−z 1
Δz + ... = ρ 1 +
−Γ
∂ρ W
∂T
+
∂ρ W
∂p
∂p
∂z
z=−z1
Δz + ....
(1.11)
As the variations in T and S are small, ∂ρ W /∂p ≈ ∂ρ/∂p and we find that as
Δz → 0
a z =
g
ρ 1
∂ρ
∂S
∂S
∂z
+
∂ρ
∂T
(
∂T
∂z
+Γ)
Δz.
(1.12)
The quantity E = −a z /(g Δz) is defined as the static stability of the water
column and with (1.12) it follows (with ρ = ρ 1 )that
E = −
1
ρ
∂ρ
∂S
∂S
∂z
+
∂ρ
∂T
(
∂T
∂z
+Γ)
.
(1.13)
When E>0 (and hence a z < 0), then the water column is statically stable since
the force on the water parcel is in the opposite direction to the initial movement.
The water parcel will return towards its initial position and in fact, oscillatory
Ex. 1.4
motion can result, with the Brunt-V¨ ais¨ al¨ a (or buoyancy) frequency N defined by
N
2 = gE.
(1.14)
Va l u e s o f E in the upper 1000 m of the open ocean are in the range 10 −6 –10 −5
m −1 . Below this depth, values of E can decrease to 10 −7 m −1 in trenches; here
∂S/∂z is small and hence E → 0 implies that ∂T/∂z = −Γ. Thus the in-situ
temperature changes at great depth are mainly due to pressure changes.
The potential temperature is the in-situ temperature corrected for the adiabatic
temperature gradient Γ and by good approximation given by ϑ = T +Γz. Hence,
E = −
1
ρ
∂σ ϑ
∂z
,
(1.15)
where the potential density σ ϑ (kgm −3 )isdefinedas
σ ϑ = ρ(ϑ, S, 0) − 1000.
(1.16)
The concept of potential temperature is important to determine whether the water
column is stably stratified or not. When the relative density σ t as in (1.5) is used,
compressibility effects are completely neglected. The difference between σ t and
σ ϑ is illustrated with the data of the WOCE A16 section in Fig. 1.7d. The profile
of σ t would indicate that the water column is statically unstable at depth, but
σ ϑ correctly indicates that it is indeed statically stable: the in situ temperature
increase at depth is only due compressibility effects.