2. Thermobaric Effect

4. Potential temperature
In situ temperature is not a conservative property in the ocean.  
Changes in pressure do work on a fluid parcel and changes its internal energy (or temperature)         compression => warming         expansion => cooling
The change of temperature due to pressure work can be accounted for
Potential Temperature: The temperature a parcel would have if moved adiabatically (i.e., without exchange of heat with surroundings) to a reference pressure.
If a water-parcel of properties (So, to, po) is moved adiabatically (also without change of salinity) to reference pressure pr, its temperature will be
     
Γ Adiabatic lapse rate:  vertical temperature gradient for fluid with constant θ
When pr=0, θ=θ(So,to,po,0)=θ(So,to,po) is potential temperature.
At the surface, θ=T. Below surface, θ<T.
Potential density: σθ=ρS,θ,0 – 1000
where
T is absolute temperature (oK)
αT is thermal expansion coefficient

6. A proximate formula: t in oC, S in psu, p in “dynamic km”
For 30≤S≤40, -2≤T≤30, p≤ 6km, θ-T good to about 6%
(except for some shallow values with tiny θ-T)
In general, difference between θ and T is small
θ≈T-0.5oC for 5km

7. An example of vertical profiles of temperature, salinity and density

8. θ and σθ in deep ocean
Note that temperature increases in very deep ocean due to high compressibility

9. Definitions Potential density: σθ=ρs,θ,0 – 1000
in-situ density anomaly:
σs,t,p = ρ – 1000 kg/m3
Atmospheric-pressure density anomaly :
σt = σs,t,0= ρs,t,0 – 1000 kg/m3
Specific volume anomaly:
δ= αs, t, p – α35, 0, p
δ = δs + δt + δs,t + δs,p + δt,p + δs,t,p
Thermosteric anomaly: Δs,t = δs + δt + δs, t
Potential Temperature:
Potential density: σθ=ρs,θ,0 – 1000

10. Static stability Simplest consideration:
light on top of heavy
Stable:
Moving a fluid parcel (ρ, S, T, p) from depth -z, downward adiabatically (with no heat exchange with its surroundings) and without salt exchange to depth -(z+δz), its property is (
, S, T+δT, p+δp) and the
Unstable:
environment (ρ2, S2, T2, p+δp).
Neutral:
(This criteria is not accurate, effects of compressibility (p, T) is not counted).

11. Buoyant force (Archimedes’ principle):
where (δV, parcel’s volume)
Acceleration:
For the parcel:
is the hydrostatic equation
(where or
, C is the speed of sound)

12. For environment:
Then
For small δz (i.e., (δz)2 and higher terms are negligible),

13. Therefore, in a neutral ocean,
Static Stability:
Stable: E>0
Unstable: E<0
Neutral: E=0 ( ) ,
Therefore, in a neutral ocean, .
Since
E > 0 means,
Note both values are negative
A stable layer should have vertical density lapse rate larger then the adiabatic gradient.

14. A Potential Problem:
E is the difference of two large numbers and hard to estimate accurately this way.
g/C2 ≈ 400 x 10-8 m-1
Typical values of E in open ocean:
Upper 1000 m, E~ 100 – 1000x10-8 m-1
Below 1000 m, E~ 100x10-8 m-1
Deep trench, E~ 1x10-8 m-1

15. Simplification of the stability expression
Since
For environment,
For the parcel,
Since
and
, Г adiabatic lapse rate,
Then
m-1

16. The effect of the pressure on the stability, which is a large number, is canceled out.
(the vertical gradient of in situ density is not an efficient measure of stability).
In deep trench ∂S/∂z ~ 0, then E→0 means ∂T/∂z~ -Г
(The in situ temperature change with depth is close to adiabatic rate due to change of pressure).
At 5000 m, Г~ 0.14oC/1000m
At 9000 m, Г~ 0.19oC/1000m
At neutral condition, ∂T/∂z = -Г < 0.
(in situ temperature increases with depth).

17. θ and σθ in deep ocean
Note that temperature increases in very deep ocean due to high compressibility

18. Note: σt = σ(S, T)
Similarly, , , ,