1. Definitions 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

2. 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).

3. 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)

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

5. 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.

6. 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

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

8. 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).

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

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

11. Г terms: 2 x 10-8 m-1 (near surface: 4 x 10-8 m-1)
(∂δS,p/∂S) is much smaller than (∂ΔS,T/∂S) (10% at 5000 m and 15% at m, opposite signs)
(∂δT,p/∂T) has the same sign as (∂ΔS,T/∂T), relatively small about 2000m, comparable below).
First approximation,
, or
(reliable if the calculated E > 50 x 10-8 m-1)
A better approximation,
(σθ ,takes into account the adiabatic change of T with pressure)
When the depth is far from the surface, σ4=σS,θ,4(p=40,000kPa=4000dbar) may be used to replace σθ.

13. In practice, E≈ -25 ~ -50 x 10-8 m-1 in upper 50 m.
a) mixed layer is slightly unstable
subtropics, increase of salinity due to evaporation, vertical overturning occurs when E ~ -16 to -64x10-8 m-1, because of the effects of heat conduction, friction, eddy diffusion, etc.
b) observational errors
Error of σt ~ 5 x 10-3
The error of σt at two depths Δσt ~ 10-2
for Δz=z1-z2=20 m.

14. (Brunt-Väisälä frequency, N)
Buoyancy frequency
(Brunt-Väisälä frequency, N)
We have known that
, (depending on δz, restoring force)
Since δz is the vertical displacement of the parcel, then or
Its solution has the form
where
(radians/s)2
N is the maximum frequency of internal waves in water of stability E.
Period:
E=1000 x 10-8 m-1, τ=10 min
E=100 x 10-8 m-1, τ=33 min
E=1 x 10-8 m-1, τ≈6hr

15. Buoyancy Frequency and ocean stratification

16. Buoyancy frequency, an example