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

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).

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)

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

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.

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

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

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).

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

Note: σt = σ(S, T) Similarly, , , ,

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 10000 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.

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.

(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

Buoyancy Frequency and ocean stratification

Buoyancy frequency, an example