First and Second Laws of Thermodynamics:dpTddhαη+=Where h=cpT i s enthal ,py _ i s entropy.

Since spTT,⎟⎠⎞⎜⎝⎛∂∂=αα , dpTTdTcTdspp ,⎟⎠⎞⎜⎝⎛∂∂−=αηFor an adiabatic process _=0d , dpTTdpTTdTc spspp ,2, ⎟⎠⎞⎜⎝⎛∂∂−=⎟⎠⎞⎜⎝⎛∂∂= ρραGiven t he coefficien tof therma l expansi on as spT T,1⎟⎠⎞⎜⎝⎛∂∂−=ρραT he adiabati c lapse rate ραρρη pTspad cTTTpT =⎟⎠⎞⎜⎝⎛∂∂−=⎟⎟⎠⎞⎜⎜⎝⎛∂∂=Γ ,2Or pTad cTgpTgzT αρηη =⎟⎟⎠⎞⎜⎜⎝⎛∂∂=⎟⎠⎞⎜⎝⎛∂∂−=ΓFor the atmosphere (idea l gas), TT1=α, padcg=ΓFor the ocean, ),,(pTsTTαα=

Thermobaric Effect

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

∫Γ+=r

o

ooooorooo

p

pdpppptSStpptS )),,,,(,(),,,( ϑθ

p

T

spp c

T

Tc

T

ραρ

ρ=⎟

⎠

⎞⎜⎝

⎛∂∂

−=Γ,

2where

αT is thermal expansion coefficientsp

T T,

1 ⎟⎠⎞⎜

⎝⎛

∂∂−= − ρρα T is absolute

temperature (oK)

A proximate formula:

2BpApt −−=θ

( )[ ]35035.0185.0104.0 −++= StA

( )3010075.0 tB −=

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 smallθ≈T-0.5oC for 5km

An example of vertical profiles of temperature, salinity and density

θ and σθ in deep ocean

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

Definitionsin-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:

∫Γ+=r

o

ooooorooo

p

pdpppptSStpptS )),,,,(,(),,,( ϑθ

Potential density: σθ=ρs,θ,0 – 1000

Simplest consideration: light on top of heavyStable:

0<∂∂

zρ

Unstable:

0>∂∂

zρ

Neutral:

0=∂∂

zρ

(This criteria is not accurate, effects of compressibility (p, T) is not counted).

ρ′ , S, T+δT, p+δp) and the

Static stability

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 (

environment (ρ2, S2, T2, p+δp).

Buoyant force (Archimedes’ principle):

gVgVgVF )( 22 ρρδρδρδ ′−=′−=

Acceleration:

ρρρ

ρδρρδ

′′−

=′′−

== 22 )(g

V

gV

M

Faz

For the parcel: ( )zg

Cp

pδρρδ

ρρρ

θ

−+=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+=′

2

1

(where zgp δρδ −= or gzp ρ−=∂∂ is the hydrostatic equation

2

1

Cp=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

θ

ρ, C is the speed of sound)

where (δV, parcel’s volume)

zC

g

zga z δρ

ρ ⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

= 2

1

zzδρρρ

∂∂

+=2

For environment:

Then

⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ +∂∂

=−

⎟⎠⎞

⎜⎝⎛ −−

∂∂+

=z

Cg

zCg

zg

zCg

zCg

zz

gazδρ

δρρ

δρ

ρ

δρ

ρδρ

ρ

2

2

2

2

1

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

Static Stability:

2

1

C

g

zzg

aE z −

∂∂

−=−=ρ

ρδStable: E>0Unstable: E<0Neutral: E=0 ( 0

12=−

∂∂

−Cg

zρ

ρ , 02p

Cg

zρρ

−=∂∂

Therefore, in a neutral ocean, 0<∂∂

zρ .

θθθ

ρρρρρ⎟⎠

⎞⎜⎝

⎛∂∂

=∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=−zz

p

ppg

C

g2

E > 0 means, θ

ρρ⎟⎠

⎞⎜⎝

⎛∂∂

<∂∂

zz

)

Since

A stable layer should have vertical density lapse rate larger then the adiabatic gradient.

Note both values are negative

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

( )pTS ,,ρρ =

zz

p

pz

T

Tz

S

Szzz δρρρρδρ ⎥

⎦

⎤⎢⎣

⎡∂∂

∂∂

+∂∂

∂∂

+∂∂

∂∂

+=+ )()(

( ) zz

p

pz

T

Tz δ

ρρρρ

ϑθ⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛∂

∂

∂

∂+⎟

⎠

⎞⎜⎝

⎛∂

∂

∂

∂+=′

Since

For environment,

For the parcel,

zz

T

Tz

S

Sδρρρρ ⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ Γ+∂∂

∂∂

+∂∂

∂∂

=′−

, Г adiabatic lapse rate,

Then

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ Γ+∂∂

∂∂

+∂∂

∂∂

−=z

T

Tz

S

SE

ρρ

ρ

1

Since gz

p

z

p ρθ

−=∂∂

=⎟⎠

⎞⎜⎝

⎛∂∂

m-1

and Γ−=⎟⎠

⎞⎜⎝

⎛∂∂

θz

T

• 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/1000mAt 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

ptSptpStptS ,,,,,, 10001000 εεεσσρ ++++=+=

Note: σt = σ(S, T)

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ Γ+∂∂

∂

∂+

∂∂

∂

∂+

∂∂

−=

⎥⎦

⎤⎢⎣

⎡Γ

∂

∂+

∂∂

∂

∂+

∂∂

∂

∂+Γ

∂∂

+∂∂

−=

⎥⎦

⎤⎢⎣

⎡Γ

∂

∂+Γ

∂∂

+∂∂

∂

∂+

∂∂

∂

∂+

∂∂

∂∂

+∂∂

∂∂

−=

z

T

Tz

S

Sz

Tz

T

Tz

S

STz

TTz

T

Tz

S

Sz

T

Tz

S

SE

ptpS

ptptpStt

pttptpStt

,,

,,,

,,,

1

1

1

εεσ

ρ

εεεσσ

ρ

εσεεσσ

ρ

θ

Similarly, pTSpSpTTS ,,,,,

1 δδδρ

α +++Δ==

,SS ∂

∂−=

∂∂ ρ

ρα

α11

, TT ∂∂

−=∂∂ ρ

ρα

α11

,

⎥⎦

⎤⎢⎣

⎡Γ

∂

∂+

∂∂

∂

∂+

∂∂

∂

∂+Γ

∂Δ∂

+∂Δ∂

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡Γ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂Δ∂

+∂∂

∂

∂+

∂∂

∂

∂+

∂∂

∂Δ∂

+∂∂

∂Δ∂

=

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ Γ+∂∂

∂∂

+∂∂

∂∂

=⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ Γ+∂∂

∂∂

+∂∂

∂∂

−=

Tz

S

Sz

T

TTz

TTz

T

Tz

S

Sz

T

Tz

S

S

z

T

Tz

S

Sz

T

Tz

S

SE

pTpSpTTSTS

pTTSpTpSTSTS

,,,,,

,,,,,,

1

1

11

δδδ

α

δδδ

α

αα

α

ρρ

ρ

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ Γ+∂∂

∂∂

+∂∂

∂∂

−=z

T

Tz

S

SE

ρρ

ρ

1