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=
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
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