Lecture 3: Thermodynamics Review, Part 1

The first and second lawsOcean thermodynamicsA simple climate model

Lecture 3:Thermodynamics Review, Part 1

Jonathon S. Wright

[email protected]

7 March 2017


The first and second lawsThe first law: energy is conservedThe second law: adiabatic processesPotential temperature

Ocean thermodynamicsOverviewDiurnal and seasonal variations in SSTVertical stability

A simple climate modelA one-box oceanA coupled two-box oceanLinearized formulation


The first law: energy is conservedThe second law: adiabatic processesPotential temperature

δQ

a fluid parcel

expansion or contraction

pdV = pd(ρ−1)

heat loss or gain

cvdT

δQ = cvdT + pd(ρ−1) = cpdT + ρ−1dp

Energy conservation: the first law



ds ≡ δQT = cp

dTT −Rd

dpp = cpdln(Tp

−Rd/cp)

Adiabatic processes: the second law

I No exchange of energy across the system boundary (δQ = 0)

I Isentropic (i.e., the entropy s is conserved)

I Reversible: the initial state can be retrieved



ds = cpd ln(Tp

− Rd/cp)= 0

θ = T(pp0

)Rd/cp

Potential temperature

Move an air parcel instantaneously from low pressure to high pressure



θ = T(pp0

)Rd/cp

I conserved for adiabaticmotion

I distribution depends ondetails of circulation

I atmosphere or ocean

I reference levels vary

90°S 60°S 30°S 0° 30°N 60°N 90°N1000

900800

700

600

500

400

300

200

100

50

Pre

ssur

e [h

Pa]

270

270

280

280

290

290

300 300

310

320

330 340 350

360380

400

450

500

600

800

1000

Zonal mean potential temperature

250

275

300

325

350

375

400

425

450

475

500

Pot

entia

l tem

pera

ture

[K]

data from CFSR

Potential temperature

http://cfs.ncep.noaa.gov/cfsr/


OverviewDiurnal and seasonal variations in SSTVertical stability

90°S 60°S 30°S 0° 30°N 60°N 90°N

0

1000

2000

3000

4000

5000D

epth

[m]

0°C

0°C

0°C

0°C

2°C

2°C 4°C

6°C

8°C

10°C 12°C

14°C16°C18°C

20°C22°C 24°C26°CZonal mean potential temperature

2

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

Pot

entia

l tem

pera

ture

[°C

]

data from ORAS4

High specific heatI second largest of all liquids

I due to hydrogen bonds in water

Large thermal inertiaI large mass × high specific heat

I small temperature range (∼ 30◦C)

I key for long-term climate stability

SalinityI freezing point, specific heat, and

latent heat of vaporization decreasewith increasing salinity

http://www.ecmwf.int/en/research/climate-reanalysis/ocean-reanalysis



1 0 1 2Difference from SSTfoundation [°C]

interface

~10µm

~1mm

~1m

~10m

Dep

th

(a) Nighttime

SSTinterface

SSTskin

SSTsubskin

SSTdepth

SSTfoundation

1 0 1 2Difference from SSTfoundation [°C]

(b) Daytime

after Kawai et al. 2007

Weak diurnal cycle

I only about 0.2–0.6◦C on average

I 1–2◦C heating on calm, sunny days, butlimited to top ∼1 m

I slight cooling at the interface at night

Latent heat fluxI important flux of energy from ocean to

atmosphere

I top millimeter of the ocean surface layer isslightly cooler because of evaporation at theocean–atmosphere interface

Sea surface temperatureI vertical gradients — SST is not just one value

http://dx.doi.org/10.1007/s10872-007-0063-0



28 29 30 31 32 33 34Temperature [°C]

1

2

3

4

5

6

7

8

Calm conditions

28 29 30 31 32 33 34Temperature [°C]

1

2

3

4

5

6

7

8

Moderate winds

28 29 30 31 32 33 34Temperature [°C]

1

2

3

4

5

6

7

8

Strong winds

28.0 28.5 29.0 29.5 30.0 30.5 31.00

2

4

6

8

10

12

Win

d sp

eed

[m s−

1 ]

45 cm depth

28.0 28.5 29.0 29.5 30.0 30.5 31.0Temperature [°C]

0

2

4

6

8

10

12

Win

d sp

eed

[m s−

1 ]

7 m depth

data from TOGA COARE

Wind influences the diurnal cyclewindy conditions can mix diurnal warming deeper into the surface layer, but reduce the magnitude ofwarming at the air–sea interface

http://uop.whoi.edu/archives/coare/coare.html



data from ORAS4

Weak seasonal cycleI stronger than diurnal cycle, but still

weaker than land areas at similarlatitudes

I annual maximum is reduced anddelayed at greater depths

I annual range of sea surfacetemperature at 40◦N in the Pacificis ∼10◦C, much less than Beijing,which is also near 40◦N

I mixed layer depth in North Pacificis deepest during winter andshallowest during summer




ρθ

z

Stable

ρθ

z

UnstableBuoyancy Force

g(ρenv − ρeq)

g[ρeq +

(∂ρ∂z

)δz − ρeq

]

g(∂ρ∂z

)δz

ρeq > ρenv

ρeq < ρenv

ρeq < ρenv

ρeq > ρenv



Buoyancy Force

g(ρenv − ρeq)

g[ρeq +

(∂ρ∂z

)δz − ρeq

]

g(∂ρ∂z

)δz

Buoyancy Oscillations

acceleration: ρeqdwdt = g

(∂ρ∂z

)δz

d2

dt2δz =

(gρeq

∂ρ∂z

)δz

d2zdt2

= −N2z

N2 = − gρeq

∂ρ∂z

N ≡ Brunt–Vaisala frequency



N2 = − gρeq

∂ρ∂z

I exist when N is real (ρ decreaseswith height/increases with depthd = −z)

I frequency depends on verticalgradient of density

I higher frequencies in upperocean thermocline (∼10 min)

I lower frequencies in deep ocean(∼2 days)

I is σt the best stability metric?

20 22 24 26 28 30 32 34 36 38 40

Density [kg m−3]

0

500

1000

1500

2000

Dep

th [m

]

Annual mean density

NH high latitudes (60-90°N)NH mid-latitudes (30-60°N)NH subtropics (15-30°N)Tropics (15°S-15°N)SH subtropics (15-30°SSH mid-latitudes (30-60°S)SH high latitudes (60-90°S)

data from ORAS4

Gravity waves




N2 = − gρθ

∂ρθ∂z 20 21 22 23 24 25 26 27 28 29 30

Density [kg m−3]

0

500

1000

1500

2000

Dep

th [m

]

Annual mean potential density

NH high latitudes (60-90°N)NH mid-latitudes (30-60°N)NH subtropics (15-30°N)Tropics (15°S-15°N)SH subtropics (15-30°SSH mid-latitudes (30-60°S)SH high latitudes (60-90°S)

data from ORAS4

Potential density (σθ)

I the density water would have ifit were raised adiabatically to thesurface

I accounts for vertical variations inpressure

I conserved for adiabatic motion

I stable: increases with depth

I neutral: constant with depth

I unstable: decreases with depth




30 31 32 33 34 35 36 37 38Salinity

0

5

10

15

20

25

30

Tem

pera

ture

[°C

]

19

20

21

22

23

24

25 26 27

28

29 30

19

20

21

22

23

24

25

26

27 28

29 30

Density at surface pressure

NH high latitudes (60-90°N)NH mid-latitudes (30-60°N)NH tropics (0-30°N)

SH high latitudes (60-90°S)SH mid-latitudes (30-60°S)SH tropics (0-30°S

Labrador SeaWeddell SeaGlobal mean σθ at ~2000m

linearized estimate:

ρ = ρ0 (1− α(T − T0) + β(S − S0))

α = 2× 10−4 kg m−3 K−1

β = 7.6× 10−4 kg m−3 psu−1

data from ORAS4

DensityI depends on T , S and p

I can usually neglect variationsin pressure at surface

I salinity effects dominate atcolder temperatures

I mixing creates denser water




(a) Temperature (b) Salinity

270 276 282 288 294 300Temperature [K]

30 32 34 36 38 40Salinity [psu]

data from ORAS4

Surface density depends on temperature and salinity

Mixing creates denser waters, especially where gradients in T and S are large.




(a) Temperature

NADW

AABW

(b) Salinity

NADW

AABW

270 276 282 288 294 300Temperature [K]

30 32 34 36 38 40Salinity [psu]

data from ORAS4

Surface density depends on temperature and salinity

Mixing creates denser waters, especially where gradients in T and S are large.




data from ORAS4

Convective instability

I occurs when N2 is negative

I convective mixing is effectiveand efficient — unstableconditions are rarely observed

I depth of convection depends onvertical gradient of σθ

Ocean deep convectionI only a few regions where surfaceσθ seasonally approaches that inthe deeper ocean

I deep convection in these regionsdrives the global overturningthermohaline circulation



A one-box oceanA coupled two-box oceanLinearized formulation

S(α)

OLR(T, ε)

D µ, TµdTdt = S(α)−OLR(T, ε)

One-box ocean modelI a planet with a well-mixed ocean

and atmosphere

I solar radiation balanced byoutgoing long-wave radiation

I effective atmospherictransmission ε

I ocean heat capacity µ = ρcpD



0 20 40 60 80 100 120Time [years]

287.5

288.0

288.5

289.0

289.5

290.0

Tem

pera

ture

One box model

70m depth

10∼20 years to reach equilibrium

One-box ocean modelSuppose ε decreases (stronger greenhouse effect)...



0 20 40 60 80 100 120Time [years]

287.5

288.0

288.5

289.0

289.5

290.0

Tem

pera

ture

One box model

70m depth500m depth

10∼20 years to reach equilibrium

∼100 years to reach equilibrium

One-box ocean modelSuppose ε decreases (stronger greenhouse effect)...



S(α)

OLR(Tm, ε)

Dm

Do

µm, Tm

µo, To

µmdTmdt = S(α)− κ(Tm − To)−OLR(T, ε)

µodTodt = κ(Tm − To)

Two-box ocean modelI surface mixed layer coupled to

deep ocean by diffusion

I NOT a realistic representation ofthe coupling between the surfaceand deep ocean

I still a useful way of exploring therole of the deep ocean in climatesensitivity



0 500 1000 1500 2000 2500 3000Time [years]

287.5

288.0

288.5

289.0

289.5

290.0

Tem

pera

ture

Two box model

surface layerdeep ocean

thousands of years to reach equilibrium...

Two-box ocean modelSuppose ε decreases...



µd(Teq+T ′)

dt = µdT′

dt = S(α)−OLR(Teq)−(∂OLR∂T

∣∣Teq

)T ′ + · · ·

≈ −βT ′, where β = ∂OLR∂T

∣∣Teq

T ′(t) = T ′0 exp (− t/τ)

T ′0 = T0 − Teq = −∆Fβ , with the initial forcing ∆F = S(α)−OLR(T0)

Linearized formulation for one-box modelTaylor series around Teq, with T ′(t) = T (t)− Teq:

This linear differential equation has the exact solution:

with τ = µ/β and T ′0 approximated by:



0 20 40 60 80 100 120Time [years]

287.5

288.0

288.5

289.0

289.5

290.0

Tem

pera

ture

One box model

500m depth500m depth (linear solution)

One-box ocean modelFull numerical and linearized solutions are almost identical



µmdT ′mdt = −βT ′m − κ(T ′m − T ′o)

µodT ′odt = κ(T ′m − T ′o)

Linearized formulation for two-box modelThe linearized form can also be applied in the two-box case:

Lecture 3: Thermodynamics Review, Part 1

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