Day(s) Time Course Instructor(s)

Monday/Wednesday

12:30pm-1:45pm

Predictability of Weather and Climate

Straus/Krishnamurthy

Tuesday10am-12:30pm Numerical Methods Schopf

Tuesday/Thursday

1:15pm-2:30pm

Statistical Methods in Climate Research DelSole

Thursday10am-12:30pm

Introduction to Dynamic Meteorology Schneider

Course Schedule

Temperature MeasurementSea Surface temperature (SST)• Bucket-sample (mercury thermometer) • Radiation thermometer

Subsurface temperatureNansen bottle• Protected reversing thermometer (±0.02K in routine use) • in situ pressure with unprotected reversing thermometer (±0.5% or ±5 m)• only a finite number (<25) of vertical points once(Mechanical) Bathythermograph (MBT) • Continuous temperature against depth (range, 60, 140 or 270 m) • Need calibration, T less accurate than thermometer (±0.2K, ±2 m)Expendable bathythermograph (XBT) • Expendeble thermister casing • dropped from ship of opportunity and circling aircraft• Graph of temperature against depth • Range of measurement: 200 to 800 m• depth is estimated from lapsed time and known falling rate

Salinity measurement methodKnudsen (Titration) method (precision ±0.02) • time consuming and not convenient on board ship• not accurate enough to identify deep ocean water mass

Electrical conductivity method (precision ±0.003~±0.001) • Conductivity depends on the number of dissolved ions per volume

(i.e. salinity) and the mobility of the ions (ie temperature and pressure). Its units are mS/cm (milli-Siemens per centimetre).

• Conductivity increases by the same amount with ΔS~0.01, ΔT~

0.01°C, and Δz~ 20 m. • The conductivity-density relation is closer than density-chlorinity• The density and conductivity is determined by the total weight of the

dissolved substance

conductivity-temperature-depth probe

In situ CTD precision: ΔS~±0.005ΔT ±0.005K Δz~±0.15%×z

The vertical resolution is high

CTD sensors should be calibrated (with bottle samples)

Subsurface driftersModern subsurface floats remain at depth for a period of time, come to the surface briefly to transmit their data to a satellite and return to their allocated depth. These floats can therefore be programmed for any depth and can also obtain temperature and salinity (CTD) data during their ascent. The most comprehensive array of such floats, known as Argo, began in the year 2000. Argo floats measure the temperature and salinity of the upper 2000 m of the ocean. This will allow continuous monitoring of the climate state of the ocean, with all data being relayed and made publicly available within hours after collection.

When the Argo programme is fully operational it will have 3,000 floats in the world ocean at any one time.

Density (ρ, kg/m3)• Determine the depth a water mass settles in equilibrium.• Determine the large scale circulation.• ρ changes in the ocean is small.

1020-1070 kg/m3 (depth 0~10,000m)• ρ increases with p (the greatest effect)

ignoring p effect: ρ~1020.0-1030.0 kg/m3 1027.7-1027.9 kg/m3 for 50% of ocean

• ρ increases with S. ρ decreases with T most of the time.

• ρ is usually not directly measured but determined from T, S, and p

Density anomaly σ

Since the first two digits of ρ never change, a new quantity is defined as

σs,t,p = ρ – 1000 kg/m3

called as “in-situ density anomaly”. (ρoo=1000 kg/m3 is for freshwater at 4oC)

Atmospheric-pressure density anomaly (Sigma-tee)

σt = σs,t,0= ρs,t,0 – 1000 kg/m

(note: s and t are in situ at the depth of measurement)

The Equation of the StateThe dependence of density ρ (or σ) on temperature T, salinity S and

pressure p is the Equation of State of Sea Water.

ρ=ρ(T, S, p) is determined by laboratory experiments.

International Equation of State (1980) is the most widely used density formula now.

• This equation uses T in °C, S from the Practical Salinity Scale and p in dbar (1 dbar = 10,000 pascal = 10,000 N m-2) and gives ρ in kg m3.

• Range: -2oC≤ T ≤ 40oC, 0 ≤ S ≤ 40, 0 ≤ p ≤ 105 kPa (depth, 0 to 10,000 m)

• Accuracy: 5 x 10-6 (relative to pure water, σt: ±0.005)• Polynomial expressions of ρ(S, t, 0) (15 terms) and K(S, t, p) (27

terms) get accuracy of 9 x 10-6.

),,(1

)0,,(),,(

ptSKp

tSptS

Bulk modulus K=1/β, β=compressibility.

1 KC , C speed of the sound in sea water.

Simple formula:(1)

kpSSbtta )()( 000 accuracy: ±0.5 kg/m3

where ρ0=1027 kg/m3, T0=10oC, S0=35 psu,

a=-0.15 kg/m3oC, b=0.78kg/m3, k=4.5x10-3 /dbar

TSPTTpTSppC )35)(,(),()()( 2048.0053.583.999 ppC

p0085.0808.0

))0683.01(068.0351.01(0708.0 Tpp ))064.01(012.0059.01(003.0 Tpp

For 30≤S≤40, -2≤T≤30, p≤ 6 km, good to 0.16 kg/m3

For 0 ≤S≤40, good to 0.3 kg/m3

(2)

where

Relation between (T,S) and σt

The temperature of density maximum is the red line and the freezing point is the light blue line

σt as a function of T and S • The relation is more nonlinear with respect to T • ρ is more uniform with S• ρ is more sensitive to S than T near freezing point • ρmax meets the freezing point at S =24.7

S < 24.7: after passing ρmax surface water becomes lighter and eventually freezes over if cooled further. The deep basins are filled with water of maximum density

S > 24.7: Convection always reaches the entire water body. Cooling is slowed down because a large amount of heat is stored in the water body

Specific volume and anomaly

Specific volume: α=1/ρ (unit m3/kg)

Specific volume anomaly:

δ= αs, t, p – α35, 0, p (usually positive)• δ = δs + δt + δs,t + δs,p + δt,p + δs,t,p

• In practice, δs,t,p is always small (ignored)• δs, p and δt, p are smaller than the first three terms

(5 to 15 x 10-8 m3/kg per 1000 m)Thermosteric anomaly:

ΔS,T = δs + δt + δs, t (50-100 x 10-8 m3/kg or 50-100 centiliter per ton, cL/t)

Converting formula for ΔS,T and σt :

Since α(35,0,0)=0.97266x10-3 m3/kg,

TStts

ts ,)0,0,35(1000

1

)0,,(

1)0,,(

3, 1097266.0

1000

1000

tTS , m3/kg

1.280105.0 , TSt

For 23 ≤ σt ≤ 28,

,26751.95, tTS accurate to 1 in cL/ton

For most part of the ocean, 25.5 ≤σt≤28.5. Correspondingly, 250 cL/ton ≥ ΔS,T ≥ -50 cL/ton

, accurate to 0.1

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

r

o

ooooorooo

p

pdpppptSStpptS )),,,,(,(),,,(

p

T

spp c

T

Tc

T

,

2where

αT is thermal expansion coefficientsp

T T,

1

T is absolute

temperature (oK)

A proximate formula:

2BpApt

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

Simplest consideration: light on top of heavyStable:

0

z

Unstable:

0

z

Neutral:

0

z

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

, 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

Acceleration:

22 )(

gV

gV

M

Faz

For the parcel: zg

Cp

p

2

1

(where zgp or gzp

is the hydrostatic equation

2

1

Cp

, C is the speed of sound)

where (δV, parcel’s volume)

zC

g

zga z

2

1

zz

2

For environment:

Then

z

Cg

zCg

zg

zCg

zCg

zz

gaz

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

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

12

C

g

z

, 0

2

C

g

z

Therefore, in a neutral ocean, 0

z .

zz

p

ppg

C

g2

E > 0 means,

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 ,,

zz

p

pz

T

Tz

S

Szzz

)()(

zz

p

pz

T

Tz

Since

For environment,

For the parcel,

zz

T

Tz

S

S

, Г adiabatic lapse rate,

Then

z

T

Tz

S

SE

1

Since gz

p

z

p

m-1

and

z

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).

ptSptpStptS ,,,,,, 10001000

Note: σt = σ(S, T)

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

Similarly, pTSpSpTTS ,,,,,

1

,SS

11

, TT

11

,

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

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,

zE t

1

, or z

E TS

,1

(reliable if the calculated E > 50 x 10-8 m-1)

zE

1

(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.

A better approximation,