1. Course Schedule Day(s) Time Course Instructor(s) Monday/ Wednesday
12:30pm-
1:45pm
Predictability of Weather and Climate
Straus/
Krishnamurthy
Tuesday
10am-
12:30pm
Numerical Methods
Schopf
Tuesday/
Thursday
1:15pm-
2:30pm
Statistical Methods in Climate Research
DelSole
Introduction to Dynamic Meteorology
Schneider

2. Temperature Measurement
Sea Surface temperature (SST)
Bucket-sample (mercury thermometer)
Radiation thermometer
Subsurface temperature
Nansen 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

3. Salinity measurement method
Knudsen (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

4. conductivity-temperature-depth probe
In situ CTD precision:
ΔS~±0.005
ΔT ±0.005K Δz~±0.15%×z
The vertical resolution is high
CTD sensors should be calibrated (with bottle samples)

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

7. 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.
kg/m3 (depth 0~10,000m)
• ρ increases with p (the greatest effect)
ignoring p effect:
ρ~ kg/m3
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

8. 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)

9. The Equation of the State
The 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.
Bulk modulus K=1/β, β=compressibility.
, C speed of the sound in sea water.

10. where ρ0=1027 kg/m3, T0=10oC, S0=35 psu,
Simple formula:
(1)
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
(2)
where
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

11. Relation between (T,S) and σt
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
The temperature of density maximum is the red line and the freezing point is the light blue line

12. Specific volume and anomaly
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
( x 10-8 m3/kg or centiliter per ton, cL/t)

13. , m3/kg For 23 ≤ σt ≤ 28, Converting formula for ΔS,T and σt :
Since α(35,0,0)= x10-3 m3/kg,
, m3/kg
For 23 ≤ σt ≤ 28,
, accurate to 0.1
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

14. 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
where
T is absolute temperature (oK)
αT is thermal expansion coefficient

15. A proximate formula: 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 small
θ≈T-0.5oC for 5km

16. An example of vertical profiles of temperature, salinity and density

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

18. 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).

19. 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)

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

21. 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.

22. 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

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

24. 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).

25. Note: σt = σ(S, T)
Similarly, , , ,

26. 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 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.