Atmosphere and ocean differences
Lesson 2
Atmosphere & ocean properties:
equations of state and density
Fluids are physically different
Review: differences b/t ocean &
atmosphere, equations of state
for atmosphere & ocean
Density
Equation of state
Density (ρ) is defined as the mass per unit volume:
mass
ρ=
Volume
With units:
.
g or
cm 3
General form:
Density is related to the specific volume, α, which is the
volume per unit mass
3
With units: cm or
g
The equation of state - The functional relationship between
the absolute pressure, specific volume and temperature for a
homogenous substance.
kg
m3
α=
Water is incompressible (atm is assumed
incompressible for many applications)
Ocean has no analogy for “latent heat” source that
atm has
Oceans are laterally bounded (by land)
Atm circulation largely forced by convection – heating
from below
Ocean circulation forced by convection – heat loss
from above – and by wind stress (no analogy in atm)
.
f ( p, α , T ) = 0
1
ρ
3
m
kg
1
Ideal Gas law – Moist
atmosphere
Ideal Gas law
The equation of state – For relatively small pressure values (on the
order of (1 atm)), there exists an analytic relationship for the
equation of state:
Due to Dalton’s law of partial pressures, we can express an ideal
gas law for the molecular constituents of the atmosphere
p = ρRT
R=
p = ρR d T *
R*
m
Where
And m is the molecular mass with units grams/mole
.
*
3
And R = 8.3143 × 10
o
Joules
Kelvin ⋅ kg ⋅ mole
2
Where Rd = 2.8705 × 10
.
is the ideal gas constant.
This equation is only valid under the conditions that:
Molecules exhibit no potential forces between each other. They only
interact through elastics collisions like billiard balls.
2. Molecules occupy no volume. They are all treated as “point”
masses
T* is the virtual temperature and is defined as the temperature a parcel of
dry air would have if its pressure and density were those of the parcel of
moist air. If W is the mixing ratio in units of gm/kg then
1.
Equation of state for the ocean
Clearly, the requirements of the ideal gas law break down in
the ocean and therefore we are stuck with the general form
of the equation of state:
f ( p, α , T ) = 0
.
Joules
is the ideal gas constant for dry air
kg ⋅o Kelvin
In practice, we are interested in examining how the density
(specific volume) of the ocean varies with Pressure,
Temperature and Salinity.
Density increases with a(n):
- increase in Salinity and Pressure
- decrease in Temperature
T* ≈T +
W
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General properties of ocean
Seawater is nearly incompressible (more than
fresh water or air), thus seawater density
depends (mostly) on salinity & temperature
More dense
Less dense
Cold water
Warm water
Salty water Fresh(er) water
Density units in kg m-3
2
3.
(from Pinet, 1998)
What are factors that control the growth and structure of the Thermocline?
4
(NORTHERN HEMISPHERE)
(from Pinet, 1998)
SALINITY
General definition - The mass of salt per unit mass of
seawater - measured in parts per thousand– It is very difficult
to actually find the amount of salt or the amount of water at a
given Temperature and pressure – Evaporation alone is not
sufficient.
Practical Salinity: Salinity measured in accordance with the
Practical Salinity Scale (conductivity scale as compared to a
standard KCL solution) and has Practical Salinity units
(PSUs).
4
(NORTHERN HEMISPHERE)
(from Pinet, 1998)
3
Major Constituents
high
temperate
subtropical
The concentrations of these
major constituents are
conservative. They are
unaffected by most biological
and chemical processes.
7.
(from Pinet, 1998)
What is the “halocline”?
What factors control the growth and structure of the halocline?
11.
(from Pinet, 1998)
(from the Navy Coastal Ocean Model)
Why is the Atlantic saltier (at the surface) than the other global oceans?
4
Effect of pressure on density
In-situ versus laboratory
θ = T − ∫ Γdp
Theoretical analysis of the ocean is important but is of little use
without the experimental results to accompany it.
There are many ways of obtaining the thermodynamics properties of
the ocean. These can broken into two main categories:
.
o
Γ=
Data off
Antarctica
Cp ≈ 4 × 104 J /(kg oC )
Γ is the
adiabatic
lapse rate
In-situ – measuring the quantities of interest at depth in the ocean.
Often, the use of a CTD can be used for these type measurements.
Laboratory – Measuring the properties of interest by taking a sample
at depth and bringing it to the surface. One must then account for
variations in pressure and, to a lesser degree, temperature, in
transporting the fluid parcel. Many chemical properties of sea-water
require this type of measurement. –
potential
in-situ
~0.1º change at 1000 m
~0.3º change at 3000 m
One must be careful though since neglect of pressure may mislead the
observer into indications of an instability when actually does not exist.
North Atlantic Deep Water (with higher potential density) over
Antarctic Bottom Water (with lesser potential density)
Γ can also be obtained from Unesco
Technical Papers in Marine Science # 44 by
Fofonoff and Millard, Unesco 1983
Ocean density representation
Ocean density representation
Given that we know density depends on salinity, temperature and pressure,
we can approximate density as a linear function of its state variables using a
Taylor series. If we expand about reference values, To, So and po we obtain
Density is actually a non-linear function of T,S and p so we must use
different techniques to get an accurate representation of density over
a wide range of state variable values
ρ (T , S , p ) ≈ ρ o (1 − α T (T − To ) + β S (S − S o ) + K ( p − p o ))
The international equation of state for seawater is a widely accepted
empirical technique to obtain accurate values of density to order 10-5.
Where
αT = −
.
βS =
K=
C
∂T
g
=
≈ 1 − 2 × 10− 4
∂p C p
db
1 ∂ρ
- Thermal expansion coefficient
ρ ∂T
1 ∂ρ
- Haline contraction coefficient
ρ ∂S
1 ∂ρ
= - Isothermal compressibility coefficient
ρ ∂p
The above representation is only useful locally (with little change in the
domain or parameters)
.
The international equation of state for seawater is not static (fixed),
and note that the Intergovernmental Oceanographic Commission has
approved a modification, TEOS-10 http://www.teos-10.org/
An amazingly detailed (179 pg) brief is hosted by the Australian government
(Commonwealth Scientific and Industrial Research Organisation) at:
http://www.marine.csiro.au/~jackett/TEOS-10/TEOS-10_Manual_08Jan2010.pdf
5
International equation of state
for seawater
International equation of state
for seawater
The international equation of state for seawater calculates density by
finding the secant bulk modulus, K, which is the slope of a segment on
a graph of density versus fractional rate of change of volume.
Its value is K = α o p = ρ p
αo − α
Solving for density in the previous equation we obtain
ρ − ρo
ρ (S , T , p ) =
ρ o (S , T , po )
1−
p
K (S , T , p )
p
.
.
K=
p o = 0,
αo
ρ
p=
p
αo − α
ρ − ρo
αo −α
=0
αo
αo − α
αo
International equation of state
for seawater
K (S , T , p ) =
There are other more recent techniques for finding the density of
seawater.
−2
+ 148.4206T − 2.327105T 2
999.842594 + 6.793952 × 10 T
+ 1.360477 × 10 − 2 T 3 − 5.155288× 10 −5 T 4
− 9.095290 × 10 −3 T 2 + 1.001685 × 10 −4 T 3
+ 3.239908 p + 1.43713× 10 −3 Tp
−4
−7
+ 1.16092 × 10 T p − 5.77905 × 10 T p
2
3
+ 8.50935 × 10 −5 p 2 − 6.12293 × 10 −6 Tp 2
.
Representing the density of
seawater
ρ (S , T ,0 ) =
19652.21
+ 8.24493 × 10 −1 S − 4.0899 × 10 −3 TS
+ 7.6438 × 10 −5 T 2 S − 8.2467 × 10 −7 T 3 S
+ 54.6746S − 0.603459TS
+ 5.3875 × 10 −9 T 4 S − 5.72466 × 10 −3 S 3 / 2
+ 1.09987 × 10 − 2 T 2 S − 6.1670 × 10 −5 T 3 S
+ 7.944 × 10 − 2 S 3 / 2 + 1.6483 × 10 − 2 TS 3 / 2
− 5.3009 × 10 T S
2
3/ 2
−3
+ 2.2838 × 10 pS
Some recent approaches are by McDougall et al (2002) or Feistel
(2003)
− 1.120083 × 10 −6 T 4 + 6.536332 × 10 −9 T 5
+ 5.2787 × 10 −8 T 2 p 2
−4
K and ρο=0 are then empirically fit to different values of salinity,
temperature and pressure. The accepted expansion of K and ρο are . .
+ 1.0227 × 10 − 4 TS 3 / 2 − 1.6546 × 10 − 6 T 2 S 3 / 2
+ 4.8314 × 10 − 4 S 2
.
McDougall T.J., Jackett D. R., Wright D. G., Feistel R., 2002, Accurate
and computationally efficient formulae for potential temperature and
density. J. Atmos. Ocean. Tech., 20, 730-741
Feistel R., 2003. A new extended thermodynamic potential of
seawater. Prog. Oceanography, 36, 249-347.
− 1.0981× 10 −5 TpS − 1.6078 × 10 −6 T 2 pS
+ 1.91075 × 10 −4 pS 3 / 2 − 9.9348 × 10 −7 p 2 S
+ 2.0816 × 10 −8 Tp 2 S + 9.1697 × 10 −10 T 2 p 2 S
6
Tables
A good java program that allows use to find the
thermodynamic properties of the ocean is found . . .
here
.
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