Chapter 3: Properties of Fluids Thermodynamic Properties of Air If two samples of air can exist in contact with each other without a change in properties, the two samples are said to have the same state. What properties define the atmospheric state?

- Temperature - Pressure - Density - Water vapor content - Other chemical consituents

Equation of state – an equation that relates properties of state to one another

Ideal gas law:

p – pressure T – absolute temperature r - density R – gas constant R=R*/M, where R* is the universal gas constant (=8.314 J mol-1 K-1) and M is the molecular mass of the gas

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pρT

= R, a constant

Composition of the Atmosphere How does the composition of the atmosphere vary in space? The Dry Atmosphere What are the constituents of the dry atmosphere?

Constituent

Molecular Weight

Fractional Concentration by Volume of Dry Air

Nitrogen (N2) 28.013 78.08% Oxygen (O2) 32.000 20.95% Argon (Ar) 39.95 0.93%

Carbon Dioxide (CO2) 44.01 380 ppm Neon (Ne) 20.18 18 ppm

Helium (He) 4.00 5 ppm Methane (CH4) 16.04 1.75 ppm

Krypton (Kr) 83.80 1 ppm Hydrogen (H2) 2.02 0.5 ppm

Nitrous oxide (N2O) 44.013 0.3 ppm Ozone (O3) 48.00 0-0.1 ppm

Water vapor is present in the atmosphere in varying concentrations from 0 to 5%. Each gas that makes up the atmosphere obeys the ideal gas law:

Partial pressure – the pressure that would be exerted by a single constituent in a mixture of gases if that constituent were present alone and occupied the same volume as the mixture of gases Dalton’s Law – the total pressure exerted by a mixture of gases that do not interact chemically is equal to the sum of the partial pressures of each constituent

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pn = ρn RnT

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p =T ρnn∑ Rn

Molecular mass for dry air: Md = 28.966 g mol-1

Gas constant for dry air:

Ideal gas law for dry air: Example: Calculate the density from the observed pressure and temperature at the ATOC weather station https://sundowner.colorado.edu/weather/atoc8/ The Moist Atmosphere Why do we care about water in the atmosphere?

- Water exists in all three phases (solid, liquid and vapor) in the atmosphere

- Energy (latent heat) is associated with water phase changes - Unlike most other atmospheric consituents water varies in space and

time - Water is what forms clouds and precipitation

Humidity variables Dew point temperature (Td): The temperature to which air must be cooled at constant pressure in order for the air to become saturated with respect to a plane surface of pure water Vapor pressure (e): The partial pressure exerted by water vapor molecules in a given volume of the atmosphere Specific humidity (q): Mass of water vapor contained in a unit mass of air Mixing ratio (r): The mass of water vapor contained in a unit mass of dry air

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Rd =

ρnR*

Mnn∑

ρ=

ρn Rnn∑

ρ=R*

Md= 287.04 Jkg−1K−1

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p = ρRdT

Converting between humidity variables

𝑒 = 𝑒!𝑒𝑥𝑝 %𝑏(𝑇" − 𝑇#)𝑇" − 𝑇$

+

where e0 = 6.112 mb, b = 17.67, T1 = 273.15 K, T2 = 29.65 K 𝑟 = !"

#$"≈ !"

# 𝑞 = !"

#

Application of the ideal gas law to a moist atmosphere: Virtual temperature

The ideal gas law for moist air: Virtual temperature (Tv): The temperature dry air would need to have if it were to have the same density as a sample of moist air at the same pressure

𝑇% = ,(1 − 𝑞) + 𝑞𝑅%𝑅"1𝑇 = (1 + 0.61𝑞)𝑇

How does the density of dry air compare to the density of moist air with the same temperature and pressure? What does this imply about the relationship between temperature and virtual temperature? Additional details about moisture variables can be found in Chapter 11.

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p = (ρd Rd + ρv Rv )T

= ρRdρdρ

+ρvρRvRd

#

$ %

&

' ( T

= ρRd (1− q) + q RvRd

#

$ %

&

' ( T

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p = ρRdTv

Static Stability Air parcel – a small mass of air which we can follow through the atmosphere First law of thermodynamics: %&

%'= 𝑐#

%(%'− 𝛼 %#

%' or 𝑑𝑄 = 𝑐&𝑑𝑇 − 𝛼𝑑𝑝

Adiabatic process – a process in which an air parcel exchanges no heat (Q) with its surroundings (environment) In an adiabatic process dQ = 0 Can T and p change for an adiabatic process? Potential temperature (q) – the temperature an air parcel would have if brought to a reference pressure (p0) (usually 1000 hPa) through an adiabatic process Potential temperature is conserved (i.e. it does not change) if an air parcel does not exchange heat with its environment!!! Derivation of the equation for potential temperature – conservation of heat Combine the ideal gas law and first law of thermodynamics to give:

Using dQ=0 in an adiabatic process we can integrate this equation:

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dQ = cpdT −RdTp

dp

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0 =cpRdT

dT − 1pdp

C =cpRdlnT − ln p

Use T=q for p=p0 to find C:

This gives us an equation for potential temperature:

Example: Calculate the potential temperature from the observed temperature and pressure at the ATOC weather station How does the pressure, temperature, and potential temperature of an air parcel change if the air parcel is lifted adiabatically in the atmosphere? Adiabatic cooling and adiabatic warming Example: Adiabatic lifting of air from Boulder to Leadville (p = 700 mb)

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C =cpRdlnθ − ln p0

cpRdlnθ − ln p0 =

cpRdlnT − ln p

ln θ cp Rd

T cp Rd

$

% &

'

( ) = ln p0

p$

% &

'

( )

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θ =T p0p

#

$ %

&

' (

Rd cp

Forces acting on an air parcel that is displaced vertically: Start with an air parcel with volume V at height z. Assume that this air parcel has the same temperature as its environment (=Tz). For both the air parcel and the environment at height z:

How does the temperature and potential temperature of the air parcel change as it is lifted adiabatically from height z to height z+dz?

The temperature of the environment at height z+dz is:

What force does this air parcel experience once it is displaced?

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Tz = θzpzp0

#

$ %

&

' (

Rd cp

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Tparcel,z+dz = θzpz+dz

p0

#

$ %

&

' (

Rd cp

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Tz+dz = θz+dzpz+dz

p0

#

$ %

&

' (

Rd cp

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F =

weight ofair displaced

"

# $

%

& ' −

weight ofair parcel

"

# $

%

& '

mass of parcel( )

=gρz+dzV − gρ parcel,z+dzV

ρ parcel,z+dzV

We can rewrite the expression for this force using the ideal gas law and noting that V of the air parcel and displaced air are equal.

Noting that qparcel,z+dz=qz:

Brunt-Väisälä frequency (N): the frequency at which an air parcel will oscillate if displaced vertically and acted upon by the restoring force arising from the buoyancy of the parcel (also known as the buoyancy frequency) What property of the environment determines the sign and magnitude of the buoyancy force? The vertical gradient of potential temperature 9"'

"(: determines the sign and

magnitude of the buoyancy force.

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F = g

pRdTz+dz

"

# $

%

& ' −

pRdTparcel,z+dz

"

# $

%

& '

pRdTparcel,z+dz

"

# $

%

& '

= g

1Tz+dz

−1

Tparcel,z+dz

1Tparcel,z+dz

= gTparcel,z+dz −Tz+dz

Tz+dz

= gθ parcel,z+dz −θz+dz

θz+dz

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F = g θz −θz+dz

θz+dz

$

% &

'

( )

= −gθdθdzdz

= −N 2dz

Dry adiabatic process – an adiabatic process in which no phase change of water occurs A dry adiabatic process does not refer to process in which no water vapor is present, only one in which the water vapor that is present does not undergo any phase changes!!!! What happens to an air parcel that is displaced vertically by a dry adiabatic process if:

Static stability classes

Stable Unstable Neutral

What happens when water vapor changes phase in an air parcel? Moist adiabatic process – an adiabatic process in which water vapor changes phase in an air parcel Moist adiabatic processes are discussed in greater detail in Chapter 11.

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dθ dz = 0

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dθ dz > 0

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dθ dz < 0

The Continuity Equation Continuity – atmospheric mass is not created or destroyed Divergence – air expands (flows outwards) Convergence – air compresses For an air parcel with volume dxdydz and a spatially varying density r the mass of air is given by:

How can we express the time rate of change of mass in this volume of air?

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Mair = ρdxdydz∫∫∫

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∂Mair

∂t=∂∂t

ρdxdydz∫∫∫ = (ρu) x −∫∫ (ρu) x+dx dydz

+ (ρv)y −∫∫ (ρv)y+dy dxdz

+ (ρw)z −∫∫ (ρw)z+dz dxdy

This can be rewritten to give the flux form of the continuity equation:

The continuity equation is a conservation equation for atmospheric mass. We can also express the continuity equation as:

What do terms (1), (2), and (3) represent physically?

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∂∂t

ρdxdydz∫∫∫ = −∂(ρu)∂x∫∫∫ +

∂(ρv)∂y

+∂(ρw)∂z

dxdydz

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∂ρ∂t

+∂(ρu)∂x

+∂(ρv)∂y

+∂(ρw)∂z

= 0

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∂ρ∂t

+ u∂ρ∂x

+ v ∂ρ∂y

+ w ∂ρ∂z

$

% &

'

( ) + ρ

∂u∂x

+∂v∂y

+∂w∂z

$

% &

'

( ) = 0

(1) (2) (3)

If density does not vary in time then:

If in addition density does not vary in space:

This relationship is known as Dines’ compensation, and requires that divergence at one location in a column of the atmosphere is exactly balanced by convergence at another location in the column.

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∂ρ∂t

= 0

⇒∂(ρu)∂x

+∂(ρv)∂y

+∂(ρw)∂z

= 0

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ρ∂u∂x

+∂v∂y

+∂w∂z

$

% &

'

( ) = 0

⇒∂u∂x

+∂v∂y

= −∂w∂z