Pat Arnott Chapter 3: Atmospheric Thermodynamics Objectives: 1.Demonstrate quantities used by Atmospheric Scientists to relate properties of air parcels.

Pat Arnott

Chapter 3: Atmospheric Thermodynamics

Objectives:

1. Demonstrate quantities used by Atmospheric Scientists to relate properties of air parcels aloft with those at the surface.

2. Develop increasingly more accurate models for the temperature, pressure, and density of air in the atmosphere.

3. Stability of air parcels.

Should be very familiar with these topics as we cover this chapter:a. Ideal gas equation applied to dry and moist air. SEE THIS AWESOME SIMULATION! b. Virtual temperature.c. Potential temperature. d. Hydrostatic equation.e. Increasingly detailed description of the temperature and pressure distribution in the atmosphere.f. SkewT logP diagrams.z. Relative humidity, absolute humidity. g. Dew point temperature.h. Wet bulb temperature.i. Equivalent potential temperature.j. Latent heat release and absorption in condensation and evaporation of water.k. Stability of air parcels.l. Indices on soundings.

Pat Arnott

Equation of State for an Ideal Gas: Air

V = volume

1. molecule size is ignorable.2. molecules don’t interact (attract or repel each

other).3. molecular collisions are like hard point like

spheres.

Most primitive, intuitive form of the I.G.L. (ideal gas law):PV = NkT

Now we manipulate to find a satisfying form of the I.G.L for analysis:

P = pressureN = # moleculesT = absolute temperature (Kelvin)k = Boltzmann’s constant = 1.38 x 10-23 Joules / (molecule K)

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Various Equivalent Forms of the I.G.L.

Note the useful bottom line form P=RT: We will use this most often.

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Partial Pressure and Ideal Gas Mixtures

EACH GAS SEPARATELY OBEYS THE IDEAL GAS LAW.

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Applications of Dalton’s Law of Partial Pressures…

What is the total pressure in the room?

What is the partial pressure due to nitrogen molecules N2?

What is the partial pressure due to oxygen molecules, O2?

What is the partial pressure due to carbon dioxide molecules, CO2?

Wait a minute… how can it be that these molecules apply pressure according to their number concentration? They don’t all have the same mass… What is going on?

Dalton's law is not exactly followed by real gases. Those deviations are considerably large at high pressures. In such conditions, the volume occupied by the molecules can become significant compared to the free space between them. Moreover, the short average distances between molecules raises the intensity of intermolecular forces between gas molecules enough to substantially change the pressure exerted by them. Neither of those effects are considered by the ideal gas model.

The fine print from Wikipedia…

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Applications of Dalton’s Law of Partial Pressures…

What is the total pressure in the room? 860 mb on 9/9/2010.

What is the partial pressure due to nitrogen molecules N2? 860 mb * 0.78 = 670 mb. Air is composed of 78% N2 molecules.

What is the partial pressure due to oxygen molecules, O2?860 mb * 0.21 = 180 mb.

What is the partial pressure due to carbon dioxide molecules, CO2?860 mb * 0.000385 = 0.34 mb.

For 10 mb water vapor partial pressure, air is about 1% water vapor.

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Kinetic Theory of Pressure (Wikipedia…)

Box of sides L

m vx

Nature is fair …On average, molecules share the burden of random kinetic energy, also known as heat. K.E.=mv2/2. On average, molecules with smaller m move faster than large m molecules.

Pressure in the kinetic theory of gases …

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Special Case: Partial Pressure of Water Vapor, e

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Virtual Temperature Tv.dry air moist air

same for bothT = temperatureP = pressureV = volumeN = # molecules

P = PDP = PD+e

Total pressure=partial pressuredue to dry air.

Total pressure=partial pressuredue to dry air + water vapor.

dry air > moist air

TWEAK … Raise the temperature of the dry air on the left to lower its density so that it is the same as the density of the moist air on the right. We have to let some of the molecules out of the box.

This raised temperature is the virtual temperature by definition. It is a useful construct because the I.G.L. for dry or moist air is written P = RD Tv .

PRT

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Virtual Temperature Tv Calculation.dry air moist air

same for bothT = temperatureP = pressureV = volume

P = PDP = PD+e

Total pressure=partial pressuredue to dry air.

Total pressure=partial pressuredue to dry air + water vapor.

Crank up the temperature of box 1, keeping pressure and volume constant (let some dry air molecules leak out), until the mass (density) of box 1 is the same as that of box 2.

PRT

box 1 box 2

From the I.G.L.

TvT

General Note:Tv>T.

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Virtual Temperature Example

Let e=10 mb

P=1000 mbT=280 K

Remember=0.622

Then Tv ≈ T [1+e(1-)/P] = T(1+0.0038) ≈ 281 K

This gives us a rough idea of the temperature increase needed to make dry air have the same density as the moist air described above.

(binomial expansion was used to show an equivalent form)

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Potential Temperature

Adiabatic compressional warming (and cooling by expansion) lends itself to predicting large-scale weather patterns, because air motions in large weather systems are, for all practical purposes, generally adiabatic in nature.

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sustainable only with diabatic heating!

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LCL: Lifting condensation level.LFC: Level of free convection.EL: Equilibrium level.CAPE: Convective available potential energy.CIN: Convective inhibition.

Solar heating, surface convergence promote parcels to the LFC: Must pass above the inversion in the CIN area.

CAPE (J/kg): 0-1000 (small) 1000-2500 (moderate) 2500-4000 (large) > 4000 (extreme).

CIN (J/kg): 0 to -25 (weak) -25 to -50 (moderate)- 50 to -100 (strong convective inhibition)

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Winds on Skew T Log P Charts

wind is 15 knotscoming from the northeast.

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Moist Adiabats: Trajectories of Saturated Parcels. Pseudoadiabats because products of condensation may fall out of the parcel as precipitation. Trace a few of them below.

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The Saturated Adiabatic Lapse Rate

Temperature Changes and Stability Inside Clouds

from

http://geog-www.sbs.ohio-state.edu/courses/G230/hobgood/ASP230Lecture17.ppt

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Temperature Changes Inside Clouds

Two processes occur simultaneously inside clouds that affect the temperature.

(1) Rising air expands, does work and cools;

(2) Condensation releases latent energy which is then stored as internal energy and warms the air inside the cloud.

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Temperature Changes Inside Clouds (Cont.)

Normally, the cooling due to the work of expansion is greater than the warming associated with the release of latent energy and its conversion to internal energy.

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Temperature Changes Inside Clouds (Cont.)

Thus, as air rises inside a cloud it still gets colder, but it does so at a slower rate than the Dry Adiabatic Lapse rate.

The rate at which rising air inside a cloud cools is called the Saturated Adiabatic Lapse Rate (SALR).

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The Saturated Adiabatic Lapse Rate (SALR)

The derivation of the equation for the SALR begins with a form of the First Law of Thermodynamics

dq = cpdT - αdp

What would we do here to get the dry adiabatic lapse rate?

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The Saturted Adiabatic Lapse Rate (Cont.)

In this case the energy gained, dq, is equal to the latent energy released when water vapor condenses inside the cloud.

dq = -Lvdws

where

Lv is the latent heat of vaporization, and

dws is the change of specific humidity of the air parcel when water vapor condenses

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The SALR (Cont.)

Substitute for dq in the First Law of thermodynamics to get

-Lvdws = cpdT – αdp

Add –cpdT + Lvdws to both sides to get

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The SALR (Cont.)

-cpdT = -αdp + Lvdws

Divide by cpdz to get

-cpdT = -αdp + Lvdws

cpdz cpdz cpdz

Since α = 1/ρ we can write this as

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The SALR (Cont.)

-dT = - 1 dp + Lvdws

dz cpρ dz cpdz

From the hydrostatic approximation

-1 dp = g

ρ dz

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The SALR (Cont.)

Substitution results in

-dT = g + Lvdws = Γs

dz cp cp dz

SALR =

cooling due to work of expansion

+warming due to latent energy released during condensation

Γdry

Γs = Γdry + Lv dws

cp dz

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The SALR (Cont.)

The saturated adiabatic lapse rate is always less than the dry adiabatic lapse rate because the cooling caused by adiabatic expansion is partially offset by the release of latent energy during condensation.

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The SALR (Cont.)

The saturated adiabatic lapse rate is quite variable!!

The magnitude of the SALR is determined by the amount of water vapor that condenses.

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The SALR (Cont.)

When warm moist air rises in a cloud, more water vapor condenses and the SALR is smaller.

When cooler, drier air rises inside a cloud, less water vapor condenses and the SALR is larger.

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cooler drier air warmer moister air

SALR = 0.9°C/100m

less water vapor condenses

more water vapor condenses

SALR = 0.5°C/ 100 m

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Unsaturated air rises and cools at the DALR

Lifting Condensation Level (LCL)

Saturated air rises inside the cloud and the release of latent energy during condensation causes it to cool at the SALR

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Stability Cases for Clouds(1) When the environmental lapse rate (ELR) is greater than the SALR,

then the air inside the cloud is unstable. Unstable air moves vertically and we tend to get tall, vertical clouds like cumulus and cumulonimbus.

z

ELRSALR T(z)

O

Tlifted > TenvTenv

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Stability Cases for Clouds (Cont.)

cumulus

cumumlo-nimbus

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Stability Cases for Clouds (Cont.)(2) When the ELR is equal to the SALR, then the air inside the cloud is

neutral.

(3) When the ELR is less than the SALR, then the air inside the cloud is stable.

z

ELR SALR T(z)

O

Tlifted < TenvTlifted

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Stability Cases for Clouds (Cont.)

There is much less vertical motion when air is neutral or stable. Thus, when air inside the clouds is neutral or stable, the clouds tend to have a flat, layered appearance. These types of layered clouds are called stratus clouds.

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Stability Cases for Clouds (Cont.)

stratus

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Stability Cases for Clouds (Cont.)

There is a special stability case that occurs when the Environmental Lapse Rate is between the Dry Adiabatic Lapse Rate and the Saturated Adiabatic Lapse Rate.

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Stability Cases for Clouds (Cont.)

For example, what if

DALR = 1.00°C/100 m

ELR = 0.75°C/100 m

SALR = 0.50°C/100 m

If the air is unsaturated ELR < DALR and the air is stable, but if the air is saturated, then ELR > SALR and the air is unstable.

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Stability Cases for Clouds (Cont.)

This special case is called conditionally unstable, because the air must be lifted until it becomes saturated in order for it to become unstable

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0000 m T = 28°C, Td = 20°C Tenv = 28°C

LCL = 800 mT = 20C°, Td = 20°C Tenv = 22°C

Air is stable

1600 m T = 16°C, Td = 16°C Tenv = 16°C

Air is neutral

2000 m T = 14°C, Td = 14°C Tenv = 13°C

Air is unstable

DALR = 1°C/100 m

ELR = 0.75°C/100 m

SALR = 0.5°C/100 m

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Stability Cases in Clouds (Cont.)

When the atmosphere is conditionally unstable it can lead to the rapid development of thunderstorms when a cold front or other weather feature lifts warm moist air in the spring.

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Stability Cases in Clouds (Cont.)

The air is stable as long as it isn’t lifted high enough, but if it is lifted until the parcel is warmer than the environment, then the air instantaneously becomes unstable and starts rising on its own.

Then thunderstorms can form rapidly.

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The Effect of Topography on Precipitation Patterns

Precipitation patterns in mountainous regions tend to be closely related to the prevailing wind direction.

Much higher precipitation amounts fall on the side of the mountains where the air is rising and it is much drier on the side where the air is sinking.

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The Effect of Topography on Precipitation Patterns (Cont.)

The process where air is forced to rise up the side of a mountain is sometimes called orographic lifting.

Pat Arnott0 meters

T = 26°C, Td = 16°CAir rises and cools at DALR

T = 16°C, Td = 16°C LCL 1000 m

Air rises and cools at SALR = 0.5°C/ 100 m

3000 mT = 6°C, Td = 6°C

Air sinks and warms at the DALR = 1°C/100 m

500 mT = 31°C, Td = 6°C

Air is warm and dry

Windward side Leeward side

wind direction

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The Effect of Topography on Precipitation Patterns (Cont.)

Rising motion causes clouds and precipitation on the windward side of the mountain range.

Sinking motion causes warm, dry conditions on the leeward side of the mountain.

The dry area on the leeward side of the mountain is called the rain shadow.

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The Effect of Topography on Precipitation Patterns (Cont.)

Since the prevailing wind direction in the middle latitudes is from the west, the western sides of the mountains along the west coast of the U.S. are the rainy sides and the rain shadows occur along the eastern slopes of the mountains.

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Find the LCL for a surface parcel: find Tw, the wet bulb temperature.

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Evaporative Cooler: Swamp Cooler.

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NASA satellite image (MODIS imager on board the Terra satellite) of a wave cloud forming off of Amsterdam Island in the far southern Indian Ocean. Image taken on December 19, 2005.

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