Oceanography/Physics 2800 Climate Change Quiz 1folkins/tests.aphys1.pdf · Oceanography/Physics 2800 Climate Change Quiz 1 January 22 2001 1. How is methane removed from the atmosphere?

Oceanography/Physics 2800 Climate Change Quiz 1

January 22 2001

1. How is methane removed from the atmosphere? (main sink) (6 points)

1. Very roughly, what is the lifetime of a methane molecule in the atmosphere? (4 points)

2. What are two sources of methane to the atmosphere? (natural or human generated) (8

points)

3. How does increasing the concentration of CO2 in the atmosphere affect the outgoing

thermal (long wave) emission to space? (5 points)

4. Why does it have this effect? (8 points)

5. How is the radiative forcing of a molecule defined? (8 points)

6. Rank the following three molecules in terms of radiative forcing per molecule (largest

to smallest): CFC’s, CO2, methane. (3 points)

7. Plot how the radiative forcing of CO2 would depend on the level of CO2 in the atmo-

sphere (starting from an atmosphere with no CO2). (8 points)

8. How would you expect increasing CO2 concentrations to affect global mean precipita-

tion? (4 points)

9. Why would you expect it to have this effect? (8 points)

10. Plot how you would expect the 13C/12C ratio to depend on depth in the ocean. (8

points)

11. What is the main reason for this variation? (6 points)

12. What is the average temperature of the earth. (4 points)

13. What would the average temperature of the earth be if there were no greenhouse

effect? (4 points)

14. What is a “window” region? (5 points)

15. Where does most of the thermal emission to space from earth come from? (Hint: 1.

clouds 2. surface or 3. atmospheric greenhouse gases). (3 points)

16. Give two factors which would affect the temperature of a planet. (8 points)

Oceanography/Physics 4500/5500 A Atmospheric Physics Quiz 2

Oct. 25, 1999

1. (i) Compare the potential temperature of an air parcel at 500 mb over the north pole

with the potential temperature of an air parcel at 500 mb over the equator. Which air

parcel would you expect to have the higher potential temperature? Explain. (2 points)

(ii) Suppose an air parcel was moving southward toward the equator from mid-latitudes.

What vertical motion would you expect, in general, this air parcel to have. Explain. (2

points)

2. This question deals with how you would you expect the height of the lifting condensation

level (LCL) of an air parcel to depend on the initial relative humidity at the surface.

(i) Draw a vertical profile of how rc depends on height in the atmosphere. (3 points)

(ii) Show on this diagram how two air parcels with initially differing mass mixing ratios at

the surface must reach the LCL at different altitudes. (4 points)

3. Air containing water vapor will be less dense than dry air. What is the percentage

difference in density between dry air and saturated air at 25 oC? You can assume that the

pressure is 105Pa and that q ≈ r is a good approximation. (6 points)

4. Air is rising within a cloud (ie. RH = 100 %). The initial pressure is 630 hPa, and the

temperature is 285 K. The height is 4 km.

(i) What is rc? (3 points)

(ii) What is θe? (You can assume that q ≈ r) (3 points)

(iii) Suppose that the air parcel eventually exits the cloud in the upper tropical tro-

posphere at an altitude at which the temperature is so cold that it can be assumed that

rc ≈ 0. What will be the potential temperature of the air parcel when it exits the cloud?

You can assume that the air parcel ascends within the cloud in a pseudo-adiabatic manner

(ie. the only source of heat in the cloud is water vapor condensation). (2 points)


Nov. 17, 1999

1. An air parcel at the surface has a temperature of 300 K. The surface pressure is 1000

mb. The relative humidity is 58 %.

(i) What is water vapor mass mixing ratio r of the air parcel at the surface (in g H2O/kg

air)? (2 points)

(ii) Using the pseudo-adiabatic chart, what is the approximate value of equivalent potential

temperature θe of the air parcel? (2 points)

(iii) Using the pseudo-adiabatic chart, what is the approximate value of the dew point

temperature Td of the air parcel? (2 points)

(iv) What is the approximate Lifting Condensation Level (LCL) of the air parcel (in km)?

(2 points)

Suppose the above air parcel is placed in an atmosphere where the temperature decreases

with height according to T (z) = 290K − Γz, where the lapse rate Γ = 4K/km.

(v) Assume that as the air parcel rises in this environment, it does not mix with the

surrounding atmosphere. What will be the height of the air parcel when it reaches its level

of neutral bouyancy? (2 points)

(vi) Approximately what will be the water vapor mass mixing ratio r at the height of

neutral bouyancy? (2 points)

2. 100 J is used to raise a 1 kg air parcel against gravity in a location where the local

gravitational acceleration g = 9.75m/sec2.

(i) What is the change in geopotential height dZ of the air parcel? (2 points)

(ii) What is the change in geometric height dz of the air parcel? (2 points)

3. A satellite in free-floating geosychronous orbit about the equator remain above a fixed

point on the equator while keeping a fixed distance above the earth. How far above the

earth’s surface is the satellite? Ignore anisotropic contributions to the earth’s gravity. (4

points)

4. For each of the following statements, indicate whether this is characteristic of the

troposphere(T) or stratosphere(S) (half point each).

(i) Air parcels tend to absorb heat at a higher temperature and lose heat at a lower

temperature.

(ii) The circulation does net work on air parcels.

(iii) Upward vertical velocities are typically 1 mm/sec.

(iv) Upward motion is approximately adiabatic or pseudo-adiabatic.

(v) The continuous formation of positive bouyancy helps maintain the circulation against

dissipation.

(vi) During a complete cycle of the circulation, parcels absorb more heat than they give

off.

(vii) Radiative heating rates are typically 1 K/day.

(viii) The circulation is mechanically driven by the breaking of planetary and gravity

waves.

(ix) Temperature increases with height.

(x) The average temperature during the descending branch of the circulation is higher than

the temperature during the ascending branch of the circulation.

Chart question, temperature of 300 K gives ec = 36 mb, or rc = 0.00224. RH = 0.58

gives r = 0.0013. From the chart this would give thetae = 340 K. LCL = 825 mb.


Dec. 1, 1999

1. (i) What is the main absorber of SW energy in the stratosphere? (1 point)

(ii) What is the main emitter of LW energy in the stratosphere? (1 point)

(iii) What is the main emitter of LW energy in the troposphere? (1 point)

(iv) What are two source of convective instability in the troposphere? (ie. mechanisms

which tend to destabilize the troposphere.) (2 points)

(v) How would you find the Level of Free Convection (LFC) of an air parcel starting from

the surface? (2 points)

(vi) Vertical velocities inside clouds are rarely as large as those predicted by the available

CAPE (Convective Available Potential Energy). What is a reason for this? (2 points)

2. How does the stability (or lapse rate) change within an air mass as it subsides (de-

scends)? Explain using a diagram. (4 points)

3. An air parcel ascends inside a cloud from 800 mb to 400 mb. During the ascent, the

temperature inside the cloud is always 1 oC warmer than outside the cloud. Estimate the

vertical velocity of the air parcel in the cloud at 400 mb assuming it started from zero

velocity at 800 mb. (6 points)

4. The surface temperature is 295 K. There is a cloud with an emissivity of 0.4 and a

temperature of 235 K at 8 km.

(i) What is the upward LW flux from the surface? (2 points)

(ii) What is the upward LW flux from the cloud above the cloud? (2 points)

(iii) What is the total (from both surface and cloud) upward LW flux above the cloud?

Assume there is no absorption or emission of LW energy in the atmosphere other than

from the cloud and the surface. (2 points)

Oceanography/Physics 4500/5500 A Atmospheric Physics Exam

Dec. 14, 1999

1. Draw (very roughly) how you would expect the N2/O2 ratio to depend on height - ie.

the region of the atmosphere where you would expect this ratio to be constant, the altitude

at which it would start changing with height, and how it would change with height. (2

points)

2. Which cloud would you ordinarilly expect to reduce by a larger amount the escape of

longwave energy to space - a cloud at 3 km or a cloud at 10 km? Explain. (2 points)

3. What are two contributors to the earth’s albedo? (2 points)

4. How does the height of an isentropic surface depend on latitude in the troposphere?

Explain with a diagram. (2 points)

5. (i) Suppose an air parcel is lifted from the surface to 12 km. The LCL is at 4 km.

Draw in a very general way how the potential temperature θ, the equivalent potential

temperature θe, and the water vapor mass mixing ratio r, of the air parcel would vary

between the surface and 12 km. Indicate height regions in which θ, θe, and r are constant,

and those in which they are increasing/ decreasing. Suppose θ = 300K and θe = 350K at

the surface. (6 points)

(ii) If in the above question, the level of neutral buoyancy was at 12 km, what is the θe of

the background atmosphere at 12 km? (1 points)

(iii) In the absence of entrainment or friction, how would you expect the velocity of the

air parcel to vary between 4 km and 12 km? (1 point)

6. Where do the moist and adiabatic lapse rates differ from each other most strongly?

Where do they become very close to each other? What is the main reason for this variation?

(2 points)

7. g is not the same at the North Pole as at the equator. How does it differ, and give a

reason for this variation. (2 points)

8. Denote in a general way the regions in which radiative heating rates in the stratosphere

are positive (warming) and those in which the radiative heating rates are negative. How

is the distribution of the heating/cooling related to the Brewer Dobson circulation? (2

points)

9. If an air mass is conditionally stable, what can you infer about the lapse rate of the air

mass? (2 points)

10. How is the net emission of longwave energy to space by the earth (surface plus

atmosphere) expected to change as greenhouse gas concentrations increase? (2 points)

11. An air parcel can experience a buoyancy acceleration upwards or downwards, even if

it is at the same temperature as its surroundings, if it has a different humidity than the

surrounding atmosphere.

(i) Suppose an air parcel has a temperature of 20 oC, and a dew point temperature of 18oC. What is the specific humidity q of the air parcel? The air parcel is at a pressure of

800 mb. (5 points)

(ii) What is the density ρ′ of the air parcel? (5 points)

(iii) Suppose the surrounding atmosphere, also at 800 mb pressure and a temperature of 20oC, has a dew point temperature of 10 oC. What is the buoyancy acceleration experienced

by the air parcel (direction and magnitude)? (5 points)

12. Air at a pressure of 600 mb, a relative humidity of 80 %, and a temperature of 260 K,

descends adiabatically to 900 mb.

(i) What is the temperature at 900 mb? (5 points)

(ii) What is the relative humidity at 900 mb? (5 points)

13. Precipitation heats the atmosphere by condensational heating. The globally averaged

precipitation rate is about 100 cm/year. Convert this to a heating rate of the atmosphere

in units of W/m2/day. (Hint 1 W = 1 J/sec.) (5 points)

14. Refer to the pseudo-adiabatic chart to obtain approximate answers. The temperature

at the surface (1000 mb) is 290 K. The dew point temperature Td = 282 K.

(i) What is the approximate water vapor mass mixing ratio r (in g water/kg dry air)? (2

points)

(ii) What is the approximate equivalent potential temperature θe of the air parcel? (2

points)

(iii) What is the approximate Lifting Condensation Level (LCL) of the air parcel (in mb)?

(2 points)

15. An air parcel reaches its level of free convection at 2 km. At that height, the air parcel

is 2 oC warmer than the environment. As the air parcel rises, its temperature decreases at

the moist adiabatic rate of Γs ≈ 6.5K/km. The temperature of the atmosphere decreases

at a rate of 6 K/km. You can assume that pressure depends on height exponentially

according to p(z) = 1000exp(−z/H), where p is in mb and the scale height H = 7 km.

(i) What will be the level of neutral buoyancy (LNB) of the air parcel? (2 points)

(ii) Estimate the upward velocity of the air parcel at the LNB if the initial upward velocity

of the air parcel at 2 km is zero, and there is no entrainment. (8 points)

14. This question involves finding the long-wave emissivity of the stratosphere. Make the

following assumptions:

- The incoming solar energy is 1370 W/m2.

- The stratosphere absorbs 10 % of the incoming solar energy, ie. solar absorbtivity as =

0.1.

- the stratosphere is in radiative equilibrium (ie. absorbed solar and longwave energy =

emitted longwave energy.)

- The average temperature of the stratosphere is 265 K.

- The longwave radiation entering the stratosphere from the troposphere can be charac-

terized by a single temperature of 220 K.

- There is no shortwave (solar) energy entering the stratosphere from below.

- The stratosphere can be characterized by a single longwave emissivity ǫ.

- There is no longwave energy entering the stratosphere from above (space is too cold).

(i) Find the long wave emissivity ǫ of the atmosphere. (6 points)

(ii) As the concentration of CO2 in the stratosphere increases would you expect the emis-

sivity of the stratosphere to go up or down? (1 point)

(iii) From your answer to (i), how would expect the temperature of the stratosphere to

respond to an increase in CO2, provided all other factors were constant? (1 point)

Physics/Oceanography 4500/5500 A Atmospheric Physics Quiz 1

October 3, 2002

1. The surface pressure is 1000 mb. Appproximately what fraction of the total atmospheric

mass is located between the 600 mb and 400 mb pressure surfaces? (1 point)

2. The specific heat of most substances tends to increase with temperature. Explain. (2

points)

3. The earth is continuously losing hydrogen molecules from its atmosphere. The earth

does, however, still have a lot of hydrogen. What has protected the earth from losing too

much hydrogen? (1 point)

4. An air parcel is moving adiabatically from the equator toward the North Pole. In

general, will the air parcel be moving upward or downward? Explain. (2 points)

5. Would you expect an increase in the average temperature of the atmosphere to increase

or decrease the gravitational potential energy of the atmosphere? Explain. (1 point)

6. A 1 kg parcel of dry air is heated isobarically at a rate of Q = 2 J/sec for 1 hour. You

can assume the air is an ideal gas.

(i) What is the change in enthalpy H of the air parcel? (2 points)

(ii) What is the change in temperature of the air parcel? (2 points)

(iii) What is the change in internal energy U of the air parcel? (2 points)

(iv) What is the work done by the air parcel? (1 point)

7. An air parcel (assume dry and ideal) rises adiabatically in the atmosphere from 1000

mb at 300 K to 500 mb.

(i) What is the temperature of the air parcel at 500 mb? (2 points)

(ii) What is the work done on the atmosphere by the air parcel? (2 points)

8. The temperature at 10 km is 220 K. What would be the temperature of this air parcel

if it were brought down adiabatically to the surface? (2 points)


October 29, 2002

1. Two air parcels are mixed isobarically.

Air parcel 1 has a mass of M1 = 2 kg, a temperature T1 = 0 oC, a pressure p1 = 1000 mb,

and a relative humidity r1 = 0.

Air parcel 2 has a dry air mass of M2d = 5 kg, a temperature T2 = 30 oC, a pressure

p2 = 1000 mb, and a relative humidity r2 = 0.8.

(i) What is the water vapor pressure e2 of air parcel 2? (5 points)

(ii) What is the density of dry air in air parcel 2? (5 points)

(iii) What is the density of water vapor in air parcel 2? (5 points)

(iv) What is the volume of air parcel 2? (5 points)

(v) What is the heat capacity Cp2 of air parcel 2 (sum of contributions from dry air and

water vapor)? (10 points)

(vi) What is the temperature Tm of a mixture of air parcels 1 and 2? (10 points)

(vii) What are the number of moles of dry air and water vapor in air parcel 2? (5 points)

(viii) What is the relative humidity rm of the mixture? (10 points)

2. What type of temperature profile maximizes the entropy of the atmosphere? (5 points)

4. The enthalpy of 1 kg of dry air is increased by 2000 J. The pressure is kept constant at

1000 mb. The initial temperature of the air parcel is 273.15 K (0 oC). What is the change

in entropy of the air parcel? You can assume dry air behaves as an ideal gas. (20 points)


November 26, 2002

The Quiz will be marked out of a total of 65 points.

1. (i) How does moist pseudoadiabatic ascent differ from moist adiabatic ascent? (5

points)

(ii) Give one reason why the buoyancy of an air parcel rising moist pseudoadiabatically in

a cloud would be different from the buoyancy of an air parcel rising moist adiabatically in

a cloud. (5 points)

2. Does the inclusion of virtual temperature effects (i.e. consideration of the effect of the

presence of water vapor on the density of an air parcel) tend to make the atmosphere more

or less stable? Explain. (5 points)

3. Assume the saturated water vapor pressure over a flat surface is 5 mb. Plot how the

saturated water vapor pressure over a droplet would vary as a function of the size of the

droplet, as the radius of the droplet varies between 0 and 1 µm. (5 points)

4. (i) Why do wet clothes hanging out to dry on a clothesline feel cold? (5 points)

(ii) Would you expect the difference between the temperatures of the wet clothes and the

atmosphere to be larger under conditions of low or high relative humidity? Explain. (5

points)

5. An air parcel at the surface has a pressure p = 100 kPa (1000 mb), a temperature

T = 30 oC, and a water vapor mass mixing ratio of w = 13 g/kg.

(i) Suppose it is lifted dry adiabatically from the surface. What is its lifting condensation

level zLCL? Draw this adiabatic lifting on the chart. (5 points)

(ii) Use the chart to estimate the θe of the air parcel. (5 points)

(iii) Estimate the dew point temperature Td of the air parcel at the surface. (5 points)

(iv) Suppose this air parcel was transported within a cloud moist adiabatically to 50 kPa

from zLFC , and then sank dry adiabatically back to the surface. What would be its surface

temperature? (5 points)

(v) You should have found that the surface temperature in the previous question was

unrealistically high. What is the main process which prevents the occurrence of such high

surface temperatures? (5 points)

6. We discussed in class how the dissolution of various solutes in water can lower the

saturated vapor pressure over water. This should imply that the saturated water vapor

pressure over the ocean is less than the saturated vapor pressure over a freshwater lake at

the same temperature. What is the percentage reduction in es over the ocean due to the

presence of salt? (Hint: There is about 3.3 kg of NaCl for every 100 kg of water in the

ocean. The molecular weight of Na is approximately 23 g/mole, that of Cl 35 g/mole, and

that of water 18 g/mole.) (10 points)

Physics/Oceanography 4500/5500 A Atmospheric Physics Final Exam

December 5, 2002; Total marks : 76

1. Is globally averaged sea level pressure always exactly constant? Discuss one factor

which might make it change, and what type of seasonal, annual, or other variation in

mean sea level pressure it would give rise to. (2 points)

2. The specific heats of gases tend to increase slightly with temperature. Why? (2 points)

3. Would you expect global warming to increase or decrease the mean height of the 900

mb pressure surface? Explain. (2 points)

4. Suppose the 400 mb pressure surface is 1.6 km above the 500 mb pressure surface. Es-

timate the mean temperature of the atmosphere between the 400 mb and 500 mb surfaces.

(4 points)

5. (i) Plot (very roughly) the Maxwell-Boltzmann distribution of molecular speeds of a gas

at -20 oC and 20 oC. (i.e. plot the frequency of occurrence versus the speed.) (2 points)

(ii) We discussed in class how the M-B distribution can be thought of as a product of two

functions. What mainly determines the shape of the distribution at low speeds? (1 point)

(iii) What mainly determines the shape of the distribution at high speeds? (1 point)

6. Suppose a nonideal gas in which the molecules have attractive forces with one another

undergoes an adiabatic free expansion. Would you expect the temperature of the gas to

decrease, increase, or stay the same? Explain. What would be the effect on the internal

energy U of the gas. Explain. (2 points)

7. There is one container of dry air with a mass of 1 kg and a temperature of 20 oC, and

another container with a mass of 4 kg and a temperature of 30 oC. The containers are

allowed to come into thermal contact. What will be the temperature of both containers

when they come into thermal equilibrium. The volumes of the two containers are kept

fixed. (4 points)

8. 1 kg of water vapor is mixed with 2 kg of dry air. What is the heat capacity Cp of the

mixture? (4 points)

9. The temperature at the surface is 30 oC. The temperature then decreases with height

at a constant rate of -7 oC/km. Suppose you released a balloon from the surface which

was filled with warm air at a temperature of 50 oC. Estimate the level of neutral buoyancy

of the balloon (i.e. how high it would rise). You can assume that the pressure of the air

inside the balloon is always the same as the pressure of the outside air, and that no heat

is transferred across the surface of the balloon. You can also assume that the air inside

the balloon is dry, and ignore virtual temperature effects. (4 points)

10. A 1 kg parcel of dry air is heated isobarically at a rate of Q = 1 J/sec. You can

assume the gas is ideal.

(i) What is the rate of temperature change (dT/dt) of the air parcel? (4 points)

(ii) What is the rate of working W done by the air parcel? (4 points)

(iii) If a parcel of sea water is heated isobarically, would you expect the ratio of working

to heating (W/Q) to be larger or smaller than in the atmosphere? Explain. (2 points)

11. An air parcel at a temperature of -30 oC has a relative humidity with respect to water

of 80 % and a relative humidity with respect to ice of 120 %. Rank the following in order

of smallest to largest Gibbs free energies (i.e. a 1 for the smallest free energy and a 4 for

the largest free energy.). (2 points)

(i) gi, the free energy of the ice phase.

(ii) gw, the free energy of the liquid water phase.

(iii) gw,d, the free energy of a tiny water droplet

(iv) gv, the free energy of the water vapor phase

12. Estimate the typical variation in boiling point temperature associated with the day

to day variation in surface pressure due to weather systems. Assume a typical low has a

minimum surface pressure of 960 mb, while a typical high has a maximum surface pressure

of 1040 mb. (4 points)

13. In the upper troposphere, where the temperature is below 0 oC, one often observes air

parcels which are supersaturated with respect to ice (i.e. a relative humidity with respect

to ice larger than 100 %). Why? (2 points)

14. Suppose you had a well mixed boundary layer extending up from the surface to 2 km.

Make a very rough plot of how you might expect potential temperature θ and water vapor

mixing ratio w to vary from the surface to 3 km. (4 points)

15. The background atmosphere has the following temperature profile: the surface tem-

perature is 30 oC; the atmosphere is dry adiabatic to 40 kPa (400 mb). Above 40 kPa, the

temperature of the atmosphere assumes a constant value equal to the temperature at 40

kPa.

(i) An air parcel at the surface has a temperature of 23 oC, and a relative humidity of 45

%. Estimate the water vapor mixing ratio w of the air parcel. (4 points)

(ii) Use the skew-T diagram to estimate the lifting condensation level pressure of the air

parcel (pLCL). (2 points)

(iii) Use the skew-T diagram to estimate the dew point temperature of the air parcel (when

it is at the surface). (2 points)

(iv) Use the skew-T diagram to estimate the equivalent potential temperature of the air

parcel. (2 points)

(v) Use the skew-T diagram to estimate the level of free convection pressure of the air

parcel (pLFC). (2 points)

(vi) Use the skew-T diagram to estimate the level of neutral buoyancy pressure of the air

parcel (pLNB). (2 points)

(vii) Use the skew-T diagram to estimate the convective inhibition energy (CINE) of the air

parcel. I am not asking for an exact result, only a graphical estimate that would be correct

to within a factor of two. In the definition of CINE, you can ignore virtual temperature

effects, (i.e. treat T and Tv as the same). Also, show graphically the region corresponding

to CINE in the skew-T diagram. Note: The problem here is one in which one is releasing

an air parcel at the surface into a background atmosphere. The surface air parcel has a

temperature which is distinct from that of the background atmosphere. (4 points)

(viii) Use the skew-T diagram to estimate the convective available potential energy (CAPE)

of the air parcel. I am not asking for an exact result, only a graphical estimate that would

be correct to within a factor of two. In the definition of CAPE, you can ignore virtual

temperature effects, (i.e. treat T and Tv as the same). (Remember dlnp = dp/p) Also,

show graphically the region corresponding to CAPE in the skew-T diagram. (4 points)

(ix) Assuming that the velocity of the air parcel at the level of free convection is zero, and

that it rises along a moist adiabat without mixing from the level of free convection to the

level of neutral buoyancy, what would you expect the upward velocity of the air parcel to

be at the level of neutral buoyancy? (4 points)


September 25, 2003

1. A 1 kg parcel of dry air is heated isobarically (constant pressure) at a rate of Q = 200

J/sec. You can assume that air is an ideal gas.

(i) What is the rate of change of enthalpy dH/dt of the air parcel? (10 points)

(ii) What is rate of change of temperature dT/dt of the air parcel? (10 points)

(iii) What is rate of change of internal energy dU/dt of the air parcel? (10 points)

(iv) What is rate of working W of the air parcel? (10 points)

(v) Suppose the same air parcel were heated with the same Q at constant volume. What

would be the rate of change of temperature of the air parcel? (10 points)

2. An air parcel at the surface (p = 1000 hPa) has a partial water vapor pressure of e = 4

hPa. What is the ideal gas constant < R > of this air parcel? (15 points)

3. Suppose that the earth was oriented wth respect to the sun so that the north pole was

always directly pointed at the sun, while the south pole was in perpetual 24 hour darkness.

The surface pressure is the same everywhere at 1000 hPa.

(i) Plot how you would expect the height of the 800 mb surface to vary in going from the

North Pole to the South Pole. (10 points)

(ii) In which direction would an air parcel on the 800 mb surface at the equator be pushed

by the pressure gradient force? Explain. (10 points)

4. (i) Suppose there existed a gas with attractive interactions. Suppose you had a container

with a wall in the middle separating it into two compartments. The gas is originally only

in one of the two compartments. Suppose you punctured the wall so that the gas freely

expanded from one container to the other. How would the temperature of the gas change?

Explain. The expansion is adiabatic (i.e. Q = 0). Explain. (10 points)

(ii) Supose the gas were ideal. How would its temperature change as it freely expanded

adiabatically from one container to both? Explain. (5 points)


October 28, 2003

1. There is a puddle of water at the bottom of a sealed container. The water vapor

pressure in the container is e. The saturated (equilibrium) water vapor pressure in es.

(i) How would the condensational flux C depend on e? Explain. (6 points)

(ii) How would the evaporative flux C depend on the temperature of the water in the

puddle. Explain. (6 points)

(ii) Suppose C is larger than E. What does that imply about the relationship between e

and es? (3 points)

2. (i) What is an irreversible process? (5 points)

(ii) Give two examples of irreversible processes in the atmosphere. (10 points)

3. (i) 10 kg of dry air at 10 oC is mixed isobarically with 2 kg of water vapor at 20 oC.

What is the temperature of the mixture? (10 points)

(ii) Suppose the volume of the mixture is 1000 m3. What is the pressure e of the water

vapor? (10 points)

4. An air parcel (assume dry and ideal) is brought adiabatically down to the surface (1000

mb) from 200 mb. Its initial temperature is 200 K. Ignore possible effects on density due

to the difference between the molecular mass of water and dry air.

(i) What is the temperature of the air parcel at the surface? (15 points)

(ii) What is the work done on the air parcel? (Hint: gas is ideal) (15 points)

5. Air is rising inside a cloud. The pressure is 500 mb. The temperature of the atmosphere

is 273 K. The temperature inside the cloud is 1 K warmer than outside. What is the

acceleration experienced by the air in the cloud? (20 points)


November 27, 2003

1. The latent heat of vaporization Lv is temperature dependent. What is one source of

this temperature dependence? (5 points)

2. An air parcel at a temperature of -30 oC has a relative humidity with respect to water

of 80 % and a relative humidity with respect to ice of 120 %. Rank the following in order

of smallest to largest Gibbs free energies (i.e. a 1 for the smallest free energy and a 4 for

the largest free energy.). (10 points)

(i) gi, the free energy of the ice phase.

(ii) gw, the free energy of the liquid water phase.

(iii) gw,d, the free energy of a tiny water droplet

(iv) gv, the free energy of the water vapor phase

3. The mixing ratio of carbon dioxide in the atmosphere is now about 375 parts per million,

on a molar basis. What would be the concentration of dissolved carbon dioxide in the

ocean? Express your answer as a molar concentration M, moles dissolved carbon dioxide

per liter of water. Assume the Henry’s Law constant for carbon dioxide is K = 35atmM−1.

(10 points)

4. A very strong wind near the surface generates a well mixed boundary layer from the

surface to 1 km. The temperature at the surface is 300 K. The water vapor pressure at

the surface is 10 mb.

(i)Would you expect clouds to be present in the boundary layer? Explain. (20 points)

(ii) Make plots of how you would expect potential temperature, water vapor pressure e,

water vapor specific humidity q, and temperature, to depend on height in this layer. (20

points)

5. Suppose the relative humidity inside a cloud is 1.005 % (i.e. a supersaturation of 0.005

%). Assume all the droplets in the cloud are composed of pure water. What is the droplet

size range in which you would expect droplets to be shrinking? (15 points)









December 12, 2003; Total marks : 114 (marked out of 110)

1. The saturated water vapor pressure over a sea salt aerosol is less than a pure water

vapor droplet of the same diameter. Explain why (i.e. what is the physical mechanism?).

(4 points)

2. The specific heat of water vapor cpv is larger than that of dry air cpd. Explain why this

might be expected. (4 points)

3. For most solids and liquids, cp ∼ cv. Explain. (Hint: why are cp and cv different for

gases?). (4 points)

4. Many houses with unheated basements have a problem with high relative humidity. In

what season would you expect this problem to be most serious? Explain. (There is no

exact answer. I am looking for a reasonable argument.) (4 points)

5. Suppose you had a thermometer, a perfectly insulating thermos, and some water. How

would you measure the wet bulb temperature Tw of an air parcel? (4 points)

6. What are three factors that affect the buoyancy acceleration B of air parcels rising

inside clouds. (i.e. in comparing parcels inside the cloud with their environment, what are

three variables that can affect B)? (6 points)

7. Outside clouds, air parcels in the atmosphere are usually undergoing slow descent across

isentropic surfaces. Why? (2 points)

8. Suppose you were stuck on a remote tropical island, and your only meteorological

instrument a very precise thermometer (0.1 oC resolution from 80 to 100 oC). Live in a

house with normal facilities ..

(i) How would you estimate the pressure decrease associated with an approaching hurri-

cane? (4 points)

(ii) Estimate the smallest pressure change that could be detected? (4 points)

9. Suppose the virtual temperature of air rising inside a cloud is about 1 oC warmer than

the environment outside the cloud from 900 mb to 100 mb. Estimate the upward velocity

air parcels at 100 mb would be experiencing, if the velocity at 900 mb is zero (1 mb = 100

Pa). (6 points)

10. A heavy rainstorm delivers a total rainfall of 50 mm in one day. Suppose the conden-

sational heat release from this rainfall production was spread over the entire depth of the

atmosphere. What is the heating rate of the atmosphere (in K/day)? The surface pressure

is 1000 mb. (1 mb = 100 Pa = 100 N/m2) (8 points)

11. Find the zLCL and TLCL of an air parcel at the surface with a temperature T = 300

K and a relative humidity of 50 %. (4 points)

12. 1 kg of air at 0 oC mixes with another kg of air at 25 oC. Both air parcels are saturated

with respect to water (1 kg refers to total mass of dry air plus water vapor). Total pressure

is 1000 mb of each air parcel before mixing and of mixture.

(i) What is the heat capacity Cp1 of the first air parcel, taking into account the differing

specific heats of water vapor and dry air. (4 points)

(ii) What is the heat capacity Cp2 of the second air parcel, taking into account the differing

specific heats of water vapor and dry air. (4 points)

(iii) Estimate the temperature resulting from an isobaric mixture of the two air parcels.

(Hints: the mixture will have a relative humidity larger than one, and water droplets will

form. Ignore the release of heat associated with the formation of these droplets. Also,

ignore the contribution to the heat capacity of the mixture arising from the presence of

water droplets. That is, assume for the purpose of calculating the heat capacity of the

mixture that there is no condensation. If you can solve this problem exactly, ignore my

hints.) (4 points)

(iv) Make an estimate the total mass of condensed water in the mixture. You can use the

approximate temperature obtained in (iii). (4 points)

(v) Estimate the correction to the final temperature that would be obtained by accounting

for condensational heating by the formation of any condensed water. (4 points)

(vi) What is a source of error in this correction? (2 points)

13. For the following questions, use the skew-T diagram provided.

(i) An air parcel at the surface has a temperature of 20 oC and a relative humidity of 27

%. Estimate its Lifting Condensation Level. (4 points)

(ii) From the diagram, estimate the (pseudoadiabatic) θe of the air parcel. Illustrate your

reasoning on the diagram. (4 points)

(iii) From the diagram, estimate the dew point temperature of the air parcel. Illustrate

your reasoning on the diagram. (3 points)

(iv) Use the diagram to estimate the adiabatic wet bulb temperature of the parcel. Illus-

trate your reasoning on the diagram. (3 points)

(v) Suppose the air parcel is placed in an atmosphere which has a surface temperature of

30 oC, and the lapse rate is dry adiabatic. What would be the Level of Free Convection

(LFC) of the air parcel? Again, show your reasoning on the diagram. (4 points)

14. The figure below represents an insulated box with two components A and B, each

containing dry air (treat as an ideal gas). They are separated by an insulating and perfectly

flexible wall, so that the pressure is equal on both sides. Initially, each compartment

measures 1 m3, and both A and B are at 1 atm (1000 mb or 1000 hPa) and 273 K. Heat

is then supplied to gas A (e.g. by means of an electrical resistance) until the pressure rises

to 10 atm. (Iribarne and Godson, Chapter II)

(i) What is the final temperature of the gas in compartment B (TBf )? (4 points)

(ii) What is the work performed on gas B (∆WB)? (4 points)

(iii) What is the final volume of the gas in compartment B (VBf )? (4 points)

(iv) What is the final temperature of the gas in compartment A (TAf )? (4 points)

(v) What is the heat ∆QA absorbed by gas A? (4 points)

For skew-T question, should get e = 6.42 mb, w = 4 g/kg, lcl = 750 mb.)


October 5, 2004

1. We discussed in class three factors that affect the buoyancy force experienced by an air

parcel inside a cloud. What are two of these? (i.e. in comparing parcels inside the cloud

with air parcels outside the cloud on the same pressure surface, what are two properties

of air parcels in the cloud that can affect their buoyancy?) (10 points)

2. Would you expect global warming to increase or decrease the mean height of the 900

mb pressure surface? Explain. (5 points)

3. Calculate the depth in meters in the ocean or a lake at which the total pressure equals

two atmospheres. Assume one atmospheric pressure = 1000 hPa, or 1000 mb. By total

pressure is meant the combined pressure from both the water and atmosphere (the pressure

you would actually feel). The density of water is approximately 1000 kg/m3. (10 points)

3. A 1 kg air parcel (assume dry and ideal) rises adiabatically in the atmosphere from

1000 mb at 300 K to 500 mb.


(ii) What is the change in internal energy U of the air parcel? (10 points)

(iii) What is the work done on the atmosphere by the air parcel? (10 points)

4. 1 m3 of water vapor at 30 oC is mixed adiabatically with 1 m3 of dry air at 20 oC. Both

gases are at 1000 hPa (1000 mb).

(i) Calculate the mass of water vapor. (10 points)

(i) Calculate the mass of dry air. (10 points)

(i) What is the heat capacity Cp of the mixture? (10 points)

(ii) What is the final temperature of the mixture? (You can assume no condensation

of water vapor occurs during mixing and that the mixing is isobaric.) (15 points)


October 28, 2004

1. A raindrop is falling through unsaturated air. What is one mehanism of irreversible

entropy production in this situation? Why does this mechanism generate entropy? (5

points)

2. How does the enthalpy of vaporization lv depend on temperature? What is the reason

for this temperature dependence? (5 points)

3. Explain in words and show mathematically that the existence of a heat flux between two

objects with a finite temperature difference always leads to a net production of entropy.

Let the temperature of the two objects be T1 and T2, and the rate of heat transfer be Q.

Treat the two objects as an isolated system. (20 points)

4. An air parcel starts off with T = 0 oC, and e = 4 mb. THis question has to be re-worded

to be clearer. Wasn’t answered very well.

(i) Suppose the air parcel is cooled isobarically. The air parcel contains particles that are

good condensation nuclei for water. There are initially no ice condensation nuclei. However

some of the supercooled water droplets do freeze after some time delay. Make a qualitative

plot of how you would expect the relative humidity with respect to ice to change with time

starting from 0 oC. Indicate on the diagram the times at which the temperature is equal to

the dew point temperature for water, the time at which the temperature equals the frost

point temperature for ice, and the time at which ice starts to form. (15 points)

(ii) Plot the relative humidity with respect to water would vary with time starting 0oC. Indicate on the diagram the times at which the temperature is equal to the dew

point temperature for water, the time at which the temperature equals the frost point

temperature for ice, and the time at which ice starts to form. (10 points)








6. The surface temperature is To = 300 K and the surface pressure is po = 1000 mb. An

air parcel at the surface has a dew point temperature Td = 290 K.

(i) What is the relative humidity of the air parcel? (10 points)

(ii) What is the relative humidity of the air parcel if it rises adiabatically to 800 mb?

should divide this question into parts: (i) calc temp (ii) find PT at 800 mb (iii) find e (iv)

find relative humidity (15 points)


November 25, 2004

1. The atmosphere is vertically well mixed from the surface (1000 mb) to 1 km (866 mb).

The temperature at the surface is 20 oC and the relative humidity is 80 %. For parts (i) -

(vi) of the question, assume that there is no cloud in this layer - i.e. calculate all quantities

as if condensation does not occur even when the relative humidity exceeds 100 %.

(i) Plot the vertical variation of temperature between the surface and 1 km. Indicate the

temperature at the ground and at 1 km. (5 points)

(ii) Plot the vertical variation of potential temperature θ between the surface and 1 km.

Indicate the potential temperature at the ground and at 1 km. (5 points)

(iii) Plot the vertical variation of water vapor mass mixing ratio w between the surface

and 1 km. Indicate the value of w at the ground and at 1 km. (10 points)

(iv) What is the value of e at 1 km? (Remember we are assuming that no condensation

occurs in the layer). (5 points)

(v) What is the value of es at 1 km? (5 points)

(vi) What would the value of relative humidity be at 1 km (again assuming there was no

condensation)? (5 points)

(vii) Now suppose that condensation and cloud droplet formation do occur in the mixed

layer as soon as the relative humidity reaches 100 %. Show in a rough way how this would

modify the profiles of θ and w between the ground and 1 km. You do not have to calculate

θ and w at 1 km, just indicate how they would change. Also indicate roughly the base of

the cloud (this also does not have to be calculated.) (10 points)

(viii) Bonus: estimate the height at which a cloud forms in this layer. (4 points)

2. The temperature at the surface is 25 oC and the water vapor mass mixing ratio w is 10

g H2O/kg dry air. Using the thermodynamic diagram, estimate (i) Td, (ii) Tw, (iii) pLCL,

and (iv) θe. Indicate on the diagram how you obtained each of your results. (5 points

each)

3. Suppose the virtual temperature of air rising inside a cloud is 1 oC warmer than the

environment outside the cloud from 900 mb to 100 mb.

(i) Calculate the CAPE of the air parcel. (10 points)

(ii) Estimate the maximum upward velocity an air parcel at 100 mb if the velocity at 900

mb is zero (1 mb = 100 Pa). (6 points)

(iii) What is one factor which lowers observed updraft velocities from the values predicted

by CAPE. (4 points)

4. Explain how you would measure the wet bulb temperature Tw of an air parcel. The air

is in a container (e.g. a piston) which maintains a constant pressure and is well insulated,

so maintains adiabatic conditions. You can add varying amounts of liquid water to the

container. You can measure the temperature of the air inside the container. The air parcel

has an initial temperature Ti and is initially unsaturated. A plot may be helpful. (15

points)


December 13, 2004

Total of 104 points. Marked out of 100

1. Prove, starting from the definition of enthalpy, that for an isobaric process, dH/dt = Q.

(4 points)

2. The global mean sea level pressure is not exactly constant. What is one reason why it

changes. (2 points)

3. The Hadley cycle is an overturning tropical circulation with three parts in which (1)

air rises in deep convective clouds, (2) sinks back to the surface, and (3) is horizontally

transported in the boundary layer to a region where deep convection is occurring.

(i) For each of these three parts, indicate whether potential temperature (θ) increases,

decreases, or stays the same. If θ increases or decreases, indicate the source or sink of θ.

(3 points)

(ii) For each of these three parts, indicate whether equivalent potential temperature (θe)

increases, decreases, or stays the same. If θe increases or decreases, indicate the source or

sink of θe. (3 points)

4. The ocean and atmosphere function as heat engines, continuously generating mechanical

(kinetic) energy from the release of heat. Which circulation (oceanic or atmospheric) does

this more efficiently? Explain. (2 points)

5. (i) Draw a plot of how the saturated water vapor pressure varies with droplet radius

for very tiny drops of pure water. (2 points)

(ii) Suppose the atmosphere was totally clean (i.e. it did not contain any aerosol liquid or

solid suspended particles). How would this affect cloud droplet formation? What impact

would this have on the frequency of cloud occurrence? (2 points)

6. (i) We discussed in class both moist adiabatic ascent and moist pseudoadiabatic ascent.

What is the main difference between these two types of moist ascent? (2 points)

(ii) In which type of ascent would you expect the temperature to decrease more slowly

with height? Explain. (2 points)

7. We discussed in class how vertical mixing near the surface tends to form boundary layer

clouds. (i) What is one cause of strong vertical mixing near the surface. (2 points)

(ii) Suppose the boundary layer is strongly vertically mixed from the surface to 1 km, and

there is a cloud between 0.5 km and 1 km. Plot how potential temperature would vary

with height between the surface and 1 km. (2 points)

(iii) Plot how equivalent potential temperature would vary with height between the surface

and 1 km. (2 points)

(iv) Plot how the specific humidity q would vary with height between the surface and 1

km. (2 points)

(v) Plot how the relative humidity r would vary with height between the surface and 1

km. (2 points)

(vi) Suppose the water vapor pressure e at the surface was 20 mb. What would be the

water vapor pressure e be at 0.5 km (950 mb)? (2 points)

8. A 1 kg parcel of dry air with a temperature of 300 K is heated isobarically at a rate of

Q = 2 J/sec for 1 hour. You can assume the air is an ideal gas. (i) What is the initial

enthalpy H of the air parcel? (2 points)

(ii) What is the final enthalpy H of the air parcel? (2 points)

(iii) What is the final temperature of the air parcel? (2 points)

(iv) What is the change in internal energy U of the air parcel? (2 points)

(v) What is the work done by the air parcel? (2 points)

(vi) Suppose the air parcel had been heated at constant volume. What would be the final

temperature of the air parcel? (2 points)

9. Which droplet has a higher specific Gibbs free energy g: a pure water droplet of 1 µm

radius or a pure water droplet of 2 µm radius? What is the origin of the difference? (2

points)

10. A 1 kg air parcel at 300 K and with a total pressure of 1000 mb has a relative humidity

of 60 %. (i) What is the water vapor pressure e of the air parcel? (2 points)

(ii) What is the mean molecular weight of the air parcel? (2 points)

(iii) What is the gas constant R of the air parcel? (2 points)

(iv) What is the virtual temperature Tv of the air parcel? (2 points)

11. An ideal gas at 300 K undergoes an free expansion from a volume of 1 m3 to 2 m3.

(i) What is the temperature of the gas after it expands? (2 points)

(ii) What is the change in internal energy of the gas? (2 points)

(iii) What is the change in entropy of the gas? (4 points)


descends adiabatically to 900 mb. (i) What is the temperature at 900 mb? (4 points)


13. (i) Estimate the number of moles (of all molecules) in the atmosphere. Assume the

average surface pressure is 1000 mb, and that the atmosphere can be assumed to have an

effective molecular weight equal to the molecular weight of dry air (Md = 28.94 g/mole).

The radius of the earth is 6.37 × 106 m. (4 points)

(ii) The current mixing ratio of CO2 in the atmosphere is roughly 380 ppmv (parts per

million by volume). Estimate the number of moles of CO2 in the atmosphere. (2 points)

(iii) Estimate the concentration of dissolved carbon dioxide in the ocean. Express your

answer as a molar concentration M (M = moles dissolved carbon dioxide per liter of water).

The Henry’s Law constant for carbon dioxide is K = 35 atm/M. (Hint: 1 atm = 1 × 106

ppmv) (4 points)

(iv) Estimate the number of moles of CO2 in the ocean. The volume of the ocean is

1.366 × 1018 m3, or 1.366 × 1021 l. (2 points)

(v) What fraction of total ocean-atmosphere CO2 is in the atmosphere? (2 points)


precipitation rate is about 100 cm/year. Convert this to a globally averaged heating rate

dT/dt of the atmosphere in units of K/day. You can assume the globally averaged surface

pressure is 1000 mb, and that the atmosphere can be treated as dry air. (4 points)

15. An air parcel at the surface (100 kPa or 1000 mb) has a temperature of 40 C and a

dew point temperature of 20 C. Using the accompanying chart:

(i) Estimate the LCL of the air parcel. (2 points)

(ii) Estimate the wet bulb temperature of the air parcel. (2 points)

(iii) What is the water vapor mixing ratio of the air parcel. (2 points)

(iv) Estimate the equivalent potential temperature of the air parcel. (2 points)

(v) What would be the water vapor mixing ratio q of the air parcel at 50 kPa (assuming

dry adiabatic ascent to the LCL followed by moist pseudoadiabatic ascent to 50 kPa). (2

points)

(vi) What would be the virtual temperature of the air parcel at 50 kPa? (2 points)

(vii) Suppose the background atmosphere had a temperature of -5 C and 0 % relative

humidity. What would be the buoyancy acceleration experienced by the air parcel? (2

points)

Physics/Oceanography 4500/5500 A Atmospheric Physics Exam

December 9, 2005 (104 points, marked out of 100)

1. It is believed that supersaturation with respect to ice is common in the upper tropo-

sphere. The evidence that this is likely to be the case is often observable from the ground.

What is this evidence, and why does it indicate supersaturation with respect to ice? (2

points)

2. We discussed in class how vertical mixing near the surface tends to form boundary layer

clouds. (i) What is one cause of strong vertical mixing near the surface. (2 points)

(ii) Suppose the boundary layer is strongly vertically mixed from the surface to 1 km, and

there is a cloud between 0.5 km and 1 km. Plot how potential temperature would vary

with height between the surface and 1 km. (2 points)

(iii) Plot how equivalent potential temperature would vary with height between the surface

and 1 km. (2 points)

(iv) Plot how the specific humidity q would vary with height between the surface and 1

km. (2 points)

(v) Plot how the relative humidity r would vary with height between the surface and 1

km. (2 points)

(vi) Suppose the water vapor pressure e at the surface was 20 mb. What would be the

water vapor pressure e be at 0.5 km (950 mb)? (2 points)

3. A 1 kg parcel of dry air with a temperature of 300 K is heated isobarically at a rate of

Q = 2 J/sec for 1 hour. You can assume the air is an ideal gas. (i) What is the change

in enthalpy ∆H of the air parcel? (2 points)

(ii) What is the final temperature of the air parcel? (2 points)

(iii) What is the change in internal energy ∆U of the air parcel? (2 points)

(iv) What is the work done ∆W by the air parcel? (2 points)

(v) Suppose the air parcel had been heated at constant volume. What would be the final

temperature of the air parcel? (2 points)


descends adiabatically to 900 mb. (i) What is the temperature at 900 mb? (5 points)



precipitation rate is about 100 cm/year. Convert this to a globally averaged heating rate

dT/dt of the atmosphere in units of K/day. You can assume the globally averaged surface

pressure is 1000 mb, and that the atmosphere can be treated as dry air. (8 points)

6. An air parcel at the surface (100 kPa or 1000 mb) has a temperature of 40 C and a

dew point temperature of 20 C. Using the accompanying chart:

(i) Estimate the LCL of the air parcel. (2 points)

(ii) Estimate the wet bulb temperature of the air parcel. (2 points)

(iii) Estimate the water vapor mixing ratio of the air parcel. (2 points)

(iv) Estimate the equivalent potential temperature of the air parcel. (2 points)

(v) Estimate the specific humidity q of the air parcel at 50 kPa (assuming dry adiabatic

ascent to the LCL followed by moist pseudoadiabatic ascent to 50 kPa)?. (2 points)

(vi) What would be the virtual temperature Tv of the air parcel at 50 kPa? (2 points)

(vii) Suppose the background atmosphere had a temperature of -5 C and 0 % relative

humidity. What would be the buoyancy acceleration experienced by the air parcel? (2

points)

7. Explain in words and show mathematically that the existence of a heat flux between two

objects with a finite temperature difference always leads to a net production of entropy.

Let the temperature of the two objects be T1 and T2, and the rate of heat transfer be Q.

Treat the two objects as an isolated system, and suppose that T1 is higher than T2. (6

points)

8. Suppose the virtual temperature of air rising inside a cloud is 1 oC warmer than the

environment outside the cloud from 900 mb to 100 mb. (i) Calculate the CAPE of the

air parcel. (6 points)

(ii) Estimate the maximum upward velocity an air parcel at 100 mb if the velocity at 900

mb is zero (1 mb = 100 Pa). (2 points)

(iii) What is one factor which lowers observed updraft velocities from the values predicted

by CAPE. (2 points)

9. 1 m3 of water vapor at 30 oC is mixed adiabatically with 1 m3 of dry air at 20 oC. Both

gases are at 1000 hPa (1000 mb). (i) Calculate the mass of water vapor. (2 points)

(ii) Calculate the mass of dry air. (2 points)

(iii) What is the heat capacity Cp of the mixture? (2 points)

(iv) What is the final temperature of the mixture? (You can assume no condensation of

water vapor occurs during mixing and that the mixing is isobaric.) (4 points)

10. Derivation of Clausius Clapeyron Relationship. Liquid water and water vapor are in

equilibrium at a temperature T and water vapor pressure es. Let the specific free energy

of the water be gw(es, T ), and the specific free energy of the water vapor be gv(es, T ).

Subject this system to a temperature change ∆T , so that the new water vapor pressure is

es + ∆es. Show that

∆es

∆T=

lvT (vv − vw)

,

where lv is the specific latent heat of vaporization, and vv and vw are the specific volumes

of the vapor and water respectively. (12 points)

11. An insulated box contains two gases, A and B, each of which can be treated as dry

air (and as an ideal gas). They are separated by an insulating and perfectly flexible wall,

so that the pressure is always equal on both sides. Initially, each compartment measures

1 m3, and both A and B are at 1 atm (1000 mb or 1000 hPa) and 273 K. Heat is then

supplied to gas A (e.g. by means of an electrical resistance) until the pressure rises to 10

atm. There is no heat transfer across the insulating wall, or the walls of the box. (Iribarne

and Godson, Chapter II) (i) What is the final temperature of the gas in compartment B

(TBf )? (2 points)

(ii) What is the work performed on gas B (∆WB)? (2 points)

(iii) What is the final volume of the gas in compartment B (VBf )? (2 points)

(iv) What is the final temperature of the gas in compartment A (TAf )? (2 points)

(v) What is the heat ∆QA absorbed by gas A? (2 points)


October 11, 2005

1. Suppose there is a temperature gradient on pressure surfaces such that temperatures

are warmer nearer the equator and colder nearer the poles. How will isentropic (constant

potential temperature) surfaces slope with respect to pressure surfaces? Explain with a

diagram, showing north and south directions. (4 points)

2. What are two sources of irreversible entropy production in the atmosphere? (Try to

give examples specific to the atmosphere.) (6 points)

3. How can one sometimes identify regions of turbulence in a radiosonde (balloon) tem-

perature profile? (5 points)

4. (i) What is the main reason it is difficult for molecules to leave the earth’s atmosphere?

(5 points)

(ii) Which types of molecules are able to leave most easilly? (5 points)

5. What are the two main sources of vertical mixing near the surface and what type of

temperature and potential temperature profiles does strong vertical mixing favour? (10

points)

6. Differential heating helps drive atmospheric circulations. Explain. (5 points)

7. Starting from the definition of enthalpy (H = U + pV ), show that Q = CpdT/dt for an

ideal gas undergoing an isobaric process. You will also need the first law, dU/dt = W +Q.

(15 points)

8. (i) 3 kg of water vapor at 30 oC is isobarically mixed with 2 kg of dry air at 10 oC.

What is the temperature of the mixture? (Assume there is no condensation of water vapor

during the mixing.) (10 points)

(ii) What is the ideal gas constant of the mixture? (10 points)

(ii) If the pressure of the mixture is 100 hPa, what is the density? (5 points)

9. A 1 kg air parcel (assume dry and ideal) rises adiabatically in the atmosphere from

1000 mb at 300 K to 500 mb.


(ii) What is the change in internal energy U of the air parcel? (5 points)

(iii) What is the work done on the atmosphere by the air parcel? (5 points)


November 17, 2005

1. Water vapor mixing ratios in the stratosphere are extremely low (about 5 parts per

million). Why? (6 points)

2. Does the latent heat of vaporization increase or decrease with temperature? What is

the origin of this temperature dependence? (6 points)

3. Make a rough plot of the dependence of saturated water vapor pressure es on temper-

ature. Also indicate the saturated vapor pressure over ice esi. (6 points)

4. Make a rough plot of how dT/dz would vary with height (between the surface and

roughly 15 km) in an air parcel which was lifted adiabatically upward from the surface.

Assume the temperature of the air parcel at the surface is 28 oC and the dew point

temperature is 20 oC. Indicate the Lifting Condensation Level on the diagram. (8 points)

5. Suppose one aerosol is twice the size of another. They are both very small, with dry

sizes on the order of a micron. Draw the Kohler curve of each one on the same plot (i.e.

equilibrium RH versus aerosol radius). Indicate the critical supersaturation of each aerosol

on the plot. (8 points)

6. How many grams of CO2 gas are in a 500 ml pop bottle that has been forced to come

into equilibrium with 1 atm (1000 mb) of CO2 gas? The Henry’s Law constant for CO2 is

K = 35 atm/M, and its molecular weight is approximately 44 g/mole. 1 M = 1 mole/liter.

(10 points)

7. At what relative humidity is the ocean in equilibrium with the water vapor in the

surrounding atmosphere? (Hint: this is close but not equal to 100 %). There is about

3.3 kg of NaCl for every 100 kg of water in the ocean. The molecular weight of Na is

approximately 23 g/mole, that of Cl 35 g/mole, and that of water 18 g/mole.) (20 points)

8. A 1 kg air parcel at 300 K and with a total pressure of 1000 mb has a relative humidity

of 40 %. (i) What is the water vapor pressure e of the air parcel? (5 points)

(ii) What is the mean molecular weight of the air parcel? (5 points)

(iii) What is the gas constant R of the air parcel? (5 points)

(iv) What is the virtual temperature Tv of the air parcel? (5 points)

(v) What is the approximate height of the Lifting Condensation Level of the air parcel?

(6 points)

(vi) What would be the relative humidity of the air parcel if it is adiabatically lifted to

900 mb? (10 points)

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