1 Goltikasa

Lapse Rate Homework

ATM OCN (Meteorology) 100

Answers for Homework 3

Summer 2004

Date Due: Thursday, 15 July 2004

The total maximum points were 100. Point distribution for each question noted below. 



Use Tables 6.5 and 6.6 of your textbook (The Psychrometric Tables), to determine the dewpoint and relative humidity when the air temperature is 15˚C and the wet bulb temperature is 12°C. [Please include appropriate units!]

Part A (6 pts. or 2 pts each)

This question involves reading (hopefully, correctly) the Psychrometric Tables. Note that the vertical columns in the Psychrometric Tables contain the Wet-bulb depression, defined as the arithmetic difference between the dry bulb (or air) temperature and the wet-bulb temperature.  Thus, you need to make this Wet-bulb depression calculation first.

Wet-bulb depression = 


Use the table for saturation vapor pressure (see bottom of 1st homework page) to calculate the following:

a. If the air temperature were 76°F and the dewpoint were 56°F, what is the relative humidity?

b. If air temperature were 62°F and the relative humidity were 80%, what is the dewpoint?

c. Which condition a. or b. would have more water vapor? Please explain!

 Dewpoint = 56°
In this case from the air temperature = 62°F, the saturation vapor pressure es = 18.96 mb can determined from the Table. Since the relative humidity is known (80%), you can solve for the actual vapor pressure, by rearranging the equation from above to yield:
e = es x RH/100% or e = 18.96 mb x [80% / 100%] = 15.17 mb.
Now enter the table to find the dewpoint corresponding to this vapor pressure of 15.2 mb -- or 55.8°F (using a simple interpolation), which can be rounded up to 56°F.  (Use this latter value for the next part c.)

        c. (6 pts.)

Both cases, a and b, have essentially the same water vapor content, 
since the dewpoint in each case is approximately 56°F; equivalently, both have a vapor pressure of approximately 15.3 mb. This problem is meant to illustrate that while the relative humidity in b is higher (80% as compared with 50% in a), the air temperature is lower, but the dewpoint is essentially the same.  Recall that the dewpoint is the best indicator of actual water vapor content of the air.  NOTE: Suggestion that b is more humid because of a higher relative humidity is incorrect!




Using the tables of Apparent Temperature Index and Heat Stress Hazards in your text book (pages 132–133), determine the apparent temperatures for the following set of air temperatures and relative humidities.  Indicate what human hazards are possible for these cases. (8 pts)

Air Temperature


Apparent temperature

Hazard to humans 




IV Fatigue possible 
III Sunstroke, heat cramps 
and heat exhaustion possible 
II Sunstroke, heat cramps 
and heat exhaustion likely 
No adverse effects 

a.    Under what condition(s) would the apparent temperature be less than the observed air temperature? (2 pts)

The Apparent temperature would be lower than the observed ambient air temperature for low relative humidity values, typically less than 50%. 

b.     Does this situation appear reasonable? Why? Explain with personal examples. (3 pts)

Yes, this situation is very reasonable since greater evaporation of perspiration (sweat) occurs when the relative humidity is low, making a person feel cooler due to the latent heat of evaporation. Any one who has traveled to the Desert Southwest or other places in the West where the moisture content of the air is usually low (low dewpoints or low relative humidities) typically would remark that they do not feel as hot as in the more humid Southeast (such as Florida), where dewpoints generally are higher. Even in Madison, days where a hot conditions associated with a west wind with lower humidities do not feel as uncomfortable as those days when a south wind would carry humid tropical air masses from the Gulf of Mexico. 

c.     How is heat lost most effectively from the human body on a hot summer day? (2 pts)

Perspiration (sweat) is evaporated from the skin and the latent heat of evaporation aids to cool the human body since approximately 590 calories per gram of water are utilized. 


(17 pts., 4 pts each for a and b, 3 pts for c, 6 pts for d)

a. Name two (2) requirements for the collision-coalescence theory:

b. Name two (2) requirements for the ice crystal (Bergeron) theory:

c. Why is the simple diffusion-condensation process not considered a plausible precipitation formation process?

The simple diffusion-condensation process would take as much as 36 to 48 hours to produce a sufficiently large raindrop from a cloud droplet - too long a time. 

d. By how many times does the volume of a typical cloud droplet (with a diameter of 20 micrometers) have to increase to form a typical raindrop (diameter of 2 millimeter)? (Please show your work for partial credit!) (Hint: 1 millimeter = 1000 micrometers)


(6 pts, 3 pts each)

Distinguish between freezing rain and ice pellets:

Freezing rain

(11 pts)

During the winter of 1995-96, Madison received four major snowstorms with total snowfall amounts for each storm totaling 4 inches or more. On 27 Nov. 1995, 7.6 inches of snow fell, with a liquid water equivalent of 1.33 inch. On 26 to 27 Jan 1996, 13.4 inches of snow fell, with a liquid water equivalent of 1.00 inch. Calculate the snow to liquid ratio (the number of inches of snow to every inch of liquid water equivalent) for each storm.

Which storm had the "fluffy stuff" (as compared with the heavy, wet snow with a high liquid content)?

The 26 to 27 Jan 1996 storm had the more fluffy snow, since the 13.4 to 1 ratio is larger than the 5.7 to 1 ratio for the storm of 27 Nov. 1995. In other words, the January storm (which had thunder and lightning) had a higher amount to each inch of snow, meaning more air space in between the ice crystals and snowflakes. 

How do these ratios for the two storms compare with the typical 10 to 1 ratio often used?

The 10 to 1 ratio is only a rough rule of thumb - since in the first case, the snow was heavier and gave a ratio that was about half the typical ratio (meaning that the standard ratio would underestimate the liquid content from the snow depth), while in the second case the snow was more fluffy and would pile to a 30% greater depth than would be estimated from the liquid equivalent (or stated differently, the liquid equivalent would be overestimated from the snow depth). 


(9 pts)
             An explanation: This exercise was originally used with an older book by  J. M. Moran that had an appendix with precipitation extremes.  While the newer book does not have such a table, the decision was made to use the tables from the links from the precipitation lecture:

A different version of the homework was created, since some of the questions could not be answered easily.  However, the older version was inadvertently posted.  This section will be graded with that in mind. EJH

Where is the greatest annual total precipitation in the United States? 

Kuki, Maui HI  (1982)
NOTE: Several other answers are also accepted:
a. Mt. Waialeale, Kauai, HI (a multi-year average) 
b. Hilo, HI  (a multi-year average of large cities)

How much precipitation was observed at this locale? 

1788 cm (704 inches) for Kuki, Maui HI 
Other locations:
a.  1199 cm (460 inches) for Mt. Waialeale, Kauai, HI (a 30-year average)
b.  325.5 cm (128 inches) for Hilo, HI

What was the greatest amount of rain that has been recorded in one day in the United States? 

1092 mm (43 inches) 

Where was this daily record rainfall recorded? 

Alvin, TX 

Approximately how long is the world record dry spell? 

14 years and 2 months Oct 1903 to Dec 1917  at Arca, Chile
In U.S., 767 days at Bagdad, CA (3 Oct 1912 - 8 Nov 1914).

What was the record amount of snow that has fallen in one day in the United States? 

157.5 cm =  62.0 inches -- in one calendar day (midnight to midnight)
192.5 cm = 75.8 inches  -- in 24-hour span

Where was this daily record snowfall recorded? 

Thompson Pass, Alaska  (29 Dec 1955) -- (calendar day record)
Silver Lake, Boulder County, Colorado (14-15 Apr 1921)  (24-hr record)

Where is the greatest annual total snowfall in the United States? 

Paradise Ranger Station, Mt. Rainier, WA (as found in the text for the 1971-72 snow year).
Since then a new "unofficial" world record has been established at Mt. Baker, WA in Washington State for the 1998-1999 snow year.

How much snow fell at this station? 

2850 cm = 1122 inches (as found in the text for Paradise RS, Mt. Rainier).
(The "unofficial" world record at Mt. Baker, WA in 1998-1999 was 1140 inches.)

Latest revision: 21 July 2004 (1205 UTC)

Produced by Edward J. Hopkins, Ph.D.Department of Atmospheric and Oceanic SciencesUniversity of Wisconsin-Madison, Madison, WI 53706hopkins@meteor.wisc.edu

URL Address: aos100/homework/s04hmk03a.html

Lapse Rate, Moisture, Clouds and Thunderstorms.

I. Introduction

The atmosphere is a three dimensional fluid. The air moves horizontally and vertically creating a mosaic of weather phenomena, and shaping the properties of climate.

Last week we learned how radiation from the sun, while interacting with the Earth's surface and the atmosphere, helps determine the vertical profile of atmospheric temperature (Fig 1). The vertical profile of temperature is an important factor in weather and climate. It controls the vertical movement of air, determines the existence of clouds and rainfall, affects visibility, and regulates the level of pollution near the Earth's surface. The subject of this lecture is to find out what are the processes that determine the vertical temperature profile, and their consequences.

One of the key factors to understand in this context is the vertical motion of air parcels, a process referred to as convection. Convection is a form of heat transfer. The vertical distribution of heat by convection affects the vertical profile of temperature in a volume of gas or liquid. It is this movement in the atmosphere that leads to the formation of clouds, removes pollutants up and away from the Earth's surface, and helps mix the atmosphere and determine the vertical profile of temperature.

To understand this lecture we should be familiar with the concepts of heat, temperature, gravity, pressure, density, and the notion of the three phases of matter and their transition.

Take away ideas and understandings

  • Convection is a from of heat transfer achieved in the atmosphere through the vertical motion of air parcels.
  • Convection moves air parcels with their content (water vapor and droplets, small particles, other gases) and thus affects visibility, cloudiness, rainfall, and levels of pollution in the air.
  • The process of convection is governed by basic physical laws (gravity and the conservation of energy), and by a fundamental relationship between three measures of the state of every gas: temperature, density, and pressure.
  • Water, in all three phases, participates in and strongly affects the convection process, leading to the formation of clouds and rainfall.
  • Water is a very important part of the climate system. In addition to its greenhouse properties, it also acts as a reservoir of heat. Water cycling through the climate system supports life and maintains a stable climate on Earth.

II. Convection - a form of heat transfer

Consider the following:

  • How does water in a kettle heat up to a boil?
  • Why is air in a room warmer near the ceiling than close to the floor?
  • Why does smoke emerge from the factory stacks and rise up in the air?
  • Why does lava ooze out of cracks in the ocean floor?
  • How do clouds form?

The answer to all of these is convection.

Convection is a form of heat transfer, as is the process of radiation which we examined in the previous lecture. Convection takes place in liquids and gases and distinguishes them from solids. It works because in a fluid, "chunks" of matter (which we will refer to as parcels) can move up or down with respect to the rest of the fluid as they are being heated or cooled, respectively.

In a kettle, the water closest to the bottom warms because it touches the hot metal base. This form of heat transfer - the transfer of heat in matter through molecular motion - requires actual contact between two objects and is called conduction. As water parcels near the bottom warm, they become less dense than the surrounding fluid (a property of liquids and gases alike - more about this later in Section 3) and rise. The heavier fluid that was not exposed to the metal base sinks to take the place of the rising parcels. In this process, heat is transferred away from the bottom into the interior of the kettle. The process continues until all the water heats up to its boiling point.

The same process occurs in the atmosphere. Here the ground acts like the metal base of the kettle, and air parcels replace the water. During the day, the ground warms up because it absorbs the Sun's radiation. Air parcels in contact with the ground warm up by conduction, become lighter than their surroundings, and rise. At night, as the ground loses its heat by radiation, the air parcels in contact with it cool and become denser than they were before. If the air is still these cold parcels will stay close to the ground and continue cooling, sometimes until fog is formed through the condensation of water vapor. Air parcels away from the ground also cool as they loose their heat by radiation, and the convection process stops.

We shall examine how convection creates clouds and rainfall later, but from its nature, it is easy to see that convection is a process that generates mixing in the atmosphere, removing not only heat but also particles that are suspended in the air away from the surface. In this way convection helps clear the air near the surface from pollutants, tiny water droplets (that make up fog and haze) etc. If convection is inhibited (as in the case of nighttime cooling in still air) particles stay close to the surface, reducing visibility.

As described above, the processes of convection seem quite intuitive to us. They are, however, governed by the laws of thermodynamics. Understanding these laws helps us quantify these processes, make predictions on the formation of clouds and fog, and explain how the vertical profile of temperature in the atmosphere is determined.

III. Thermodynamic properties of dry air - adiabatic temperature change

The equation of state - ideal gas law:

Dry air is air that contains no water. We will address the thermodynamics of "moist" air later. The state of a parcel of dry air is described by three properties: temperature (T, expressed in °K, where 273°K = 0°C), pressure (p, force per unit area, expressed in Newtons/m2) and density (ρ, the mass of a unit volume, in Kg/m3).  In a gas these properties are related by a relatively simple physical law called the ideal gas law (ideal because it is not exact, albeit quite accurate for most applications in meteorology). This law states that:

p = ρR T

R is a coefficient, called the gas constant. It does not depend on either p, ρ, or T. The gas constant depends only on the composition of gases that make up the air (every gas has its own gas constant). Since this composition (for dry air) is roughly constant throughout most of the atmosphere (see Table 1), R of air is constant and equal to 287 Joules/(kg °K).

To understand the equation of state, assume that you have a fixed mass of air enclosed in a container with rigid walls (hence with fixed volume). If we warmed the container, say by putting it over a flame, the temperature of the air (i.e., kinetic energy of the air molecules) will rise and the pressure (i.e., the force exerted by these molecules on the container walls) will increase. The density of the air will not change since we are not increasing the amount of gas in the container nor the volume of the container. The ideal gas equation states that the increase in pressure is directly proportional to the increase in temperature.

Lets replace the rigid wall of the container with flexible ones, that are allowed to stretch freely if the pressure inside rises above that on the outside. In that case, when we raise the temperature, the pressure inside will remain constant (and equal to the outside pressure), but the container's volume will increase. This means that the density will decrease (because the mass inside does not change). The ideal gas law states that the density decrease will be inversely proportional to the increase in temperature.

The first law of thermodynamics and adiabatic expansion:

Let us remove the flame that heated our flexible walled container, and put it in a chamber where the pressure can be controlled from the outside, lowered or raised at will. What will happen to the density of our air parcel when we lower the pressure surrounding our container? What will happen to its temperature?

Here too the pressure on both sides of the flexible container walls will equalize - as the outside pressure drops, the container will expand and the pressure inside will drop by the same amount. The density of the air parcel in the container will decrease as well, in agreement with the ideal gas law. But what the ideal gas law can not tell us is what will happen to the temperature. To find that out we need to consider the first law of thermodynamics - a physical law that extends the principle of conservation of energy to include the concepts of heat and work.

In thermodynamics the simplest form of energy conservation is the balance between internal energy (the kinetic energy of the body's internal molecular motion - directly proportional to its temperature), and the amount of heat added to the body minus the work done by the body on its surroundings.

As our air parcel expands in response to the lowering of the outside pressure, the force of its internal pressure is moving the walls of the container outwards. When a force is moving an object over a given distance it does work. Thus the expanding air parcel does work on its surroundings. Thiswork must come at the expense of internal energy (remember, heat is neither added nor taken away from the parcel in this experiment). Thus the molecular motion within the parcel will slow down, and the parcel's temperature will drop.

The expanding parcel will experience not only lowering of its pressure and density, but also of its temperature. All three state variables: pressure, density, and temperature will remain in balance as described by the ideal gas law. The process described above is called adiabatic expansion, implying the change in parcel density without the exchange of heat with its surroundings, and its consequential cooling. The opposite will occur when the parcel is compressed. Adiabatic compression leads to warming.

The mathematical expression of the first law of thermodynamics is beyond the scope of this class. A more complete treatment of this subject can be found in Philander (1998), Appendix 4.

IV. Atmosphere under gravity - hydrostatic balance.

Hydrostatic balance

In the vertical direction, gravity is by far the most important external force acting on the atmosphere. It is the reason for the existence of this crucial envelop of gases around the Earth.

The atmosphere does not collapse under the downward pull of gravity because of the energy embedded in the movement of the air molecules. This movement creates the force of pressure which counters the gravitational pull on the atmosphere. The balance between the force of pressure and gravity is the hydrostatic balance.

To find the expression for the hydrostatic balance, we first note that atmospheric surface pressure is due to the weight of the entire atmospheric column above. As we ascend, there is less of an atmosphere above us, and hence the pressure drops. Consider a column of gas Δz meters tall suspended somewhere in the atmosphere (here Δ symbolizes an interval or difference). The reason this column of air does not "fall" down under the pull of gravity is because the pressure acting on its bottom surface is higher than the pressure acting on its top surface. The pressure difference Δp exactly balances the weight (per unit area) of the column. Stated in mathematical terms this balance is written as:

Δp = - ρ g Δz

where g is the acceleration due to gravity = 9.8 m/s2.

The drop of pressure with height and adiabatic cooling of rising air.

Pressure drops as we ascend in the atmosphere, but it does not do so linearly (that is the drop in pressure is not proportional to the height increase? Why is that? The hydrostatic balance provides the clue: density does not remain constant with height!

In water (oceans), density stays close to constant as depth increases because water is not very compressible. Therefore water pressure in the ocean tends to increase linearly with depth. As a gas, air is highly compressible. As pressure decreases with height, the molecules of air are free to move further apart from one another and the density decreases. Close to the ground, where gravity causes the air to be rather compressed under the weight of the entire atmosphere above, a small change of altitude results in a large drop in pressure (roughly 100 Pa for every 8 meters of ascent, see homework problem #2). This means that near the surface the change in density with height will also be large. As height increases and density drops, the rate at which pressure drops with height decreases. This in turn affects the rate of density change with height; it will also decrease. At very high elevations, pressure and density decrease very very slowly with height, causing the gaseous envelop of earth to stretch to a few hundred kilometers above ground.

The functional dependence of pressure (and density) on height is called exponential (see Philander, 1998 - Appendix 1, for an explanation of exponential change)

We can now combine the thermodynamic laws with the effect of gravity on pressure. Using the equation of state, the first law of thermodynamics, and the hydrostatic equation we can find that the rate of adiabatic temperature change in an ascending air parcel (also termed the adiabatic lapse rate and denoted Γd) is constant:

Γd = - ΔT / ΔZ = 9.8 °C/km

Note that Γd is defined as the negative of the actual temperature change, so that Γd is the amount of cooling that the rising parcel experiences. Sinking air will warm at the same rate as it is being compressed by the increasing pressure.

V. The stability of dry air - dry convection.

The stability of air under vertical displacement is determined by the outcome of a small change in an air parcels elevation:

  • If the environment (the surrounding atmosphere) is such that vertically displaced parcels continue to rise on their own, even when the lifting exerted on them stops, the environment is referred to as unstable.
  • If vertically displaced parcels sink back to their initial elevation after the lifting ceases, the environment is stable.
  • If vertically displaced parcels remain where they are after being lifted, the environment is neutral.

The stability of the atmosphere can be compared to the stability of an object (a ball) under gravity. In the figure above 3 balls are shown in 3 different position on flat and curved surfaces. These 3 states are analogous to the three different states of stability in the atmosphere. Graphic by Yochanan Kushnir.

There could be at least two simple reasons for the initial displacement of an air parcel:

  1. An initial warming, through contact with the ground, raises the parcel's temperature, lowers its density, and makes it initially lighter than its environment (as in convection in a kettle of water).
  2. A vertical push due to an air current hitting an obstacle (such as a mountain).

What determines stability is the difference in density between the rising parcel and the environment. At the same pressure density differences are determined by temperature differences (ideal gas law). The rate of change of temperature with height in a dry air parcel - the adiabatic lapse rate - is fixed: 9.8 °C/km, but the rate of change with height in the surrounding atmosphere varies from place to place and time to time. The measured local vertical profile of temperature in the air is called the environmental lapse rate.

If the environmental lapse rate is less than the adiabatic lapse rate, rising parcels will quickly become colder than their surroundings (even if they started up a bit warmer) and sink. In Fig 2 we depict an environment where the lapse rate is 7 °C/km, smaller than the adiabatic lapse rate. The parcel starts initially a few degrees warmer than the environment and is thus able to rise. But by cooling faster than the environment, it reaches a level (at the height of of 2 km) where it is colder and denser than the surrounding air and stops its ascent.

If, on the other hand, we are faced with the situation shown in Fig 3 the consequences of an initial "lift" are quite different. The environmental lapse rate is 10 °C/km - higher than the adiabatic lapse rate - and the parcel continues to rise throughout the entire column. At the tropopause the lapse rate changes and becomes 5 °C/km. The parcel then quickly becomes colder than the environment and stops its ascent.

The following table summarizes the subject of stability for dry air:

Table 2: Stability of rising air

Lapse Rate *Stability
Γenvironment > ΓdUnstable
Γenvironment = ΓdNeutral
Γenvironment < ΓdStable

* Remember that lapse rate measures the rate of cooling.

Convection is controlled by atmospheric stability. It can be forced initially by the warming of air parcels near the surface, but the vertical profile of temperature determines whether convection will be deep (that is penetrating to high elevation as in Fig 3) or shallow as in Fig 2.

The process of convection causes air warmer than its environment to rise, and cold air to sink. Convection thus transports heat upward, making it warmer aloft than it would be in the absence of convection. However, in the atmosphere, in contrast to what happens in a kettle filled with water and set on the stove to boil, convection does not lead to a temperature profile that is uniform with height. This is because the rising air cools adiabatically. Dry convection will tend to create an adiabatic temperature profile in which the temperature falls at a rate of 9.8 °C per kilometer. However, since the atmosphere generally contains water vapor, convection is usually more complicated than that, as we shall see in the next section.

VI. Water in the atmosphere - thermodynamic properties of moist air.

Importance of water in the climate system.

Water exists in the atmosphere in all three phases: gas (vapor, mixed with other gasses in the air), liquid, and ice (droplets and ice crystals suspended in the air as clouds).  Atmospheric water plays an extremely important role in the climate system due to three outstanding properties:

  1. Water vapor is an absorber of infrared radiation: As we saw in a previous lecture, water vapor creates a greenhouse effect that helps maintain a life sustaining climate on Earth.
  2. Water vapor acts like a reservoir of heat: It take 540 Calories to evaporate a gram of water (see below). This latent heat of vaporization is much larger than for most other common substances. Once water vapor condenses, it releases that amount of heat into the environment. As we shall see later, this fact alone makes atmospheric water a primary agent in transporting heat throughout the climate system (horizontally and vertically).
  3. In its condensed phase in the atmosphere, as water droplets which form clouds, water absorbs infrared radiation and, more importantly, reflects it back into space. Clouds cover more than 70% of Earth's surface and thus play a primary part in determining the Earth's albedo (Fig 4).

Moist air is air that contains water in vapor form. The moisture in the air is usually referred to as humidity. The average concentration of vapor in the atmosphere is 0.48% (Table 1). Another way to appreciate the amount of water in the atmosphere is to note that if all of it was condensed and made to cover the Earth uniformly, it would make a layer of liquid 1 inch thick. Air can not carry unlimited amounts of water. Even in the most humid situations the concentration of vapor in the atmosphere can not exceed a few percent. The colder the air, the less vapor it can hold (see below in the discussion of saturation).

The largest source of water in the climate system is the world ocean. Water evaporates from the ocean surface to mix in the air. Wet or forest covered land surfaces are secondary sources of atmospheric water. The highest concentrations of vapor are found near the surface in the tropics. The concentration drops quite rapidly with height, and half the way up the tropical troposphere it is a fraction of what it is near the surface. Vapor concentration also falls off rapidly as we move north or south of the tropical belt, and it is generally higher over the oceans than it is over land.

Describing water vapor amount in the atmosphere.

There are a few ways to measure the concentration of water vapor in the atmosphere.

  • Vapor pressure (denoted e): is the partial pressure of water vapor molecules in the atmosphere. Partial pressure is a term in thermodynamics of gas mixtures (in our case - air). We can break down the air pressure into the pressure each of its individual gas constituents would exert, had all the others been removed. The pressure in an air parcel is the sum of the partial pressures of all the constituents. The smaller the concentration of a gas in the mixture, the lower its partial pressure. However, since molecules of different constituents have different mass, the partial pressure is not directly proportional to the molecular concentration.

    The concept of vapor pressure is important for understanding the processes of evaporation and saturation. If we hold a parcel of air still over a flat water surface, water molecules will escape the surface and start mixing with the other gases in the air parcel. This is evaporation - it can happen even if the liquid is not at its boiling temperature. Evaporation can only go on until the maximum amount of water vapor that air can hold is reached. At this point, the pressure that the water molecules exert as they are trying to escape the liquid is equaled by the partial pressure of water in the air parcel, called the saturation vapor pressure. Saturation is a process of equilibrium where water molecules cross back and forth across the boundary between water and air, maintaining a fixed concentration in the air. The saturation vapor pressure is a function of temperature (Fig 5).

    Figure 5 requires some explanation. It is based on measurements of vapor pressure as a function of temperature, taken in a forest environment. When you examine the figure, notice that there is an upper limit to how high vapor pressure can be at any given temperature. At that limit the air is saturated. If we trace this upper limit we will obtain an exponential curve describing the dependence of saturation pressure on temperature. This is the so called: Clausius-Clapeyron relationship, named after two physicists who helped formulate this dependence using thermodynamic theory. The forest air measured in Figure 5 was not always saturated, as is obvious from the points lying under the maximum value. Moreover, it is the lowland measurements more than the ridge top values that reached saturation. Can you guess why?

  • Relative humidity: is the ratio of actual vapor pressure to saturation vapor pressure (expressed as % if multiplied by 100). This is a common way to indicate air humidity. Because perspiration plays a very important function in maintaining body temperature, relative humidity figures into consideration of the degree of comfort we have when following our daily activities.
  • Mixing ratio: is the mass of water vapor in grams per kilogram of air. This is the most common way to indicate air humidity in scientific applications. At the Earth's surface, mixing ratio varies from ~18 gm/kg in the tropics to less than 2 gm/kg near the poles.
  • Dew point temperature: is yet another way to express the vapor content of an air parcel. The dew point temperature gets its name from the process leading to the formation of dew. In the early morning hours before sunset, when the air is still, and the ground is cool (compared to its day time temperature) because it radiated its heat into the atmosphere and outer space, the air in immediate contact with it cools too by conduction. Since the air's ability to hold vapor decreases with temperature (Fig 5), any vapor in excess of the saturation value is rejected and condenses as droplets on the ground or its cover (see below for more discussion of phase changes of water). Following this natural process we define the dew point as the temperature at which the vapor in a cooled parcel of air begins to condense. The dew point can be either lower than (if the air is not saturated) or equal to (if the air is saturated) the actual temperature. The bigger the difference between the actual air temperature and the dew point, the drier the air is.

Phase changes of water.

Phase changes are the transition between different states of a substance. They are accompanied by the absorption or the release of heat. In the normal conditions that exist in the climate system, some substances can be found in only one state (most atmospheric gases, for example). Water can be found in all 3 states.

The liquid-vapor phase transition in water takes up (or gives out) 540-600 calories/gm (= 2.25 to 2.5 x106 Joules/kg) (the exact amount depends on temperature). This heat is known as the latent heat of vaporization/condensation. At the sea-air boundary, water coexists as vapor and liquid. Unless the air is saturated, water evaporates continuously from the liquid side of the interface. This process draws heat from the evaporating liquid and cools it. Alternatively, if vapor condenses (as in clouds), the surrounding air is warmed.

In the cold polar oceans, liquid water and ice are in equilibrium with each other. The heat required to melt ice into water is much less than that required to turn water into vapor. In melting water we need 80 calories/gm (so called the latent heat of melting). This heat is returned in the process of fusion (when water freezes).

Water vapor can also be in equilibrium with ice. In this case, molecules of water can cross the boundary between the ice surface into the air, just as they do over a water surface. The transition between the solid phase and the vapor phase is called sublimation. When ice turns directly into vapor (sublimation) the heat required per gram of ice is the sum of the latent heat of melting and the latent heat of vaporization - a total of 620-680 calories/gm.

Table 2: Phase Changes of Water

Direction of Phase ChangeThermodynamic Effect
going to lower energy phase
heat is released
(warms air)
going to higher energy phase
heat is absorbed
(cools air)

Stability of moist air - moist convection.

The largest differences in behavior between moist and dry air thermodynamics is in the cooling process encountered under lifting of air parcels. This is because when air containing water vapor is lifted up it begins to cool at the dry adiabatic lapse rate. But when it reaches its dew point temperature, saturation occurs, and water droplets begin to condense inside the rising parcel, forming a cloud. With condensation begins the process of latent heat release within the parcel (see Table 2 above). The altitude at which saturation occurs is termed the lifting condensation level (LCL) and defines the cloud base. The LCL depends on the initial relative humidity of the parcel. Dry air (with low relative humidity) must be lifted higher to saturate. Clouds will form more easily the more humid the rising air is.

When air continues to rise above LCL, water continues to condense. This is because the condensation is just enough to continue keeping the parcel saturated (you can't condense more than is necessary to reach saturation). The process steadily releases latent heat, warming the rising air parcel, to partly offset the adiabatic cooling. As a result, the saturated parcel cools more slowly than a dry parcel. The moist adiabatic lapse rate is typically about 6.5 °C/km (compare to 9.8° in a dry parcel). Unlike the dry adiabatic lapse rate the moist one is not constant, because the dependence of saturation on temperature is exponential (see above section 6.2), and the colder the air gets, the less water condenses per degree of cooling (confirm that with the aid of Fig 5). The colder the rising parcel gets, the closer the moist adiabatic lapse rate gets to the dry adiabatic rate.

The presence of water vapor in the air complicates the calculation of stability that we examined earlier in section 5. To examine stability and the likelihood for convection in realistic situations we need information on both the temperature and moisture profiles in the atmosphere. Thus these two are routinely measured with balloon carried instrumentation packages. The process of figuring out the stability of moist air is broken down to the following stages:

  1. A lifted air parcel cools at the dry adiabatic lapse rate until it reaches LCL.
  2. LCL is found by considering the parcel's initial relative humidity and temperature. This calculation assumes that, as it rises, the parcel's moisture content remains unchanged (that is, it does not mix with the surrounding air - a valid assumption if the lifting occurs fast enough).
  3. Once LCL is reached, the lapse rate changes abruptly to moist adiabatic and continues to be so until the lifting stops. The moist rate has to be continually adjusted to compensate for its change with temperature.
  4. Throughout this process we continue to compare the parcels lapse rate to the environmental rate. As is the case with dry convection, the air will begin to convect freely once the temperature of the rising parcel becomes higher than the temperature of the environment**.

Overall moist air is more unstable than dry air because of the release of latent heat involved in moist convection. Since latent heating warms the rising air parcel, it is easier for it to become warmer than the environment and thus unstable. In Fig 6 the atmospheric lapse rate is identical to that shown earlier in Fig 2. The difference is in the fact that the air is now assumed close to being saturated at the surface. When it is warmed slightly near the ground, it begins to ascend and condenses. From there it cools at the moist rate instead of the dry one. At the elevation of 1 km it is already 3.5°C warmer than the environment, and continues to be so after that.

Meteorologists do not have to resort to many calculations to figure all this out because they are equipped with smart "thermodynamic diagrams" which are used as graph paper to plot the balloon measurements. The figuring out of stability becomes a relatively simple comparison between the environmental lapse rate (measured) and lines of dry and moist adiabats included in the original diagram.

Importance of moist convection.

Clouds and rain are the visible aspect of the convection process. Fig 6 is an example of how thunderstorms (Fig 7) form. Here the vertical motion was caused by differential warming (that is heating one parcel more than the neighboring ones, and the instability it triggers.

As indicated above, the role of water in the atmosphere is extremely important. Consider the transport of heat from surface to upper troposphere that we discussed under dry convection. The presence of water makes the process of convection more efficient. Here are the components of heat transfer involved in moist convection: At the surface Conduction removes heat through contact with cooler overlying air, and Evaporation removes the latent heat required to push the vapor into the air. Convection acts to remove heat gain through both these processes from the surface up in the form of sensible heat (heat stored in the molecular motion of the air parcel - its internal energy) and latent heat (heat stored in the evaporated water molecules, ready to be released as soon as the air cools enough aloft). In doing so moist convection heats up the upper troposphere more efficiently than dry convection (that only moves sensible heat up). Once it reaches the upper troposphere the heat is released into space as radiation, balancing the incoming solar heating. In regions where moist convection exists, the temperature profile is less steepthan in dry convecting regions. In the tropics, where moist convection is always active, the vertical profile of temperature closely follows a moist adiabatic lapse rate.

It is important to note that convection is not the only reason for vertical motion, and we do not have to reach instability for clouds to form. Air can rise also if it is moving horizontally and pushed against a mountain barrier or a boundary between air masses of different densities (so called fronts). In the latter case, the lighter air will be pushed over the denser one. Another reason for ascent is the existence of internal disturbances in the air such as waves and turbulence. In all these cases clouds will form if the rising air is cooled to its dew point through the mechanical ascent - even if conditions for instability are not achieved. Clouds that form in stable air however, have different form than clouds that form through convection in unstable air. Stable air clouds tend to be layered and are referred to as stratus clouds. Clouds that form in unstable air are tower like, stretching vertically more than horizontally. These are referred to as cumulus clouds. All clouds are suppressed by downward motion.

Moist convection communicates between the sea surface (from where water originally evaporates) and the upper atmosphere, allowing a "dialog" between the ocean state and the atmosphere - a subject for later discussion. Moist convection also returns water back to the Earth's surface, balancing what was lost by evaporation and transpiration.

The hydrological cycle.

Water continually recycles through the climate system as shown in Fig 8. The cycle consists of reservoirs (places where water is stored, e.g., ocean, atmosphere, sea ice, groundwater) and fluxes (transfers between reservoirs, e.g., evaporation, precipitation, runoff). Vertical motions control the fluxes into and out of the atmosphere.

Horizontal fluxes are also important. Over the oceans evaporation (denoted E) is larger than precipitation (P). Over land P exceeds E. To counter the imbalance, the atmospheric circulation must transport vapor from maritime to continental locations. Runoff (rivers) returns the excess land precipitation to the oceans.

Summary Slides

All figures by Yochanan Kushnir

  1. Forms of heat transfer.
  2. Properties of air parcels and the Ideal Gas Law (Equation of State).
  3. Implications of Ideal Gas Law.
  4. Adiabatic cooling.
  5. Hydrostatic balance.
  6. Atmospheric stability.
  7. Atmospheric stability (continued).
  8. Stability of moist air.
  9. Water in the atmosphere.

Text by Yochanan Kushnir, 2000.

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