Self-study questions – Ch. 4

Hydrostatic Equation

The following questions are drawn from pp. 83-87:

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1. Be able to explain and/or reproduce part or all of the derivation of (4.9).

2. Under what conditions is (4.10) a reasonable approximation to the relationship between pressure and density any liquid or gas? Are those conditions usually met in the free atmosphere? In the ocean? Inside a 1-km thick ice sheet?

3. What specific assumption allows us to recast the hydrostatic equation in the form given by (4.15)? Can this form be used in the ocean?

Gravity and geopotential

The following questions are drawn from pp. 88-91: Click to show/hide questions

1. Give the two major reasons why the acceleration due to gravity at mean sea level varies with latitude.

2. At a particular level above the Earth’s surface, the acceleration due to gravity is only 95% of the standard value at sea level. At that level, how many vertical meters (as measured with a ruler) must one ascend in order to achieve a change of one geopotential meter?

3. Explain why meteorologists prefer to use geopotential height rather than geometric height to measure altitudes in the atmosphere.

Hypsometric equation

The following questions are drawn from pp. 92-93:

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1. Be able to derive the hypsometric equation from the hydrostatic equation and the ideal gas law.

2. A layer of the atmosphere bounded by two pressure levels has an initial mean virtual temperature of 10\(^\circ\)C. If that layer is warmed by 5\(^\circ\)C, by what percentage does the thickness of the layer change?

3. Imagine that, on a particular day, the 300 hPa level over some part of the ocean is found at a constant altitude of 9.0 km.
a) If the sea level pressure in this case is 1000 hPa, what is the mean virtual temperature of the atmosphere below the 300 hPa level?
b) If a nearby column of the atmosphere is 5\(^\circ\)C warmer than the answer you gave to (a), what is the corresponding sea level pressure at the bottom of that column?

4. An airplane flying at an altitude of 100 gpm above sea level measures a temperature of 8.0\(^\circ\)C, a specific humidity of 10 g/kg, and a pressure of 990 hPa. Assuming that the temperature and humidity below the aircraft aren’t too different from those at flight level, determine the pressure at sea level.

Pressure Profiles

The following questions are drawn from pp. 93-103:

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1. For each of the following cases, be able to derive the relationship between pressure and altitude, starting with the hydrostatic equation:
a) Constant density atmosphere,
b) Isothermal atmosphere,
c) Constant lapse rate atmosphere.

2. In the case of the isothermal atmosphere, be able to explain the meaning of the scale height H and what it specifically tells you about the rate of change of pressure with altitude.

3. Be able to explain the significance of the autoconvective lapse rate. Lapse rates that are greater than the autoconvective lapse rate exhibit what unusual property? Why do they almost never occur over deeper layers of the atmosphere? When DO they sometimes occur, and what is one visible manifestation that can accompany such a lapse rate?

4. In words, how does one generalize from the constant lapse rate atmosphere to one that has a different lapse rate over a succession of different layers of the atmosphere? (Note: I won’t ask you to derive this case on the exam, though you did a simple example of this in Problem 4.8).

5. We have looked at the vertical separation between a pair of pressure levels via the hypsometric equation. We have also looked at pressure as a function of altitude for different model atmospheres. These are all different ways of looking at the relationship bewteen pressure and altitude, and all are ultimately rooted in the hydrostatic equation and therefore in some ways equivalent. When would you choose one over the other, or does it even matter?

Standard atmosphere

The following questions are drawn from pp. 111-117:

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1. State the assumptions that go into the altimeter equation, and explain the role of the altimeter setting for aviation.

2. A pilot sets out on a flight with a starting altimeter setting of 30.01 inches, and he ascends to an altimeter-indicated altitude of 2,000 m. En route, the altimeter setting value decreases to 29.93 inches, but he fails to make the adjustment and continues flying at the indicated altitude. Assuming a standard atmosphere, what is his new actual altitude?

3. Be able to explain what density altitude is. What does a high density altitude imply about the actual density of the air and for the performance of an aircraft? In particular, what potential hazard is created at an airport by high density altitude?

4. The observed air density at an airport on a particular day is 0.916 kg/m^3. From Appendix E.2, what is the approximate density altitude?

Vertical structure of pressure/height fields

The following questions are drawn from pp. 118-122:

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1. Make the sketches that illustrate the list of relationships described starting at the bottom of p. 120.