Lesson 18
Hypsometric Equation
Objectives:
By the end of this lesson, students will be able to:
1. Derive the hypsometric equation using the hydrostatic equation and the equation of state for moist air.
2. Describe how temperature affects the thickness of atmosphere between any two pressure surfaces.
3. Understand the concepts of station pressure, altimeter setting, and mean sea level pressure.
I. Hypsometric Equation
A. Assumption: A moist atmosphere in hydrostatic equilibrium. Therefore, the hydrostatic equation applies:
B. Using the moist air equation of state and solving for density:
C. Substitute the expression for density into the hydrostatic equation:
D. Separate variables for integration (P and z):
E. Integrate
1. and are considered constant with respect to pressure and can be taken outside of the integral.
2. varies with pressure, since T varies with height in the atmosphere. We take it outside of the integral, and account for the variation with pressure by finding an average value for the layer:
3. Integrate from z1 to z2 and p1 to p2:
.
.
Where is the average virtual temperature in the atmospheric layer.
(NOTE: This is not a simple arithmetic average (see text p. 327), but for conceptual purposes and close estimates, an arithmetically-averaged virtual temperature in the layer will suffice.)
F. Reverse the positions of the pressures to cancel the negative sign, and:
is the hypsometric equation.
1. It describes the thickness (ΔZ) of the layer of atmosphere between p1 and p2.
a. For two given pressure surfaces, this thickness is directly
proportional to the in the layer.
2. Note that p1 is the pressure value (at z1, lower in the atmosphere)
and p2 is the pressure value (at z2, higher in the ` atmosphere).
3. A layer (between two given pressure surfaces) of cold air will have a
thickness compared to a layer of warm air.
Figure:
G. Practical Applications of the Hypsometric Equation
Reference on-line data at fnmoc.navy.mil and weather.uwyo.edu
1. 1000 to 500 mb atmospheric layer thickness chart is used by meteorologists to determine temperature advection (cold air and warm air advection) in the atmosphere. The thickness values on this chart are calculated using the hypsometric equation.
2. The 700 mb level temperatures are often used to approximate the average (virtual) temperature in the atmospheric layer between 1000 mb and 500 mb.
3. By observing the 1000 to 500 mb thickness values and the 700 mb temperature at the same location, and comparing those same values at a different location, the relationship between thickness and temperature in the hypsometric equation is seen.
4. Exercise: Experience has shown that a 1000 mb to 500 mb atmospheric layer thickness of 5400 m often corresponds to the rain / snow cutoff line for low elevation inland stations. What is the average virtual temperature of the 1000 mb to 500 mb layer when the thickness is 5400 m?
5. Station Pressure:
a. The pressure that is observed at a specific elevation and is the true barometric pressure of a location.
i. Higher elevations above sea level experience lower pressure since there is less atmosphere on which gravity can act.
6. Mean Sea Level Pressure:
a. In order for pressure values on a surface weather map to be meaningful to a meteorologist, pressure measurements must be standardized to a common reference level. Otherwise, high elevation locations would consistently read low pressures compared to low elevation sites, and isobaric analysis would be meaningless.
b. Exercise: Rearrange the hypsometric equation to solve for a reference pressure at to sea level, where:
p1= reference pressure at sea level (psea level)
p2=pressure at the station location = pstation
The reference level is taken as sea level (z = 0).
Δz=Zstation – Zsealevel = Zstation – 0 = Zstation
Figure:
Solving for psea level:
.
i. The equation uses an average temperature vice an average virtual temperature.
ii. Since it is difficult to measure a mean virtual temperature between a high elevation station and sea level, the following estimates an average temperature between the station and sea level, using the standard lapse rate of 6.5 K/km to estimate the temperature at sea level:
where
6. Altimeter Setting:
a. Aircraft altimeters are really pressure sensors that convert pressure to a height measurement using the hypsometric equation.
i. The altimeter is set using a location’s MSL pressure value so that the aircraft altimeter scale will indicate the altitude (above mean sea level) of the aircraft on the ground at that location.
ii. The altimeter setting is an attempt to remove elevation effects from pressure readings using standard conditions.
iii. Calculation is identical to that for MSL pressure:
b. Exercise: The station pressure is 900 mb. The station height above sea level is 100 m. The station temperature is 273.15 K. What is the altimeter setting?
8. Note that the above equation in iii. is just one of several different algorithms that exist for the calculation of altimeter setting and MSL pressure. Some algorithms use a temperature averaged over the past 12 hours, to account for diurnal variations.