pV = nRT
where p is pressure, V is volume, n is amount of the gas in moles, R is the universal gas constant, and T is temperature.
The formulation often used for the atmosphere is:
p = ρRdT
where ρ is the density (mass per volume) and Rd is the specific gas constant for dry air.
One way to handle moisture in the air is to use the Virtual Temperature. The virtual temperature will be explained further when we discuss moisture in the atmosphere. The virtual temperature is typically slightly higher than the actual temperature, and is denoted by T* in equations. So the ideal gas law applied to the atmosphere becomes:
p = ρRdT*
The value of Rd in SI units is 287. When doing calculations using the ideal gas law, remember to convert all the units to SI units. It is a common mistake to fail to convert pressure in mb to pressure in Pa.
Next we consider the hydrostatic balance in the atosphere. We observe that air does not plummit to the ground under the force of gravity, nor does it fly into space due to the pressure of the air below it. Two forces are nearly in balance at any point in the atmosphere: the pressure force of the air below, and the force applied from the mass of air above (under the acceleration due to gravity).
When we consider the function of f(z) = p we can determine the rate of change of this function from basic physics. The derivation is in Wallace and Hobbs Ch. 3, and was also presented in class. It gives the Hydrostatic Equation which is one of the fundamental equations in meteorology:
dp/dz = -ρg
The equation gives that the rate of change of pressure as z (geopotential height) changes is a function of the density of the air. Near the ground, where the density is high, the rate of change is thus high. High in the atmosphere, where the density is low, the rate of change is thus lower.
For example, going from the surface to 100m above the surface, the pressure might change 10mb (not the actual value). But going from 20,000m to 20,100m might result in a change of pressure of only 1 or 2mb. These are just example numbers, and we would like to be able to calculate the actual pressure change as height is changed. We do this below.
It is important to remember the hydrostatic equation is derived from a balance of two forces, and the assumption is thus made that no other forces are acting on the air. In the real atmosphere, this assumption is rarely true. But the other forces are small compared to the hydrostatic balance forces.
To determine the actual change in pressure given a change in height,
the hydrostatic equation is combined with the ideal gas law and
integrated. The derivation is provided in Wallace and Hobbs Ch. 3
and is also available online in various places, such as Wikepedia.
It is important to note that during the integration, the constants
Rd and g are removed from the integral.
T*
is also removed from the integral by assuming that
T*
changes linearly with height, and then taking the average of
T* denoted by
The result is the Hypsometric Equation:
z2 - z1 = (Rd∗
Given a fixed pressure layer -- say, 1000mb to 500mb -- this equation shows the thickness of that layer is a function of the mean virtual temperature.
It is also important to note that this equation shows that as we go upward in the atmosphere, the height changes as the log of pressure! This is why many charts, such as the SkewT chart, plot pressure logarithmically.
Example: The 1000mb to 500mb layer has a mean virtual temperature of 5°C. What is the thickness of this pressure layer?
First convert to SI units. 5°C is 278.15K. 1000mb is 100000Pa. 500mb is 50000Pa. Note that Rd is 287, and g is 9.8. Putting the values into the hypsometric equation we get:
(287∗278.15/9.8)∗ln(100000/50000) = (z@500 - z@1000) = 5643m