Raindrops and the Doppler Effect

 

Editor’s note: As part of the preparations for the upcoming Marine ARM GPCI Investigations of Clouds (MAGIC) field campaign, principal investigator Ernie Lewis discusses how radars use the Doppler effect to determine raindrop sizes and speeds.

This illustration of the Doppler effect shows the change of wavelength caused by the motion of the source (in this case, raindrops).
Most of us know that the Doppler effect pertains to the change in frequency of a wave emitted by or scattered from a moving object. Our familiarity with this phenomenon is predominantly with sound waves, but the effect is the same for any wave. When a siren, for example, is moving toward us, the pitch (i.e., the frequency of the sound) is greater, whereas when it is moving away from us the pitch is lower—this is the Doppler effect in a nutshell. The amount by which the pitch is greater or lower, called the Doppler shift, is related to the speed of the object and to the speed of sound. Similarly, for radars, the amount by which the frequency of the radio waves reflected from a moving object changes depends on the speed of the object and the speed of propagation of radio waves, which is the speed of light.

Radio waves consist of oscillations that occur a given number of times every second, which by definition is the frequency of the wave. Each of these oscillations propagates at the speed of light toward the receiver, where they will be detected at a later time that is determined by the distance to the object and the speed of light. Because all oscillations travel the same distance and at the same speed from the object to the receiver, the receiver detects the same number of oscillations every second as are being created by the object. In other words, it detects the wave at the same frequency at which it was emitted.

For the situation in which the object is moving toward the radar receiver, the same number of oscillations is being created every second, but each successive oscillation occurs closer to the receiver, and takes less time to travel to the receiver than the previous one. As the motion of the object toward the radar results in more oscillations being received by the radar every second, the frequency is higher. If the object is moving away from the radar the oscillations will be received less often, and the frequency will be lower.

How big are raindrops?

The Ka-band ARM zenith pointing radar (KAZR), shown here on Gan Island, is one of the radars that will be deployed for the MAGIC field campaign.
The Doppler effect is employed by the ARM radars to determine the sizes of raindrops. This may at first seem puzzling, as the magnitude of the Doppler effect depends on the speed of an object, not its size. The speed at which a drop is moving toward or away from the radar might not be the same as the speed at which it would normally fall because of updrafts and downdrafts in clouds (and in the atmosphere in general). In the simplest case, the Doppler signal measured by a vertically pointing radar consists of frequency shifts, each shift corresponding to a given speed. By employing the relation between this speed and raindrop size, the Doppler signal can be related to the raindrop sizes.

How fast do water drops fall?

For drops near the surface of the Earth, the following approximate values will give an idea of the speeds involved.

  • The terminal velocity of a cloud drop, with typical diameter 20 millionths of a meter (approximately one thousandth of an inch), is one centimeter (~1/2 inch) per second.
  • For drops comprising drizzle, which are perhaps ten times as large, it is 3/4 of a meter (2 feet) per second.
  • Small raindrops, with diameters of one millimeter, fall at 4 meters (13 feet) per second, and large raindrops, with diameters 5 millimeters, fall at 9 meters (30 feet) per second (20 mph).

Another way to look at this is to consider the times required to fall (in still air) a distance of ten meters, the height of a three-story building. Approximate values are fifteen minutes for cloud drops, fifteen seconds for drizzle drops, two second for small raindrops, and one second for large raindrops.

Not only do we know the relation between raindrop size and terminal velocity, we also know how strongly raindrops of a given size reflect radio waves back to the radar. This information means that from the strength of the Doppler signal at a given frequency shift we can determine how many raindrops of the corresponding size are in the volume of air sampled by the radar.

The sizes of the raindrops, plus the number of drops of each size, comprise an important quantity in meteorology known as the drop size distribution (DSD). If the DSD is known, we can calculate the rainfall rate, as we know how much water is in each size of raindrop, how many raindrops of each size there are, and how fast drops of each size are falling.

–Ernie Lewis, MAGIC principal investigator