METEO 300
Fundamentals of Atmospheric Science

9.2 Watch these air parcels move and change.

satellite image of north america showing water vapor in atmosphere
Water vapor in the atmosphere over North America showing the behavior of different air parcels as they interact.
Credit: NOAA

The water vapor image from the GOES 13 satellite, above, indicates different air masses over the United States. As we know from Lesson 7, the water vapor image actually shows the top of a column of water vapor that strongly absorbs in the water vapor channel wavelengths, but it is not a bad assumption to think that there is a solid column of moister air underneath the water vapor layer that is emitting and is observed by the satellite. In a single snapshot, it is not possible to see what happens to the air parcels over time. But if we look at a loop, then we can see the air parcels moving and changing shape as they move.

Watch a Loop

Visit this website to see a loop. Pick any air parcel with more water vapor in the first frame and then watch it evolve over time. What does it do? Maybe it moves; it spins; it stretches; it shears; it grows. Maybe it does only a few of these things; maybe it does them all.

We can break each air parcel’s complex behavior down into a few basic types of flows and then mathematically describe them. We will just describe these basic motions here and show how they lead to weather.

blue oval
An air parcel
Credit: W. Brune

Assume that we have an air parcel as in the figure above. We focus on motion in the two horizontal directions to aid in the visualization (and because most motion in the atmosphere is horizontal) but the concepts apply to the vertical direction as well. If the air parcel is moving and does not change its orientation, shape, or size, then it is only undergoing translation (see figure below).

blue ovals with green and black arrows representing translation as described in caption text.
Air parcel undergoing translation in x-direction (top) and at 45o (bottom). Black arrows are the wind field; green arrows are the motion of the air parcel. The orientation, shape, and size of the air parcel does not change as it is translated.
Credit: W. Brune

The air parcel can do more than just translate. It can undergo changes relative to translation, and its total motion will then be a combination of translation and relative motion. Let’s suppose that different parts of the air parcel have slightly different velocities. This situation is depicted in the figure below.

blue oval with green arrows on a graph representing relative motion as described in caption text.
Air parcel with relative motion for two different points in the air mass separated by dx in the x direction and dy in the y direction and with different velocities at each point.
Credit: W. Brune, after R. Najjar

If we consider very small differences dx and dy, then we can write u and v at point (x+dx,y+dy) as a Taylor series expansion in two dimensions:

u( x o +dx, y o +dy )u( x o , y o )+ u x dx+ u y dy v( x o +dx, y o +dy )v( x o , y o )+ v x dx+ v y dy                                 ______     ___________                              translation   relative motion This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[9.1]
 

We see that u(xo,yo) and v(xo,yo) are the translation, and the relative motion is expressed as gradients of u in the x and y directions and gradients of v in the x and y directions.

There are four gradients represented by the four partial derivatives. Each can be either positive or negative for each partial derivative.

u x This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. is the following change in velocity in the x direction:
 
change in velocity in the x direction as described in the text above
v y This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. is the following change in velocity in the y directions:
 
arrows pointing up and down to represent change in y velocity as described in the text
u y This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. is the following change in velocity in the y direction:
 
arrows pointing right and left to represent change in velocity as described in the text

v x This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. is the following change in velocity in the x direction:

 
arrows pointing up and down to represent x velocity as described in the text

Note that a partial derivative is positive if a positive value is becoming more positive or a negative value is becoming less negative. Similarly, a negative partial derivative occurs when a positive value is becoming less positive or a negative value is becoming more negative. Be sure that you have this figured out before you go on.

Watch this video (2:38) for further explanation:

Click here for a transcript for Partials Velocity Distance Video