Published on *METEO 300: Fundamentals of Atmospheric Science* (https://www.e-education.psu.edu/meteo300)

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.

Visit this website to see a loop [1]. 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.

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).

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.

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 _{o}* +

$$\begin{array}{l}u\left({x}_{o}+dx,{y}_{o}+dy\right)\approx u({x}_{o},{y}_{o})+\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy\hfill \\ v\left({x}_{o}+dx,{y}_{o}+dy\right)\approx v({x}_{o},{y}_{o})+\frac{\partial v}{\partial x}dx+\frac{\partial v}{\partial y}dy\hfill \\ \text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\hfill \\ \text{translationrelativemotion}\hfill \end{array}$$[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@FE30@

We see that *u(x _{o},y_{o})* and

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

$\frac{\partial u}{\partial x}$ is the following change in velocity in the *x* direction:

$\frac{\partial v}{\partial y}$ is the following change in velocity in the *y* direction:

$\frac{\partial u}{\partial y}$ is the following change in velocity in the *y* direction:

$\frac{\partial v}{\partial x}$ is the following change in velocity in the *x* direction:

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 [3]

**Links**

[1] http://mp1.met.psu.edu/~fxg1/SAT_US/anim16wv.html

[2] mailto:MathType@MTEF

[3] https://www.e-education.psu.edu/meteo300/sites/www.e-education.psu.edu.meteo300/files/Transcripts/METEO%20300%3A%20Partials%20Velocity%20Distance.html