Generally, air velocities change with distance in such a way that more than one partial derivative is different from zero at any time. It turns out that any motion of an air parcel is a combination of five different motions, which can be represented by pairs of partial derivatives of velocity. One is translation, which we have already discussed. A second is a deformation of the air parcel, called stretching, which flattens and lengthens the air parcel. A third is another deformation of the air parcel, called shearing, which twists the air parcel in both the x and y direction. A fourth is pure rotation, called vorticity. A fifth enlarges or shrinks the parcel without changing its shape, called divergence. Let’s consider each of these alone, even though more than one is often occurring for an air parcel.
For these cases, we will assume that the partial derivatives are all positive. We could have all of the partial derivatives be negative and the directions would be different but the conclusions would be the same.
Translation simply moves the air parcel without stretching it, shearing it, rotating, or changing its area. There are no partial derivatives of velocities involved with translation.
Stretching deformation is represented by . u gets more positive with positive x and more negative with negative x (so that the derivative is always positive), making the parcel grow in the x direction. In the other direction, -v gets more negative with positive y and more positive with negative y (so that the derivative is always negative), making the parcel shrink in the y direction (see figure below). However, the total area of the air parcel will remain the same if . Shown in the figure is positive stretching deformation; negative stretching deformation occurs when the parcel is stretched in the y direction.
Shearing deformation is represented by . In this case, v gets more positive with positive x and more negative with negative x, resulting in the air parcel part at lower x getting pushed down, and the air parcel part at higher x getting pushed up. At the same time, u gets more positive with positive y and more negative with negative y, resulting in the air parcel part at lower y getting pushed to lower x and the air parcel part at higher y getting pushed to higher x (see figure below). The total area of the air parcel remains the same after the shearing occurs. Shearing deformation is positive when the air parcel stretches in the southwest/northeast direction and contracts in the southeast/northwest direction. It is negative when it stretches in the southeast/northwest direction and contracts in the southwest/northeast direction.
Both of these deformations cause stretching in one direction and contraction in the direction at right angles. In both cases, these motions cause air from different parts of the air parcel to come toward each other.
These deformations result in weather fronts. The air coming together is called frontogenesis.
Vorticity is represented by . Vorticity is special, and because it is special, it is represented by a Greek lower-case letter, zeta (ζ). In this case, the air parcel does not get distorted if and does not change area. It simply rotates (see figure below).
This difference in partial derivatives may look familiar to you.
Vorticity follows the right-hand rule. Note that vorticity is positive if the rotation is counter-clockwise and is negative if the rotation is clockwise. Vorticity describes the behavior of counter-clockwise rotation around low-pressure areas and clockwise rotation around high-pressure areas (in the Northern Hemisphere, but opposite in the Southern Hemisphere). Lows and highs are responsible for a lot of weather.
Divergence is represented by . Divergence is also special, and because it is special, it is represented by a Greek lower-case letter, delta ( ). When the divergence is positive, the air parcel grows (i.e., its area increases) (see figure below). If the divergence is negative, then the air parcel shrinks (i.e., its area decreases). Strictly speaking, is the horizontal divergence because it describes a change in parcel area projected onto a horizontal plane. Adding ∂w/∂z to the horizontal divergence gives the 3-D divergence.
The divergence can be written in vector notation:
Watch this video (1:56) for further explanation: