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Weather Observation and Analysis
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Chapter 14. VORTICITY
14.1 Curl Like divergence and gradient, curl involves derivatives of the components of a vector. Like gradient, curl is a vector. The mathematical r way of writing the curl of some vector v is r ∇×v r Even if the vector v is entirely horizontal, the curl is a fully three- dimensional vector.
The definition of curl (in three dimensions) is most clearly written
in the form of a determinant, as follows:
i j k ∂ ∂ ∂ r ∇×v = ∂x ∂y ∂z u v w Here we have explicitly assumed that the vector in question is the velocity, so the three velocity components appear on the bottom row.
If you know what a determinant is, great. If you don’t know what a determinant is or how to compute in three dimensions, don’t worry. You
can get by with knowing what the three components of the curl are:
r ⎛ ∂w ∂v ⎞ ⎛ ∂u ∂w ⎞ ⎛ ∂v ∂u ⎞ ∇×v = ⎜ ⎜ ∂y − ∂z ⎟i + ⎜ ∂z − ∂x ⎟ j + ⎜ ∂x − ∂y ⎟k ⎟ ⎜ ⎟ ⎠⎝ ⎠⎝ ⎝ ⎠ Why do they call it the curl? Because it measures the tendency of the vector field (in this case, the velocity) to rotate. Consider, for ATMO 251 Chapter 14 page 1 of 16 example, the vertical component of curl, the last term in the preceding equation. For an entirely two-dimensional world, this is the only component of curl that is non-zero. Suppose now that you are looking down on a low pressure center, with its associated cyclonic (counterclockwise) circulation. If the wind is in geostrophic balance, we know that the divergence is zero, but what about the curl?
ATMO 251 Chapter 14 page 2 of 16 To proceed, let’s check out the sign of the vertical component of curl. East of the circulation center, the wind ought to be from south to north (a positive v), while west of the circulation center, the wind ought to be from north to south (a negative v). So v changes in the x direction;
specifically, it increases with increasing x. That means that the derivative of v with respect to x is positive.
North of the circulation center, the wind should be blowing from east to west, while south of the circulation center, the wind should be blowing from west to east. So to the south u is positive and to the north u is negative. Thus the derivative of u with respect to y is negative, since u decreases with increasing y.
Taking stock, we have the vertical component of curl equal to a positive term minus a negative term. Since minus a minus is a plus, the vertical component of curl must be positive overall.
You can confirm the same thing for the other two components.
Imagine that you’re looking at the x = 0 plane from a position on the positive x side. Suppose you see counterclockwise rotation of the wind. If you work out the signs of the various derivatives, you’ll find that the x component of curl is positive. Repeat the procedure with counterclockwise rotation in the y plane, and you get a positive y component of curl.
To remember all this, go back to the gig ‘em rule. Point the thumb of your right hand in the direction of the component of curl. The vector pattern that produces a curl component in that direction is given by the fingers of your right hand if you are forming the shape of a gig ‘em. The vectors are oriented (or the wind is blowing) along your fingers, pointing from the hand along the curve of the fingers to the tips of your fingernails.
ATMO 251 Chapter 14 page 3 of 16
14.2 Vorticity So far we’ve used the term curl exclusively. In meteorology, or in fluid mechanics in general, the curl of the wind or flow field (that is, the curl of the velocity) gets a special name: the vorticity.
Meteorologists are actually a little bit lax about the term. When they say “vorticity”, they are generally referring only to the vertical component of the curl of the wind. When the full three-dimensional curl is meant, meteorologists generally refer to the “vorticity vector”. We will follow this convention here. For the rest of this chapter, the vorticity means the vertical component of the curl of the velocity.
Vorticity has a crucial physical interpretation. The vorticity is a measure of the “spin” of the wind about a vertical axis, with counterclockwise spin being positive. Imagine you stick a four-pronged propeller into the air, with the spin axis of the propeller vertical. We’ll take the Cartesian coordinate system as being oriented so that the four prongs are sticking directly north, south, east, and west, respectively.
Think about the two prongs sticking east and west. For the wind to try to cause them to spin counterclockwise, you’d want the northward component of wind to be larger on the east prong than on the west prong.
In other words, the derivative of v in the x direction would need to be positive, and the larger the derivative, the stronger the turning impulse.
As for the two prongs sticking north and south, to get them to turn counterclockwise you’d want the eastward component of wind to be larger on the south prong than the north prong. So the derivative of u in the y direction would need to be negative, and the more negative the better.
Thus, the larger the vorticity, the greater the tendency of the wind to spin a propeller.
ATMO 251 Chapter 14 page 4 of 16There are lots of different wind patterns that would cause a propeller to spin and that therefore have vorticity. We’ve considered only one example so far, that of wind rotating in a circle. Vorticity that results from horizontal variations in the direction of the wind is called curvature vorticity. For positive curvature vorticity, the wind must be curving counterclockwise, or toward the left facing downwind. Negative curvature vorticity would have the wind curving in the other direction.
The other type of vorticity is called shear vorticity. Here, the shear refers to horizontal shear rather than the vertical shear we discussed in the last chapter. Imagine a perfectly straight wind field, and let’s say it’s blowing from west to east. If the wind speed changes in the north-south direction, so that (for example) the westerly winds to the north are weaker
ATMO 251 Chapter 14 page 5 of 16than the westerly winds to the south, we have horizontal shear. But notice that this situation corresponds to one of the propeller cases, so that the winds on the north and south prongs of the propeller would act to cause the propeller to turn. The north-south wind, being zero everywhere, has no effect, so we have a counterclockwise-rotating propeller and therefore we must have positive vorticity. The basic rule is that if you are facing downwind and the winds are weaker to your left (and stronger to your right), you have positive shear vorticity. If winds are weaker than your right, the shear vorticity is negative.
The total (vertical component of) vorticity is the sum of the shear vorticity and the curvature vorticity. Sometimes you have both simultaneously; sometimes they add together and sometimes they cancel.
The atmosphere, even if the wind is not blowing, is rotating counterclockwise (in the Northern Hemisphere). That’s because the Earth itself is rotating, and the atmosphere is rotating with it.
Including the effect of the Earth’s rotation gives us the absolute vorticity. The absolute vorticity has two contributing terms: the vorticity associated with the wind and the vorticity associated with the spin of the Earth. We call these two contributing terms the relative vorticity and the planetary vorticity.
The planetary vorticity is exactly equal to the Coriolis parameter f.
The relative vorticity is usually represented as the Greek letter ζ. So the absolute vorticity is ζ+f.
ATMO 251 Chapter 14 page 6 of 16
14.3 Vorticity and Convergence There are several reasons to care about vorticity. One simple reason is that high vorticity implies strong cyclonic circulation centers or strong troughs. If cyclonic vorticity is concentrated along a line, it marks the windshift associated with a front. (Fronts always have cyclonic vorticity.) But perhaps the most important reason is related to the relationship between vorticity, convergence, and vertical motion, which lets us diagnose the intensification of cyclonic circulation centers or troughs, and their converse, anticyclonic circulation centers or ridges.
It would be great if we could diagnose changes in pressure directly.
Since pressure is proportional to the weight of air in the overlying air column, one might think there’s some hope of diagnosing temperatures and thereby diagnosing pressures. But that’s not very workable, because horizontal temperature advection and vertical motion tend to act to oppose each other, and the net temperature change is difficult to determine.
ATMO 251 Chapter 14 page 7 of 16We can also diagnose the forces acting on the wind, and thereby diagnose changes in the wind. But to do that, we would need to be able to predict changes in the pressure gradient force, so we have the same problem as if we’re trying to diagnose pressure.
Vorticity, however, is fairly easy to diagnose. There are two basic techniques, the first involving convergence and vertical motion, the second involving something called potential vorticity. First, we’ll talk about the connection between vorticity, convergence, and vertical motion.
In Chapter 9, the connection between convergence, divergence, and vertical motion was first discussed, and they go hand in hand. A brief review: if there’s large-scale upward motion, there must be convergence in the lower troposphere and divergence in the upper troposphere.
Conversely, if there’s large-scale downward motion, there must be divergence in the lower troposphere and convergence in the upper troposphere.
Now for the connection between all this and vorticity:
Dζ r = −( f + ζ )(∇ h • v ) + other stuff Dt This equation states that the rate of change of vorticity following an air parcel is proportional to the absolute vorticity times the negative of the divergence (i.e., the convergence), plus some other stuff. The “other stuff” would be rather ugly if I wrote it all out, but fortunately, the larger the horizontal scale, the smaller the other stuff. Once we get to weather systems larger than a few hundred kilometers or so, the other stuff is completely negligible and the divergence and convergence controls the vorticity.
Consider changes to vorticity in the lower troposphere. If there’s a low pressure center, it will necessarily have cyclonic relative vorticity, which is positive. If the low intensifies, the vorticity must increase. The above equation states that, for the vorticity to increase, there must be convergence and therefore aloft there must be upward motion. If, instead, the low weakens, there must be divergence present and downward motion aloft.
High pressure centers have anticyclonic relative vorticity. This is negative in the Northern Hemisphere, but the absolute vorticity (the sum of the relative vorticity ζ and the planetary vorticity f) will almost always be positive in the Northern Hemisphere. So, for the high pressure center to intensify, the vorticity must decrease. Hence there must be divergence present, and therefore downward motion present aloft.
Notice that if the issue is the intensification or weakening of a cyclonic disturbance (a trough) in the upper troposphere, you still need convergence, but since that convergence would be taking place in the
ATMO 251 Chapter 14 page 8 of 16upper troposphere it would mean downward motion in the middle troposphere.
The basic rule is as follows: mid-tropospheric upward motion implies that the relative vorticity will become more positive (or less negative) at low levels and less positive (or more negative) aloft, and midtropospheric downward motion implies that the relative vorticity will become more negative (or less positive) at low levels and less negative (or more positive) aloft.
14.4 Predicting Vertical Motion Using the principles described in the previous paragraph, we can use vertical motion to diagnose intensification and weakening of weather systems. All that’s left is to predict vertical motion!
In past chapters, we’ve encountered several indicators of vertical motion. One is friction: friction produces cross-isobar flow and convergence within lows, while the cross-isobar flow is outward and divergent around highs. However, frictionally-induced vertical motion is the one kind of vertical motion that doesn’t lead directly to intensification of large-scale weather systems, because the friction that causes the vertical motion also slows down the wind. (That’s one of the things hidden in the “other stuff”.) Another is the effect of curvature and ageostrophic winds around upper-level troughs and ridges. Upward motion is expected downstream of troughs and upstream of ridges, while downward motion is expected downstream of ridges and upstream of troughs. So, for example, if a lowlevel cyclone is located downstream of a trough, one would expect to find upward motion and thus low-level convergence and intensification of the cyclone.
Another is the effect of accelerations within a jet streak. Upward motion is expected beneath the right-hand side of the entrance region and beneath the left-hand side of the exit region, while downward motion is expected beneath the left-hand side of the entrance region and beneath the right-hand side of the exit region.
Another is the effect of intensification or weakening of fronts. If a front is becoming stronger, there should be upward motion on its warm side, while if a front is weakening, there should be upward motion on its cold side.