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thunderstorms and tornadoes. It is not endorsed by anyone other than
me. This tutorial has benefitted from comments by my colleague, Dr.
Robert Davies-Jones, but that does not imply that he endorses or
underwrites these contents. That is, if there are any errors or
problems, they are mine alone. Some minor changes/corrections have
resulted from e-mails sent to me by Mike Branick, Matt Bunkers, and
Prof. Bob Terrell (at Cornell University). Thanks!`

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Vorticity is a topic that can be plagued with confusion and
misunderstanding, although it has considerable power in helping to
understand the processes that go on in a real vortex, such as a
tornado. Vorticity is a property of the *flow* of air (or any
other fluid, for that matter). Air at rest has some temperature and
dewpoint, but it has no vorticity. Mere motion does not guarantee
that the air has vorticity, however. Moving fluids acquire properties
that allow someone to describe that flow in great detail. In order to
develop a deep understanding of these properties, you have to delve
into the discipline known as *fluid dynamics *. This subject can
involve quite sophisticated mathematics, but for the purposes of this
primer, I'm going to try to keep the mathematics to a minimum. I'm
going to include equations for the benefit of those who can use them,
but I hope to be able to achieve some success at explaining the basic
concepts even to those who know no mathematics at all. You can be the
judge of my success. Mathematical diversions can be skipped over by
the mathematically-challenged.

Vorticity is one of four quantities that are called the
*kinematic * properties of fluid flow. The wind flow is
described as a *vector * quantity; 2-dimensional vectors require
two numbers to describe them fully; 3-dimensional vectors need three
numbers, etc. Let us define the wind field as a function of position
in a coordinate system; for simplicity, let the coordinate system be
the standard 3-dimensional Cartesian system (*x* ,*y*
,*z* ), where (*x* ,*y *) are the two horizontal
coordinates and *z * is the vertical coordinate. Thus, since the
wind vector **V** can vary according to its position in space, it
is also denoted as **V**(*x* ,*y* ,*z* ). Vectors
will be indicated in boldface characters. Let **i**, **j**, and
**k** denote unit vectors of position in the 3-dimensional
coordinate system (see Fig. 1),

Fig. 1. Illustration of the basic definitions of the wind in a 3-dimensional Cartesian coordinate system.

corresponding to the *x* , *y* , and *z*
directions, respectively. The wind vector **V**, therefore, has
three components: *u* , *v* , and *w* . We also can
describe **V** as **V**_{h}+*w* **k**, where
**V**_{h} is the *horizontal* wind vector and
*w* **k** is the vertical wind vector.

For the *horizontal* (2-dimensional) wind, then, it takes
*two* quantities to describe that vector. The mathematically
correct way is to use the components I've already described and
illustrated. However, horizontal wind also can be described in
another way, that might be more familiar: *wind speed and wind
direction *. When meteorologists describe the wind in this way,
they refer to the direction *from which * the wind is blowing. A
wind from north to south is called a *northerly * wind, for
example. In Fig. 2,

Fig. 2. An example showing two ways to define the wind: by (u,v) components and by speed and direction.

the components are *u *= 3, *v* = 4. This means that the
speed of the wind is *V* = 5. The direction is shown as a ~ 37 deg ... but this is not the way a
meteorologist would refer to it. Instead, the *meteorological*
wind direction is described as b = 180 deg
+ a = 217 deg ... the wind is from the
south-southwest. If 180 deg + a is greater
than 360 deg (say a = 280 deg) then the
meteorological wind direction b = a - 180 deg = 100 deg. As shown in Fig. 2, the
wind speed and direction are related directly to the *u* - and
*v*
-components.^{[1]}

Vorticity is a 3-dimensional *vector*, similar to the wind
field itself. It's associated with the spatial variation of the wind
field - the wind changes from point to point in space, but not
*all* such changes produce vorticity. Since the 3-d vorticity is
a vector, it has components, just as the wind does: If **w** is the 3-d vorticity vector then denote
its three Cartesian coordinate components as (*x,h,z*). Each of the three components
represents the vorticity associated with flow in a 2-d plane normal
to the component. The *x*-component of the vorticity (*x*) is perpendicular to the *y-z* plane,
the *y*-component of the vorticity (*h*) is perpendicular to the *x-z* plane,
and the *z*-component of the vorticity (*z*) is perpendicular to the *x-y* plane.
Positive values of the vorticity components follow the right-hand
rule (see below). Details of this are found in
Mathematical Diversion -1. Meteorologists,
in general, are most accustomed to talking and thinking about the
*vertical* component of vorticity (*z*).

Vorticity in its 2-dimensional forms results from two different types of wind variation: curvature of the flow, and differences in wind speed in a direction perpendicular to the flow. When looking at a 2-d vortex, it's easy imagine how the curvature of the flow produces vorticity, as in Fig. 3.

Fig. 3. An illustration of circular flow, showing the vorticity associated with this flow. The thin lines show the flow direction and the bold line indicates the sense of rotation associated with the vorticity. It is assumed that there is no shear vorticity in this flow, except at the exact center of the circular flow, where there isinfiniteshear (the flow must go to zero atr=0, implying aninfinitevorticity).

For circular flow, at least, pure curvature vorticity is physically unrealistic (obviously). Real atmospheric vortices (in 2-d flow) have a combination of shear and curvature vorticity (see below).

However, as suggested by Fig. 4 ,

Fig. 4. An illustration of straight flow (no curvature), but with speed shear, showing the vorticity associated with this flow. As in Fig. 3, the thin lines show the flow direction and the bold line indicates the sense of rotation associated with the vorticity.

variations in wind speed can also produce vorticity. The figure
indicates the sort of rotation that is implied by this form of wind
shear, but the fluid need not actually be rotating this way. Fluids
do not behave as solids do - imagine rolling a pencil in between the
palms of your hand. Since a pencil is a solid object, it can only
rotate between your hands. However, *fluids* have another
"option" ... to see this, imagine pushing on a book that has been
glued to the top of a table. What happens is shown in
Fig. 5.

Fig. 5. Schematic showing the shearing action associated with a book, glued to a table top, subjected to a shearing force.

The pages of the book can slide past each other and the book is
"sheared" without rotation. In a very similar way, the layers of a
fluid can slide past each other when subjected to shearing forces.
This produces vorticity in the flow, but *without* closed
rotation!

Now, picture a 2-d Cartesian coordinate system that is set up such
that one axis is aligned with the flow at some point in the fluid,
and the other axis is perpendicular to the first. In meteorology,
this is referred to as "natural" coordinates. The two different
contributions to vorticity are easily seen in natural coordinates.
The shearing contribution is shown in
Fig. 6a, whereby vorticity arises
from changes in wind speed *V* in the direction perpendicular to
the flow (at the central point). The curvature contribution shown in
Fig. 6b is derived from changes in
wind direction a along the flow (at the central point).

Fig. 6. Schematic in natural coordinates, defined at the point indicated by the small circle at the base of the dark vector; thes-direction is always pointed in the direction of the flow and then-direction is pointed to the right and perpendicular to thes-direction. In (a), the flow is straight, but speed increases to the right; in (b) the flow is curved but the speed is not changing in the direction of the flow.

It is most common for the 2-d flow to have *both *shear and
curvature at the same time, so it can happen that the contribution to
2-d vorticity from shear is of the *opposite *sign as the
contribution from curvature, just as it is possible that they have
the *same *sign. In fact, as a special case, the flow can form a
closed circle even though the total vorticity is zero! This happens
when the shear term is of the opposite sign to, and is exactly the
same magnitude as the curvature term. Even stranger, if the shear
magnitude is large enough, the total vorticity could have the
opposite sign to the curvature vorticity! Imagine a Rankine Combined
Vortex (see Fig. 8, below). It is
discussed in Mathematical Diversion - 2
that outside of the so-called Radius of Maximum Wind
(*r _{o}*) in the Rankine Combined Vortex, the tangential
velocity decreases in proportion to 1/

For 2-d flow (say, at the surface), the vorticity vector points either up or down, perpendicular to the surface. The voriticity sign convention follows what is called the "right-hand rule": hold the right hand on the paper with the thumb in the "hitchhike" position; if the fingers on the right hand curve in the direction associated with the vorticity (as in Figs. 3 and 4), then when the thumb is pointing up, the vorticity is positive. If the fingers curving in the sense of rotation force the thumb to point downward, the vorticity is negative.

It's relatively easy to extend this concept to three dimensions -
the 3-dimensional vorticity vector includes contributions from three
mutually perpendicular surfaces: a horizontal surface (*x , y
*), and two vertical surfaces (*x , z *) and (*y , z *).
The total 3-dimensional vorticity vector is the vector sum of these
three. It also follows the same right-hand rule: if the right-hand's
thumb points in the direction of the vorticity vector, the fingers
curl about that thumb in a way that shows the sense of rotation
produced by that 3-dimensional vorticity.

Consider the following example of the vertical component of
vorticity (*z*), shown in
Fig. 7.

Fig. 7. A horizontal flow field, with the winds indicated by meteorological "wind barbs" instead of vectors. A positive vertical component of vorticity (z) is indicated by the stippling, getting larger as the stippling gets denser; the lines with arrows are subjectively-drawnstreamlines(see discussion below).

Wind barbs are shown instead of wind vectors ... the notation is
conventional. For this system, it can be seen that the peak vorticity
is to the southwest of the center of circulation, where the curvature
is high and the shear contribution to the vorticity is of the same
sign as the curvature. Flow streamlines have been sketched in by hand
... *streamlines are defined to be lines that are everywhere
parallel to the wind*** **(strictly speaking, they are
everywhere *tangential* to the wind flow). Where the streamlines
have high curvature, the vorticity is high owing to the curvature
contribution. For instance, note the zone wrapped around the vortex
where the streamlines make an abrupt turn; this coincides with a long
zone of high vorticity (along a trailing cold front-like structure).
On the other hand, to the northeast of the center of circulation
(near the edge of the figure), there is a region of high vorticity
due to the shear contribution. For these particular data, the shear
term by itself does not create very much high vorticity, unless it
coincides with contributions from the curvature term, whereas there
are several vorticity maxima associated with strong flow curvature.
This is not always the case, but depends on the flow situation under
consideration.

A common problem in understanding the role of vorticity in flow
curvature is that the vorticity field is what is called a *Galilean
invariant * property of the flow. Simply put, Galilean invariance
means that it you can change to any coordinate system moving with a
constant velocity and it does not change the vorticity. However, the
partitioning of the contributions to vorticity from shear and
curvature is *not* a Galilean invariant property (see Viudez and
Haney 1996). The amount of curvature we see in the flow can depend on
the motion of the coordinate system.

To see this, consider the following simple example. Suppose we have a Rankine Combined Vortex, as shown in Fig. 8.

Fig. 8. Schematic showing the effect of adding increasingly strong constant background flows to a Rankine vortex. The vortex core is shown by the stippling, the blue arrows indicate the vortex flow, the red arrows denote the background flow, the green arrows show the sum of the vortex and the background winds, and the streamlines show the combined flow.

In this vortex, the flow is perfectly symmetric and circular, with
contributions from both shear and curvature. However, in Fig. 8b, a
background flow (that is constant everywhere) is added to the vortex,
with that background flow being pure westerly with a speed equal to
the speed at the radius of maximum winds in the pure vortex flow.
Streamlines have been added to indicate the flow of the sum of these
two. In Fig. 8c, the background westerly flow speed is twice that of
that of the maximum vortex flow, whereas in Fig. 8d, the background
westerly flow is three times that of the maximum vortex flow. If
considered in a coordinate system traveling *with * the
background flow in Figs. 8b-d, all the flows would be identical to
that of Fig. 8a. However, it can be seen that by adding this
background flow, the curvature of the resultant flow in a fixed
coordinate system (i.e., *not * moving with the background flow)
decreases as this background flow increases. In all cases, the
*vorticity * remains the same, because the background flow has
no vorticity!

In situations with mesocyclones as seen on a Doppler
radar,^{[2]} it is
common for the flow in a ground-relative sense to be very asymmetric
(perhaps with stronger inbound flow than outbond flow or vice-versa),
because the vortex is moving. For a moving vortex, the flow on the
side of the vortex that is going in the direction of storm motion,
its ground-relative flow is the sum of the vortex flow and its
motion. On the opposite side of the vortex, the flow is the
difference between the vortex flow and the motion, so it is
relatively slower. In a coordinate system moving *with* the
vortex, the vortex flow typically is much more symmetric.

I already have defined streamlines to be lines everywhere tangent
to (i.e., parallel to) the wind flow. Remember that most
meteorologists are accustomed to dealing with the *vertical
component * of the vector vorticity. However, the vertical
component of the vector vorticity typically is the *smallest* of
the three
components.^{[3]}
That is, the variation of wind with height usually is much larger
than the variation of wind along a more or less level surface (like
on a surface of constant pressure). The *horizontal* part of the
vorticity has two components, just like the horizontal wind has two
components. Like the wind vector, the 3-dimensional vorticity vector
is dominated by its horizontal part whenever the flow field can be
said to be in near-hydrostatic balance (as discussed in the
Mathematical Diversion - 1). Therefore, it
should be relatively easy to imagine a field of vorticity vectors
that might look very similar to a field of wind vectors, but which in
fact are horizontal vorticity vectors **w**_{h} (which have both *x*- and
*y*-components) *derived* from the wind. As shown in
Mathematical Diversion - 3, the horizontal
vorticity vectors are dominated by the vertical wind shear of the
horizontal wind. In fact, under the hydrostatic assumption, the
horizontal vorticity vector is very nearly perpendicular to the
vertical wind shear vector. However, there is no guarantee what the
relationship is between the direction of the horizontal *wind*
vector on some surface and that of the associated horizontal
*vorticity* vector, which depends on the *vertical shear *
vector of the horizontal wind.

When the horizontal vorticity vector and the horizontal wind
vector are parallel, the horizontal vorticity is said to be
*streamwise* , and when they are perpendicular, the horizontal
vorticity is said to be *crosswise* . When the horizontal wind
and vorticity vectors are parallel but point in opposite directions
(antiparallel), the horizontal vorticity is said to be
*antistreamwise* .

As noted in
Doswell
(1991), the notions of *helicity * can be developed from
this point rather easily. That is not the point of *these*
notes. Rather, I now want to move on to the idea of *vortex
lines*. If streamlines are everywhere tangent to the velocity
vectors, it seems straightforward to draw upon this analogy and
picture lines that are everywhere tangent to the *vorticity*
vectors. From a strictly two-dimensional viewpoint, this is
moderately interesting, but the real value comes from the realization
that there also is a vertical component to the vorticity vector.

In the same way that streamlines point in the direction of the wind flow, the vortex lines point in the direction of the vorticity vectors. The same old "right-hand rule" applies: if the right-hand thumb points in the direction of the vortex line, the sense of rotation is given by the fingers of the right hand.

When it comes to tornadoes and mesocyclones in storms, we're
considering the rotation about a *vertical* axis. That is, the
vertical component of the vorticity is what's important even though,
on large scales associated with a thunderstorm's *environment*,
the horizontal component is larger. Mesocyclones and tornadoes are
not synoptic-scale! Recall that I've indicated the vertical component
is often much smaller than the horizontal component in situations
where hydrostatic balance
applies.^{[4]} If
we make the approximation that the storm inflow and updraft simply
acts on the *environmental * horizontal vorticity, an
interesting picture begins to emerge.

If the originally horizontal vortex lines can be tilted into the vertical by some mechanism, then the vortex lines develop a vertical component, and so the vertical component of vorticity can be changed simply by tilting of the existing purely horizontal vorticity. It's widely accepted that this tilting of horizonal vorticity is the primary mechanism for creating mid-level mesocyclones in supercells (e.g., see Brooks et al. 1994)

Suppose the flow is such that the low-level wind shear is
producing only crosswise vorticity. For example, a pure change of
wind speed with height (wind direction remains constant) produces
only crosswise vorticity. This is analogous to the case shown in Fig.
5 with the sheared book. Vortex lines are all horizontal in the
environment.^{[5]}
Such a case is depicted in Fig. 9.

Fig. 9. Schematic showing the development of counter-rotating vortices on either side of an updraft from tilting the vortex lines associated with purecrosswisevorticity. Notice the sense of rotation indicated by the circles around the vortex lines, following the right-hand rule. This schematic is in the Northern hemisphere, where positive vertical vorticity iscyclonic, indicated by the "C" and negative vertical vorticity isanticyclonic, indicated by the "A".

On either side of the updraft, the vortex lines are tilted upward by the updraft, such that on one side of the updraft, a cyclonically rotating vertical component of vorticity is created. On the opposite side of the updraft, of course, an anticyclonically rotating vertical component of vorticity is found.

What happens when the vorticity is purely streamwise? This is illustrated in Fig. 10;

Fig. 10. As in Fig. 9, except that it is a schematic showing the development of a cyclonically rotating updraft from tilting the vortex lines associated with purestreamwisevorticity.

rather than producing a cyclonic-anticyclonic couplet of vortices
flanking the updraft, what one finds is a helically rotating updraft.
Note the difference between Fig. 10 and Fig. 9: the vortex line is
shown going up in Fig. 10 but not coming down - this is an important
difference between streamwise and crosswise tilting situations. A
vortex line brought up in the updraft in the streamwise case
eventually turns downward *somewhere* downstream, but this isn't
shown in Fig. 10. It turns out that if we define some area on a
horizontal plane, the vorticity about the vertical in that plane is
related to the number of vortex lines inside that area. Thus, the
more densely packed the vortex lines passing through some plane, the
greater the vorticity. This is illustrated in
Fig. 11, where the vortex lines are
rather densely packed in one region as they pass through the plane.

Fig. 11. Schematic showing vortex lines passing through a quasi-horizontal surface.

It's a law of fluid dynamics that vortex lines must form closed
loops (Fig. 12). They can
*appear* to terminate on a solid surface (as suggested by a
tornado, for instance), but since the flow at a solid surface must
equal zero (for a real fluid with finite viscosity), these vortex
lines actually must diverge close to that solid surface and
eventually loop back around on themselves - at least in principle. In
a real, turbulent fluid, the picture is far more complicated than can
be imagined.

Fig. 12. Schematic showing (a) vortex lines "terminating" on solid surfaces, or (b) forming a closed loop.

This can have some interesting consequences in trying to picture
the vortex line structures near the surface - as noted above, the
flow can still have horizontal vorticity very near the surface (see
Warsi 1993; p. 22). Thus, a vortex line approaching the surface in a
viscous fluid must become increasingly horizontal. There are no other
options, at least in the simple world of kinematics. This law has
some pretty obvious implications for atmospheric vortices, such as
mesocyclones and tornadoes. What we see as a funnel aloft (or a
Tornado Vortex Signature [TVS] on a Doppler radar) does *not*
have vortex lines that begin and end somewhere in the air - since
that's not allowed, what must be happening is shown in
Fig. 13.

Fig. 13. Schematic showing the vortex lines associated with a TVS aloft.

The vortex lines that cluster somewhere aloft must diverge above and below the intense vortex, always forming closed loops but perhaps seeming to end near the surface. The figure also shows the vortex lines ending at the tropopause, but their actual behavior might be even more complex there than near the surface.

**What we call "touchdown" of a tornado is actually an
intensification of the vortex at the ground**, which can be viewed
as a clustering of the vortex lines. Nothing material is actually
coming down as the tornado "touches down" - this process might more
properly called a "spin-up" or "surface intensification." The intense
part of a vortex aloft can build both upward and downward, as shown
in Fig. 14.

Fig. 14. As in Fig. 13, except that the vortex has intensified both upward and downward and now is a tornado.

Apparently, tornadic vortices can intensify first at the surface (Fig. 15) and then build upward.

Fig. 15. Schematic showing vortex lines in a situation where the vortex intensifies first at low levels.

After some time period, Fig. 15 could evolve into something resembling Fig. 14; that is, it also can evolve into a tornado through a deep layer. Thus, the intensified vortex that we call a tornado might be seen to develop upward from the surface, downward from aloft, or both upward and downward.

` `

`Note``: Comments and questions can be sent to me
at
cdoswell@earthlink.net.`

Bluestein, H.B., 1992: *Synoptic-Dynamic Meteorology in
Midlatitudes . Vol. I: Principles of Kinematics and Dynamics *.
Oxford University Press, 431 pp.

Brooks, H.E., C.A. Doswell III, and J. Cooper, 1994: On the
environments of tornadic and non-tornadic mesocyclones. *Wea.
Forecasting*, **9**, 606-618.

Doswell, C.A. III, 1984: A kinematic analysis of frontogenesis
associated with a nondivergent vortex. *J. Atmos. Sci* .,
**41**, 1242-1248.

______, 1991:
A
review for forecasters on the application of hodographs to
forecasting severe thunderstorms. *Nat. Wea. Dig* .,
**16**, No. 1, 2-16.

Viudez, A., and R.L. Haney, 1996: On the shear and curvature
vorticity equations. *J. Atmos. Sci*., **53**, 3384-3394.

Warsi, Z.U.A., 1993: *Fluid Dynamics. Theoretical and
Computational Approaches.* CRC press, 683 pp.