C  =  V • dl  =  V (cos a) dl  =  (u dx + v dy)

as illustrated above. Note that the nonbold V is the magnitude of the bold vector V velocity. Note that a dot product simply multiplies the parallel components of 2 vectors, as illustrated (u is parallel with the x-axis, and v parallel with the y-axis). The symbol denotes a line integral, meaning that integration is performed around the entire closed curve. Recall that an integral is another form of summation. Thus, the equation above is simply a summation of the components of wind parallel with (tangent to) the curve. Mathematical convention is that circulation is defined as numerically positive when counterclockwise (as shown) and negative when clockwise. Recalling flows around Highs and Lows, you should see that circulation around a normal Northern Hemisphere Low is thus positive, and around such a High is negative.

Relative Vorticity

Solving the equation for circulation is very difficult for such a shape as above, so let's consider a simpler one :

Supposing this as an infinitesimally small plane horizontal area (which d's denote), the circulation around it is :

dC  =  u dx + (v + (¶v/¶x) dx) - (u + (¶u/¶y) dy) - v dy  =  (¶v/¶x - ¶u/¶y) dx dy

obtained performing the line integral around the area to direction shown. ¶ denotes a partial derivative. Noting that the differential area dA = dx dy and considering the limiting value of dA approaching 0 (infinitesimally small area shown vanishes), the equation can be written :

lim _dA ® 0 dC/dA  =  ¶v/¶x - ¶u/¶y  =  w_z =  z

w_z or z : vertical component of relative vorticity

This states that the vertical component of relative vorticity equals circulation ÷ area @ the limit of area approaching 0 (i.e., at a point - P in the diagram). Though the point circulation shown occurs in the x-y (horizontal) plane, it is considered a vertical component (z-component) because the circulation is positive (using the right hand rule) around the vertical (z) axis.

Doing similar calculations for circulation around the x &; y axes (in the y-z & x-z planes) and recalling that the velocity vector V = i u + j v + k w yields the horizontal components of relative vorticity :

x-component : lim _dA ® 0 dC/dA  =  ¶w/¶y - ¶v/¶z  =  w_x
y component : lim _dA ® 0 dC/dA  =  ¶u/¶z - ¶w/¶x  =  w_y

Combining the components in 3-dimensional space and using the definition of cross-products and the del operator (Ñ), you may see that :

lim _dA ® 0 dC/dA  =  i w_x + j w_y + k w_z  = Ñ × V  =  w

w : Relative vorticity vector

For which dA is the infinitesimal area normal (perpendicular) to w, around which the total circulation occurs. Thus strictly speaking, relative vorticity w is the curl of the velocity vector.

Note that the vertical component of relative vorticity w_z is given the special symbol z. This is done because synoptic scale horizontal winds are typically about 100-1000 times stronger than vertical winds (for example, 20 m/sec compared with 5 cm/sec aloft), so the large scale vertical component is dominant (and thus most often used). This is not always so for mesoscale and microscale phenomena such as supercell thunderstorm circulations. z is often simply called "vorticity", with the understanding that its vertical component is meant. I believe z was chosen because it is the Greek letter corresponding with the letter z, referring to the vertical axis.

Typical magnitudes

Similar with divergence, synoptic and large mesoscale relative vorticity magnitudes are typically about 10^-4 sec^-1. A rough estimation can be made using typical wind speeds around an idealized synoptic cyclone of typical scale :

Computing the partial derivatives as finite differences (this is not strictly correct, but does provide an idea of orders of magnitude), mks units used :

z  @   Dv/Dx - Du/Dy  =  {((10) - (-15))/700000} - {((-12) - (12))/700000}  =  49/700000  =  7 × 10^-5 sec^-1

Synoptic scale magnitudes tend to be greater aloft, where winds are typically stronger, than near the surface; though localized magnitudes can be much greater in near-surface circulations (do such a calculation for a tornado or hurricane eyewall, for example).

Important ideas

Though relative vorticity is defined for point locations as illustrated, air circulation around a region is the sum of vorticities at all points contained therein :

Mathematically,

(u dx + v dy)  =  (¶v/¶x - ¶u/¶y) dx dy

relating circulation and relative vorticity as described above. (This is for the vertical component, but other components can be done similarly.) This is a useful relation for air circulations such as those typically occurring around Lows & Highs, thru trofs and ridges aloft, hurricanes, and tornadoes, among others; and equations above are valid for a defined outer boundary involving these.

Relative vorticity has a perpendicular characteristic compared with divergence (they can't truly be perpendicular because one is a scalar and the other a vector). Note that :

Divergence  =  D  = Ñ • V
Relative vorticity =  w  =  Ñ × V

The only mathematical difference being that one is a dot product of the del operator with wind velocity, the other a cross product. Magnitudes of dot products and cross products do indeed result from multiplication with perpendicular vectors. Thus, a purely divergent flow does not circulate, and a purely circulative flow is nondivergent * (because they contain no common component). Thus, a force parallel with the wind is associated divergence, and a force or stress perpendicular with the wind is associated with relative vorticity :

I mention "or stress" because relative vorticity is nonzero in a straight but sheared flow :

Though no force would be necessary for sustaining this flow, a shear stress exists. An object embedded in the flow such as the pinwheel shown would experience this stress and its rotation would be the response. Many people consider such a rotation in an environment with great vertical wind shear a very important contributor to development of tornado circulations :

Differential vertical velocities in a storm's vicinity can tilt an initially strong horizontal relative vorticity component, contributing to development of a strong vertical vorticity component (strong circulation in the horizontal plane), which then can be augmented via vertical stretching (the mechanism for which I hope I can discuss later).

* A misconception ?

Some people explain that a purely vortical flow is nondivergent and a purely divergent flow nonvortical showing that Ñ • (Ñ × V) = 0, and that Ñ × (Ñ • V) is non-existent; thus stating that the divergence of the vorticity is 0 and the vorticity of the divergence is non-existent. I do not think this is true because Ñ • (Ñ × V) is not vorticity divergence, just as Ñ • V is not "velocity divergence", as is sometimes called. It is air divergence - the air diverges, not the velocities - velocities simply illustrate how the air moves. I can indeed show vorticity vectors which diverge; but air in a purely vortical flow does not diverge (thus Ñ • (Ñ × V) = 0).

Text and embedded images are copyright of Joseph Bartlo, though may be used with proper crediting.

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