Weather Observation and Analysis
John Nielsen-Gammon
Course Notes
These course notes are copyrighted.
If you are presently
registered for ATMO 251 at Texas A&M University, permission is hereby
granted to download and print these course notes for your personal use.
If you are not registered for ATMO 251, you may view these course notes,
but you may not download or print them without the permission of the
author.
Redistribution of these course notes, whether done freely or for
profit, is explicitly prohibited without the written permission of the author.
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
way of writing the curl of some vector
v
r
is
v
r
×
∇
Even if the vector
is entirely horizontal, the curl is a fully three-
dimensional vector.
v
r
The definition of curl (in three dimensions) is most clearly written
in the form of a determinant, as follows:
w
v
u
z
y
x
v
∂
∂
∂
∂
∂
∂
=
×
∇
k
j
i
r
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:
k
j
i
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
−
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
−
∂
∂
=
×
∇
y
u
x
v
x
w
z
u
z
v
y
w
v
r
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

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*