book14vorticity - Weather Observation and Analysis John...

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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
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