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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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
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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
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This note was uploaded on 04/02/2008 for the course ATMO 251-501/50 taught by Professor Alcorn during the Fall '07 term at Texas A&M.

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book14vorticity - Weather Observation and Analysis John...

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