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Unformatted text preview: 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 9. VERTICAL MOTION 9.1 Divergence in Two and Three Dimensions The gradient symbol, encountered earlier in these notes, is really a vector operator. Gradient itself does not have a magnitude and direction, but the gradient of something does. A vector operator such as the gradient operator (or del, for short) is something that represents a mathematical operation but that can be manipulated like a vector. These manipulations always are performed using the vector components as manipulation tools. The components of the del operator in three dimensions are + + = z y x k j i So + + = z T y T x T T k j i is just the vector formed when each component of del operates on temperature. Another quantity that can be created through mathematical manipulation of the gradient vector is divergence. The divergence of some vector field, or div for short, is the dot product of del with that vector field. The word divergence itself, when not followed by the ATMO 251 Chapter 9 page 1 of 22 clarifying statement of the vector field to which it applies, is assumed to mean the divergence of the vector velocity. Remember from Chapter 7 that the dot product of two vectors can be expressed as the sum of the products of each of the components. When dotting something with the del operator, each component of del operates on the corresponding components of the other vector. Thus the divergence of a vector is the derivative with respect to x of the x component, plus the derivative with respect to y of the y component, plus (if the vector is three- dimensional) the derivative with respect to z of the z component. In three dimensions, using component notation, the divergence of the velocity vector is: z w y v x u v + + = r In two dimensions it would be: y v x u v h + = r Perhaps you can now see how del works in vector-form equations. For the dot product, each component of del is combined with the corresponding components of the other vector....
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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.
- Fall '07